DISS. ETH NO. 21329
On the nature of pleiotropy
and its role for adaptation
A dissertation submitted to
ETH ZURICH
for the degree of
Doctor of Sciences
presented by
ROBERT POLSTER
Dipl. Biomath., Ernst Moritz Arndt University Greifswald
born 28th of January, 1984
citizen of Germany
Accepted on the recommendation of
Prof. Sebastian L. Bonhoeffer, examiner
Dr. Frédéric Guillaume, co-examiner
Prof. Jérôme Goudet, co-examiner
2013
ii
Contents
Summary
vii
Zusammenfassung
1
Introduction
1.1 The GP map and pleiotropy
1.2 Modularity . . . . . . . . . .
1.3 The G-matrix . . . . . . . .
1.4 Outline of the thesis . . . .
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I
The nature of pleiotropy
2
The extent of pleiotropy in S. cerevisiae and its explanatory role for evolutionary rate
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Why is modularity important? . . . . . . . . . . . . . . . . . . . .
2.1.2 Previous studies on yeast gene evolution . . . . . . . . . . . . . .
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 The Ohya data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The Brown data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 The GO data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Results from gene-based modularity measures . . . . . . . . . . .
2.4.2 Comparison with previous studies . . . . . . . . . . . . . . . . . .
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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III
5
Exploring effects of modular pleiotropy
The relatedness of GP map and G-matrix
3.1 Introduction . . . . . . . . . . . . . . .
3.2 Methods . . . . . . . . . . . . . . . . .
3.2.1 G-matrix comparison . . . . .
3.3 Results . . . . . . . . . . . . . . . . . .
3.3.1 Random mutation effects . . .
3.3.2 Correlated mutation effects . .
3.3.3 Summary of results . . . . . .
3.4 Discussion . . . . . . . . . . . . . . . .
3.5 Appendix . . . . . . . . . . . . . . . .
3.5.1 GP maps used for simulations
29
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Why modularity is advantageous: a G-matrix approach
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Modularity in the G-matrix . . . . . . . . . .
4.1.2 Selection on G . . . . . . . . . . . . . . . . . .
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Generating G-matrices . . . . . . . . . . . . .
4.2.2 Summary statistics of G . . . . . . . . . . . .
4.2.3 Random skewers . . . . . . . . . . . . . . . .
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Eigenanalysis . . . . . . . . . . . . . . . . . .
4.3.2 Summary statistics of G . . . . . . . . . . . .
4.3.3 Random skewers . . . . . . . . . . . . . . . .
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Quantifying genetic constraints . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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Evolutionary consequences of pleiotropic effects in HIV
Evolutionary capacity for drug resistance in HIV
5.1 Introduction . . . . . . . . . . . . . . . . . . . . .
5.2 Material and methods . . . . . . . . . . . . . . .
5.2.1 The HIV dataset . . . . . . . . . . . . . .
5.2.2 PCA and correlation matrix comparison
5.2.3 Selection skewers . . . . . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 PCA and matrix comparison . . . . . . .
5.3.2 Selection skewers . . . . . . . . . . . . . .
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v
5.4
5.5
5.6
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Discussion . . . . . . . . . . . . . . . . . . . . . .
5.4.1 PCA . . . . . . . . . . . . . . . . . . . . .
5.4.2 Matrix comparison . . . . . . . . . . . . .
5.4.3 Hierarchical clustering . . . . . . . . . . .
5.4.4 Selection skewers . . . . . . . . . . . . . .
5.4.5 Perspectives . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Antiretroviral drugs . . . . . . . . . . . .
5.6.2 G-matrices and eigendecompositions . .
5.6.3 Correlations between drug environments
5.6.4 Bootstrap results from G-matrix statistics
Epistasis modifies pleiotropic effects in HIV
6.1 Introduction . . . . . . . . . . . . . . . . . . . .
6.2 Materials and methods . . . . . . . . . . . . . .
6.2.1 The HIV dataset . . . . . . . . . . . . .
6.2.2 Pleiotropy measurements . . . . . . . .
6.3 Results . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Pleiotropy and epistasis . . . . . . . . .
6.3.2 Drug class effects . . . . . . . . . . . . .
6.3.3 Mutation frequencies . . . . . . . . . . .
6.4 Discussion . . . . . . . . . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . .
6.6 Appendix . . . . . . . . . . . . . . . . . . . . .
6.6.1 Random expectations . . . . . . . . . .
6.6.2 Results for different significance levels
Final discussion and outlook
7.1 The nature of pleiotropy . . . . .
7.2 The modular genetic architecture
7.3 Evolutionary lessons from HIV .
7.4 Outlook . . . . . . . . . . . . . . .
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Bibliography
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Acknowledgments
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vi
Summary
A gene is called pleiotropic, when it affects multiple phenotypic characters. Act several pleiotropic genes on the same set of traits, we speak of modular pleiotropy. This
property builds the basis for a modular genetic architecture of an organism.
In this thesis I investigate how modular pleiotropy influences the evolvability of
organisms and the adaptation of populations to new environments. For this, it is
necessary to understand the genetic architecture of an organism and incorporate biological findings into theoretical models. In the past, many models were grounded on
biological assumptions that recently have been found to be inappropriate. Therefore,
I develop models and apply analyses in light of new biological insights to improve
our understanding of evolutionary factors. First, I examine the nature of pleiotropy
and its role in explaining the rate of molecular evolution among genes in the model
organism Saccharomyces cerevisiae. The results on the distribution of pleiotropic degree
and on the structure of the GP map are then incorporated into theoretical models of
population dynamics. Using stochastic simulations I try to better understand what
the driving forces of evolution are. Finally, I study the role of pleiotropy and epistasis in HIV by analyzing a large dataset of in-vitro fitness measurements of viruses
sampled from infected patients. In this context I characterize constraints in resistance
evolution in HIV.
The contributions of this thesis in the field of evolutionary population genetics
comprise extended knowledge about the role of pleiotropy for modularity and hence
for evolvability and adaptation. Most importantly, I show that pleiotropy leads to
strong evolutionary constraints in HIV, and that epistasis can relax these constraints
by modifying the pleiotropic degree of mutations and increasing the genetic variation of the virus. This work also emphasizes the difficulties of drawing general
conclusions about evolutionary principles. Even analyses of several datasets from the
same species can lead to different conclusions about the evolutionary potential of the
species. Therefore, it is important to develop theoretical models for testing evolutionary hypotheses under a variety of possible parameters and validate these models with
empirical data and specific biological experiments. This is not only crucial for understanding the evolutionary past but also for predicting future developments, especially
in the context of species extinction and climate change.
vii
viii
Zusammenfassung
Ein Gen ist pleiotrop, wenn es mehrere phänotypische Merkmale beeinflusst. Wir
sprechen von modularer Pleiotropie, wenn verschiedene pleiotrope Gene für die Variation in derselben Gruppe von Phänotypen verantwortlich sind. Diese Eigenschaft
bildet die Grundlage für eine modulare genetische Architektur eines Organismus.
In dieser Dissertation untersuche ich, wie modulare Pleiotropie die Evolution
und Anpassung von Organismen an neue Umweltbedingungen beeinflusst. Dafür
ist es unabdingbar die genetische Architektur eines Organismus zu verstehen und
gewonnene Erkenntnisse über die Biologie in theoretische Modelle einzubauen. Viele
Modelle beruhen auf biologischen Annahmen, die sich als unzutreffend erwiesen
haben. Daher entwickle ich Modelle und wende Analysen an, die neue biologische Erkenntnisse berücksichtigen, um unser Verständnis von evolutionären Faktoren
zu verbessern. Zunächst untersuche ich, wie häufig Pleiotropie im Modellorganismus
Hefe (Saccharomyces cerevisiae) vorkommt und welche Rolle sie für die Geschwindigkeit
der molekularen Evolution von Genen spielt. Die Erkenntnisse über die Verteilung
der Pleiotropie und die Struktur der genetischen Architektur werden dann in theoretische Modelle zur Populationsdynamik eingebaut. Unter Verwendung von stochastischen Simulationen werde ich versuchen herauszufinden, welches die treibenden Kräfte
der Evolution sind. Abschliessend studiere ich die Rolle von Pleiotropie und Epistase
in HIV, indem ich einen umfangreichen Datensatz analysiere, in dem die Vermehrung
von Viren unter dem Einfluss von Medikamenten gemessen wurde. In diesem Zusammenhang charakterisiere ich ebenso die Möglichkeiten für die Evolution von Resistenzen in HIV.
Der Beitrag dieser Arbeit im Gebiet der evolutionären Populationsdynamik umfasst erweiterte Erkenntnisse über die Rolle von Pleiotropie für Modularität und damit
für die Entwicklungsfähigkeit und Anpassung von Arten. Ein wesentliches Ergebnis
ist, dass Pleiotropie zu starken evolutionären Einschränkungen in HIV führt, und dass
Epistase diese Einschränkungen durch Modifikation der Pleiotropie von Mutationen
verringern kann und die genetischen Variabilität des Virus erhöht. Diese Arbeit unterstreicht auch die Schwierigkeiten, allgemeine Schlüsse über evolutionäre Prinzipien
zu ziehen. Selbst Analysen von mehreren Datensätzen der gleichen Spezies können
zu unterschiedlichen Schlussfolgerungen über das evolutionäre Potenzial dieser Art
ix
x
führen. Daher ist es wichtig, theoretische Modelle zum Testen evolutionärer Hypothesen unter einer Vielzahl von möglichen Parametern zu entwickeln und diese Modelle
mit empirischen Daten und spezifischen biologischen Experimenten zu validieren.
Dies ist nicht nur entscheidend für das Verständnis der evolutionären Vergangenheit,
sondern auch für die Prognose künftiger Entwicklungen, vor allem im Kontext des
Artensterbens und des Klimawandels.
1 | Introduction
Evolutionary theory is concerned with the way evolution works. It describes the principles and factors that influenced the development of simple unicellular life forms to
complex species like humans over hundreds of millions of years. By understanding
these processes, it is possible to also predict future evolutionary trajectories, which is
not only important for the fate of endangered species, but especially for our own wellbeing. Gaining knowledge about the relatedness of species and the heredity of genetic
material was the first step. Today we know a lot about the genetic and developmental system, species interactions and the biological history in general. Sophisticated
mathematical tools have been developed to mimic the processes we observe in biological systems and to explain mysteries we are often not able to observe but which
must covertly operate in the background. However, there are still many open questions. How is the genetic architecture of an organism organized? How do new species
evolve? How do new phenotypic characters arise? Under which circumstances is a
population likely to adapt to a changing environment or dies out? The answer for
many of these questions depends on specific assumptions made about the studied
system, and general conclusions can often not be drawn. On the other hand, there
are concepts that are universally valid and play key roles in evolutionary population
genetics.
Before I describe the goals of this thesis, I will shortly introduce the most important
concepts that it is based on.
1.1
The GP map and pleiotropy
The genotype-phenotype (GP) map is the mapping of genes on traits and describes
the genetic architecture of an organism (Fig. 1.1). The idea that each gene of an organism’s genome has an effect on each phenotypic trait is still used in many theoretical
studies on mutational fitness effects (Orr, 2000, 2006; Otto, 2004; McGuigan et al.,
2011). However, through experimental measurements it recently became obvious that
universal pleiotropy does not seem to be a property of living creatures. Pleiotropy,
the ability of a gene to affect several traits, seems to be very restricted and has a distribution which is L-shaped (e.g., Dudley et al., 2005), i.e., many genes only affect one
1
2
or a few traits and only some genes have an influence on a large number of phenotypes. One of the models for studying mutations and adaptation is Fisher’s geometric
model (Fisher, 1930) which assumes universal pleiotropy, i.e., a mutation in one gene
has an effect on all phenotypic traits of the organism. Many results derived from this
model might not mirror the natural dynamics of adaptation if this assumption is not
fulfilled. It is believed that the genetic architecture greatly influences the evolvability, the ability of a species to evolve, and hence also the rate of molecular evolution
(Hansen, 2006). Several constraints result from pleiotropic effects, e.g., if selection
favors one of the affected traits, the other traits might develop into a disadvantageous
direction. Another example comes from the study of aging: expression of a certain
gene might increase fitness in young individuals, but decrease fitness later in life.
Figure 1.1. An example for a GP map. Circles denote genes and squares symbolize traits.
An arrow indicates that the gene affects the trait. This not only means that expression of the
gene leads to a certain phenotype, but also that variation is the gene (e.g., through mutation
or differential expression) leads to variation in the appearance of the trait. Genes 3, 5, 6 and 7
are pleiotropic, where gene 5 has a pleiotropic degree of three.
1.2
Modularity
Another recent finding is that GP maps are often modular (Wang et al., 2010, e.g.),
which is highly associated with varying pleiotropy among genes. Modular pleiotropy
describes a genetic architecture in which a set of genes tends to have pleiotropic effects on a common set of traits, but few and weaker effects on other traits (Wagner
et al., 2007), see Fig. 1.2. What we do not completely understand yet is how this
modular genetic architecture influences the evolvability of an organism. This is one of
the questions I want to answer. A modular architecture of the GP map supports the
evolvability of one functional/developmental part without interfering with the rest of
the organism. In fact, modularity may also hinder evolution if environmental conditions change such that a combination of traits that are currently in different modules
3
would be favorable for adaptation to the new environment (Griswold, 2006). It is not
clear yet how pleiotropy and modularity are linked. One hypothesis is that modularity evolves by removing pleiotropic effects among genes (Wagner and Altenberg,
1996). So far, we learned from theoretical studies that evolvability is maximized by
variable pleiotropic effects (Hansen, 2003) and that an organism has its highest fitness
at an intermediate pleiotropy level (Orr, 2000). We also learned that biological networks are modular (Wagner et al., 2007) and that modularity favors the evolution of
plasticity through reduction in pleiotropic constraints between alternative phenotypes
(Snell-Rood et al., 2010). Whether all these findings are compatible with real world
evolutionary dynamics is another question, because most of them rely on strong assumptions. I will relax some of these assumptions and incorporate recent biological
findings on pleiotropy and modularity to expand existing and develop new models
to investigate the role of the genetic architecture for adaptation.
Another interesting question arising in this context concerns the relationship between
modularity and the complexity of an organism. Higher organisms are more complex
than simple unicellular creatures (Szathmary and Smith, 1995). Based on Fisher’s geometric model which relates high complexity with high pleiotropy, Orr (2000) found
that increased complexity comes with a certain cost in terms of adaptation disadvantage. However, modularity seems to significantly reduce this cost of complexity
(Welch and Waxman, 2003). In this work, I will further investigate this phenomenon.
Figure 1.2. An example of a modular GP map. See Fig. 1.1 for annotation and comparison.
1.3
The G-matrix
An important concept in population genetics is the additive genetic variance-covariance
matrix — short G-matrix. The G-matrix depicts the patterns of genetic correlation between the phenotypic traits of an organism. For n traits it is a symmetric n × n-matrix
4
G = (σij ), where σij is the covariance between traits i and j, for i 6= j, and the additive
genetic variance of trait i, for i = j (Fig. 1.3). G aims at describing the constraints
residing in the genetic architecture of an organism. Moreover, it seeks to make predictions about the traits’ (directional and correlational) response to selection. Each
trait has a certain genetic variance which selection can act on. Additionally, traits can
be correlated due to pleiotropy or linkage among genes. In this way, G summarizes
the functional interactions of traits and is able to determine the rate and direction of
the response to selection.
For selection to work, a trait has to vary in its phenotypic appearance. Only variable traits can respond to directional selection and thus evolve. This variation is
usually maintained by a mutation-selection-drift balance. Often traits do not evolve
independently but show a certain degree of correlation with other traits. These covariances influence the direction and magnitude of response by placing constraints
onto the adaptive potential of traits. The degree of covariance between traits depends
on the alleles contributing to this mutual variation. Since allele frequencies change
during the course of evolution, G is expected to change as well. It is not clear how
exactly G evolves and over which time span it can be assumed to be constant (Steppan
et al., 2002; Jones et al., 2003; Mezey and Houle, 2003; Arnold et al., 2008), but since I
look at short-term evolution, this problem is not apparent in this work.
Figure 1.3. Two-dimensional visualization of the G-matrix. Black dots symbolize the breeding
values for two phenotypic traits for each individual of the sampled population. The G-matrix
is drawn as the ellipse that harbors 95% of these dots and has a major and a minor axis of
variation. In mathematical terms, it is a matrix with the trait variances in the diagonal and
the covariance between the traits in the off-diagonals.
1.4
Outline of the thesis
It becomes more and more evident that the genotype-phenotype map is modular and
that mutational effects do not have a universal impact on the phenotypes. Thus, in
5
many previous theoretical models on evolvability and adaptation of populations, the
key assumption on the extent of pleiotropic effects do not hold. Therefore, there is
need for new theoretical models that integrate recent biological findings into population genetics frameworks to make studies on evolvability and adaptation more
realistic. It is interesting to see how predictions change if strong assumptions are
relaxed and biological observations are included into theoretical models. With this
project I follow the prosperous road of previous theoretical studies on the dynamics
of adaptation and evolvability by considering recently found biological facts, namely
the modular structure of the GP map caused by modular pleiotropy.
The first part of this thesis deals with an initial study on the extent of pleiotropy
in the model organism yeast. I then go into more detail by considering modular
pleiotropy as an important factor of the genetic architecture and show that it greatly
improves our understanding of the link between GP map and G-matrix. The last part
is an application of the methods used in the previous chapters on a dataset of HIV
fitness measurements.
In Part I of the thesis (Chapter 2) I investigate the effect of modular pleiotropy
on the rate of molecular evolution of genes. For this purpose I use data from genedeletion experiments in the yeast Saccharomyces cerevisiae. Since it is a very difficult
task to detect the modules of a real-world network (Orman and Labatut, 2009; Danon
et al., 2005), I try to find gene-based measures, which involve gene-trait interactions
and account for the modular pleiotropy without actually knowing the full modular
structure of the GP map. I then test these measures on simulated networks and real
data and link it to measurements of fitness and evolutionary rates of genes. The
aim is to explain more of the variation in evolutionary rate than the pure pleiotropy
does, because the place of a gene within the genetic architecture could influence its
evolutionary rate.
Key Question: How is the variation in gene-based modularity measures linked to variation in
the gene’s rate of evolution?
The second part of the thesis (Chapters 3-4) is concerned with how features of
the GP map get translated into the G-matrix and how modularity on the level of the
G-matrix helps to increase the adaptive potential of a population. First, I create a set
of different GP maps that contain features like pleiotropy, modularity and asymmetry
in module composition, and simulate G-matrices for populations exhibiting these GP
maps. I investigate how robust the relatedness between GP map and G-matrix is and
whether it is possible to detect the modeled features of the GP map in the corresponding G-matrix.
Key Question: Does modular pleiotropy in the GP map translate into a modular G-matrix?
Chapter 4 then follows up on that by studying the evolvability of modular Gmatrices. The goal of this simulation study is to find hints on the evolution of modularity and the interaction of modularity, pleiotropy and organismal complexity influencing the adaptation of populations to changing environments. This part aims to
6
explain how the genetic architecture influences the ability of an organism to evolve
and to describe the parameters that support or hinder evolution.
Key Questions: How important is modularity for evolvability? What are the driving factors?
Which parameters influence evolvability the most?
In Part III (Chapters 5-6) I investigate pleiotropy and the resulting constraints on
resistance evolution in HIV based on a large dataset of in-vitro fitness measurements of
viruses in different drug environments. This data provides information about fitness
effects of single and double mutations and makes it possible to study pleiotropy and
epistasis together. The viral strains were derived from HIV-infected patients and
tested for resistance in the presence of 15 different single antiviral drugs. First, I
use the dataset to analyze the correlational structure of drug environments to make
predictions about the adaptive potential of this hypothetical HIV population.
Key Question: Is the genetic variation in HIV ruled by main or epistatic effects of mutations?
Are there constraints on resistance evolution?
Further, I investigate how epistasis modifies the pleiotropic degree of mutations.
Two mutations exhibit an epistatic interaction when the variation at one locus changes
the phenotypic effect of genetic variation at another locus. Through these interactions
a mutation can broaden its range of affected traits. The models used in the previous
chapters assumed additivity of genetic effects and hence ignored epistatic interactions.
This data provides information about the role of epistasis in shaping the GP map.
Epistasis is abundant in HIV and predominantly positive (Bonhoeffer et al., 2004). I
scrutinize whether this leads to an increase in the trait repertoire of mutations and
discuss the consequences for evolvability of the virus.
Key Question: How do pleiotropy and epistasis interact and what are the consequences for
phenotypic variation and evolvability?
l end the thesis with a summarizing discussion in which l critically evaluate the
findings in this work and propose directions for further development and validations
of the models discussed.
I | The nature of pleiotropy
7
2 | The extent of pleiotropy in
S. cerevisiae and its explanatory
role for evolutionary rate
Abstract
Genotype-phenotype maps are often modular, but it is difficult to exactly determine
the modules of real networks and allocate all modules of traits that a gene plays a
role in. Trying to overcome this issue we develop gene-based modularity measures
to explain the effect of modular pleiotropy on evolvability. We test these measures on
six different datasets of the budding yeast Saccharomyces cerevisiae and show that this
leads to very contradictory results. Of particular interest are correlations and partial
correlations with evolutionary rate and fitness values. We also discuss the problems
and limitations of this approach.
9
10
2.1
Introduction
The idea that each gene of an organism’s genome has an effect on each of its phenotypic traits stems from the time when R.A. Fisher introduced his geometric model of
adaptation (Fisher, 1930) and assumed universal pleiotropy. This assumption is still
used in many theoretical studies on mutational fitness effects, but in recent years it became apparent that universal pleiotropy is not at all a property of living creatures. As
mentioned in the general introduction, only some genes have an influence on a large
number of phenotypes. This is at least valid for model organisms like S. cerevisiae, S.
elegans and M. musculus where quantitative studies on genetic effects (gene-deletion
experiments and QTL studies) have been carried out (Dudley et al., 2005; Ohya et al.,
2005; Ostrowski et al., 2005; Sonnichsen et al., 2005; Ericson et al., 2006). However,
it is still not clear if this observation applies universally or is very different for other
organisms.
Another recent finding is that genotype-phenotype (GP) maps are often modular (Wang et al., 2010), which is highly associated with varying pleiotropy among
genes. What we do not know is, how this modular genetic architecture influences the
evolvability (ability to adapt) of an organism. The crucial problem lies in finding the
modules of a network and although different approaches exist (Orman and Labatut,
2009), most of them fail. This is mainly due to overlapping modules and measurement noise in experimental studies. To overcome this issue we developed different
gene-based measures of modular pleiotropy that rely on the information we find in
the GP map.
In this project we study the available results of yeast gene-deletion experiments,
with an important extension compared to previous analyses. Besides a broad comparative study, we seek for a gene-based measure of modularity, which might explain
more of the variation in fitness and especially evolutionary rate than the pure measure of pleiotropy does. The general idea is that pleiotropy is associated with a cost of
complexity (Orr, 2000), because the more traits a gene affects, the more likely it is that
a mutation is disruptive for one of the traits. However, if pleiotropy is not random, but
restricted to functional or otherwise related units, this cost could be weaker. Therefore, a measure of modular pleiotropy might be a better predictor for evolutionary
constraints, like the rate of molecular evolution of genes.
2.1.1
Why is modularity important?
It is believed that the genetic architecture greatly influences the evolvability of an organism and hence also its rate of molecular evolution. Several constraints result from
pleiotropic effects, but a modular architecture of the GP map could attenuate these
constraints. Modularity supports the evolvability of one module without interfering a
11
lot with the rest of the organism. But it may also hinder evolution into other directions
than proposed by this architecture.
It is yet not clear how pleiotropy and modularity interact. One theory is that
modularity evolves by removing pleiotropic effects among genes (Wagner and Altenberg, 1996). Thus, a reduction of the overall pleiotropy of the genes would result
in an increased modular architecture and an increase in pleiotropic degree would decrease the level of modularity and increase the level of integration. All this requires
the assumption that genes initially had a high pleiotropic degree (or even universal
pleiotropy) and evolution was acting in a way to reduce integration to build quasiindependent functional or developmental modules. This process is called parcelation
(Mezey et al., 2000). On the other hand, one can view this problem from the opposite
perspective. Modularity could have evolved by adding links between genes and traits,
assuming that each gene initially only affected one single trait. This puzzle still needs
to be solved. However, the truth might be more complicated than the two possibilities
presented above.
From theoretical studies we know so far that evolvability is maximized by variable pleiotropic effects (Hansen, 2003), that an organism has its highest fitness at
an intermediate pleiotropy level (Orr, 2000), that biological networks are modular
(Wang et al., 2010) and modularity favors the evolution of plasticity through reduction in pleiotropic constraints between alternative phenotypes (Snell-Rood et al., 2010).
Whether all these findings are compatible with real world evolutionary dynamics is
another question. However, what we can conclude from it, is that the modular structure of the GP map should also play an important role in the rate of evolution on the
single gene level.
2.1.2
Previous studies on yeast gene evolution
Saccharomycec cerevisiae is a fungal model organism that has been extensively studied.
Therefore, a lot of data is publicly available that we can use to study pleiotropy and
the evolutionary rate of genes.
Several factors influence the rate of substitution (or rate of molecular evolution).
Nucleotide substitutions can be synonymous (S – no change in amino acid, presumably neutral) or non-synonymous (N – change in amino acid, possibly experience
selection). Here we use dN/dS 0 as a measure of evolutionary rate of genes, which
quantifies selection pressures by comparing the rate of substitutions at synonymous
sites (dS) to the rate of non-synonymous substitutions (dN). dN/dS > 1 means that
natural selection promotes changes in the protein sequence, whereas a ratio less than
one indicates purifying selection. Hirsh et al. (2005) proposed an adjusted measure of
dS, denoted dS 0 , which controls for the influence of codon preference.
Until today it remains unknown by what factors evolutionary rate is determined.
Towards solving this question, Drummond et al. (2006) carried out a large correla-
12
tion and principal component analysis on yeast evolution. They took a gene-deletion
dataset from Deutschbauer et al. (2005) with 568 genes and 12 different growth media and combined it with seven different yeast gene characteristics (expression, CAI
(codon adaptation index), length, dispensability, degree (protein-protein interactions),
centrality and abundance (number of protein molecules per cell)) to explain the variation in evolutionary rates (taken from Wall et al. (2005)) of these genes. A principal
component regression revealed that one component comprising expression, CAI and
abundance explained 43% of the variation in dN and dS, respectively. The dN/dS 0
regression was nearly indistinguishable from that of dN, because dS 0 has been purged
of the influence of selection on synonymous sites. Furthermore they argued that it
is unlikely to find other determinants that explain the remainder of evolutionary rate
variation, since stochastic heredity probably introduces a high level of noise. Additionally, measurement noise and nonlinearity of predictor variables might reduce the
true R2 . Finally, they conclude that translational selection is the main driver for the
rate of evolution: the more predicted translation events, the slower evolution (Drummond et al., 2006).
Already some years earlier Pal et al. (2001) found a strong correlation between
expression level and evolutionary rate in yeast genes. They received dN values by
comparing homologs of S. cerevisiae and C. albicans and linked this information with
expression values measured by Holstege et al. (1998). For 29 genes, located within
one functional class, the Spearman’s correlation was -0.485. Later, Pal et al. (2006)
used Wall’s evolutionary rate measures on a much larger dataset, revealing again a
strong relationship between expression level and rate of protein evolution (R2 = 0.29),
which confirmed the finding of Drummond et al. (2006), that expression level is a
major predictor of the rate of evolution of a gene.
Dudley et al. (2005) conducted a gene-deletion experiment under 22 environmental
conditions and proved the distribution of pleiotropic degrees of the genes to be Lshaped, i.e., many genes have only an influence on one or two traits and only a few
genes influence many traits. Ensuing, several studies used the Dudley dataset for a
measure of pleiotropy. Salathe et al. (2006) and He and Zhang (2006) for example
examined how pleiotropy of a gene is related to its annotated Gene Ontology (GO)
categories (Ashburner et al., 2000). They reported pleiotropy to only correlate with
’biological processes’ and concluded that gene pleiotropy is likely due to multiple
biological processes in which the gene participates. Salathe et al. also found a weak,
but significant correlation between the number of GO biological processes of a gene
and its rate of evolution among 2291 yeast genes (Spearman’s ρ = -0.087, P = 3 ∗ 10−5 ),
as well as between pleiotropy and dN/dS 0 of 404 genes (Spearman’s ρ = -0.115, P
= 0.021) (Salathe et al., 2006). This was also confirmed by He and Zhang with their
own measures of evolutionary rate from a comparison of homologs in S. cerevisiae and
S. bayanus (r = -0.12, P < 10−11 ) (He and Zhang, 2006).
13
In recent years more and also larger datasets of gene-deletion experiments became
available. Ohya et al. (2005) carried out a study with 501 morphological parameters like cell shape and actin cytosceleton, and determined 2378 mutant strains that
exhibited differences from wild-type cells in at least one of 254 statistically reliable
parameters. Brown et al. (2006) observed deletion mutants under 51 environmental
stresses and measured fitness ratios of 4281 mutant strains compared to a control
strain. A similar approach was carried out by Hillenmeyer et al. (2008) who exposed
the mutant strains to over 400 chemicals and diverse environmental stresses.
Cooper et al. (2007) used the Ohya dataset to investigate the relationship between
pleiotropy and fitness effects of gene deletions, using fitness measurements from
Steinmetz et al. (2002) and the evolutionary rate values of Wall. Their main results
were that pleiotropy strongly correlates with fitness (ρ = -0.292, P < 10−9 ), but only
very weakly correlates with dN (ρ = -0.061, P = 0.007).
Recently Wang and Zhang (2009) invented a measure called ’gene importance’,
which is denoted as the amount of fitness reduction a gene deletion is causing under a
certain experimental condition. It is based on the experimental results of Steinmetz and
Hillenmeyer. They found a Spearman’s correlation of -0.219 between gene importance
and evolutionary rate (which was measured as in He and Zhang (2006)).
Overall, it seems that expression level is the main contributor to the rate of molecular evolution, but there are many other factors that play a significant role, like abundance, codon adaptation index, gene importance and pleiotropic degree. It is not clear
why pleiotropy is highly correlated with fitness, but only slightly (if at all) with rate of
evolution. On the other hand, we have a reverse phenomenon with expression level:
high correlation with evolutionary rate, but no correlation with fitness (see Results).
2.2
Methods
Instead of taking simply the pleiotropic degree of a gene, we thought about a more
sophisticated measure. It is known that most real-world networks are modular, i.e.,
the network is organized in more or less independent subgroups (Wang et al., 2010).
In case of a genotype-phenotype map this means that a number of genes influence
the same group of traits and only a few other traits are affected that do not belong
to this group. The advantage appears in the evolvability of these modules being
fairly independent from the rest of the organism. However, a crucial problem lies
in finding the modules of a network. Although different approaches exist, most of
them fail to exactly recover all modules. This is mostly due to partially overlapping
modules, measurement noise and also the fact that many biological networks are
strongly interconnected.
To overcome this issue, we developed different gene-based measures of network
integration that rely on the information residing in the GP map. Let T be the set of all
14
traits t and G the set of all genes g. With tg we denote a trait effected by a gene g and
gt is a gene influencing trait t. We then define the following:
pleiotropic degree (PD):
PDg = |{tg ∈ T}|
trait degree (TD):
T Dt = |{gt ∈ G}|
P
−1
tg ∈T T Dtg
GIg = PDg ∗
∗
PDg
gene index (GI):
modularity coefficient (MC):
influence (IN):
MCg = PDg /GIg
P
INg = tg ∈T T D1 t
essentiality (ES):
ESg = INg /PDg
effective complexity (EC):
ECg = PDg /ESg
P
T Dt
|T|
t∈T
g
The pleiotropic degree is just given by the number of traits a gene effects and the
trait degree denotes the number of genes that influence a certain trait. The gene index
(GI) is a weighted pleiotropy measure. It is larger than PD if the affected traits on
average have a smaller trait degree than the traits of the whole GP map. Thus, it
takes into account how valuable the gene is for its traits. The modularity coefficient is
the ratio of average affected trait degree and average overall trait degree and showed
properties of a clustering coefficient in simulations. It is a similar measure as GI,
but independent from pleiotropy and indicates whether a gene has a higher or lower
influence on its traits than an average gene in the network. The influence (IN) is
the sum of the importance of a gene for its traits. A gene is very important, when
few other genes affect the same traits. The essentiality normalizes the influence by
the pleiotropic degree, therefore it is the average influence on its traits. Finally, the
effective complexity is the ratio of pleiotropic degree (PD) and essentiality (ES), or
the square of PD divided by the influence. Its significance becomes clearer in the
following example.
(a)
(b)
(c)
(d)
Figure 2.1. Four examples of how a gene could be imbedded in the genetic architecture.
Circles are genes and squares are traits. Gene A affects a trait alone. Gene B shares its two
traits with other genes. Gene C has a higher pleiotropy than B. And gene D affects two traits
alone, but shares a third trait with many other genes.
15
Example: Let’s consider different simple architectures of how a single gene
could be part of a larger network (Figure 2.1) and calculate the six values
for each of these situations, assuming that the average trait degree of the
network is T D.
Gene A
PDA = 1
GIA = 1 ∗ 1−1 ∗ T D = T D
MCA = 1/T D
INA = 1 ∗ 1 = 1
ESA = 11 /1 = 1
ECA = 1/1 = 1
Gene B
PDB = 2
GIB = 2 ∗ 3.5−1 ∗ T D = 47 T D
MCB = 2/ 47 T D = 72 /T D
7
7
INB = 24
∗ 2 = 12
≈ 0.6
1
7
1
≈ 0.3
ESB = ( 3 + 4 )/2 = 24
7
48
ECB = 2/ 24 = 7 ≈ 7
Gene C
PDC = 3
GIC = 3 ∗ 3−1 ∗ T D = T D
MCC = 3/T D
INC = 13 ∗ 3 = 1
ESC = ( 31 + 13 + 13 )/3 = 31
ECC = 3/ 31 = 9
Gene D
PDD = 3
GID = 3 ∗ 3−1 ∗ T D = T D
MCD = 3/T D
∗ 3 = 15
≈ 2.1
IND = 15
21
7
1
1
1
ESD = ( 1 + 7 + 1 )/3 = 15
≈ 0.7
21
63
ECD = 3/ 15
=
≈
4.2
21
15
The four genes have different levels of pleiotropy and there is variation in
how other genes affect the same traits. Especially the differences between
genes C and D are worth noticing. Both genes have the same pleiotropy (3),
but find themselves in very different situations. Nevertheless, for both genes
we obtain the same gene index (GI = T D), which is also equal to gene A, and
the same modularity coefficient (MC = 3/T D). With respect to the modular
aspect, one would intuitively expect that the values have to be different for
genes C and D. For this reason we developed the essentiality measure (ES),
which accounts for the relative influence a gene has on each its traits. As
expected, all measures incorporating ES differ for genes C and D and it is
also intuitive that the essentiality of gene C for its traits equals 1/3, because
each trait is affected by three genes. For gene D the situation is completely
different, since it exclusively affects two traits, and one trait is affected by
many other genes. Therefore the mean impact of gene D on its traits is
very high (≈ 0.7). An interesting observation can be made with respect to
the effective complexity measure (EC). For gene C the value is equal to the
number of edges in the (sub)network (9) and for gene D, although having
the same number of edges in its network, this value is only half as large (≈
4.2). This is caused by the second trait being affected by many genes, which
reduces the modular complexity of this (sub)network.
Although these examples probably over-simplify real-world scenarios they
give a good impression about what the differences in our measures are.
16
Values for all measures are calculated for the genes of S. cerevisiae based on five different experimental studies and the GO annotations (Table 2.1). We only considered
genes that have significant effects on at least one trait.
Additionally, we extend the influence measure by weighting the influence of a
gene (a) with fitness values (fwIN) and (b) with expression values (ewIN), and also
calculated the weighted essentiality and effective complexity from that. That means
we consider how strong the influence of a gene on its traits is, by weighting these
interactions with the expression level or the fitness effect of the gene.
For each dataset we obtained the corresponding GP map, a binary transition matrix of genes and traits. These matrices served as bases for all calculations. Finally,
we link our gene-based measures with values of fitness (Steinmetz et al., 2002), rate of
evolution (Wall et al., 2005) and expression level (Holstege et al., 1998) and perform
regression analyses to estimate how much of the variation in dN/dS’, expression level
and fitness can be explained by our measures. For this, we will use the R2 value of
a linear model. Additionally, principal component regression (with ’pls’ package in
R (Mevik et al., 2011)) is used to estimate how much of the variation in dN/dS’ and
fitness is explained by several gene-based measures together. We do correlation and
partial correlation analyses (with ’corpcor’ package in R (Schäfer et al., 2013)) to test
for the association between our measures and rate of evolution and fitness. Since
most of our measures depend on the pleiotropic degree, we perform partial correlations to account for the influence of PD on the correlation to obtain a PD-independent
association of the measure with dN/dS’ or fitness.
Study
Ericson [Ericson et al. (2006)]
Dudley [Dudley et al. (2005)]
Ohya [Ohya et al. (2005)]
Brown [Brown et al. (2006)]
Hillenmeyer [Hillenmeyer et al. (2008)]
Gene Ontology (GO) [Ashburner et al. (2000)]
# Genes # Traits
293
774
2059
1933
4676
5888
18
22
248
31
417
2086
Table 2.1. Experimental studies (and their sizes) used in this project.
The Ericson and Dudley datasets comprise tables of significant growth defects for
deletion mutations in certain stress environments. Thus, this is already preprocessed
data and we could directly infer the transition matrix from it. The Ohya data consists of two tables, one with growth measurements of 501 morphological parameters
(133 cell shape, 96 Actin (cytoskeleton), 272 nucleus) for 126 wild-type strains. The
other one comprising all gene-deletion mutants with measurements on the same set
of morphological parameters. Taking the wild-type measurements as reference dis-
17
tribution for each parameter, we obtained the GP map of this dataset by calculating
which measurements of the deletion mutants significantly differed from the means
of these distributions (before, we filtered the data and only retained normally distributed parameters in the wild-type dataset). The Brown and Hillenmeyer datasets
consist of preprocessed tables of gene-deletion mutant strains measured in 51 stress
environments (Brown) and 419 chemicals and diverse environmental stresses (Hillenmeyer). In both studies they calculated the fitness effects with log-ratios of treatment
intensity versus mean control intensity (Brown), respectively mean control intensity
versus treatment intensity (Hillenmeyer). For the Brown data we determined the mutant strains with significant growth defects in each environment. The final transition
matrix was obtained after merging highly correlated traits (as reported in Zhou et al.
(2009)), resulting in 31 final traits. The Hillenmeyer dataset contained additional Pvalues, already indicating the significance of the fitness effects measured. We used
these values to obtain the GP map. Gene Ontology is a database where genes are
annotated with functional categories. We took only the biological processes as traits
since this is the only category that correlates with the pleiotropic degree of yeast genes
in experimental studies (Salathe et al., 2006).
2.3
Results
As mentioned before, for each of the six datasets on yeast gene association, we build
the corresponding GP map and calculate our gene-based modularity measures. Then
we link these measures for each dataset to independent measures of gene expression
(Holstege et al., 1998), gene’s evolutionary rate (Wall et al., 2005) and gene’s fitness
effects in complete medium YPD (Steinmetz et al., 2002) and perform regression analyses.
A first comparison of the corresponding GP maps reveals that they all have similar
L-shaped gene degree (pleiotropy) distributions (Fig. 2.2) with mean values between
2.44 and 13.47 (medians between 2 and 6), but very different trait degree (TD) distributions (Fig. 2.3). This is probably due to differences in network size, but is also
caused by the disparity of traits measured in the experimental studies (stress environments, morphological parameters, GO annotations). Interestingly, the fraction of
pleiotropic genes (PD>1) lies between 53% and 58% for the networks with few traits
(Ericson, Dudley and Brown) and between 70% and 77% for the networks with many
traits (Ohya, Hillenmeyer and GO).
2.3.1
General Observations
First of all, we investigate correlations among some of the measurements. Since most
of the factors that we analyze here are not normal distributed, it is more appropri-
18
DUDLEY
max = 14
mean = 2.44
median = 2
0
0
50
100 150 200 250 300
100
max = 8
mean = 2.45
median = 2
50
absolute frequency
150
ERICSON
0
2
4
6
8
10
0
5
10
800
max = 16
mean = 2.69
median = 2
400
400
300
200
200
0
0
100
absolute frequency
500
max = 63
mean = 5.58
median = 3
0
5
10
15
20
0
10
15
20
GO biological processes
max = 142
mean = 13.47
median = 6
max = 30
mean = 4.26
median = 3
0
0
500
1000
100 200 300 400 500 600
5
1500
HILLENMEYER
absolute frequency
20
BROWN
600
600
OHYA
15
0
10
20
30
pleiotropic degree
40
50
0
5
10
15
20
25
30
pleiotropic degree
Figure 2.2. Distributions of pleiotropic degree in the six datasets. All distributions are Lshaped with only few genes affecting 10 or more traits.
19
1.5
max = 264
mean = 85.68
median = 49
0.0
0.5
1.0
1.5
1.0
0.5
0.0
absolute frequency
max = 136
mean = 39.83
median = 32
2.0
DUDLEY
2.0
ERICSON
0
50
100
150
0
50
100
250
300
3.0
1.5
6
2.0
2.5
max = 312
mean = 167.71
median = 154
1.0
4
0
0.0
0.5
2
absolute frequency
8
max = 268
mean = 46.35
median = 26
0
50
100
150
200
250
0
100
150
200
250
300
max = 1244
mean = 12.03
median = 3
0
0
1
100
2
200
3
300
4
500
GO biological processes
max = 334
mean = 117.37
median = 113
5
50
400
6
HILLENMEYER
absolute frequency
200
BROWN
10
OHYA
150
0
50
100
150
200
trait degree
250
300
350
0
20
40
60
80
100
trait degree
Figure 2.3. Distributions of trait degree (number of genes a trait is affected by) in the six
datasets. In contrast to the distributions of pleiotropic degree, they differ substantially in
shape and mean.
20
ate to use the non-parametric Spearman’s correlation coefficient instead of Pearson’s
correlation. A quick comparison also showed that they mostly give slightly different
results. When speaking about correlation we will further always refer to Spearman’s
rho (ρ).
The measures of fitness, evolutionary rate (dN/dS’) and gene expression are pairwise correlated as follows:
dN/dS’ ∼ expression
dN/dS’ ∼ fitness
fitness ∼ expression
ρ
P-value
-0.495
0.105
0.005
< 2.2e-16
5.33e-07
0.7526
As reported before, expression level highly anti-correlates with rate of evolution.
But also the fitness values measured by Steinmetz et al. (2002) under YPD conditions
show a slight but significant correlation with dN/dS’. There is no correlation between
fitness and expression.
We would also expect that highly pleiotropic genes have higher expression levels,
since they are involved in more biological processes and therefore need to be more
abundant. Looking at the association between gene expression (EX) and pleiotropic
degree (PD), we find a small but significant positive correlation for half of the datasets.
Study
Correlation between PD and EX
ρ
P-value
Ericson
Dudley
Ohya
Brown
Hillenmeyer
GO (biological processes)
-0.077
0.059
0.077
0.068
-0.006
0.087
0.2259
0.1192
0.0011
0.0007
0.7610
4.35e-10
None of our measures significantly correlates with evolutionary rate or fitness
in the Ericson, Dudley and Hillenmeyer datasets, whereas for the data of Ohya and
Brown and for the GO network we find many significant correlations of the genebased measures with dN/dS’ and fitness. Since the Ericson and Dudley datasets are
fairly small, one could argue that by chance the wrong set of genes was used for
the experiments. However, this assumption can be rejected by sub-sampling from the
larger datasets of Ohya and Brown, where smaller gene sets reduce the significance but
not the correlation coefficients. It is also not clear why there seem to be no significant
relationships in the Hillenmeyer dataset, the largest experimental one in our study.
One explanation could be that the experiments are very noisy or somehow biased.
Another one could lie in the way the data was preprocessed. Thus, we focused on
analyzing the Ohya, Brown and GO networks in more detail.
21
2.3.2
The Ohya data
In this dataset, only PD and EC correlate significantly with dN/dS’. All measures
except ES significantly correlate with fitness. Partial correlations, where we control
for the effect of PD on the other measures, still result in significant correlations of our
measures with fitness (although the correlation coefficient mostly decreases) and in no
significant correlation of EC with dN/dS’. When we control the expression-weighted
measures for expression value (EX), none of them correlates with the rate of evolution
anymore, but still significantly correlates with fitness.
Correlation
ρ
P-value
controlled for PD
ρ
P-value
controlled for EX
ρ
P-value
dN/dS’ ∼ PD
dN/dS’ ∼ GI
dN/dS’ ∼ MC
dN/dS’ ∼ IN
dN/dS’ ∼ ES
dN/dS’ ∼ EC
dN/dS’ ∼ ewIN
dN/dS’ ∼ ewES
dN/dS’ ∼ ewEC
-0.098
-0.042
-0.035
-0.050
0.023
-0.102
-0.264
-0.288
0.205
0.0017
0.1810
0.2577
0.1061
0.4688
0.0010
< 2.2e-16
< 2.2e-16
1.15e-10
0.044
-0.034
0.038
0.043
-0.044
-0.257
-0.276
0.275
0.1617
0.2702
0.2174
0.1728
0.1561
2.22e-16
< 2.2e-16
< 2.2e-16
-0.011
0.018
-0.056
0.7245
0.5858
0.0794
fitness ∼ PD
fitness ∼ GI
fitness ∼ MC
fitness ∼ IN
fitness ∼ ES
fitness ∼ EC
fitness ∼ ewIN
fitness ∼ ewES
fitness ∼ ewEC
-0.296
-0.148
-0.091
-0.171
0.016
-0.281
-0.222
-0.110
-0.054
< 2.2e-16
3.96e-11
4.89e-05
1.79e-14
0.4722
< 2.2e-16
< 2.2e-16
4.45e-06
0.0261
0.108
-0.095
0.094
0.082
-0.096
-0.063
-0.065
0.066
1.41e-06
2.04e-05
2.80e-05
0.0003
1.85e-05
0.0092
0.0068
0.0063
-0.222
-0.111
-0.053
< 2.2e-16
3.45e-06
0.0282
Except from essentiality (ES) and expression-weighted effective complexity (ewEC)
all measures have a negative correlation with evolutionary rate and fitness. Overall,
PD explains 1% of the variation in dN/dS’ and about 8.5% of the variation in fitness.
All other measures cannot independently explain more, but they can explain at least
a small portion of the variation in fitness.
2.3.3
The Brown data
In this dataset, all correlations are higher and more significant compared to the Ohya
dataset. The signs of the correlation coefficients are the same as in the previous data.
The fitness-weighted IN, ES and EC have a much smaller correlation coefficient
with dN/dS’ than the expression-weighted measures and surprisingly, the correla-
22
Correlation
ρ
P-value
dN/dS’ ∼ PD
dN/dS’ ∼ GI
dN/dS’ ∼ MC
dN/dS’ ∼ IN
dN/dS’ ∼ ES
dN/dS’ ∼ EC
dN/dS’ ∼ ewIN
dN/dS’ ∼ ewES
dN/dS’ ∼ ewEC
dN/dS’ ∼ fwIN
dN/dS’ ∼ fwES
dN/dS’ ∼ fwEC
-0.117
-0.094
-0.053
-0.094
0.0345
-0.124
-0.412
-0.437
0.291
-0.145
-0.106
0.020
0.0003
0.0034
0.0985
0.0034
0.2851
0.0001
< 2.2e-16
< 2.2e-16
< 2.2e-16
6.23e-06
0.0010
0.5279
fitness ∼ PD
fitness ∼ GI
fitness ∼ MC
fitness ∼ IN
fitness ∼ ES
fitness ∼ EC
fitness ∼ ewIN
fitness ∼ ewES
fitness ∼ ewEC
fitness ∼ fwIN
fitness ∼ fwES
fitness ∼ fwEC
-0.395
-0.285
-0.245
-0.290
0.167
-0.453
-0.249
-0.054
-0.168
-0.294
-0.095
-0.179
< 2.2e-16
< 2.2e-16
< 2.2e-16
< 2.2e-16
1.98e-13
< 2.2e-16
< 2.2e-16
0.0256
4.42e-12
< 2.2e-16
3.01e-05
3.90e-15
controlled for PD
ρ
P-value
controlled for EX
ρ
P-value
0.046
-0.051
0.046
0.049
-0.040
-0.428
-0.433
0.413
0.1583
0.1171
0.1518
0.1287
0.2130
< 2.2e-16
< 2.2e-16
< 2.2e-16
-0.124
-0.052
-0.081
0.0002
0.1127
0.0142
0.243
-0.257
0.234
0.235
-0.252
-0.048
-0.030
0.020
< 2.2e-16
< 2.2e-16
< 2.2e-16
< 2.2e-16
< 2.2e-16
0.0495
0.2176
0.4150
-0.249
-0.054
-0.168
< 2.2e-16
0.0254
3.03e-12
tions with fitness are equally high or even smaller than for the unweighted measures
of IN, ES and EC. The expression-weighted ones are again highly correlated with
dN/dS’. We should mention here that correlations of ±0.4 can be considered high
for such kind of noisy data. When we control for expression level, only ewIN is still
significantly correlated with evolutionary rate and thus even has a bit larger impact
than PD. The significant correlation of the expression-weighted measures with fitness
is lost when controlling for PD.
In this dataset, PD explains 1.4% of the variation in dN/dS’ and about 15.6% of the
variation in fitness. Controlling ewIN for expression still explains 1.5% of the variation
in the rate of evolution. The other measures at least explain a highly significant
proportion of fitness variation (5-6% each). A principal component regression reveals
that all simple factors (without weighting) together explain only 2.9% of the variation
in dN/dS’ and 19.7% of the variation in fitness. When taking the expression-weighted
factors, this can be increased to 23.6% for dN/dS’ and to 20.2% for fitness.
23
2.3.4
The GO data
In contrast to analyzing experimental data which can be very biased towards individual labs or experimental settings, it can be beneficial to use broader databases like GO
for the analysis. We take the functional categories assigned to each gene by Gene Ontology as traits. Especially the information about biological processes very well describe
the properties of the genes with respect to functional expression, as reported previously (He and Zhang, 2006). Therefore we confine our analyses on this sub-class. All
our measures correlate significantly with evolutionary rate, even when controlled for
pleiotropy. Correlations with fitness give a similar picture, although the correlations
are smaller. However, here some significances disappear when controlling for PD.
Correlation
ρ
P-value
controlled for PD
ρ
P-value
controlled for EX
ρ
P-value
dN/dS’ ∼ PD
dN/dS’ ∼ GI
dN/dS’ ∼ MC
dN/dS’ ∼ IN
dN/dS’ ∼ ES
dN/dS’ ∼ EC
dN/dS’ ∼ ewIN
dN/dS’ ∼ ewES
dN/dS’ ∼ ewEC
-0.271
-0.200
0.123
-0.237
-0.182
0.107
-0.326
-0.300
0.250
< 2.2e-16
< 2.2e-16
1.22e-11
< 2.2e-16
< 2.2e-16
3.23e-09
< 2.2e-16
< 2.2e-16
< 2.2e-16
-0.053
0.067
-0.062
-0.070
0.082
-0.208
-0.212
0.223
0.0036
0.0002
0.0007
0.0001
6.02e-06
< 2.2e-16
< 2.2e-16
< 2.2e-16
-0.195
-0.145
0.069
< 2.2e-16
5.77e-15
0.0002
fitness ∼ PD
fitness ∼ GI
fitness ∼ MC
fitness ∼ IN
fitness ∼ ES
fitness ∼ EC
fitness ∼ ewIN
fitness ∼ ewES
fitness ∼ ewEC
-0.181
-0.163
0.131
-0.151
-0.110
0.073
-0.175
-0.152
0.118
< 2.2e-16
< 2.2e-16
< 2.2e-16
< 2.2e-16
9.12e-13
2.42e-06
< 2.2e-16
< 2.2e-16
2.72e-13
-0.067
0.079
-0.023
-0.020
0.035
-0.072
-0.074
0.085
1.53e-05
3.51e-07
0.1311
0.1891
0.0245
8.74e-06
4.73e-06
1.71e-07
-0.175
-0.152
0.118
< 2.2e-16
< 2.2e-16
2.25e-13
Like for the Ohya data, we have no fitness measurements in this dataset and therefore cannot calculate fitness-weighted measures. The expression-weighted values
highly correlate with evolutionary rate and fitness, even if controlled for pleiotropy or
expression. However, the partial correlation coefficients are smaller than the correlations of PD with dN/dS’ and fitness. In contrast to both previous datasets, the signs
of the correlations for MC, ES and EC changed.
In the GO annotation dataset the pleiotropic degree (here the number of biological
processes a gene is involved in) explains 7.4% of the variation in evolutionary rate
and 3.3% of the variation in fitness. A principal component regression shows that all
simple factors together explain 8.8% of the evolutionary rate variation and only 6.5%
of the variation in fitness. Taking the expression-weighted factors instead leads to an
24
explanation of 12.7% of evolutionary rate variation and of 5.8% of fitness variation. In
contrast to the results of the Brown data, this shows that the weighting by expression
level does not strongly influence the association.
2.4
2.4.1
Summary of results
Results from gene-based modularity measures
It is obvious that our measures GI, MC, ES, IN, EC and also the weighted factors fwES,
fwIN, fwEC, ewES, ewIN and ewEC highly depend on PD (or expression level in the
case of expression-weighted measures). Therefore the partial correlations are mostly
not better than the correlations of PD with evolutionary rate and fitness. However,
some measures still seem to make a significant contribution. On the other hand we
did not expect to find one independent factor, because all calculations based on the
GP map include the degree of a node to some extent. We rather aimed at expanding the pure value of pleiotropy with some additional information contained in the
network structure. In the Ohya and Brown datasets the effective complexity (EC) has
an equally high or even higher correlation with evolutionary rate and fitness than
the pure pleiotropic degree, although the differences are only marginal. The fitnessweighted values did not lead to increased correlations. The expression-weighted values have a much higher correlation with evolutionary rate than any simple measure,
but not with fitness. On the other hand expression level alone has a higher correlation with dN/dS’ than any of these measures. A principal component regression
reveals that the new measures only add 1-2% to the explanation of evolutionary rate
variation.
Removing non-pleiotropic genes (PD=1) from the analysis has very contradictory
effects. In the Brown dataset it results in an almost unchanged correlation coefficient of
PD with dN/dS’ (-0.117 → -0.134), but significantly reduces the correlation between
PD and fitness (-0.395 → -0.22). This means, genes affecting only one trait do not
contribute to PD explaining the variation in rate of evolution but they do greatly
contribute to PD explaining the variation in fitness. On the other hand, in the GO
network the opposite effect is observed. Removing monotropic genes significantly
decreases the correlation between PD and dN/dS’ (-0.273 → -0.159), whereas the
correlation between PD and fitness does not change much (-0.181 → -0.139). Again,
we see that although we are only looking at a single organism, different datasets can
give rise to very different results and conclusions. This could be due to the differences
in the datasets. The Brown data comprises measurements of change in organismal
fitness in several stress environments when a gene is deleted, and in the GO data
base we find associations with biological processes for a gene. Therefore, it is possible
25
that the pleiotropic degree of a gene has very different associations with evolutionary
rate and fitness, depending on the traits defined to measure pleiotropy.
2.4.2
Comparison with previous studies
We confirmed the strong role of expression level in determining the speed of evolution. Analyzing the GO network, we found a much higher correlation between the
number of biological processes and evolutionary rate than Salathe et al. (2006) (-0.273
instead of -0.087). This could be due to the constant enhancement of the GO database,
which might probably lead to different datasets over a period of four years.
Compared to the study of Cooper et al. (2007), for the Ohya dataset we found a
similar correlation of pleiotropy with fitness (≈ -0.29), but a slightly higher correlation
with evolutionary rate (-0.10 instead of -0.06). Nevertheless, some datasets failed to
produce any significant results (Ericson, Dudley and Hillenmeyer), and for the Brown
data we even obtained higher correlations.
The question is where these large differences come from. One point is that small
datasets lack the statistical power to find significant associations. Also, if one only
looks at a subset of a large network it is possible to miss the overall trend. On the
other hand in very large datasets it could be possible that there are some clear trends
in parts of the network, but these may cancel out when looking at the whole dataset.
For example, most genes with high pleiotropy might evolve slowly, but if we fail
to measure the traits these genes affect, they get assigned a small pleiotropy in the
experiment and therefore ’destroy’ the overall trend. Our work shows that one has to
be very careful with drawing general conclusions from analyses of experimental data,
since these results are highly dependent on the experimental setting and might only
reflect a special case under special conditions.
2.5
Discussion
We initially posed the question if we could integrate the modular information of the
GP map into a gene-based measure and if such a measure could explain more of the
variation in evolutionary rate than the pure pleiotropic degree of a gene. Our analyses
have shown that the results highly depend on the quality and content of the datasets.
Generally, the proportion of variation in evolutionary rate explained by our measures
is only slightly larger than what is explained by pleiotropy and expression level alone.
This could have different reasons: (1) None of the measures is perfectly suited to really
reflect the modular structure of the network, (2) the datasets used here give a very
non-precise picture of the real GP map of S. cerevisiae, which is nearly impossible to
obtain, (3) the experimental data usually contains a high degree of noise and they
are biased due to many constraints in the experimental measurements, which can
26
result in false positive as well as false negative correlations, and (4) also combination
and comparison of different experiments is problematic due to diverse experimental
settings and conditions resulting in different levels of precision.
The intuition is, that genes involved in many biological functions tend to be more
conserved and thus evolve more slowly than genes with low pleiotropy. This indeed
seems to be the case, as indicated by the negative correlation between pleiotropy and
dN/dS’. However, the relationship is rather weak and there might be factors besides
the pure number of affected traits, that influence the rate of gene evolution. Our motivation for considering modular pleiotropy as such a factor stems from the finding that
modularity can reduce the cost of complexity (Welch and Waxman, 2003). Although
in theory, we considered important properties of the GP map, in real biological networks it might be much more complicated to unwind the modular structure down to
the single gene level. On the other hand, it might be that the data analyzed in this
project is simply not suited to investigate this hypothesis more thoroughly. To do so,
it would be more appropriate to design an experimental study specifically tailored for
this question. The best way would be to obtain all measures of interest from the same
system. This would include specifying the traits suited to build a modular GP map,
measuring fitness effects of genes on these traits and determining the rate of molecular evolution of the genes over a certain period of time. Ideally, one would need a
genetic architecture that is highly integrated in one part and modular in another part,
such that it would be possible to compare the evolutionary rate of genes in the two
sub-networks.
Hansen (2003) hypothesized that evolvability is maximized by variable pleiotropy.
That mutations indeed have variable effects among genes, and are not universally distributed, is undoubtfully the case as several experimental studies in yeast, nematodes
and mice have shown (Wang et al., 2010). Whether this pattern evolved to maximize
evolvability is still another question not answered yet.
A large number of other problems and questions arise in this context: Results are
highly dependent on experimental design and measurement accuracy. Small effects
are difficult to detect. What is not detected and how important is this? Deleting a
gene could have a different influence on its traits compared to other types of mutations (substitutions, insertions, deletions, etc). Gene deletion influences all traits,
mutation might only influence one or two. Gene deletion experiments exclude essential genes, but mutations in essential genes might still have fitness effects, different
from killing the cell caused by deletion. One could ask whether experimental surveys tend to detect interactions more often in highly expressed proteins as proposed
earlier (Bloom and Adami 2003, 2004; Fraser and Hirsh 2004), leading to a true but
biologically irrelevant pleiotropy-dN relationship. It should also be kept in mind that
not all variation has a pure genetic basis. There might be other sources of variation
influencing the phenotype: transcription, mRNA-transport, translation, protein fold-
27
ing, localization, interactions and histone modifications. All of this potentially affect
the analysis and conclusions we drew from the data investigated in this project.
28
II | Exploring effects of
modular pleiotropy
29
3 | The relatedness of GP map
and G-matrix
Abstract
The genotype-phenotype (GP) map and the genetic (co-)variance matrix (G) are two
well-known interrelated concepts in population genetics that are at the heart of fundamental research in evolutionary biology. Still, we do not know much about their
relationship. How well is the G-matrix determined by the GP map? How do features
of the GP map get translated into the G-matrix and do similar GP maps lead to similar
G-matrices? Can we infer the underlying GP map given we know the G-matrix? We
investigate these questions by constructing 21 different GP maps covering features
like pleiotropy, modularity, mutation effects and asymmetric structure. Simulating
populations on these GP maps gives rise to G-matrices that we analyze with different summary statistics (e.g., eigenanalysis) and matrix comparison methods (e.g.,
Krzanowski Subspace Comparison, random skewers). We find that similar GP maps
usually give rise to similar G-matrices, provided that mutational pleiotropic effects
are correlated. Under this condition, pleiotropy and modularity increase the robustness of G-matrix prediction. On the other hand, modules in the G-matrix correspond
to modules in the GP map, if they are of different sizes.
31
32
3.1
Introduction
Genetics text books teach us that variation at the gene level (in terms of polymorphisms, repeat elements, translocations, etc.) leads to variation at the phenotypic
level (e.g., eye color or quantitative traits, like wing shape). This variation can be
maintained by several forces, like mutation, neutral drift, competition or parasites,
but can also be depleted by strong selection, population bottlenecks or inbreeding.
There is great interest in finding the exact associations between genetic loci and phenotypic traits, not only for answering evolutionary questions, but especially in medical research to identify the genetic basis of human diseases. If these associations are
known, they can be summarized by a GP map. It describes the mapping of genes
or mutations onto phenotypic traits and can be used to predict how genetic variation
translates into phenotypic variation. Typically, the GP map can be constructed from
genome-wide association studies or mapping of quantitative trait loci (QTLs). Unfortunately, these methods are quite laborious and often lack precision or have high
false positive and false negative rates. It is usually much easier to observe phenotypes
and measure traits. Therefore, a lot of research is conducted on the evolution of traits
without actually considering the genetic basis of their variation. One popular concept
in this field is the genetic variance-covariance matrix G. The G-matrix summarizes
the genetic variances of traits and covariances between traits without providing information about the genetic architecture producing these variations. Admittedly, this
is a very useful instrument for studying population dynamics, i.e., to predict how a
population evolves. The drawback is, that it can lead to wrong conclusions if we do
not consider the underlying genetic basis. This is important, since the G-matrix might
change over time, because of changes in allele frequency, drift, selection or migration.
This in turn might change the conclusions from theoretical and empirical studies of
G-matrices.
The two concepts, GP map and G-matrix, are seldom considered together and we
usually either know which genes affect which traits without having an idea about the
genetic constraints that result from this. Or we know how traits covary and change
together without being able to name the exact reason for the correlations. To fully
understand how the genetic architecture enhances or constrains evolution, we need
information about both, the GP map and the correlations between traits. To be able
to omit assumptions about how genes affect traits, we would ideally also need information about the nature of fitness effects of alleles and mutations. Without this
information it is difficult to assess how GP map and G-matrix are related. Will similar
GP maps always result in similar G-matrices? Can we infer any parameters of the
GP map by knowing the G-matrix of a population? Previous work studied the stability and evolution of the G-matrix (Schluter, 1996; Steppan et al., 2002; Jones et al.,
2003; Griswold et al., 2007; Guillaume and Whitlock, 2007; Arnold et al., 2008; Eroukhmanoff and Svensson, 2011; Jones et al., 2012) and compared G-matrices (Mezey
33
and Houle, 2003; Calsbeek and Goodnight, 2009; Roff et al., 2012; Aguirre et al., 2013).
Recently, Berner (2012) investigated the stability of G given several different GP maps.
In his paper he modeled 4 traits affected by up to 28 loci with random allelic effects.
With this setup he studied the effects of pleiotropy and asymmetry in the genetic
basis of traits on the orientation of G’s eigenvectors and the response to selection on
the first trait. One of the major conclusions of this paper is that the orientation of the
eigenvectors has no meaningful relationship with the genetic architecture. However,
there are at least two important criticisms: (1) with random allelic effects, pleiotropy
is hidden and does not create trait correlations, and (2) important findings of previous studies are ignored. For instance, Jones et al. (2003) already showed that size,
shape and orientation of G are rather unstable in absence of mutational correlation.
In this sense, Berner’s work does not provide any new insights, although he explicitly
modeled both, GP maps and the respective G-matrices.
Allelic and mutational effects are usually not random but shaped by selection on
developmental processes and functional integration. As a consequence, traits are usually neither independent units nor completely integrated by universal pleiotropy, but
show some level of modularity. Therefore, we extent previous studies in several ways.
First, we create GP maps with a larger number of traits (10) and loci (100) for three
reasons: (1) most previous work is based on 2-loci – 2-traits models, (2) this dimensionality is closer to many experimental genotype-phenotype association studies, and
(3) it is necessary for including modularity as a feature of the GP map. Second, we
explicitly model different levels of mutational correlation to allow pleiotropy to create
correlations between traits. And third, this is the first study looking at modularity in
this context. Mezey and Houle (2003) investigated the role of modules for the similarity of G-matrices, but their 2-loci – 2-allele model is rather artificial and very restricted.
They find that two G-matrices will always have common principal components when
the two populations have modules in common. However, the modules have to have
exactly the same direction, which is extremely unlikely in real biological systems.
We show that features of the GP map can be preserved in the G-matrix and under
some circumstances the eigenvectors of G reflect the underlying genetic architecture.
3.2
Methods
We construct 21 different GP maps with 10 traits and 100 loci and with differing levels
of pleiotropy, modularity and asymmetry in structure. These can be divided into three
modularity categories: one module (5 GP maps), two modules (7 GP maps) and three
modules (9 GP maps). In the 1-module series all loci affect the same set of traits, but
we decrease pleiotropy from 10 (full pleiotropy) to 2 in steps of two. In the 2-module
series six of the seven GP maps have perfectly separated modules, i.e., loci only affect
traits within and not outside their module. In one case we added between-module
34
effects. And also in 6 out of 7 GP maps there is full pleiotropy within modules. Only
in one case we set the pleiotropic degree of each locus to two. Otherwise the GP
maps differ in module composition, i.e., symmetric/asymmetric distribution of loci
and traits to the two modules. We differentiate between the gene pool size (number of
loci in the module) and the trait pool size (number of traits in the module). Therefore
we create asymmetry in trait pools, gene pools, and both. The 3-module series is
similar to the 2-module series, but the additional module makes more combinations
of asymmetries possible. For a visual representation of the GP maps see Appendix.
The GP maps are numbered sequentially, therefore the last digit in the names does
not have any particular meaning.
To receive corresponding G-matrices we use an additive model of genetic effects,
i.e., each trait value is given by the sum of the genetic effects of the loci affecting the
trait. Therefore we do not consider any epistasis or dominance. Genetic effects for
each locus are drawn from a multivariate normal distribution with mean zero and
covariance matrix M, given as

1

σ
M =  2,1
 ...
σp,1
σ1,2
...
···
...
..
..
.
···
.
σp,1−p

σ1,p
.. 
. 
,
σ1−p,p 
1
(3.1)
where p is the pleiotropic degree of the locus (number of traits affected) and σi,j
is the correlation between the traits. We set the correlation between genetic effects
,rµ , to 0 (no correlation – random effects), 0.3 (weak correlation), 0.6 (intermediate
correlation) and 0.9 (strong correlation). The variance for each trait is set to 1 (diagonal elements). The additive genetic variance-covariance matrix G is then calculated
from a simulated population of 100 individuals that obtain their trait values by the
procedure explained above. By this design, the amount of genetic variation for each
trait is simply given by the number of loci affecting it. Finally, we analyze G-matrices
from 1000 replicates of each GP map. First, we perform an eigendecomposition of the
G-matrices to investigate the distribution of eigenvalues and the loadings of the traits
on the eigenvectors. Eigenvectors of a G-matrix are orthogonal axes of a new trait
space, where each eigenvector is a linear combination of the original traits. The first
eigenvector accounts for the largest amount of variation in the data, the second eigenvector accounts for the largest amount of the remaining variation, and so on. The
eigenvalues of a G-matrix describe the amount of genetic variation present in each
eigenvector. Eigenvalues close to zero indicate phenotypic space without genetic variation, i.e., constrained evolutionary directions. What is usually found for G-matrices
estimated from natural populations is an L-shaped distribution of eigenvalues with a
large leading eigenvalue and many small eigenvalues close to zero. This is often indicating a large module that harbors a size component (with traits like length, height,
35
weight, etc.). We specifically investigate modularity in the GP map as a factor shaping
the G-matrix. Modularity in the G-matrix can for example be detected by looking at
the correlations between traits. If there are large correlations within a set of traits
but only small correlations with other traits, this set can be considered a module.
Another hint on modularity is given by the trait loadings on the eigenvectors of G.
The loadings are the contributions of the traits to the principal components. Since the
first eigenvector explains the largest proportion of genetic variation, we would expect
that traits of the largest module, with respect to the gene pool, have highest loadings
on this first eigenvector (provided that all loci have the same mutational variation).
A third possibility for detecting modularity is provided by the selection skewer approach. Using the Lande equation ∆z̄ = Gβ (Lande, 1979), one can predict the change
of the mean trait values (∆z̄) due to a certain selection vector (β). We apply selection
on the first trait only (β = (1, 0, ..., 0)) to demonstrate the effect of this method.
Finally, pairwise G-matrix comparisons are conducted to test (i) the robustness of Gmatrix estimates given the same GP map, and (ii) the (dis-)similarity between matrices
with different underlying genetic architectures. These comparison methods are Roff’s
T test, co-inertia analysis, Krzanowski subspace comparison and the random skewers
approach.
3.2.1
G-matrix comparison
Roff’s T test is an element-wise test that determines the difference between two matrices based on the differences between matrix elements at the same position (Roff
et al., 1999; Bégin and Roff, 2001). Thus, it provides a measure for the difference in
covariances between traits. Co-inertia analysis is a multivariate method that identifies
trends or co-relationships in multiple datasets. Compared to other correlation analysis methods it is very flexible and allows the analysis of large datasets. It calculates
the shared structure of two hyper-spaces, where the co-inertia between two hyperspaces is the sum of squares of the covariances between all variables pairs (Dolédec
and Chessel, 1994). From this, one can obtain a coefficient of correlation between two
datasets. Krzanowski subspace comparison (KSC) provides a measure of shared geometry of matrices (Krzanowski, 1979). It focuses on the space containing variation
and how this is oriented. Thus, it tests whether the subspace containing most of the
variation is in common between matrices. It is a very fast method that simply uses
the eigenstructure of matrices. Random skewers are a number of selection gradients
individually applied to a given G-matrix to infer the average response to selection
(Cheverud, 1996). As described above, we use Lande’s equation to apply 10,000 random selection gradients β to G, track the responses to each of them and calculate
the mean phenotypic change of each trait ∆z̄. The entries of the vector of selection
gradients β are randomly drawn from a uniform distribution between -1 and 1. β is
then normalized to a magnitude of 1 (|β| = 1). This means that the fitness optimum
36
lies somewhere on a circle with radius 1 around the centre of G. Two matrices are
challenged with the same set of random skewers and we calculate the correlation of
their responses as a measure of similarity in evolvability.
3.3
3.3.1
Results
Random mutation effects
Setting the mutational correlation of genetic effects to zero is equivalent to simulating random effects. Under this assumption, trait correlations are also random with
average zero (Fig. 3.1), but with extremes at -0.4 and +0.4. Therefore, there will be no
correlated responses of other traits when we select on the first trait, even if modules
are explicitly modeled (Fig. 3.2c). The magnitude of the response depends on the gene
pool size (amount of genetic variation). For most G-matrices we observe an almost
linear decline of eigenvalues. Asymmetry leads to a slight L-shape of the eigenvalue
distribution (Fig. 3.2b). The more asymmetric the GP map is, the more L-shaped the
eigenvalue distribution gets. For the 1-module cases all traits contribute equally to
the loadings on the eigenvectors. For the modular GP maps (2- and 3-module cases)
the corresponding G-matrices show no differences in loadings when gene pools are
of same size. Larger modules have higher loadings on all eigenvectors (Fig. 3.2d-f).
rµ = 0
rµ = 0.3
rµ = 0.6
rµ = 0.9
1.0
Traits
0.8
0.6
0.4
0.2
Traits
0.0
Figure 3.1. Heat map of average G-matrices derived from GP map 3mod.9 (see Fig. 3.2a) based
on different levels of mutational correlation. Red colors indicate high correlation between
traits and light colors indicate no or low correlation.
We tested the robustness of G-matrix estimates from each GP map by sampling
1000 replicates. Given random genetic effects, there is quite a high variance in the
matrix comparison measures for all GP maps (Fig. 3.4). Also, except from KSC, there
are largely no differences between GP maps, that is, no GP map seems to enhance
robustness of G (Fig. 3.3). The average co-inertia correlation is 0.78 and the average
number of shared dimensions is 3 out of 5. One trend we do see, is that decreasing
37
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(e)
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(f)
Figure 3.2. G-matrix statistics for 1000 replicates derived from GP map 3mod.9 (a) with
random mutation effects. Eigenvalues decline almost linearly (b), response to selection on
trait 1 is only seen for trait 1 (c) and loadings correspond to module sizes, but no differences
between the eigenvectors (d-f). Compare to Fig. 3.5.
pleiotropy decreases Roff’s T, but also decreases the number of shared dimensions.
On the other hand, G-matrices share more dimensions when there is one dominating
module (large gene and trait pools) (Fig. 3.3b). The random skewer responses have
all very high correlations (mean = 0.916) and there is almost no effect of pleiotropy or
modularity (Fig. 3.3c). Only G-matrix replicates of GP maps with asymmetric gene
pools show a slightly higher response correlation (mean = 0.935).
Comparing G-matrices of different GP maps reveals high similarities in most measures, irrespective of their structure (Fig. 3.6). Roff’s T is usually very small (between 0.01 and 0.06) and seems to decrease with increasing modularity and increasing
pleiotropy. The co-inertia correlation and the number of shared dimensions vary more
than the other two statistics. KSC lies on an intermediate level (2 - 4 out of 5) and
there is a greater similarity when the largest gene pools of two GP maps are of similar size. Across G-matrices from different GP maps the random skewer correlations
1.0
0.9
0.8
0.6
0.7
rµ = 0.9
rµ = 0.6
rµ = 0.3
rµ = 0
0.5
Mean Co−intertia
38
1mod.1 1mod.3 1mod.5 2mod.2 2mod.4 2mod.6 3mod.1 3mod.3 3mod.5 3mod.7 3mod.9
GP maps
4
3
1
2
rµ = 0.9
rµ = 0.6
rµ = 0.3
rµ = 0
0
Mean KSC
5
(a)
1mod.1 1mod.3 1mod.5 2mod.2 2mod.4 2mod.6 3mod.1 3mod.3 3mod.5 3mod.7 3mod.9
GP maps
1.00
0.95
0.90
0.85
rµ = 0.9
rµ = 0.6
rµ = 0.3
rµ = 0
0.80
Mean RS correlation
(b)
1mod.1 1mod.3 1mod.5 2mod.2 2mod.4 2mod.6 3mod.1 3mod.3 3mod.5 3mod.7 3mod.9
GP maps
(c)
Figure 3.3. Means of within-GP map G-matrix comparisons for all GP maps. Colors indicate
different levels of mutational correlation. Dashed vertical bars separate the modularity classes.
Lines are drawn to guide the eye.
0.015
0.005
0.010
rµ = 0.9
rµ = 0.6
rµ = 0.3
rµ = 0
0.000
Co−intertia variation
39
1mod.1 1mod.3 1mod.5 2mod.2 2mod.4 2mod.6 3mod.1 3mod.3 3mod.5 3mod.7 3mod.9
GP maps
0.1
0.2
0.3
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rµ = 0.6
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0.5
(a)
1mod.1 1mod.3 1mod.5 2mod.2 2mod.4 2mod.6 3mod.1 3mod.3 3mod.5 3mod.7 3mod.9
GP maps
8e−04
4e−04
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rµ = 0.6
rµ = 0.3
rµ = 0
0e+00
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(b)
1mod.1 1mod.3 1mod.5 2mod.2 2mod.4 2mod.6 3mod.1 3mod.3 3mod.5 3mod.7 3mod.9
GP maps
(c)
Figure 3.4. Variances of within-GP map G-matrix comparisons for all GP maps. Colors indicate different levels of mutational correlation. Dashed vertical bars separate the modularity
classes. Lines are drawn to guide the eye.
40
are generally very high. Contrary to the within replicates comparison, asymmetry in
gene pools leads to lower correlations between different GP maps.
3.3.2
Correlated mutation effects
Correlations in the mutational effects also create correlations among traits with the
same genetic basis. Therefore, we obtain modular G-matrices with correlations among
traits of the same module and (on average) no correlation among traits of different
modules (Fig. 3.1). The magnitude of (random) correlation between traits of different modules is equal to that simulated with random mutation effects. The average
magnitude of correlation between traits of the same module is equal to the correlation between mutational effects, which we specify (0.3, 0.6 or 0.9). The correlations
are slightly smaller when we allow intermodule effects and decrease with decreasing
pleiotropy. There is a correlated response of traits belonging to the same module as
the selected trait (Fig. 3.5c). The magnitude of response depends on the strength of
trait correlation, pleiotropy and gene pool size.
Only for very low pleiotropy, the eigenvalues of G decline linearly. In the 1-module
cases, increasing pleiotropy leads to a larger first eigenvalue. The others are small
and close to zero for large mutational correlations. In the 2- and 3-module cases,
the eigenvalue distribution is L-shaped with most eigenvalues close to zero for large
mutational correlations (Fig. 3.5b). The leading eigenvalues correspond to module
sizes (i.e., one dominating module – one dominating eigenvalue, two equal modules
– two large eigenvalues). In the 1-module cases there are no differences in eigenvector loadings, neither across GP maps, nor across eigenvectors. Only for the largest
mutational correlation (rµ = 0.9) the variance in loadings among replicates increases
with decreasing pleiotropy. For the modular GP maps the eigenvector loadings of G
largely coincide with the modules in the GP map. There are no differences in trait
loadings when modules (gene pools) are of equal size or pleiotropy is very low. When
modules have different sizes, traits of the largest module give most contribution to the
loadings on EV1. Similarly, the largest loadings on EV2 come from traits of the second
largest module and traits of the third largest module have highest loadings on EV3
(Fig. 3.5d-f). In the 3-module cases there are almost no differences between EV2 and
EV3 when the two smaller modules are equal. This pattern is most pronounced with
the highest mutational correlation. For smaller rµ the loadings on EV2 and EV3 do
not always correspond to module sizes. Often, the patterns in EV3 are similar to those
in EV1 or EV2.
When the GP map is not modular (1-module cases) even high mutational correlations do not necessarily increase robustness of G-matrix estimates. For instance, the
mean co-inertia correlation decreases for non-modular GP maps with high pleiotropy
(and variance increases) (Figures 3.3a and 3.4a). For these 1-module cases, the matrix
comparisons within the 1000 replicates of the same GP map result in values similar to
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Figure 3.5. G-matrix statistics for 1000 replicates derived from GP map 3mod.9 (a) with
mutational correlation rµ = 0.9. One leading eigenvalue, two following and the rest being
close to zero (b). Response to selection on trait 1 results in a response of trait 1 and a large
correlated response of traits in the same module (c). Loadings correspond to module sizes
and differ between eigenvectors – largest module has highest loadings on EV1, second largest
module has highest loadings on EV2 and smallest module has highest loadings on EV3 (d-f).
Compare to Fig. 3.2.
matrices generated with random effects (see above). The only difference can be seen
in the random skewer approach. The response correlation increases with increasing
pleiotropy and a large mutational correlation leads to a smaller variance among replicates. However, the whole picture changes when we analyze modular GP maps. For
increasing mutational correlation the co-inertia correlations increase and the variances
in these values across replicates decrease (Figures 3.3a and 3.4a). Only GP maps with
low pleiotropy are almost blind to these improvements of robustness. They do not
differ much from the cases without modules or without mutational correlation. Furthermore, replicate G-matrices from GP maps with one dominating module have the
largest T values, whereas low pleiotropy leads to small T values (not shown). The
42
highest similarity in shared dimensions is found for GP maps with asymmetry in
gene pool (but not one large dominating module) and the lowest number of shared
dimensions is found for GP maps with equal gene pool sizes (Fig. 3.3b). The random
skewer response correlation increases with increasing mutational correlation and the
variance among replicates decreases (Figures 3.3c and 3.4c). GP maps with asymmetric gene pools lead to larger response correlations and those with low pleiotropy
lead to smaller correlations. Generally, we still see high variances in G-matrix comparison measures among replicates, but these variances are decreasing for increasing
mutational correlation. Co-inertia correlations and shared dimensions increase with
increasing rµ . Low pleiotropy leads to small values for Roff’s T but also low shared dimensions. Differences between G-matrices are smallest with asymmetric asymmetries
(many genes affecting few traits and few genes affecting many traits) and G-matrices
share more dimensions for asymmetric gene pools, but not one dominating module (large gene and trait pool). In summary, modularity, pleiotropy, asymmetry and
mutational correlation increase robustness in G-matrix estimates.
Comparing G-matrices from different GP maps largely reveals that similar features
in the GP maps also lead to similar G-matrices. The different comparison measures focus on different aspects of G. There is a high co-inertia correlation between G-matrices
with same (or similar) trait distributions. However, gene pool size is unimportant. We
also find fairly high correlations when at least one module (trait pool size) is similar
and between the low pleiotropy cases and 1-module cases. Roff’s T on the other hand
is smallest (i.e., the difference between matrices is smallest) when GP maps have
similar gene pool distributions. Overall, similarity seems to increase together with
modularity. Unequal asymmetries cause high differences. The same is true for the
number of shared dimensions and the random skewer correlations. For all measures,
higher mutational correlations enforces similarities and differences.
We use Newman’s modularity (Newman, 2006) as a measure for GP map integration to have a criterion for comparing GP maps. It computes a value between 0
and 1 as the ratio of within to between module connections in the map, where values close to 0 indicate no modularity and values close to 1 a high degree of modular
subdivision. We compare G-matrices derived from GP maps with similar Newman
modularity and find that they generally do not show a high level of similarity. We
see only low to medium co-inertia correlations, high T values and intermediate KSC
values and response correlations. The larger the mutational correlation, the more pronounced are these dissimilarities. On the other hand, if we compare G-matrices from
GP maps that are visually similar, they show a higher degree of similarity. Co-inertia
correlations of GP maps with same structure are very high, Roff’s T is rather small,
there are quite many shared dimensions (around 4 of 5) and response correlations
are also mostly very high. In this case, greater mutational correlation leads to higher
similarities.
43
rµ = 0
rµ = 0.3
rµ = 0.6
rµ = 0.9
GP maps
1.0
0.8
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Co−inertia
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1.0
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RS correlation
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GP maps
1.0
0.8
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Roff's T
0.0
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GP maps
1.0
0.8
0.6
0.4
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GP maps
KSC
0.0
(d)
Figure 3.6. Heat maps for comparison measures co-inertia correlation (a), random skewer response correlation (b), Roff’s T (c) and Krzanowski subspace criterium (d) between G-matrices
derived from different GP maps based on different levels of mutational correlation. From each
GP map we estimated one G-matrix and did pairwise matrix comparison between all of them.
Red colors indicate high correlation between GP maps and light colors indicate no or low
correlation.
44
Co−inertia correlation
RS correlation
1.0
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2mod.5
2mod.4
2mod.3
3mod.3
3mod.7
3mod.2
3mod.1
0.8
0.6
0.4
0.2
(a)
(b)
Roff's T
KSC
1.0
0.6
0.4
0.2
0.0
1.0
1mod.5
1mod.2
2mod.5
1mod.3
2mod.4
1mod.1
1mod.4
2mod.2
2mod.3
3mod.5
3mod.2
2mod.1
3mod.9
3mod.4
3mod.6
2mod.6
2mod.7
3mod.7
3mod.8
3mod.3
3mod.1
0.8
0.6
0.4
0.2
0.0
1mod.5
1mod.2
2mod.5
1mod.3
2mod.4
1mod.1
1mod.4
2mod.2
2mod.3
3mod.5
3mod.2
2mod.1
3mod.9
3mod.4
3mod.6
2mod.6
2mod.7
3mod.7
3mod.8
3mod.3
3mod.1
1mod.2
1mod.1
2mod.3
2mod.4
3mod.4
3mod.6
2mod.2
2mod.6
2mod.1
1mod.3
2mod.5
1mod.4
3mod.3
3mod.7
3mod.1
3mod.9
3mod.2
3mod.5
3mod.8
2mod.7
1mod.5
1mod.2
1mod.1
2mod.3
2mod.4
3mod.4
3mod.6
2mod.2
2mod.6
2mod.1
1mod.3
2mod.5
1mod.4
3mod.3
3mod.7
3mod.1
3mod.9
3mod.2
3mod.5
3mod.8
2mod.7
1mod.5
(c)
0.8
2mod.6
2mod.2
2mod.1
1mod.1
1mod.2
1mod.3
1mod.4
3mod.9
3mod.8
1mod.5
2mod.7
3mod.6
3mod.5
3mod.4
2mod.5
2mod.4
2mod.3
3mod.3
3mod.7
3mod.2
3mod.1
0.0
2mod.6
2mod.1
2mod.2
3mod.2
2mod.7
1mod.3
1mod.4
1mod.1
1mod.2
1mod.5
3mod.5
3mod.6
3mod.4
3mod.9
2mod.3
2mod.5
2mod.4
3mod.3
3mod.8
3mod.7
3mod.1
2mod.6
2mod.1
2mod.2
3mod.2
2mod.7
1mod.3
1mod.4
1mod.1
1mod.2
1mod.5
3mod.5
3mod.6
3mod.4
3mod.9
2mod.3
2mod.5
2mod.4
3mod.3
3mod.8
3mod.7
3mod.1
(d)
Figure 3.7. Heat maps with hierarchical clustering for comparison measures co-inertia correlation (a), random skewer response correlation (b), Roff’s T (c) and Krzanowski subspace
criterium (d) between G-matrices derived from different GP maps with a mutational correlation rµ = 0.9. From each GP map we estimated one G-matrix and did pairwise matrix
comparison between all of them. Red colors indicate high correlation between GP maps and
light colors indicate no or low correlation.
0.8
0.6
0.4
0.2
0.0
45
3.3.3
Summary of results
Random mutation effects do not allow for good associations between GP map and
G-matrix. In these cases, differences in the structures of GP maps will generally not
translate into differences in the corresponding G-matrix. Also the robustness of Gmatrix estimates from a single GP map is quite low. We do see effects of pleiotropy
and asymmetry in gene pools for some statistics. Lower pleiotropy leads to smaller
T values and fewer shared dimensions, and asymmetric gene pool sizes lead to a
higher random skewer response correlation among replicates of the same GP map
but a lower response correlation between G-matrices from different GP maps.
The main effect of mutational correlation is an amplification of the structural features of GP maps through phenotypic correlation. Modularity in G-matrices will only
appear with robust trait correlations, and these in turn are responsible for a more
accurate comparison of G-matrices. It is possible to detect the modular structure of
G by tracking the correlated responses to selection on a single trait, and by analyzing
the eigenvalues and trait loadings on the first eigenvectors of the matrix. Especially
with high mutational correlations, the modules in GP can be identified by trait loadings on the eigenvectors of G. Overall, an increase in rµ increases all similarities and
dissimilarities between G-matrices and it stabilizes G.
3.4
Discussion
The value of G-matrix approaches has been questioned recently (Berner, 2012). The
author criticized that the information provided by the eigenvectors of G does not help
understanding the genetic architecture. However, this was purely based on theoretical
models with questionable assumptions. Mutational effects are usually not random
and pleiotropy tends to occur in a modular fashion. We included these two important
facts in our simulations to provide a more realistic framework for investigating the
relationship between GP map and G-matrix. Additionally, the eigenvectors of G are
not the only determinants of the characteristic of the G-matrix. Therefore, we also
applied matrix comparison methods.
We agree with Jones et al. (2003) and Berner (2012) that G-matrix estimates are
rather unstable in the absence of mutational correlations. On the one hand, the variances of matrix comparison measures are quite large within estimates from the same
GP map, and on the other hand estimates of G from different GP maps cannot be
well differentiated. However, we also confirm Berner’s result that asymmetry in the
distribution of loci among the gene pools enhances the stability of G and helps differentiating G-matrices based on different GP maps. As Jones et al. (2003) suggested,
mutational correlation greatly increases robustness of G. This is usually associated
with large evolutionary constraints, because the genetic variation is compressed in
one single dimension. We demonstrate in the next chapter, that modularity can relax
46
these constraints and enhance the adaptive potential of a population. The important
new insights of the focal chapter are the role of modularity for G-matrix stability
and for the association between GP map and G-matrix. We find that modularity
can increase robustness of G-matrix estimates by increasing the mean and decreasing
the variance of matrix comparison measures. Also, modularity in the GP maps enhances the similarities and dissimilarities between G-matrices. If we allow mutational
correlation to create structural features (like modules), G-matrices will reflect these
features. Hence, two G-matrices will be more similar, when the underlying GP maps
are similar and they will differ more when the underlying GP maps are rather distant.
Most importantly, we show that the eigenvectors of G can be used to draw meaningful conclusions about the genetic architecture. A requirement for this is mutational
correlation and heterogeneity in module sizes.
Our results are still based on rather artificial GP maps and the improvements we
find largely depend on the magnitude of mutational correlation. Nevertheless, we
are confident that our assumptions are valid and realistic. For example, in chapter
6 of this thesis we find large mutational correlations for pleiotropic effects on drug
susceptibility in HIV. It has also been shown that GP maps are modular (Wang et al.,
2010), and probably not all modules have the same size. Therefore the association
between GP map and G-matrix can be high and valuable conclusions about the genetic
architecture and its evolutionary potential can be drawn from the G-matrix.
47
3.5
3.5.1
Appendix
GP maps used for simulations
Figure 3.8. GP maps with one module. All 100 loci (elliptic shape) affect the same set of
10 traits (quadratic shape). The GP maps differ in their level of pleiotropy, indicated by p,
decreasing from complete pleiotropy (p = 10) to low pleiotropy (p = 2).
48
Figure 3.9. GP maps with two modules and different distributions of loci and traits among the
modules (asymmetries in trait and gene pool size). GP map 2mod.6 includes a low degree of
inter-module connections (dashed arrows) and 2mod.7 is a case of low pleiotropy. Otherwise,
there is complete pleiotropy within modules and no connections between modules.
49
Figure 3.10. GP maps with three modules and different distributions of loci and traits among
the modules (asymmetries in trait and gene pool size). GP map 3mod.7 includes a low degree
of inter-module connections (dashed arrows) and 3mod.8 is a case of low pleiotropy. Otherwise, there is complete pleiotropy within modules and no connections between modules.
50
4 | Why modularity is advantageous:
a G-matrix approach
Abstract
To adapt to a new or changing environment, a population has to show some variation
in the characters under selection. Often, the ability of one trait to adapt is constrained
by other traits with which it covaries. We are interested in the adaptive potential given
by the G-matrix, and specifically how modularity influences this adaptive potential.
With the help of stochastic simulations we investigate how modular G-matrices affect
the response to selection. Does modularity increase evolvability? And which factors
does modularity exactly act on? In this study we describe how in theory a modular
organization on the phenotypic trait level influences adaptation. We need to understand this to tackle a rather ambitious goal, namely to explain how higher (complex)
organisms evolved and still adapt to environmental challenges. In this work we show
that a modular G-matrix is beneficial for adaptation, because it reduces the constraints
caused by genetic correlations. It brings the population closer to the fitness optimum
by increasing the probability that the response to selection is aligned with the selection
gradient. Furthermore, we suggest that knowing the effective number of dimensions
of the G-matrix is not sufficient to explain the variation in genetic constraints.
51
52
4.1
Introduction
One of the big questions in evolution is how higher organisms can evolve in spite of
their complex multivariate architecture. Correlations between phenotypic traits create
evolutionary constraints. A mutation that is beneficial for one trait might be detrimental for others. Based on Fisher’s geometric model (Fisher, 1930) it was shown that the
more complex an organism is, the more constraints it has to face and the harder it is to
react to environmental changes and adapt to a new fitness optimum. This is known as
the ’cost of complexity’ (Orr, 2000). However, there is growing evidence that the ’universal pleiotropy’ assumption of Fisher’s model is violated and genes usually affect
only a few phenotypic traits (Dudley et al., 2005; Su et al., 2010; Wang et al., 2010). This
fact might at least influence our (theoretical and experimental) estimates of complexity (e.g. Chevin et al., 2010; Lourenço et al., 2011), or can even relax the assumption
of a high cost of complexity (Wagner et al., 2008). An alternative loophole from this
complexity dilemma could be provided by modularity (Welch and Waxman, 2003;
Griswold, 2006; Wagner et al., 2007; Wang et al., 2010). In biology, modularity appears
on several levels: from cellular organization to developmental processes. In the focal
context the underlying architecture is the genotype-phenotype map (GP map), which
consists of a set of genes and a set of traits. The GP map is modular when gene-trait
connections are predominantly within modules (i.e., between elements of the same
module) rather than between modules. Traits are clustered into modules when a set
of traits is highly correlated with each other and not, or only weakly correlated with
other traits. In theory, the advantage of modularity is described as the ability of one
module to evolve without interfering with other traits. Because the traits of a module
usually share similar functions or developmental processes, it is more likely that a
beneficial mutation in a gene affecting one of those traits also has a beneficial effect
on other traits in the same module. This is also known as modular pleiotropy (see
Wagner et al., 2007).
The assumption we make in this work is that the underlying genetic architecture
(GP map) creates a modular G-matrix. This requires that the GP map already possesses a modular structure acquired by modular pleiotropy. We have shown in the
previous chapter that patterns in the GP map can be reflected by the same patterns
in the G-matrix. The transition from GP map to G-matrix can be explained via the
mutational variance matrix M. When considering no other causes than mutation for
covariance between traits, G is proportional to M. Assuming that the GP map translates into a G-matrix where structures like modularity are conserved, G becomes a
handy tool to look at variation in correlation patterns among traits. In addition, once
the G-matrix is obtained, its simple algebraic form makes it mathematically easy to
work with.
There are three main types of studies that deal with G-matrices. First, pure theoretical/methodological approaches that aim at explaining the nature of G itself, i.e.,
53
how it evolves and by which factors its shape and adaptive potential is influenced
(Phillips and Arnold, 1989; Arnold et al., 2001; Jones et al., 2003; Mezey and Houle,
2003; Jones et al., 2007; Guillaume and Whitlock, 2007; Arnold et al., 2008; Guillaume,
2011). Second, methodologically motivated studies where theoretical principals are
developed and finally applied to real data to either reveal properties of natural populations or to compare different estimates of genetic constraints, or both (Lande, 1979;
Schluter, 1996; Cheverud, 1996; Hansen et al., 2003; Klingenberg et al., 2004; Hine and
Blows, 2006; Hansen and Houle, 2008; Marroig et al., 2009; Agrawal and Stinchcombe,
2009; Kirkpatrick, 2009; McGuigan and Blows, 2010). Third, there are some studies
that are more experimentally motivated and estimate G from natural populations to
identify differences between species (e.g. Colautti and Barrett, 2011; Eroukhmanoff
and Svensson, 2011). Our work will find its place in the first category. The important
difference is that we explicitly model G-matrices with different degrees of modularity, i.e., we construct a sequence of genetic architectures that might reflect the path of
evolution towards more complex organisms through increased modularity.
There is no doubt that modularity exists and that it plays a role in evolution. We
are, however, far from understanding the real significance of modularity. For example, how does it interact with other forces of evolution, i.e. selection, migration,
mutation or drift? Evidence for modularity is extensive. Examples include the division of the mouse mandible into the ascending ramus and the alveolar region (e.g.
Mezey et al., 2000), functional modules in protein networks (e.g. Campillos et al.,
2006), modular structures in GP maps of yeast, nematode and mouse (Wang et al.,
2010) and modules in other biological and also social networks (see e.g. Newman,
2006; Wagner et al., 2007). However, besides finding and describing modules in natural systems, most theoretical research on modularity has so far focused on how it
arises and evolves (Wagner and Altenberg, 1996; Lipson et al., 2002; Hansen, 2003).
This is a very interesting and important question itself, but it usually does not answer
in which way modularity exactly acts on the process of adaptation. The G-matrix
concept is a promising attempt into this direction.
We will first introduce the concept of a modular G-matrix. In the methods section
we describe our stochastic simulations and the parameters we use to characterize G
and its adaptive potential. These parameters will include summary statistics like the
effective number of dimensions, respondability and autonomy, as well as measures
of selection response like the angle between direction of selection and direction of
response, the distance to the optimum after selection and the evolvability. Finally, we
present results and discuss their relevance with respect to contributions in the field of
theoretical evolution.
54
4.1.1
Modularity in the G-matrix
In biology, modularity describes the concept that organisms or intra-organismal organizations like metabolic pathways are composed of (quasi-) independent modules.
Since the G-matrix reflects the properties of the genotype-phenotype map in terms of
correlations between phenotypic traits, we can also apply the concept of modularity
here.
A modular G-matrix can be divided into blocks (modules) of correlated traits,
where traits within a module are highly correlated to each other, but between traits of
different modules there is no correlation. The benefit from such an architecture is that
a block of traits can evolve independently from other traits, i.e., without interfering
with other parts of the organism.

0.3
0.1
G1 = 
0.1
0.1
0.1
0.3
0.1
0.1
0.1
0.1
0.3
0.1

0.1
0.1

0.1
0.3


0.3 0.1 0
0
0.1 0.3 0
0

G2 = 
0
0 0.3 0.1
0
0 0.1 0.3
G1 is a matrix without modularity (the amount of covariation is the same
between all pairs of traits). G2 is modular, with two blocks of two traits each.
When correlations among traits are very high, we expect modular G-matrices to
lead to a ’better’ response to selection than completely integrated G-matrices, because
they are less constrained in their adaptive potential. Better here means closer to the
fitness optimum.
4.1.2
Selection on G
The response to selection can be described by the multivariate extension of the breeder’s
equation (Lande, 1979)
∆z̄ = Gβ,
(4.1)
where β is the vector of selection gradients and ∆z̄ is the response to selection in
terms of phenotypic change in mean trait values. Thus, each trait’s response to selection depends on the directional selection acting on itself and the correlated selection
acting on the other traits.
The direction of G with the highest genetic variance is called the genetic line of
least resistance and is usually termed gmax (Schluter, 1996). This corresponds to the
major principal component (largest eigenvector) of G. Response to selection will be
favored in this direction and the greatest response is expected when selection is acting
in the same direction as gmax . Selection acting in another direction will result in a
displaced response, i.e., it will take a direction which is between β and gmax . This
55
will lead to a curved trajectory of adaptation and a longer time until the optimum is
reached, compared to a straight trajectory when gmax and β are aligned. It should be
mentioned that we assume an adaptive landscape (Simpson, 1953) with a single fitness
optimum and β is the vector pointing to this optimum. Thus, the population under
investigation is not at the fitness optimum and experiences directional selection.
4.2
Methods
In order to understand how modularity influences the adaptive potential of a species,
we simulate several types of matrices with different properties. We are most interested
in comparing G-matrices with different number of modules, and especially comparing non-modular to modular matrices. However, we also investigate the effects of
different degrees and types of variances and covariances among traits and various
sizes of G. The focus of this work lies on two different dimensions and two different approaches. The dimensions used to describe G are correlation and modularity.
Surely, these two might act jointly, but we separate them in our framework by keeping
the within-module correlation constant. The effects of trait correlations are already
well studied. Here, we explicitly look at the effect of modularity at a given level of
correlation. The approaches to characterize the adaptive potential of G are static (summary statistics) and dynamic (random skewers (Cheverud, 1996)). Summary statistics
capture the genetic variation of the traits and the extent to which correlations among
them confine the response to selection. However, they do not take the direction of
selection into account. Thus, they cannot tell us whether the potential properties of
G really apply in nature, i.e. whether or not selection acts in the direction where
the highest evolvability or the largest constraints are expected. The random skewers
approach, on the other hand, is sensitive to the number of traits, i.e. it takes the organismal complexity into account. It also cannot make predictions about the direction
of selection, but it reveals the average response to any selection gradient.
4.2.1
Generating G-matrices
In this work, most G-matrices are generated such that they have equal sized modules
with traits within modules being correlated and traits between modules being uncorrelated. For comparison we additionally generate a G-matrix with two modules of
different size. In what follows, we will distinguish the two 2-module cases as ’2d’
(two different) and ’2e’ (two equal). See Figure 4.1 for a visualization. Depending on
the number of traits n it is of course possible that the modules can have unequal sizes,
e.g., for n = 10 and 4 modules there would be two modules with two traits each and
two modules with three traits each. Note that a G-matrix without any correlations
56
among traits can be seen as a matrix without modules, but also as a matrix with n
modules where each trait is totally independent.
Figure 4.1. Different modular G-matrices. The shaded areas indicate that the traits within
these ’blocks’ are correlated, whereas traits in white blocks are uncorrelated. 1 One module:
all traits are correlated to each other (total integration). 2d Two modules of different size: the
first module contains 1/3 of the traits and the second module contains the remaining 2/3. 2e
Two modules of equal size: 1/2 - 1/2. 3 Three modules of equal size: 1/3 - 1/3 - 1/3. 4 Four
modules of equal size: 1/4 - 1/4 - 1/4 - 1/4.
For the genetic variances (diagonal elements) we mainly choose a constant value
of 0.2 for all traits (homogeneous variance). For a comparison we also generated Gmatrices with higher and lower trait variances. It is reasonable to use homogeneous
variances since the traits usually have different measurement units and have to be
scaled prior to analysis (see for example Hansen and Houle, 2008). However, we also
implement a scenario with some variation in the genetic variances (heterogeneous
variance). The genetic covariances between pairs of traits (i, j) within modules depend on the given level of correlation (set to cor ∈ {0, 0.25, 0.5, 0.75, 0.99}). They are
√
√
calculated as covij = cor ∗ σii ∗ σjj , where σii and σjj are the genetic variances of
traits i and j. The covariances between pairs of traits from different modules are set
to zero. Because we only take positive values for the correlation, it follows that all covariances in these G-matrices are greater or equal to zero. This is a valid assumption
since most genetic correlations are positive (Roff, 1996).
To obtain a perfectly modular G-matrix, one gene would not be allowed to affect
two traits that are part of different modules, because this would create correlation
between these traits. This is of course not what we see in nature, where we usually find at least some degree of correlation between most pairs of traits, i.e., a lot
of ’background’ correlation. We account for this by also generating some G-matrices
with low levels of between-module correlation. In this alternative scenario we generate random G-matrices from a Wishart distribution. This leads to symmetric, positive
semi-definite matrices with positive and negative covariances. To create modules, we
rearrange the values of the random G such that the covariances with highest absolute
values are distributed among the within-module trait pairs. With this design we produce modular random G-matrices that basically have the same structure as the ones
described above, but with additional low background correlation for between-module
trait pairs. However, to answer the main questions on how modularity in principle
affects the response to selection, it is valid to assume a simplified genetic architecture
with distinct, non-overlapping modules.
57
For all generated G-matrices, we only allow those that are positive semi-definite,
which is a requisite for a covariance matrix, because negative eigenvalues have no biological meaning. This criterion is always true for random G-matrices from a Wishart
distribution and for G-matrices with homogeneous trait variances. For the other cases
(especially the rearranged matrices), only between 1 in 10 to 1 in 1,000 of the generated
G-matrices are positive semi-definite. For large matrices (n >> 10) it is sometimes impossible to fulfill this criterion.
4.2.2
Summary statistics of G
This is the static part of the G-matrix analysis. Only the matrix itself has to be known
to calculate the summary statistics. The G-matrix is well characterized by its eigenvectors and eigenvalues λ. The leading (largest) eigenvector and its corresponding
eigenvalue describe the first principal component of the matrix, which is the major
axis of genetic variation gmax . However, from the eigenvalues several other interesting theoretical properties of G can be calculated. Kirkpatrick (2009) proposed two
measures, effective dimensionality nD and maximum evolvability emax . nD describes the
number of independent dimensions that selection can effectively act on and ranges
between 1 (all genetic variation lies in a single dimension) and n (all traits have equal
amounts of variation and are uncorrelated). emax refers to a combination of traits with
maximum genetic variation for proportional change. Three other summary statistics,
proposed by Hansen and Houle (2008) under the scenario of random selection gradients β, are average measures of conditional evolvability c̄, respondability r̄ and autonomy
ā. These statistics are valid under the assumption that the mean length of the selection
gradient equals one (E[|β|] = 1). The conditional evolvability c̄ is a general measure
of genetic constraints of all directions in phenotypic space and has the unconditional
evolvability (or simply evolvability) as its upper bound. This means that it measures
the response to directional selection along β, given that there is stabilizing selection
on all other directions in phenotypic space. The conditional evolvability relates the
degree of modularity directly to evolvability (Hansen, 2003). In general, evolvability
is defined as the distance in phenotypic space that a population can cover along the
direction of selection. The respondability r̄ is the length of the response vector and
measures how rapidly the population will respond to directional selection. r̄ has the
unconditional evolvability as its lower bound. The autonomy ā describes how integrated the selected trait combination is with the rest of the measured phenotype, i.e.,
the degree to which evolvability is reduced by conditioning on traits under stabilizing selection. ā ranges from 0 (most eigenvalues close to 0) to 1 (all traits have equal
variance and are uncorrelated). In this sense, the autonomy can also be calculated as
ā = c̄/ē, where ē = E[λ] is the mean evolvability. Conversely, the conditional evolvability can be described as the product of its evolvability and its autonomy: c̄ = ē ∗ ā.
58
The analytical solutions for nD and emax from Kirkpatrick (2009) and approximations for c̄, r̄ and ā from Hansen and Houle (2008) are given in Table 4.1.
Table 4.1. Summary statistics calculated from the eigenvalues of G and measures of selection
response.
Summary statistic
effective dimensionality
calculation
P
nD =
i λi
λ1
√
λ1
conditional evolvability c̄ = H[λ] 1 + 2I[1/λ]
n+2 2 p
I[λ ]
respondability r̄ = E[λ2 ] 1 − 4(n+2)
2I[λ]I[1/λ]
I[λ]+I[1/λ]−1+H[λ]/E[λ]+ (n+2)
H[λ]
autonomy ā = E[λ] 1 + 2
,
n+2
P
where λ = (λ1 , ..., λn ) are the eigenvalues of G, E[x] = n
i=1 xi /n is the arithmetic
mean, H[x] = 1/E[1/x] is the harmonic mean and I[x] = Var[x]/E[x]2 is the mean
standardized variance.
maximum evolvability emax =
Selection response
calculation
angle between ∆z̄ and β (in ◦ ) θ = arccos
angle between ∆z̄ and gmax (in ◦ )
∆z̄ β
|∆z̄||β|
180
π
∆z̄ gmax
γ = 90 − abs 90 − arccos |∆z̄||gmax | 180
π
distance to optimum dopt = |β − ∆z̄|
evolvability
e = ∆z̄ β,
where |x| is the euclidean norm of the vector x and abs(x) is the absolute value of
x.
4.2.3
Random skewers
Random skewers are a number of selection gradients individually applied to a given
G-matrix to infer the average response to selection (Cheverud, 1996). As described
above, we use Lande’s equation (Eq. 4.1) to apply 10,000 random selection gradients
β to G, track the responses to each of them and calculate the mean phenotypic change
of each trait ∆z̄.
The entries of the vector of selection gradients β are randomly drawn from a uniform distribution between -1 and 1. β is then normalized to a magnitude of 1 (|β| = 1).
This means that the fitness optimum lies somewhere on a circle with radius 1 around
59
G. Here we measure the responses to random selection gradients that can point into
any direction in phenotypic space. This might not reflect the general selection regime
that most natural populations are exposed to, but we usually do not know β and it is
hard to estimate. In this work, we aim at describing populations that might experience rapid changes of selection gradients. Those populations or organisms that best
respond to changing selection regimes should be favored.
optimum
dopt
Trait 2
e
z
θ
gmax
G
Trait 1
Figure 4.2. Measures of selection response. Selection (β) leads the population to the fitness
optimum. The response (∆z̄) after one generation of random mating is displaced, because
gmax does not point into the direction of selection. After one round of selection we measure
the respondability (|∆z̄|), the angle between ∆z̄ and β (θ), the angle between ∆z̄ and gmax (γ),
the distance to the optimum (dopt ) and the evolvability (e).
Additionally we calculate the angle between ∆z̄ and β (called θ) and the smaller
angle between ∆z̄ and gmax (called γ) to measure the alignment of the response to
selection with the direction of selection and the direction of highest variance. This
provides information on the average direction the response will take with respect to
the underlying genetic architecture and the direction of selection. Strong constraints
in the genetic architecture will on average lead to a larger angle between the direction
of selection and the direction of response. From ∆z̄ we also calculate the evolvability
(length of the projection of ∆z̄ on β) as well as the distance to the optimum after
selection dopt (distance between ∆z̄ and β) (see Figure 4.2 for a visualization and
Table 4.1 for the calculations). For different G-matrices we compare all the above
mentioned properties.
60
4.3
Results
In what follows, we will only present the results for the modular G-matrices without
any background correlation (as described in Fig. 4.1). We obtain qualitatively similar
results for the rearranged random G-matrices, but these do not display the findings
as clearly as the cases with distinct modules.
4.3.1
Eigenanalysis
The number of modules in G is reflected by its number of leading eigenvalues (Fig. 4.3).
This becomes more obvious for larger n, but is also detectable for small n when the
covariances between traits are high. However, with increasing module number the
leading eigenvalues become less distinguishable.
4
7 10
1
4
7 10
4
7 10
0.6
0.2
0.0
0.2
0.0
1
4 modules
0.4
0.6
3 modules
0.4
0.6
0.2
0.0
0.2
0.0
1
2e modules
0.4
0.6
2d modules
0.4
0.6
0.4
0.2
0.0
eigenvalues
1 module
1
4
7 10
1
4
7 10
Figure 4.3. Distribution of Eigenvalues. n=10, homogeneous trait variances of 0.2 and correlations of 0.25 among traits. ’2d’ means two modules of different size and ’2e’ means two
modules of equal size (see Fig. 4.1).
Since the eigenvalues describe how much of the variation is explained by the corresponding principal component (eigenvector), it is clear that the individual modules
build principal components of joint variation and are thus visible as distinct eigenvalues. What is not obvious from the plotted eigenvalue distributions but can easily be
seen from our data, is that the eigenvectors capture the traits within the modules; i.e.,
in the 1-module case all traits contribute equally to the largest eigenvector, whereas in
the 4-modules case only the traits within a module contribute to corresponding eigenvector. As expected, modules with the same size have the same eigenvalues, whereas
different sizes can make a huge difference in eigenvalues. From the simulations, we
observe that for matrices with heterogeneous variances among traits, higher variances
as well as higher covariances make the leading eigenvalues more distinguishable from
the others.
61
4.3.2
Summary statistics of G
As expected, high genetic correlations constrain adaptation in most situations, because they decrease the effective dimensionality nD , the conditional evolvability c̄
and the autonomy ā. However, we find that modularity has a positive effect on nD
and c̄ (Fig. 4.4). With an increasing number of modules each trait is less constrained
by other traits that influence its adaptation (more modules lead to smaller modules
in our framework), which results in a higher conditional evolvability and a higher
number of effective dimensions overall. It is surprising that the autonomy does not
increase with modularity. We expected that fewer constraints per trait increase the independence of each trait. It might well be that this is an artifact of the formula given
by Hansen and Houle (2008). We here used the revised version of the formula published in a corrigendum. When calculating ā from the formula given in the original
paper, we find an increase in autonomy with increasing modularity.
On the other hand, correlations increase the maximum evolvability emax and the
respondability r̄, which are in turn reduced by modularity. For the maximum evolvability it makes sense from the definition of emax , which only depends on the first
eigenvalue. This is largest for G-matrices with one module and decreases with an increasing number of modules (see Figure 4.3). This means that the potential response
to selection is higher for a single module compared to many modules, because there
is only one dominating principal component that selection can act on. With more
modules, the genetic variance is shared between the principal components and each
component explains less of the variance than a single component in a G-matrix with
only one module does. Consequently, the average response to selection is smaller for
G-matrices with many modules, because the genetic variance that is available in the
direction of selection is smaller. Admittedly, the maximum evolvability is only realized when the selection gradient β is aligned with the line of least resistance gmax .
All other directions of β lead to a highly reduced evolvability.
Generally, these results are robust among different levels of complexity (= size
of G) and hold for different levels of genetic variation, but are mostly weak or not
observable for low variances and very strong for high variances (not shown). Additionally, the observed trends in emax and r̄ are strongest for the highest degree of
covariance between traits, whereas in nD it is the smaller levels of correlation where
modularity has the strongest effect. These results are qualitatively the same, for both,
homogeneous and heterogeneous variances and covariances among traits and trait
pairs (results not shown).
4.3.3
Random skewers
Like for the summary statistics of G, an increasing genetic correlation has mostly
negative effects. It leads to a larger angle between ∆z̄ and β and a smaller angle
3
4
1
2d
2e
3
4
autonomy [a]
8
6
4
2
0
2d
2e
3
4
1
2d
2e
3
4
1
2d
2e
3
4
0.20
1
0.10
0.00
2e
conditional evolvability [c]
1.2
0.8
0.4
2d
0.50
1
0.0
0.35
0.20
respondability [r]
maximum evolvability [emax]
correlation = 0
correlation = 0.25
correlation = 0.5
correlation = 0.75
correlation = 0.99
0.8
n = 10, variance = 0.2
0.4
effective dimensionality [nD]
62
modules
modules
Figure 4.4. Summary statistics of G. Matrices have 10 traits and different degrees of correlation
between traits. For all plots we generated G-matrices with homogeneous variances among
traits (0.2). Lines are drawn to guide the eye.
between ∆z̄ and gmax . This means that the response to selection is rather aligned with
the direction of highest variance than with the direction of selection when correlations
between traits are strong. Also, the distance to the optimum after selection becomes
larger for higher correlation. However, modularity has a positive effect on adaptation.
θ slightly declines with rising modularity, whereas γ strongly increases.
When G is composed of modules, the probability that β is aligned with one of
these components (modules) increases with increasing number of modules. That is
why we see a decline in the average angle between ∆z̄ and β for G-matrices with
the same degree of covariance but increasing modularity. A similar, but converse
1
2d 2e
3
4
evolvability [e]
modules
60
0 20
1
2d 2e
3
4
1
2d 2e
3
4
0.23
4
0.20
3
0.17
angle(Δz,gmax) [γ]
60
0 20
2d 2e
1.00
1
0.90
0.80
dist to opt [dopt]
angle(Δz,β) [θ]
63
modules
Figure 4.5. Response to selection. G-matrices have 10 traits and homogeneous variance (0.2)
for different degrees of correlation between traits. Dots stand for the average values from
10,000 random selection gradients. Color code as in Fig. 4.4
explanation accounts for the trends we see for the average angle between ∆z̄ and
gmax . High modularity leads to a less constrained response to selection even in a
high correlation scenario. Surprisingly, the evolvability is independent of the number
of modules and the degree of correlation.
Finally, the most important observation is that increasing modularity brings the
population closer to the fitness optimum, because it provokes that ∆z̄ is rather aligned
with β than with gmax . Like for the summary statistics, all trends hold for homogeneous as well as heterogeneous variances and covariances among traits (not shown),
and trends are stronger for smaller n and higher trait variances.
4.4
Discussion
Putting all this together clearly indicates that modularity indeed has a positive effect
on the adaptive potential. When comparing two populations, one with a fully integrated and the other with a modular G-matrix, the population with one large module
64
will have an adaptive advantage if selection is aligned with gmax , but the population
with a highly modular genetic architecture bears the potential for larger evolvabilities in all other directions. The question is in which direction β usually points. In
the regime of environmental changes we would expect that β is rather not aligned
with gmax . Under this assumption, modularity increases the effective dimensionality
and the average conditional evolvability. It leads to a larger average angle between
the direction of response and gmax and a smaller average angle between ∆z̄ and β,
which means that the response is more often aligned with the direction of selection
and more independent from the direction of highest variance. All these facts lead to
a faster approachment to the fitness optimum due to modularity.
What is not obvious at first sight, is the stability of evolvability among very different genetic architectures. The evolvability, measured as the projection of the response
vector ∆z̄ on β, is independent of the number of modules in G and the correlation
between traits and is always equal to the average genetic variance of the traits. This
means that the average distance that the population evolves along the direction of
selection is always equal to the genetic variance of the traits, irrespective of the shape
of G. However, the distribution of evolvabilities changes with the number of modules
in G. It is almost exponential for G-matrices with one big module and almost normal for G-matrices with many modules (not shown), whereas the mean always stays
the same. This can be explained by the distribution of the eigenvalues of G. Hansen
and Houle (2008) pointed out that the evolvability reduces to an eigenvalue of G if
the selection gradient β points in the direction of the corresponding eigenvector. The
G-matrix with one large module has one large leading eigenvector (gmax ) and n − 1
small eigenvectors. Thus, the probability that β is aligned with one of the smaller
eigenvectors is much higher than an alignment with gmax . Therefore, most evolvabilities are very small and only a few are high, which leads to an almost exponential
distribution of evolvabilities for the 1-module case. On the other hand, the eigenvalues of a G-matrix with many modules are much more evenly distributed. This in
turn leads to an almost normal distribution of evolvabilities, because β will in most
cases be aligned with one of the intermediate eigenvectors. And since the mean of the
eigenvalues is similar to the mean of the genetic variances for all structures of G, the
distribution of evolvabilities will also always have the same mean. Consequently, the
specific evolvability of a population mainly depends on the direction of selection in
multivariate phenotypic space. Modularity only increases evolvability under specific
conditions, i.e., when selection is acting on one of the components with low genetic
variance.
The question we addressed relates to the puzzling fact that complex organisms
exist. We assume that very simple organisms were the first to colonize the earth
and are thus the ancestors of today’s variety of complex and less complex creatures.
Therefore, there must have been a line of evolution that led from simple to complex
life forms. If modularity has played a role in this process it must have co-evolved with
65
complexity. Either, a higher modularity (i.e., more modules or stronger modules) had
allowed new traits to arise without a big cost, or an increased complexity had to be
followed quickly by an increased modularity to be successful. Either way, one would
rather expect module size to be more or less constant while complexity increases. We
tackled this from a slightly different perspective (constant complexity and increasing
modularity). However, the proportional change of module number, module size and
organismal complexity is still captured in our framework, which makes it a valid
method to approach the above mentioned question.
Whether modularity is a key concept in the evolution of higher organisms can of
course not be answered by theoretical studies. We can only give hints on why and
how it might work. Our findings and those of other theoretical researchers in this
field have to be proven in natural systems.
A recent study used the statistics defined by Hansen and Houle (2008), as well
as the random skewers approach to analyze variance-covariance matrices of mammalian skulls (Marroig et al., 2009). They investigated the P matrices of skulls of
15 mammalian orders with 35 morphological landmarks as traits. The phenotypic
variance-covariance matrix P can be used as a substitute for G when they are sufficiently similar, which was shown to be the case in another study by the same research
group (Porto et al., 2009). Additionally to our measures, they used a factor called
’morphological integration index’ r2 , which is the average of squared correlation coefficients between traits, and is supposed to reflect the level of modularity in the
cranium. The study revealed a strong relationship of the natural logarithm of r2 with
the angles θ and γ and with autonomy, as well as a weak but significant correlation
with conditional evolvability. This is in very good agreement with our hypotheses, if
we believe that low values of r2 correspond to high degrees of modular organization.
Together, the two studies by that group (Marroig et al., 2009; Porto et al., 2009) provide a very nice example of how enhanced organismal complexity (from rodents to
primates) goes hand in hand with a decline in overall trait integration, which is a hint
for an advancement in the modular structure of the genetic architecture. Additionally, they show that this improvement in modularity has very likely acted in a way as
predicted by our simulations.
This is apparently one of the few studies that measured a sufficiently large number
of traits in order to be able to detect differences in integration patterns among species.
Often, only some hands full of characters are investigated because of technical or
feasibility reasons. Moreover, most of these traits are often of morphological nature
(e.g., wing shape, skull landmarks) and therefore size dependent, which makes them
clustered in a big module that evolves as a component of size. Hence, it is important
to note that we might miss a great deal of the hole picture by observing only a small
part of an organism. Measuring more traits (and more diverse traits) could lead to
finding more modularity and to a greater exploration of the genetic patterns that
influence evolution.
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4.4.1
Quantifying genetic constraints
One of the issues is to quantify genetic constraints. It is not only about knowing
the genetic variances and covariances, because in a multivariate context the genetic
architecture is much too complex. That is why several measures have been developed to characterize adaptive constraints of the G-matrix. Kirkpatrick (2009) applied
his summary statistics to five different datasets of cattle, mice, fish and flies and
found that the effective dimensionality was in a small range between 1 and 2 although the numbers of traits measured were quite different (between 4 and 21). On
the other hand, he found a large variety of maximum evolvabilities ranging from 0.033
in Drosophila bunnanda to 1.2 in Melanotaenia eachamensis (rainbow fish). Hansen and
Houle (2008) found very similar evolvabilities and conditional evolvabilities among related Drosophila species, which were higher than expected. In an earlier study Hansen
et al. (2003) showed that the short-term evolvability of D. scandens blossoms is substantially constrained by genetic correlations among floral characters. This has also
been assessed with a measure of conditional evolvability.
These examples show that there is no consensus about how to measure genetic
constraints. Research in this direction has focused on different approaches to quantify constraints: the number of zero eigenvalues, the angle between ∆z̄ and gmax ,
the angle between ∆z̄ and β, the conditional evolvability or the effective number of
dimensions (nD ). However, there are many ways to calculate these properties, which
makes it difficult to compare estimates from different studies. For example, Martin and Lenormand (2006) estimated values of the effective number of traits for five
different species based on a model that predicts the distribution of fitness effects of
mutations. They found very low values and little variation among viruses, bacteria,
yeast, nematodes and fruit flies. On the other hand, Tenaillon et al. (2007) found
very high values of phenotypic complexity (they call it the number of genetically
uncorrelated phenotypic traits contributing to an organism’s fitness) for two viruses,
estimated with a model based on drift load. There are many other ways to calculate
nD (see e.g. Hine and Blows, 2006). So, it seems that these estimates capture different
aspects of constraints. But what does nD really tell us? The effective number of dimensions is a measure of the number of independent trait combinations that selection
can effectively act on. It is supposed to capture the constraints of the genetic architecture and is therefore an important parameter in influencing the response to selection.
It is, however, not clear how much of the variation in selection response nD is able
to explain, and whether calculating this value is sufficient to draw conclusions about
the adaptive potential of an organism. We checked for this by correlating nD with the
other summary statistics and selection response measurements. For this we used the
framework of generating modular matrices with background correlation as described
above in section 4.2.1. In this way we made sure that no other factors (like correlation)
are influencing the relationships between the different measures. In this scenario it is
67
particularly difficult to get positive definite matrices (only about 3 in 1,000 fulfill this
criterium). For this reason we only generated 100 matrices of each of the five types:
totally random; rearranged such that two modules of different size exist; rearranged
such that two modules of same size exist; rearranged such that three modules of same
size exist; and rearranged such that four modules of same size exist. We found that nD
is highly correlated with emax (Spearman 0 s ρ = −1, p < 2.2∗10−16 ) and γ (ρ = 0.9835,
p < 2.2 ∗ 10−16 ), intermediately correlated with dopt (ρ = −0.3985, p < 2.2 ∗ 10−16 ),
and not significantly correlated with any other measure of selection response. This
means that the effective dimensionality is a good measure to predict the maximum
evolvability, the average angle between gmax and the response, and the distance to
the optimum after one round of selection (at least to some extent), but it cannot explain the variation in all the other measures of genetic constraints. We believe that
the autonomy of the traits and the conditional evolvability are two factors that play
an important role in influencing the response to selection. However, since ā = c̄/ē
and ē is constant, it is sufficient to use either autonomy or conditional evolvability
as an additional measure of genetic constraints. Both significantly correlate with θ
(ρ = −0.3598, p < 2.2 ∗ 10−16 ) and γ (ρ = −0.1856, p = 3 ∗ 10−5 ), but nD is a much
better predictor for γ. Additionally, θ, the angle between selection and response, is
much more strongly correlated with dopt than nD is (ρ = 0.7221, p < 2.2 ∗ 10−16 ). In
summary, we suggest that both, static and dynamic approaches should be applied to
describe the characteristics of a G-matrix. Most of the variation in correlation patterns
is captured by calculating the effective number of dimensions, the conditional evolvability and the angle between the direction of selection and direction of response.
4.5
Conclusions
Our simulations show that modularity is beneficial for adaptation. It increases the
probability that the response to selection is aligned with the selection gradient, it
reduces the constraints caused by genetic correlations and it brings the population
closer to the fitness optimum.
But what prevents evolution from creating perfect modularity? Our simulations
suggest that the best architecture is the one where all traits are independent and build
separate one-trait modules. However, we do not observe this in nature. One reason
might be, that it is not always beneficial to have the highest theoretical adaptive potential. An obvious explanation is that completely independent traits cannot respond to
selection coordinately, i.e., it would prevent a faster evolution of several similar traits
that could be tuned at once. Other reasons may include correlations caused by linkage disequilibrium (which cannot be broken down by recombination), observation of
transient stages of evolution (’we are just not there yet’), or that G changes during the
68
course of evolution and the best architecture at the optimum is different from the one
away from the optimum.
There are many studies on modularity, complexity, evolvability and selection in
natural systems, but to our knowledge none that combines these four key ideas of
evolution. It would be nice to see such a study in the future.
Finally we have shown that the best strategy to describe the constraints of real
G-matrices is to calculate the effective dimensionality nD , the average autonomy ā or
the average conditional evolvability c̄, and the average angle θ between the selection
gradient and the corresponding response from a random skewers approach.
III | Evolutionary consequences of
pleiotropic effects in HIV
69
5 | Evolutionary capacity for drug
resistance in HIV
Abstract
An important step in fighting a disease is to understand the pathogen that causes
it. With growing availability of experimental data and more sophisticated computational techniques we can now get more insights and a deeper understanding of
the mechanisms that drive a disease and the coevolution between host and parasite.
In this work we use data on HIV-1 fitness measurements and corresponding fitness
prediction models for main effects of individual mutations and epistatic effects of
double mutations to uncover different aspects of the HIV fitness data. We tackle two
main questions: (1) How is the mutational variation in HIV characterized and does
it differ between main effects, epistatic effects and whole sequence fitness in different drug environments? And (2) can we find a drug combination that minimizes the
risk of selecting for resistance mutations? We construct genetic covariance matrices
and apply principal component analysis (PCA), hierarchical clustering and selection
skewers approaches. In general, the data reflects the correlational structure that we
expect from our knowledge about the three different drug classes (protease inhibitors
(PIs), (non-)nucleoside reverse transcriptase inhibitors ((N)NRTIs)). The information
contained in single mutations and double mutations is very similar to that from the
whole sequence data. The selection analysis reveals that the lowest respondability is
achieved by selecting only on few drugs of the NRTI class.
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72
5.1
Introduction
The major problem with HIV treatment is the rise of drug resistance strains, which
escape the suppression of the virus by pharmaceutical drugs. These resistance strains
randomly evolve by mutation at particular sites in the RNA sequence and are able
to replicate in the presence of anti-viral treatment which has fatal consequences for
the patient in terms of disease progression (Clavel and Hance, 2004). The issue about
drug treatment, not only in the fight against HIV, is, that the medication works against
the wild type strains of the parasite, and the higher the dose, the more of them are
killed (neglecting side effects). But at the same time drug treatment is selecting for
resistance mutations, and the higher the dose, the stronger the selective force. Since
the awareness of the existence of resistance there are ongoing debates about the best
strategy for disease treatment (Gulick, 2010). Usually, the loss of efficacy of one drug
leads to the development of a new drug which tries to interfere with the parasite in a
different way than before, e.g., blocking another active site of a protein or disturbing
the entry mechanism. However, this approach has its limitations with respect to interference possibilities and research resources. The accumulation of drug variants led
to the emergence of combination therapy. Taking two or more drugs in an alternating
fashion might at least decelerate the spread of resistance strains (Balzarini, 1999).
Due to its high rate of mutation and replication (Dougherty and Temin, 1988;
Nowak, 1990; Ho et al., 1995), the ability for recombination (Levy, 1988) and the
resulting high rate of adaptation it is still not possible to heal HIV infections. Over 20
antiretroviral drugs have been developed to help the immune system kill viruses or
impede their replication (De Clercq, 2009). Nowadays, combination therapy is usually
able to suppress viral load to a degree that does not cause death anymore. But as with
many antibiotics, we encounter the problem of resistance.
Resistance is a mechanism developed to escape from environmental threats. In
medicine and public health, resistance has been observed since the invention of first
pharmaceuticals. Often only a single point mutation is needed to confer resistance
(e.g., Mégraud, 1998; Jackson et al., 2000; Weill et al., 2003). That such resistant variants fixate and spread is usually observed only a few years after introduction of a
new drug (Vincent et al., 1995). Therefore, it is important to understand the mechanisms and evolution of resistance and to find out what we can do against it. There are
some mutations in HIV that are known to confer resistance to one or more antiviral
drugs. Combination therapy tries to avoid that such mutations are selected in a patient. However, the impact of multiple drugs on the mutational landscape of the virus
is poorly understood. Pleiotropic effects and epistatic interactions of mutations can
influence the expected outcome of drug therapy in an unpredictable manner. Drug
resistance mutations can act in different ways, usually changing the part of the protein used as a drug target, such that the drug is unable to bind. When a mutation
leads to structural changes that prevent more than one drug from binding to its target
73
site we speak of multidrug resistance caused by a single mutation, which means that
this drug has pleiotropic effects on resistance to different drugs. It is also possible
that these pleiotropic effects are mediated by the presence of other mutations through
epistatic interactions. That positive epistasis is prevalent and leads to multidrug resistance has been shown in a recent study on antibiotic resistance in E. coli (Trindade
et al., 2009).
The approach we pursue comes from a different perspective. Using data on viral
fitness we are able to construct genetic covariance matrices. These give us an impression of how the mutational variation in HIV corresponds to phenotypic variation in
different drug environments. The idea is to find the direction in phenotypic space
(i.e., a drug combination) which accounts for a very low amount of genetic variation.
When selection acts in a direction where there is no genetic variation, it will result in
a null response. This means that resistance mutations cannot arise when the selective
pressure is aligned with a direction in which there is no room for fitness improvement caused by mutations. This sounds like an easy task, but since we are dealing
with a medical problem here, there are many complications to take into account. The
prerequisite for medical treatment is that the patient does not suffer more from the
drug than from the disease itself. To this end it is essential that a new drug has as few
side effects as possible. When we think of a new drug as a cocktail of existing drugs,
we might want to use as few as possible, for two reasons. First, the more different
drugs we use the more possible side effects will cumulate. And second, the interactions between different drugs are poorly understood and they may cancel each others
positive treatment effects out or even create negative effects.
But how do we select against the emergence of resistance? The answer is: we
cannot really. Medical treatment works by putting the population of parasites into
an environment where it can not or hardly survive. In this way the population is
far away from the fitness optimum and there is a strong selection pressure on those
variants that have a higher fitness than the mean of the population. Since the creation
of mutations is a random process, it is just a matter of time until resistance emerges.
The question is: can the population survive until a beneficial mutation occurs and
takes over or will it die out before? For us, the question reads: is there enough
mutational variation in the direction of selection? Or more precisely, can we apply a
drug treatment such that the direction of selection is aligned with a component of the
phenotypic space where there is very little (or no) variation in the mutational space?
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5.2
5.2.1
Material and methods
The HIV dataset
Our analyses are based on fitness measurements of 70’081 patient-derived HIV-1 sequences in a drug free environment and in presence of 15 different individual antiretroviral drugs (Petropoulos et al., 2000). Fitness was measured as replicative capacity by transferring the full gene for the protease and most of the gene for the reverse
transcriptase (amino acids 1-305) of the viral sequence to an HIV-derived test vector,
undergoing routine drug resistance testing. Fitness of one sequence is quantified as
the number of infectious progeny virus produced by the patient-derived virus. The
15 drugs belong to three different classes: protease inhibitors (PIs), nucleoside reverse
transcriptase inhibitors (NRTIs) and non-nucleoside reverse transcriptase inhibitors
(NNRTIs), see Appendix 5.6.1.
Additionally to the fitness values of the whole sequences we have access to estimated fitness values of individual mutations and pairwise epistatic interactions
(Hinkley et al., 2011). This dataset consists of fitness values of 1859 amino acid substitutions (mutations) in 16 conditions (one drug free and 15 drug environments) and
of the pairwise epistatic interactions of these 1859 mutations. Thus, we have a differentiation of main effects (hereafter ME) and pairwise epistatic effects (hereafter EE) of
mutations, irrespective of the genetic background.
For the principal component analysis, hierarchical clustering and matrix comparisons we use the whole raw data including all 16 environments to make use of the
complete variation in the data. For the selection skewer analyses we exclude the drug
free environment to only investigate the genetic variation caused by drug treatment
and to find the drug combination with lowest selection response.
5.2.2
PCA and correlation matrix comparison
We built genetic variance-covariance matrices (G-matrices) for each of the three datasets, given by the absolute fitness effects of mutations in the different drug environments. We will further refer to them as GSEQ (sequence data), GME (main effects) and
GEE (epistatic effects). These G-matrices are then used for principal component analysis (PCA). The aim of PCA is to reduce the dimensionality of the data by compressing
it to the most informative components. To this end, we can identify trait combinations with large amount of variance and investigate how mutations collectively affect
combinations of traits. Note that the ME and EE data can be considered independent,
but both are derived from the sequence data. We test here whether main and epistatic
effects contain the same information with respect to the mutational variation in the
16 environments, or alternatively reflect different dynamics not seen in the sequence
data. We also use the ’pvclust’ R package for hierarchical clustering (Suzuki and Shi-
75
modaira, 2006; R Development Core Team, 2012) to find the correlational structure
among the drug environments and scrutinize the PCA results. Further, a comparison of correlation matrices is carried out to unravel differences in the correlational
structure of drug environments. For this purpose, we calculate the effective number
of dimensions nD and the maximum evolvability emax (Kirkpatrick, 2009) and apply
four matrix comparison methods. (1) Roff’s T test is an element-wise test that determines the difference between two matrices based on the differences between matrix
elements at the same position (Roff et al., 1999; Bégin and Roff, 2001). Thus, it provides a measure for the difference in correlations between drug environments. (2)
Co-inertia analysis calculates the shared structure of two hyper-spaces, where the coinertia between two hyper-spaces is the sum of squares of the covariances between all
variables pairs (Dolédec and Chessel, 1994). From this, one can obtain a coefficient
of correlation between two datasets. (3) Krzanowski subspace comparison provides a
measure of shared geometry of matrices (Krzanowski, 1979). It focuses on the space
containing variation and how this is oriented. Thus, it tests whether the subspace
containing most of the variation is in common between matrices. And (4) the random
skewers approach, which computes the correlation between two matrices based on
their response to the same set of 10’000 random selection skewers (Roff et al., 2012). It
is a measure of the degree of similar evolutionary reaction to selection of two populations. These comparative analyses are executed on the correlation matrices instead of
covariance matrices, because the datasets have different scales and are thus normalized to be comparable. To calculate the covariance and correlation matrices we use
the R functions cov() and cor() with default settings.
Additionally, we run bootstraps to test for the significance of the covariance matrix statistics. Doing within-dataset bootstrapping provides us with an estimate of the
variation of the comparison statistics when two datasets are assumed to be equivalent. We then test whether the observed statistics between datasets are part of the
within-dataset confidence intervals or not. To obtain confidence intervals for the
within-dataset matrix comparisons, we draw k mutations/sequences with replacement, calculated the correlation matrix from this re-sampled data and computed the
four comparison measures between 10’000 correlation matrix replicates. Here, k is
the number of mutations/sequences in the original datasets (k = 1 0 859 in ME data,
k = 1 0 094 0 918 in EE data, k = 62 0 736 in sequence data). To test whether ME or EE is
more similar to the sequence data, for each bootstrap replicate of the sequence data,
we compute the statistics with a bootstrap replicate of ME and EE. Repeating this
10’000 times provides us with a mean and confidence interval for the difference for
each statistic. We then test whether the bootstrapped difference is different from zero
and whether the observed difference is different from what would be expected by
chance.
76
5.2.3
Selection skewers
We use the selection skewers approach (Calsbeek and Goodnight, 2009) to find the
drug combination that leads to the worst selection response. We here define the worst
response in terms of the smallest value in respondability, i.e., the smallest response
vector (see below). To achieve this, we calculate covariance matrices and apply to
them selection skewers, that are designed in the following way: a skewer is a 15dimensional vector where each position in the vector stands for the selection pressure
caused by applying one of the 15 drugs. We generate skewers with ’1’ at positions
with drugs applied and ’0’ at positions with drugs that are not applied. This gives
us 32’767 different selection skewers (drug combination cocktails) reaching from single drug treatment to having a cocktail with all 15 drugs. In each combination the
contained drugs are used in same proportions and the selection skewers are then
normalized to unit length such that the strength of selection is always the same. We
could of course use the random skewers approach (Cheverud, 1996) to generate selection skewers in any direction of phenotypic space, but to make it simpler and actually
more realistic in terms of medical drug treatment, we only consider selection in positive direction. The multivariate breeder’s equation (Lande, 1979) provides us with a
mathematical tool to calculate the response of a population to a given selective force:
∆z̄ = G · β.
For each selection skewer β we track the response ∆z̄ and calculate the following
properties:
The angle between ∆z̄ and β:
−1
θ = cos
∆z̄ · β
k∆z̄kkβk
→ How well is the response aligned with the direction of selection?
The angle between ∆z̄ and gmax :
−1
γ = cos
∆z̄ · gmax
k∆z̄kkgmax k
→ How well is the response aligned with the direction of highest variance in the Gmatrix?
Respondability (Hansen and Houle, 2008), the length of the response vector:
respondability = k∆z̄k
77
→ How far does the population mean move due to selection?
Evolvability (Hansen and Houle, 2008), defined as the projection of ∆z̄ on β:
evolvability = ∆z̄ · β
→ How far does the population mean move in the direction of selection?
With these response measures we can describe the G-matrices and their adaptive
potential and also compare the matrices to each other.
Additionally, what we are interested in is the response in the direction of selection
and how constrained that response is. To measure evolutionary constraints we use
the following quantity: R = Rc /Ru (Agrawal and Stinchcombe, 2009), where Rc is the
response along β (what we call evolvability) for the constrained case and Ru is the
response along β in the unconstrained case (computed as the response vector when
setting all covariances to zero in the covariance matrix).
R = βT Gβ/βT Gu β,
with Gu being the diagonal matrix of G and T indicating vector transposition. Computing that ratio thus standardizes the response by the amount of genetic variance of
the traits under selection.
5.3
5.3.1
Results
PCA and matrix comparison
Although the data points are widely scattered, the Principal Component Analysis
(PCA) reveals that the three drug classes can clearly be separated given the first two
components (Fig. 5.1). This means that different viral sequences usually have similar
replicative capacities in presence of drugs of the same drug class and rather different
replicative capacities among drug classes. These two largest components explain 71%
of the variation in the sequence data (ME: 86%, EE: 83%). The patterns are very similar among the three datasets. Because environments are highly correlated, the first
component of the ME and EE data is composed of all drugs with almost equal loadings (Appendix 5.6.2). Therefore the biplot of second and third principal component
highlights the separation of drug classes even more. In the sequence data three drugs
of the NRTI class (D4T, DDI, TFV) do not contribute to the first three components,
which causes the difference to the other two datasets in the right hand plots.
The visual differentiation of drug classes is confirmed by the angles between the
drug vectors in the PCA plots. The angles within drug classes are much smaller than
78
Figure 5.1. Principal component analysis of the sequence data (SEQ), the main effects only
(ME), and the epistatic effects only (EE). Grey dots indicate the relative positions of data
points (fitness of sequences/mutations) within the two principal components (PCs) and arrows symbolize the positions of the original traits (drug environments) in this new phenotypic
space. Arrows closest together have highest correlation, i.e., the respective drug environments
have similar effects on viral fitness. Drug classes are highlighted with colors (ND = no-drug
condition).
between drugs of different classes. The values only differ slightly between datasets
(not shown).
Hierarchical clustering leads to similar results among the three datasets (Fig. 5.2).
All show distinct clusters for PIs, NRTIs and NNRTIs, but unexpectedly NRTIs and
NNRTIs do not build a large cluster. Instead, NNRTIs are separated from the other
two classes. The clustering of the sequence data differs from the other two insofar as
the drug-free environment is placed within the NNRTI class and not within the NRTI
79
class. The hierarchical structure within the NNRTIs is identical for all three datasets,
whereas there are differences within the other two classes. Most nodes are highly
supported by bootstrapping.
Hierarchical clustering of sequence data
Hierarchical clustering of single mutations
Hierarchical clustering of double mutations
au bp
edge #
au bp
edge #
98 99
14
0.04
100 100
100 100
13
11
Distance: euclidean
Distance: euclidean
Cluster method: ward
Cluster method: ward
Cluster method: ward
(a)
(b)
TFV
100 100
8
ZDV
DDI
D4T
3TC
ABC
Distance: euclidean
100 100
100 100
4 2
NODRUG
AMP
LPV
RTV
SQV
IDV
100 100
100 100
100
100 100 9 100 100
100
7
6
100 100
5
3
1
NFV
EFV
0
NVP
DLV
100 100
12
100 100
10
DDI
3
D4T
3TC
ABC
TFV
ZDV
96 100
8
100 100
NODRUG
LPV
95 100
9
100 100
5
RTV
NFV
AMP
SQV
IDV
EFV
NVP
DLV
100 100
6
100 100
90 75
10
7 94 82
100 100 4 99 99
2
1
2
97 100
12
100 100
11
100 100
14
4
Height
0.03
100 100
13
0.00
NVP
100 100
10
100 100
6
0.01
0.02
NODRUG
ZDV
ABC
TFV
D4T
3TC
DDI
IDV
RTV
SQV
LPV
AMP
NFV
0
85 82
11 85 82
9
100 100
100 100
2 1
DLV
100 100
12
EFV
100 100
13
100 100
8 86 86
100 100
99 100 7 99 99
5 4
3
Height
6
20000
100 100
14
10000
Height
8
30000
0.05
10
0.06
au bp
edge #
(c)
Figure 5.2. Hierarchical clustering of sequence data (a), main effects (b) and epistatic effects
(c). The red numbers are approximately unbiased (au) p-values, the green numbers are bootstrap probabilities (bp). The red rectangles indicate clusters highly supported by the data.
The NNRTIs have the highest mutational variation and the lowest mutational variation is found in the NRTIs D4T, DDI and TFV (Appendix 5.6.2). In general, the
correlations in the main effects and epistatic effects data are much higher than in the
sequence data. The average correlation between two drug environments is 0.50 in the
sequence data and 0.75 respectively 0.67 in the ME and EE data. This is caused by
higher correlations between drugs of different classes in ME and EE (Table 5.1), which
makes the drug classes also less well distinguishable than in the sequence data (see
also Appendix 5.6.3). Although the mean correlation is not very different between
ME and EE, all pairs of drugs have a higher correlation in the ME data.
Table 5.1. Mean correlations within and between drug classes.
PIs
PIs
NRTIs
NNRTIs
0.89
Sequence data
NRTIs NNRTIs
0.46
0.62
0.24
0.34
0.92
PIs
0.93
Main effects
NRTIs NNRTIs
0.74
0.83
0.62
0.66
0.90
PIs
0.90
Epistatic effects
NRTIs NNRTIs
0.68
0.79
0.46
0.54
0.87
We applied four different methods for correlation matrix comparison. Roff’s T test
is an element-wise comparison and calculates the difference between elements of two
matrices. Co-inertia analysis tries to find a co-structure between two datasets and
calculates a coefficient of correlation. The Krzanowski subspace comparison (KSC)
searches for the shared geometry of two correlation matrices. We reduced the analysis
80
to 7 dimensions, which correspond to the first 7 principal components and explain at
least 95% of the variation in the data. And the random skewers approach tests for the
similarity in selection response of two populations. For these comparisons, correlation
matrices instead of covariance matrices were used, because the values of the three
datasets are not on the same scale. The different ways of matrix comparison all reveal
a similar picture and indicate a high similarity between the three correlation matrices
and therefore also between the correlational structures of the three datasets. However,
although ME and EE are extremely similar, all three matrix comparison methods
suggest that the sequence data has more similarity to EE than ME; but differences are
minor (Table 5.2).
Table 5.2. Correlation matrix comparison methods applied to the ME, EE and sequence (SEQ)
data.
Method
Roff’s T test (difference)
Co-inertia analysis (correlation)
KSC (shared dimensions, out of 7)
Random skewers (correlation)
ME vs. EE
ME vs. SEQ
EE vs. SEQ
0.070
0.976
6.811
0.991
0.240
0.886
5.868
0.915
0.173
0.908
5.966
0.947
To test for the significance of these measures, we performed a bootstrap analysis (described in the Methods section). For the pairwise matrix comparisons we get
very different results among the comparison statistics (Appendix 5.6.4 Tab. 5.10). The
observed values of Roff’s T are all within the 99% confidence intervals (CI) of their
bootstrap values. The observed co-inertia correlations are all larger than the CIs, the
observed KSC values of ME vs. EE and ME vs. SEQ are larger and the observed value
of EE vs. SEQ is smaller than their CI. For the random skewers correlation, the value
for ME vs. EE is larger, for ME vs. SEQ smaller and for EE vs. SEQ within the 99%
confidence interval. More interesting is the question, whether the observed matrix
comparison values display real differences between the matrices or whether these differences can also occur by chance. Our results of within-dataset bootstraps reveal
that all observed measures between two correlation matrices are significantly more
different than all bootstrapped within-data measures, with one exception: the number of shared dimensions between ME and EE is larger than the number of shared
dimensions within the ME bootstraps (Appendix 5.6.4 Tab. 5.11). This means that the
differences between our three datasets are largely significant.
To test whether the differences in matrix comparison measures between ME vs. SEQ
and EE vs. SEQ are significant, we calculated confidence intervals for each statistic by
comparing a bootstrap replicate of the sequence data with bootstrap replicates of ME
and EE respectively and computed the difference between them. The bootstrapped
differences are all significantly different from zero (P < 0.05). The observed differ-
81
ences in Roff’s T and random skewer correlation are within the bootstrapped confidence intervals and the observed differences in co-inertia correlation and KSC are
smaller than the bootstrapped CIs (Appendix 5.6.4 Tab. 5.12).
Table 5.3. Effective number of dimensions nD and maximum evolvability emax for the ME,
EE and sequence (SEQ) data.
Statistic
ME
EE
SEQ
nD
emax
1.29
3.52
1.42
3.36
1.88
2.92
Before continuing with the selection skewer approach to find the drug combination causing the smallest evolutionary response, we look at two simple statistics of
genetic covariance and correlation matrices that summarize how genetic variation is
distributed. They are based on the eigenvalues (λ) of the matrix which account for
the genetic variation
P present in the principal components. The effective number of
dimensions (nD =
λi /λ1 ) indicates in how many
dimensions the genetic variation
√
lies. And the maximum evolvability (emax = λ1 ) is proportional to variance of the
combination of traits (principal component) accounting for most of the genetic variation.
We find that the HIV sequence data has 1.88 effective dimensions, which means
that most of the genetic variation present among the 16 environments can be summarized by marginally 2 main components that selection can act on (Tab. 5.3). Therefore,
the evolutionary space is greatly constrained. Even larger constraints are found when
we consider only the main effects of mutations or only the epistatic effects of pairwise
mutations. On the other hand, the ME data shows the largest maximum evolvability,
because its first principal component accounts for more variation than in the other
two datasets. However, this evolvability is only realized when selection acts exactly
on this component, otherwise evolvability is low. These values lie all within the 99%
bootstrap confidence intervals (Appendix 5.6.4 Tab. 5.13).
5.3.2
Selection skewers
Number of selected traits
We divide the selection skewers into groups given by the number of drugs they contain and investigate the response to these different groups of skewers. There are clear
trends in the response measures dependent on the number of drugs selected for. We
find that θ and γ decrease for an increasing number of drugs in the selection skewers.
Respondability and evolvability increase when more drugs are involved (Fig. 5.3).
Shown in the figure are only the results for the sequence data, but the ME and EE
data reveal the same patterns.
82
40
80
Since we are interested in the worst selection response (the direction with least
genetic variation), we can already see that the fewer drugs the cocktail contains, the
worse (smaller) is the according response. Intuitively, this is somehow surprising, but
it is explained by the fact that the first principal component (capturing the greatest
variance) is build up almost equally be all environments (see Appendix 5.6.2). Thus,
selecting on all drugs leads to the largest response.
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θ (in degrees)
●
1
3
5
7
9
11
13
Selected drugs
15
1
3
5
7
9
11
Selected drugs
Figure 5.3. Selection response dependent on the number of drugs selected for in GSEQ .
Drugs selected for
Selecting on drugs of the NNRTI class on average leads to significantly smaller values of γ (e.g. γEFV = 2.4◦ , γDDI = 2.7◦ , Wilcoxon-test: W = 116877150, P <
2.2 ∗ 10−16 , the over-line denotes the mean value), which means that the response
vector is more closely aligned with gmax , the direction of highest variance. NNRTIs
also create significantly larger respondabilities (e.g., respondabilityEFV = 2.5∗10−6 ,
respondabilityDDI = 2.4 ∗ 10−6 , Wilcoxon-test: W = 150484237, P < 2.2 ∗ 10−16 )
83
and evolvabilities (e.g., evolvabilityEFV = 1.9 ∗ 10−6 , evolvabilityDDI = 1.8 ∗ 10−6 ,
Wilcoxon-test: W = 150033670, P < 2.2 ∗ 10−16 ). Since we are looking for the worst response, including drugs of the NNRTI class in the drug cocktail is thus not preferred.
Finding constraints on resistance evolution
We looked for those selection skewers (β) that cause the worst response in the focal
HIV population and counted the number of appearances of each of the 15 drugs in
these β that lead to the worst responses. We first define the worst response by the
lowest value in respondability (smallest response vector), i.e., the smallest change in
population mean due to selection. Note that this is the unconstrained response.
For each covariance matrix we plot the cumulative appearance of each drug in
the selection vectors that lead to the 1000 worst responses (Fig. 5.4). The responses
are ordered by the value of respondability. It appears that the three datasets differ
in some aspects. In the sequence data three NRTIs (DDI, D4T and TFV) appear in
more than half of the focal selection vectors, while there are no drugs with such a
high count in ME and EE. However, as for the sequence data drugs DDI and D4T
are also most frequent in ME and EE which suggests that selecting on these drugs
generally leads to a very small selection response in HIV. In the sequence data and EE
data NNRTIs appear late and stay at low frequency, whereas they start to contribute
early on in the ME data. Nevertheless, all datasets suggest that NNRTIs should not
be contained in the drug cocktail to achieve a bad response.
EE
0
200
400
600
800
1000
Worst responses
200
100
300
400
500
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
PI
NRTI
NNRTI
0
200
100
0
100
200
300
300
400
400
500
500
ME
0
Selection skewer appearance
Seq
0
200
400
600
800
1000
Worst responses
0
200
400
600
800
1000
Worst responses
Figure 5.4. Appearance of drugs in the selection skewers (β) that lead to the 1000 worst
respondabilities in the three covariance matrices. Of interest are those drugs that most often
lead to very small selection responses.
84
For the sequence data, the skewers leading to the 1000 worst responses contain
a maximum of seven drugs and a mean of 4.12. These numbers are even lower for
the ME data (max=4, mean=3.30) and the EE data (max=5, mean=3.65). Randomly
choosing 1000 skewers will on average (among 10000 replicates) result in a maximum
of 13.4 drugs and a mean of 7.5 drugs. Thus, for all three datasets the selection
skewers that lead to the worst responses contain significantly less drugs than expected
by chance (P < 2.2 ∗ 10−16 ). To test whether the observed number of appearance of
each drug is significant given the distribution of selection skewers, we re-sampled the
drugs for each selection skewer giving each drug the same probability being included
in the skewer. With 10000 replicates we obtain a confidence interval for the expected
value of appearances of each drug in 1000 selection skewers and can thus calculate
the probability that the observed value falls into this interval or not. For the sequence
data, only IDV, LPV and SQV fall into the expected interval, all other drugs have
significantly higher or lower appearances. For the ME data, IDV, LPV, NFV, SQV and
ZDV are not significantly different from the expected appearance. And for the EE
data, AMP and RTV are not significantly different from the random expectation and
3TC and ABC are only weakly significantly different (P < 0.05). That means, those
drugs we mentioned before as causing bad responses (e.g., DDI, D4T) or not causing
bad responses (e.g., the three NNRTIs) all have significantly different appearances
in the observed selection skewers as we would expect from a random collection of
selection skewers.
350
300
200
150
100
200
400
600
Worst R
800
1000
PI
NRTI
NNRTI
0
50
0
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
250
300
100
0
50
150
200
250
300
250
200
150
100
50
0
Selection skewer appearance
EE
350
ME
350
Seq
0
200
400
600
Worst R
800
1000
0
200
400
600
800
1000
Worst R
Figure 5.5. Appearance of drugs in the selection skewers (β) that lead to the 1000 worst R
values in the three covariance matrices. Of interest are those drugs that most often lead to
very small constrained selection responses.
85
Finally, our interest is on the constrained evolutionary response. Instead of looking
for the worst respondability, we do the analysis with the smallest values of R, i.e.,
the evolvability standardized by the amount of genetic variance of the traits under
selection. Similar to respondability, R increases with the number of traits selected.
That means the smallest response is achieved by selecting on few drugs. However,
the appearances of drugs in the skewers causing the smallest values of R is very
different to what we saw for respondability (Fig. 5.5). Especially the NNRTIs rank
much higher than before, which is probably due to their high genetic variance that
the measure of R is normalized by. Only the result for the sequence data is relatively
similar to that for respondability. At least, the first two highest ranked drugs are
equal. However, for ME and EE the picture changes a lot. Interestingly, if we just
look at the 600 worst R values in the ME data, all drugs appear equally often in the
selection skewers. The reason could be the very strict relationship between R and the
number of selected traits in ME (similar to the bottom left graph in Figure 5.3, but
with very small variances around the means). Probably, all skewers selecting on a
small number of drugs are included in this set and therefore all drugs have the same
frequency at that point. Overall, NNRTIs are the main contributors for low constraint
selection responses in ME and EE.
5.4
5.4.1
Discussion
PCA
Principal component analysis makes it possible to identify trait combinations with
the largest amount of variation. Our PCA results for the sequence data confirm previous findings that used part of this data (Martins et al., 2010). With access to main
and epistatic effects of mutations we show that they resemble the pattern seen for the
whole sequence data. Despite the very high correlations between environments, the
three drug classes can be fairly well separated by principal component analysis. This
suggests an underlying modular genotype-phenotype map. In different drug environments mutations affect viral fitness in a similar fashion when the drugs belong to
the same class of drugs. For the ME and EE data, most of the variation is captured
by the first principal component, with all drugs contributing almost equally to PC1.
When we apply common principal component (CPC) analysis based on the Flury hierarchy (Phillips and Arnold, 1999), we find that the three datasets have only the first
principal component in common (results not shown).
5.4.2
Matrix comparison
Since covariance matrices might differ in various aspects, like single elements, rank,
structure, shape or orientation, we also applied matrix comparison tests. Roff’s T test,
86
co-inertia analysis, Krzanowski subspace comparison and random skewers all reveal
a similar picture, namely that the differences between the three correlation matrices
are minor. Especially between ME and EE data there is a very high similarity, i.e.,
single and double mutations characterize the correlational relationship between drug
environments in similar ways. The results suggest that the epistatic effects reflect the
whole sequence data slightly better than the main effects do, but it is difficult to quantify how much epistasis adds to the information we obtain from the main effects. In
the original study of this data by Hinkley et al. (2011) the model including epistasis
substantially outperformed a model without epistasis in terms of predictive power
(average increase of 18.3% across all environments). They showed that epistasis is
crucial for determining realistic fitness values and even found a significant correlation between epistasis and protein secondary structure. In the following chapter we
provide a more detailed view on the role of epistasis and demonstrate that it modifies
the pleiotropic degree of mutations.
The high correlations between drug environments constrain the adaptive potential of the virus. An allele doing well in the presence of one drug usually also does
well in the presence of another drug. On the other hand, there are only few possibilities for compensation of deleterious effects, since a mutation with negative fitness
effect in one drug environment is usually also deleterious in other drug environments.
Therefore, we also looked for the drug combination that has the highest proportion of
antagonistic mutations (i.e., which have at least one positive and one negative fitness
effect across all environments). For two drugs we find that EFV and LPV together
cause the highest amount of antagonism, and also the lowest amount of positive
mutations in both drugs. For a combination of three drugs TFV adds to the two
mentioned before. Not surprisingly, these three drugs belong to three different drug
classes. However, about 22% of mutations still have positive fitness effects in all three
drug environments.
5.4.3
Hierarchical clustering
Hierarchical clustering largely confirms the results of principal component analysis.
Drug classes cluster in the trees. Most of the splits into distinct clusters are highly supported by bootstrapping. What should be reflected in the clustering trees is the difference or similarity in chemical composition and mode of action of the different drugs.
As mentioned before, there is agreement with respect to the drug classes which have
different targets (protease versus reverse transcriptase) and different chemical compositions (nucleoside versus non-nucleoside analogues). Within NNRTIs, DLV is hardly
used in practice (de Béthune, 2010), which is reflected in the trees by being separated
from EFV and NVP. These two are often used in combination therapy. Within NRTIs,
one distinguishes at least three sub-classes: DDI, D4T and 3TC are dideoxynucleosides (De Clercq, 2009), ABC is a carbocyclic nucleoside (Ogden and Skowron, 2006)
87
and TFV is a nucleotide analogue (all others are nucleoside analogues) (Fung et al.,
2002). The first antiviral drug ZDV was even developed as an anti-cancer agent initially in the 1960s (Saunders, 2000). The only obvious analogy in our trees is the
clustering of DDI and D4T. 3TC should belong to the same cluster but is placed further apart and clusters with ABC in the trees of the ME and EE data. ZDV and TFV
are chemically quite different, but they seem to have very similar modes of action
since they cluster together in two of the hierarchies. Within PIs, NFV is the only nonpeptidomimetic inhibitor (Wlodawer, 2002). LPV and RTV have identical cores and
AMP is supposed to be chemically similar to SQV. AMP has been withdrawn from
market in 2004 because its prodrug fosamprenavir proved to be better (Brunton et al.,
2006). At least in the trees from the ME and EE data LPV and RTV are clustered together, otherwise the described relationships are not really reflected in the hierarchies.
Overall, there is much more agreement than disagreement between the pharmaceutical characteristics of drugs and our hierarchical clusterings based on HIV fitness data.
Hence, fitness effects of mutations reflect, at least to some extent, the similarities in
drug design.
5.4.4
Selection skewers
Selection skewers and random skewers are widely used to investigate the evolvability
of a population represented by a genetic covariance matrix. The response of a population to a selective force depends on the ability to improve fitness in the traits that
selection is acting on. Our selection skewer analysis of unconstrained response reflects findings from investigations of the raw data and results from the PCA. We find
the highest mutational variation in the NNRTIs and the lowest mutational variation
in the NRTIs D4T, DDI and TFV. These three NRTIs also do not contribute to the first
three principal components of the covariance matrix of the sequence data. Therefore
it is not surprising that the viral population shows the smallest respondability when
selecting in the direction of these traits. D4T, DDI and TFV have a much larger contribution to the worst respondabilities and the NNRTIs a much smaller contribution
than expected by chance. Thus, this analysis strongly suggests that the three mentioned NRTIs have the greatest potential to keep viral adaptation at a minimum. It
also suggests that the NNRTIs lead to large responses and possibly increase the risk
of emergence of resistance strains. This hypothesis is supported by a study on evolutionary dynamics of complex transmission networks composed of multiple resistant
strains from San Francisco, USA (Smith et al., 2010). They built a theoretical model to
reproduce the observed dynamics over the past 20 years and predict a wave of NNRTIresistant strains emerging over the next 5 years in San Francisco in the risk group of
men having sex with men. This demonstrates that NNRTIs seem to bear the potential
for enhanced resistance evolution. However, when we look at constrained selection
response, the NNRTIs contribute much more to limited responses. This is because we
88
normalize by the genetic variance of the drugs selected for, which is high for NNRTIs.
This causes a difference in the results between the three datasets. Nevertheless, TFV
and DDI still have the highest rank in the sequence data, which we believe to be more
trustworthy since it comprises all the genetic variation in HIV. Therefore, these two
NRTIs potentially qualify for preventing new mutations from arising and therefore
reduce the emergence of resistance. We have to point out that these results have to be
treated with caution. There are several limitations that cannot be accounted for here:
(1) The sequence data comes from a diverse group of patients and does not reflect
a single viral population within one patient. (2) We do not know which drugs the
sampled viruses have been exposed to, therefore it is not clear in how much selection
has already shaped the mutational landscape. (3) The fitness prediction model for
main and epistatic effects has a high predictive power, but cannot completely explain
the variation in the fitness data (Hinkley et al., 2011).
Generally, our analyses show that selecting on fewer drugs leads to a smaller selection response. The reason for this analytical result is the composition of principal
components in the genetic covariance matrix. The first component, comprising the
largest amount of variation in the data, is build up by almost all drugs in similar proportions. We therefore see the greatest selection response when including all drugs
in the selection skewer. It is however unclear (and difficult to test) whether this reflects in-vivo treatment of patients. Since we normalized our selection skewers to
unit length, all of them symbolized equal strengths of selection, independent of the
number of different drugs contained in the ’cocktail’. Therefore we do not assume a
higher dose when applying more drugs, but a smaller portion of each single drug at a
constant dose. Hence, the pure toxic burden should not be a problem. Rather, the unknown interactions of drugs and the larger number of different ways of interference
with the human body might cause problems. Apart from that, it might well be that a
drug cocktail with 15 different drugs (of very low dose each) fails to repress the virus
and leads to an increased chance for the virus to develop resistance mutations.
5.4.5
Perspectives
The available data can be explored even further. For example, to circumvent limitation
(1) mentioned above, that the data does not represent the viral population of a single
patient, one could use a within-host model of viral evolution. Simulated within-host
populations from founder strains present in the existing data can be investigated for
their level of genetic variation and compared to the present analysis.
Furthermore, for a better validation of the estimated fitness effects and their significance, it might be possible to re-run the generalized kernel ridge regression algorithm
of Hinkley et al. (2011) with additional artificially generated neutral mutations. This
would give an idea about the magnitude of effects assigned to intrinsically neutral
89
mutations and thus would provide a confidence interval for the significance of ’real’
effects.
5.5
Conclusions
Facing the spread of multidrug resistance, it is important to develop strategies to
fight back. This work presents a way of analyzing experimental data that is based on
real-life pathogenic samples to investigate the genetic basis of adaptation to drug environments and offers suggestions for future treatment strategies. Covariance matrix
analysis offers opportunities to investigate the variational and correlational structure
of a population based on genetic data and makes it possible to predict the population’s response to selectional forces. It can help medical research to foresee evolutionary changes and implement appropriate actions.
Generally, these analyses suggest that mutations for drug resistance might not
trade-off between environments. That means selection for resistance on one drug
also favors evolution of resistance for other drugs. This statement could be verified
by investigating the actual fitness effects of mutations and calculate the probability
that a mutation is beneficial in another drug when it was selected in one drug. If
that probability was high and believed it to be a general characteristic of resistance
mutations, it would be bad news for fighting resistance. Therefore it is important to
continue and intensify research in this direction to see how general the assumptions
are and how they change over time.
90
5.6
5.6.1
Appendix
Antiretroviral drugs
Full name
Abbreviation
Drug class
Amprenavir
Indinavir
Lopinavir
Nelfinavir
Ritonavir
Saquinavir
AMP
IDV
LPV
NFV
RTV
SQV
Protease
inhibitors
Lamivudine
Abacavir
Stavudine
Didanosine
Tenofovir
Zidovudine
3TC
ABC
D4T
DDI
TFV
ZDV (formerly AZT)
Delavirdine
Efavirenz
Nevirapine
DLV
EFV
NVP
Nucleoside reverse
transcriptase inhibitors
Non-nucleoside
reverse transcriptase
inhibitors
91
5.6.2
G-matrices and eigendecompositions
Table 5.4. Covariance/correlation matrix of sequence data. Variances in diagonal, correlations
in lower triangle.
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
356.2739027
0.8614661
0.9283482
0.7761564
0.8863663
0.8323464
0.3482103
0.4854613
0.5174188
0.4236019
0.3975924
0.4826635
0.1985093
0.2150910
0.2418322
0.2751442
343.9453805
447.4232518
0.9281061
0.9224489
0.9397147
0.9199751
0.3767393
0.5074805
0.5500339
0.4653536
0.4423281
0.5159511
0.2319454
0.2462213
0.2743863
0.3459927
364.4017631
408.2577910
432.4691804
0.8239929
0.9428882
0.8694758
0.3563058
0.4923488
0.5177971
0.4165125
0.4015056
0.4989939
0.2006760
0.2197502
0.2485871
0.2582357
367.6254991
489.6277095
429.9970116
629.6928522
0.8840389
0.8996006
0.4003790
0.5061985
0.5399076
0.4887504
0.4440960
0.5015239
0.2493414
0.2563686
0.2843811
0.4002571
392.1521313
465.9128171
459.6075450
519.9781521
549.4127186
0.9021825
0.3788744
0.5039477
0.5331570
0.4521723
0.4247427
0.5074386
0.2335397
0.2487461
0.2774379
0.3223800
326.9001291
404.9060990
376.2305166
469.7135440
440.0101463
432.9501764
0.3772056
0.5117544
0.5500488
0.4576255
0.4385774
0.5206845
0.2288417
0.2447710
0.2739813
0.3137698
216.3685514
262.3376838
243.9276432
330.7471863
292.3514140
258.3792398
1083.7299074
0.8522277
0.3618069
0.6312852
0.2036925
0.3419473
0.2392942
0.2710896
0.3025679
0.2830898
184.6706653
216.3365332
206.3487137
255.9979702
238.0598233
214.6010301
565.4151703
406.1650237
0.6659965
0.7694061
0.5128664
0.6315108
0.2921518
0.3418388
0.3740726
0.3305657
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
98.7436070
117.6314586
108.8709408
136.9804493
126.3511578
115.7164100
120.4237580
135.7056720
102.2232165
0.7758799
0.8331262
0.8162307
0.3464230
0.3695667
0.3880009
0.5470597
68.3206152
84.1093548
74.0129211
104.7981121
90.5639292
81.3637719
177.5775563
132.4978348
67.0303022
73.0136090
0.5916698
0.5169515
0.4224737
0.4158832
0.4272061
0.6696545
65.9315192
82.1990499
73.3554170
97.9048609
87.4657967
80.1730255
58.9112959
90.8068234
74.0028726
44.4165027
77.1837777
0.8397027
0.3224476
0.3427688
0.3523672
0.5179060
169.3775542
202.9024083
192.9266152
233.9781894
221.1323291
201.4247990
209.2854482
236.6199858
153.4288606
82.1241817
137.1538274
345.6517272
0.2663284
0.3275540
0.3500181
0.3140753
135.8045304
177.8224021
151.2566434
226.7775651
198.4044418
172.5820010
285.5183697
213.4032977
126.9471022
130.8407257
102.6747209
179.4643705
1313.6592246
0.8947817
0.9010614
0.4416497
148.8657852
190.9702242
167.5666106
235.8901896
213.7894567
186.7495416
327.2307222
252.6115910
137.0087342
130.3028596
110.4193030
223.2970678
1189.1568423
1344.5018850
0.9571365
0.3826771
178.5670492
227.0475656
202.2326498
279.1645678
254.3959867
223.0155019
389.6534188
294.9186821
153.4626208
142.8020510
121.1026474
254.5687108
1277.5881515
1372.9341950
1530.3482125
0.3759698
255.8298269
360.5163574
264.5408369
494.7687225
372.2341264
321.6095420
459.0748346
328.1767911
272.4634512
281.8717018
224.1365706
287.6417493
788.5297117
691.2131048
724.5140979
2426.5964187
Table 5.5. Eigenvectors (EV) and eigenvalues (λ) of GSEQ , excluding the drug-free environment.
EV1
EV2
EV3
EV4
EV5
EV6
EV7
EV8
EV9
EV10
EV11
EV12
EV13
EV14
EV15
3TC
ABC
AMP
D4T
DDI
DLV
EFV
IDV
LPV
NFV
NVP
RTV
SQV
TFV
ZDV
-0.25
-0.18
-0.16
-0.09
-0.08
-0.43
-0.46
-0.19
-0.18
-0.23
-0.50
-0.21
-0.18
-0.06
-0.15
-0.23
-0.16
-0.26
-0.07
-0.04
0.34
0.34
-0.30
-0.30
-0.34
0.33
-0.33
-0.29
-0.04
-0.13
0.81
0.37
-0.15
0.01
0.10
-0.09
-0.05
-0.18
-0.18
-0.17
-0.04
-0.20
-0.16
-0.01
0.03
0.27
-0.24
0.06
-0.33
-0.10
0.14
-0.02
0.08
0.08
0.11
-0.01
0.11
0.05
-0.34
-0.75
-0.00
0.02
-0.12
0.11
0.14
0.77
-0.39
0.01
-0.13
0.17
-0.37
-0.06
0.03
0.11
0.03
-0.01
0.08
0.51
0.02
-0.01
0.22
-0.11
-0.09
0.41
-0.65
-0.09
0.15
-0.20
-0.02
0.01
0.08
-0.15
-0.07
-0.17
-0.28
0.03
-0.66
-0.01
0.02
-0.02
0.61
0.03
0.03
-0.07
0.23
0.25
-0.42
-0.12
-0.32
-0.44
0.11
0.27
0.02
0.07
-0.01
-0.34
0.12
-0.09
-0.05
0.48
0.01
-0.07
-0.06
0.07
0.13
-0.05
-0.05
0.13
0.14
0.31
0.08
0.22
-0.87
0.09
-0.04
0.08
-0.18
0.69
-0.02
-0.04
-0.01
0.00
-0.20
-0.13
0.37
0.01
-0.52
-0.09
0.02
0.08
-0.25
0.65
-0.04
-0.34
-0.36
0.08
0.02
0.20
0.15
0.14
-0.07
-0.28
-0.14
-0.28
0.05
-0.11
0.26
0.15
-0.14
-0.11
0.02
0.01
-0.65
-0.30
0.18
-0.02
0.56
-0.01
-0.09
0.02
0.01
-0.06
-0.29
0.12
0.08
-0.00
0.01
-0.55
0.70
0.21
-0.00
-0.20
0.09
-0.11
0.01
-0.02
-0.09
0.03
0.35
0.19
0.01
0.00
0.08
-0.13
-0.01
-0.01
0.04
-0.04
-0.86
0.24
-0.06
-0.03
0.01
-0.68
0.70
-0.02
-0.02
0.01
0.03
-0.02
0.02
-0.02
0.03
-0.06
0.22
λ
4875.9
2376.7
959.5
325.4
163.8
132.0
61.9
54.3
45.8
39.6
29.6
26.4
14.4
12.1
7.1
92
Table 5.6. Covariance/correlation matrix of ME data. Variances in diagonal, correlations in
lower triangle.
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
0.000000372
0.917488437
0.936808057
0.900933611
0.933356982
0.891165954
0.744168999
0.764505542
0.755888245
0.776426426
0.699347429
0.658266992
0.600655916
0.615733892
0.605858267
0.818611734
0.000000320
0.000000327
0.954317935
0.963131230
0.954933662
0.931943935
0.758919404
0.771770389
0.767866466
0.789618464
0.711835588
0.671061783
0.616399047
0.635000193
0.624870725
0.833816911
0.000000330
0.000000316
0.000000334
0.925384427
0.957561298
0.910320776
0.754845018
0.770881450
0.760708030
0.781565008
0.709130321
0.669780374
0.608058866
0.622281048
0.612120719
0.824348872
0.000000298
0.000000299
0.000000290
0.000000294
0.948482919
0.932525227
0.766739747
0.779611630
0.775364744
0.800777326
0.717480424
0.679751828
0.624683401
0.639427841
0.630993460
0.849744550
0.000000334
0.000000321
0.000000325
0.000000302
0.000000345
0.924749351
0.762815845
0.775801333
0.769479017
0.792962278
0.716667951
0.678161968
0.631579626
0.643360124
0.635745345
0.834214234
0.000000313
0.000000307
0.000000303
0.000000292
0.000000313
0.000000332
0.744517423
0.763479037
0.763834200
0.782134949
0.708487213
0.671898163
0.616381753
0.633304390
0.620302189
0.826622786
0.000000258
0.000000246
0.000000248
0.000000236
0.000000254
0.000000243
0.000000322
0.923417401
0.762318547
0.849000849
0.695754487
0.663323384
0.633802653
0.664022408
0.659333714
0.820302111
0.000000251
0.000000237
0.000000240
0.000000227
0.000000245
0.000000236
0.000000282
0.000000289
0.875198335
0.915245047
0.835038348
0.791775057
0.656230115
0.680993463
0.669795496
0.837473478
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
0.000000232
0.000000221
0.000000222
0.000000212
0.000000228
0.000000222
0.000000218
0.000000237
0.000000254
0.926680219
0.900336774
0.848543959
0.653859580
0.659597916
0.641757803
0.846725769
0.000000226
0.000000216
0.000000216
0.000000208
0.000000223
0.000000215
0.000000230
0.000000235
0.000000223
0.000000228
0.843263202
0.766316689
0.688562178
0.695752101
0.682093169
0.894378422
0.000000238
0.000000227
0.000000229
0.000000218
0.000000235
0.000000228
0.000000221
0.000000251
0.000000253
0.000000225
0.000000312
0.900853749
0.658467507
0.661205274
0.636339981
0.777792733
0.000000252
0.000000240
0.000000243
0.000000231
0.000000250
0.000000243
0.000000236
0.000000267
0.000000268
0.000000229
0.000000315
0.000000393
0.633793822
0.655975815
0.631581164
0.709386244
0.000000251
0.000000242
0.000000241
0.000000232
0.000000254
0.000000244
0.000000247
0.000000242
0.000000226
0.000000226
0.000000252
0.000000272
0.000000470
0.886607747
0.868505088
0.693994736
0.000000269
0.000000260
0.000000258
0.000000249
0.000000271
0.000000262
0.000000270
0.000000263
0.000000238
0.000000238
0.000000265
0.000000295
0.000000436
0.000000514
0.946678165
0.709894994
0.000000258
0.000000249
0.000000247
0.000000239
0.000000261
0.000000249
0.000000261
0.000000251
0.000000225
0.000000227
0.000000248
0.000000276
0.000000415
0.000000474
0.000000487
0.699080696
0.000000196
0.000000187
0.000000187
0.000000181
0.000000192
0.000000187
0.000000182
0.000000176
0.000000167
0.000000167
0.000000170
0.000000174
0.000000186
0.000000200
0.000000191
0.000000154
Table 5.7. Eigenvectors (EV) and eigenvalues (λ) of GME , excluding the drug-free environment.
EV1
EV2
EV3
EV4
EV5
EV6
EV7
EV8
EV9
EV10
EV11
EV12
EV13
EV14
EV15
3TC
ABC
AMP
D4T
DDI
DLV
EFV
IDV
LPV
NFV
NVP
RTV
SQV
TFV
ZDV
-0.24
-0.24
-0.27
-0.22
-0.22
-0.28
-0.30
-0.26
-0.26
-0.25
-0.29
-0.27
-0.26
-0.24
-0.26
-0.05
-0.05
-0.26
-0.05
-0.03
0.44
0.48
-0.24
-0.25
-0.21
0.47
-0.23
-0.22
0.00
0.03
-0.11
-0.27
0.19
-0.32
-0.20
0.15
0.18
0.20
0.19
0.16
0.20
0.20
0.18
-0.44
-0.52
0.70
0.41
-0.07
-0.02
0.23
-0.09
-0.02
-0.08
-0.08
-0.06
0.03
-0.09
-0.09
-0.25
-0.41
-0.06
-0.01
-0.00
0.14
0.18
0.79
-0.30
-0.02
-0.01
-0.01
-0.41
0.00
-0.01
0.07
-0.25
-0.37
-0.08
-0.03
0.49
0.46
-0.26
0.10
0.01
-0.08
0.03
0.17
-0.05
0.05
0.18
-0.51
-0.05
0.03
0.70
0.02
0.01
-0.02
0.01
-0.19
0.24
-0.36
0.04
0.05
-0.52
0.05
-0.05
0.01
-0.03
-0.40
-0.03
-0.01
0.02
-0.26
0.31
0.27
0.23
0.25
0.24
-0.64
0.14
-0.06
-0.05
-0.06
0.12
0.24
0.14
0.09
-0.57
-0.10
-0.13
0.01
0.49
-0.02
0.09
-0.48
0.26
0.05
0.10
0.09
-0.30
-0.23
0.02
-0.39
-0.11
-0.07
-0.10
0.38
-0.05
0.28
0.60
-0.26
0.03
-0.04
0.36
-0.09
-0.02
-0.00
0.02
0.11
-0.64
0.59
-0.03
-0.10
-0.27
0.09
-0.01
-0.08
0.15
0.09
0.02
-0.08
0.04
-0.04
0.48
0.25
0.01
0.04
-0.81
-0.01
-0.04
-0.00
-0.52
0.73
-0.04
-0.34
0.16
0.01
0.01
-0.06
-0.02
0.08
-0.01
0.12
-0.01
-0.15
0.03
-0.05
0.34
-0.02
0.56
-0.71
0.03
0.03
0.02
-0.10
0.00
-0.01
0.13
-0.02
-0.07
-0.15
-0.01
0.01
0.03
-0.06
0.08
0.00
-0.01
0.65
-0.43
-0.56
-0.00
0.25
0.01
-0.01
0.03
λ [∗10−7 ]
39.94
5.37
2.89
1.41
0.72
0.51
0.44
0.28
0.26
0.25
0.18
0.14
0.13
0.11
0.08
93
Table 5.8. Covariance/correlation matrix of EE data. Variances in diagonal, correlations in
lower triangle.
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
0.000007942
0.879978958
0.915237673
0.854371755
0.894701924
0.844856101
0.638942157
0.690752589
0.691146167
0.698703871
0.646235804
0.630303826
0.463604013
0.448439266
0.454499843
0.729037369
0.000005858
0.000005579
0.931366927
0.941047344
0.926995023
0.901715903
0.654428330
0.706587434
0.709798737
0.717887485
0.663607374
0.643169763
0.475740962
0.457301821
0.465286497
0.757570696
0.000006459
0.000005509
0.000006272
0.884786874
0.933394857
0.872681627
0.644399743
0.697741172
0.698038912
0.701550700
0.649091573
0.637508357
0.462541206
0.447465564
0.454855607
0.731002563
0.000005294
0.000004887
0.000004872
0.000004834
0.914130366
0.896488185
0.660219988
0.708219602
0.713170693
0.727560110
0.667754751
0.642260963
0.484158661
0.462383004
0.470388867
0.772978164
0.000006657
0.000005781
0.000006171
0.000005306
0.000006971
0.883281308
0.657031405
0.705949536
0.704920960
0.715023076
0.660645398
0.642220426
0.478715204
0.459766927
0.466474603
0.750956116
0.000005427
0.000004855
0.000004982
0.000004493
0.000005316
0.000005196
0.645226061
0.703954805
0.712954885
0.718690558
0.661220676
0.643911732
0.471774303
0.451626495
0.459533991
0.750589749
0.000005420
0.000004652
0.000004857
0.000004369
0.000005221
0.000004427
0.000009059
0.902387014
0.694304763
0.809192515
0.622773098
0.610658446
0.519677855
0.508404511
0.519541385
0.747681005
0.000004890
0.000004192
0.000004389
0.000003911
0.000004682
0.000004030
0.000006822
0.000006309
0.822205679
0.885055671
0.771701087
0.743937599
0.545581243
0.541431372
0.547966458
0.795792696
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
NODRUG
0.000003904
0.000003360
0.000003504
0.000003143
0.000003730
0.000003257
0.000004188
0.000004139
0.000004017
0.889471950
0.884884934
0.832527020
0.546766050
0.534410041
0.534687480
0.838440785
0.000003664
0.000003155
0.000003269
0.000002977
0.000003513
0.000003048
0.000004532
0.000004137
0.000003317
0.000003463
0.813584427
0.721182467
0.587225075
0.565834825
0.565866675
0.884488528
0.000004508
0.000003879
0.000004023
0.000003634
0.000004317
0.000003730
0.000004639
0.000004797
0.000004389
0.000003747
0.000006126
0.880616832
0.558982134
0.550101810
0.543338400
0.798004191
0.000005392
0.000004611
0.000004846
0.000004287
0.000005147
0.000004455
0.000005579
0.000005672
0.000005065
0.000004074
0.000006616
0.000009214
0.526829283
0.534644408
0.531406703
0.711823023
0.000005215
0.000004485
0.000004624
0.000004249
0.000005045
0.000004293
0.000006244
0.000005470
0.000004374
0.000004362
0.000005522
0.000006384
0.000015934
0.843942467
0.841433844
0.618905518
0.000005825
0.000004979
0.000005165
0.000004686
0.000005595
0.000004745
0.000007054
0.000006269
0.000004937
0.000004854
0.000006276
0.000007481
0.000015528
0.000021247
0.929652178
0.585202734
0.000005576
0.000004784
0.000004959
0.000004502
0.000005362
0.000004560
0.000006808
0.000005992
0.000004665
0.000004584
0.000005854
0.000007022
0.000014622
0.000018655
0.000018952
0.581967523
0.000002647
0.000002305
0.000002358
0.000002189
0.000002554
0.000002204
0.000002899
0.000002575
0.000002165
0.000002120
0.000002544
0.000002784
0.000003183
0.000003475
0.000003264
0.000001660
Table 5.9. Eigenvectors (EV) and eigenvalues (λ) of GEE , excluding the drug-free environment.
EV1
EV2
EV3
EV4
EV5
EV6
EV7
EV8
EV9
EV10
EV11
EV12
EV13
EV14
EV15
3TC
ABC
AMP
D4T
DDI
DLV
EFV
IDV
LPV
NFV
NVP
RTV
SQV
TFV
ZDV
0.25
0.22
0.24
0.18
0.17
0.35
0.42
0.20
0.21
0.19
0.39
0.23
0.19
0.21
0.25
0.16
0.15
0.28
0.12
0.10
-0.34
-0.47
0.24
0.26
0.21
-0.43
0.26
0.22
0.12
0.15
-0.30
-0.31
0.29
-0.24
-0.19
0.07
0.09
0.24
0.27
0.20
0.10
0.28
0.19
-0.35
-0.46
0.71
0.31
-0.04
-0.13
0.11
0.02
-0.04
-0.03
-0.04
-0.02
0.02
-0.03
-0.03
-0.34
-0.50
-0.04
-0.04
-0.02
0.02
0.04
0.86
-0.36
-0.01
-0.02
0.01
-0.34
-0.00
0.00
0.04
-0.05
-0.04
0.02
0.02
0.07
0.11
-0.02
0.67
-0.01
-0.01
-0.00
-0.71
0.00
-0.00
0.07
-0.15
0.25
-0.04
0.35
-0.37
-0.41
0.07
0.10
-0.11
0.13
-0.20
-0.16
0.01
-0.22
-0.29
0.51
0.13
-0.06
-0.66
-0.21
-0.22
0.02
0.07
0.25
-0.12
0.34
-0.09
0.09
0.35
-0.18
0.28
-0.03
0.02
0.34
0.04
0.02
0.00
0.01
-0.16
-0.30
-0.05
-0.01
-0.46
0.72
-0.18
0.07
0.24
-0.23
0.20
-0.40
-0.23
-0.02
-0.01
0.06
-0.32
0.29
0.01
-0.12
-0.04
0.64
-0.18
-0.03
0.20
-0.21
-0.28
-0.19
0.01
-0.01
-0.23
0.32
-0.53
0.01
0.23
0.41
0.36
-0.15
-0.28
0.57
-0.01
-0.25
-0.11
0.02
-0.00
0.39
0.25
0.10
-0.01
-0.53
-0.14
0.04
-0.04
0.31
-0.54
-0.13
0.20
0.02
-0.01
0.00
0.25
0.48
-0.20
-0.00
-0.46
0.05
0.09
-0.02
0.06
0.15
0.02
0.59
-0.76
0.02
0.01
0.06
-0.09
-0.01
-0.01
0.03
-0.01
0.03
-0.17
0.00
-0.03
0.07
-0.03
0.08
-0.00
-0.00
0.69
-0.43
-0.55
0.00
0.16
-0.00
-0.02
0.02
λ [∗10−6 ]
87.05
22.55
6.79
4.54
3.15
1.41
1.30
1.17
0.70
0.55
0.50
0.46
0.44
0.27
0.23
94
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
ND
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
ND
Correlations between drug environments
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
ND
5.6.3
AMP
IDV
LPV
NFV
RTV
SQV
3TC
ABC
D4T
DDI
TFV
ZDV
DLV
EFV
NVP
ND
1.0
0.8
0.6
0.4
0.2
SEQ
ME
EE
0.0
Figure 5.6. Heatmaps of the correlation matrices of sequence data (SEQ), main effects (ME)
and epistatic effects (EE) for the 15 drug environments and the drug-free condition (ND).
Drugs are ordered by class: PIs (AMP, IDV, LPV, NFV, RTV, SQV), NRTIs (3TC, ABC, D4T,
DDI, TFV, ZDV) and NNRTIs (DLV, EFV, NVP).
95
5.6.4
Bootstrap results from G-matrix statistics
Table 5.10. Between-data comparison. Observed correlation matrix comparisons (obs) and
means and confidence intervals (boot) from 10’000 bootstrap replicates of the ME, EE and
sequence (SEQ) data. Values are computed for bootstraps from two different datasets. T =
Roff’s T, Co-I = co-inertia correlation, KSC = Krzanowski subspace criterium, RS = random
skewer response correlation.
Statistic
ME vs. EE
ME vs. SEQ
EE vs. SEQ
T obs
0.0699
0.2401
0.1729
T boot
0.0701
[0.0699, 0.0703]
0.2401
[0.2399, 0.2403]
0.1730
[0.1729, 0.1730]
Co − I obs
0.9759
0.8860
0.9077
Co − I boot
0.9708
[0.9705, 0.9712]
0.8823
[0.8820, 0.8826]
0.9076
[0.9075, 0.9077]
KSC obs
6.8110
5.8679
5.9664
KSC boot
6.6823
[6.6779, 6.6867]
5.8579
[5.8559, 5.8600]
5.9681
[5.9677, 5.9686]
RS obs
0.9910
0.9146
0.9475
RS boot
0.9903
[0.9902, 0.9903]
0.9150
[0.9148, 0.9153]
0.9477
[0.9475, 0.9478]
96
Table 5.11. Within-data bootstraps. Means and confidence intervals from 10’000 bootstrap
replicates of ME, EE and SEQ. Values are computed for bootstraps from the same datasets. T
= Roff’s T, Co-I = co-inertia correlation, KSC = Krzanowski subspace criterium, RS = random
skewer response correlation.
Statistic
ME
EE
SEQ
T boot
0.0135
[0.0134, 0.0136]
0.0034
[0.0034, 0.0035]
0.0046
[0.0045, 0.0046]
Co − I boot
0.9909
[0.9907, 0.9911]
0.9996
[0.9996, 0.9997]
0.9997
[0.9997, 0.9997]
KSC boot
6.7558
[6.7505, 6.7610]
6.9954
[6.9953, 6.9956]
6.9969
[6.9968, 6.9969]
RS boot
0.9988
[0.9988, 0.9989]
0.9999
[0.9999, 0.9999]
0.9999
[0.9999, 0.9999]
Table 5.12. Is EE more similar to SEQ than ME? Means and confidence intervals from 10’000
bootstrap replicates. Values are computed for bootstraps from ME and EE with the same
bootstrap from SEQ. T = Roff’s T, Co-I = co-inertia correlation, KSC = Krzanowski subspace
criterium, RS = random skewer response correlation.
Statistic
(ME vs. SEQ) - (EE vs. SEQ)
observed difference
0.0671
[0.0669, 0.0673]
0.0672
Co − I
-0.0253
[-0.0256, -0.0250]
-0.0217
KSC
-0.1102
[-0.1123, -0.1081]
-0.0985
RS
-0.0326
[-0.0329, -0.0323]
-0.0329
T
97
Table 5.13. Means and confidence intervals for the effective number of dimensions nD and
maximum evolvability emax from 10’000 bootstrap replicates of the ME, EE and sequence
(SEQ) data.
Statistic
ME
EE
SEQ
nD obs
1.2888
1.4182
1.8805
nD boot
1.2886
[1.2875, 1.2896]
1.4180
[1.4177, 1.4183]
1.8804
[1.8798, 1.8810]
emax obs
3.5235
3.3588
2.9169
emax boot
3.5239
[3.5225, 3.5253]
3.3591
[3.3588, 3.3594]
2.9170
[2.9165, 2.9175]
98
6 | Epistasis modifies pleiotropic
effects in HIV
Abstract
Pleiotropy and epistasis are two players influencing the mutational variation of an
organism. But how do they interact? Does epistasis change the pleiotropy of a mutation? We use data on HIV-1 fitness measurements to study fitness effects of amino
acid substitutions in different drug environments. To this end, we make use of fitness
prediction models for effects of individual mutations and epistatic effects of double
mutation to uncover important aspects of the genetic architecture providing information about evolvability and constraints residing in the mutational variation of the
virus. We find that main effects and epistatic interactions can target very different
sets of traits, which leads to a high variability of HIV with respect to the mutational
response to different drug encounters. Epistasis increases the trait repertoire of a
mutation. We also find that even highly pleiotropic mutations often have either only
positive or only negative fitness effects among the different drug environments, which
suggests a high level of cross-resistance of those mutations with positive effects. On
the other hand, the data shows a large number of mutations with negative (main)
fitness effects that should be eliminated by selection and are probably compensated
by positive epistasis. Overall, epistasis seems to be the major player in the dynamics
of adaptation of HIV.
99
100
6.1
Introduction
The Human Immunodeficiency Virus (HIV) is one of the major infectious diseases
that mankind encounters since three decades. Unfortunately, until today it is not
possible to heal HIV infections. The main reasons are the high rate of mutation and
replication (Dougherty and Temin, 1988; Nowak, 1990; Ho et al., 1995), the ability for
recombination (Levy, 1988) and the resulting high rate of adaptation of this RNAvirus, which makes it very difficult for the human immune system to clear. There are
awareness and prevention campaigns to avoid transmission in the first place. This is
quite successful and has led to a decrease of the number of new infections per year
(UNAIDS, 2011), but medicine and public health have the duty (and of course the
interest) to help those already infected, also in the hope to decrease the infection rate.
Therefore, over 20 antiretroviral drugs have been developed to help the immune system kill viruses or impede their replication (De Clercq, 2009). Generally, these drugs
(especially taken in combination) reduce early mortality and can even prolong the
lives of infected people to more than the average 10 years after infection. But as with
most diseases, we encounter the problem of resistance.
It is not always clear how resistance arises. There are some mutations in HIV that are
known to confer resistance to one or more antiviral drugs (Bennett et al., 2009). Combination therapy tries to avoid that such mutations are selected in a patient. However,
the impact of multiple drugs on the mutational landscape of the virus is poorly understood. Pleiotropic effects and epistatic interactions of mutations can influence the
expected outcome of drug therapy in an unpredictable manner. Drug resistance mutations can act in different ways, usually changing the part of the protein used as a
drug target, such that the drug is unable to bind. When a mutation leads to structural
changes that prevent more than one drug from binding to its target site we speak of
multi-drug resistance caused by a single mutation, which means that this mutation
has pleiotropic effects on resistance to different drugs. It is also possible that these
pleiotropic effects are mediated by the presence of other mutations through epistatic
interactions (see Fig. 6.1). That positive epistasis is prevalent and leads to multi-drug
resistance has been shown in a recent study on antibiotic resistance in E. coli (Trindade
et al., 2009).
Pleiotropy (one genetic locus affecting several traits) and epistasis (non-additive
interaction between loci) are features of the genetic architecture that can favor or constrain adaptation. Baatz and Wagner (1997) found that the response to selection is
strongly influenced by hidden pleiotropic effects. Whether this leads to an acceleration or an inhibition of the strength of response, depends on allele frequencies. Based
on Fisher’s model of adaptation, Orr (2000) claimed that complex organisms adapt
more slowly than simple ones, because the probability that a random mutation is favorable decreases with the number of phenotypic dimensions. This has led to the ’cost
of complexity’-hypothesis, which rests upon the assumption of universal pleiotropy
101
Figure 6.1. A possible genetic interaction of mutations leading to epistatic effects. Mutation A
alone only affects trait 1 (A). Mutation B alone only affects trait 3 (B). But when both mutations
occur on the same genome, they might lead to an additional effect on trait 2 (C). This means
that epistasis would increase the pleiotropy of mutations A and B. When the fitness effect of
this epistatic interaction is negative, it can be maladaptive for the organism and thus puts a
genetic constraint on the selection of available mutations.
(every gene affects all traits). Other theoretical work has shown that a large fraction
of fixed mutations have deleterious pleiotropic effects (Griswold and Whitlock, 2003)
and that the fraction of overall beneficial alleles is halved by pleiotropy (Otto, 2004).
That pleiotropy does not have to lead to such strong constraints has been found in several recent theoretical and empirical studies, reviewed by Wagner and Zhang (2011).
For instance, Welch and Waxman (2003), Martin and Lenormand (2006), Chevin et al.
(2010) and Lourenço et al. (2011) analyzed generalized versions of Fisher’s model, relaxing some of the strong assumptions. They find that correlated traits and restricted
(also called partial or modular) pleiotropy strongly reduce the cost of complexity by
changing the distribution of fitness effects of mutations. Overall, there is no unique
answer to the question how pleiotropy and genetic correlations affect the response to
selection. Already Gromko (1995) showed that different combinations of pleiotropic
effects can lead to the same value of genetic correlation, and reversely that genetic
correlation is not a reliable predictor of pleiotropic constraint, because it depends on
the number of positive, negative and one-trait-only effects. The nature and strength of
selection also has a large impact on the response. That epistasis can change pleiotropy
has been reviewed in a recent study by Pavlicev and Wagner (2012). They suggest that
so-called relationship QTLs (rQTLs) can influence the pleiotropy of a regular quantitative trait locus (QTL) by modifying the gene product allocated to one of the traits.
Those rQTLs do not need to have direct effects themselves but serve as a mechanism
for genetic variation for pleiotropy and correlation. This hypothesis is supported by
102
several QTL studies on the mouse mandible (Cheverud et al., 2004; Wolf et al., 2005,
2006; Pavlicev et al., 2008, 2011). In our work, we identify modifiers of pleiotropy and
test hypotheses claimed in Pavlicev and Wagner (2012).
Increasing data for pleiotropy measurements becomes now available, but the picture of pleiotropy is still very incomplete and might vary among species. The data we
are using to infer pleiotropy is different from previous studies that measured S. cerevisiae deletion mutant morphological phenotypes (Ohya et al., 2005) and their growth
in environmental stress conditions (Dudley et al., 2005; Ericson et al., 2006), E. coli
mutant fitness in different carbon sources (Ostrowski et al., 2005), or embryogenesis
traits in C. elegans treated with RNA interference (Sonnichsen et al., 2005). In contrast,
we characterize the mutational variation for fitness in HIV. We use a large dataset
of HIV type 1 (HIV-1) sequences together with estimates of fitness effects of single
mutations and double mutations. The disentanglement of fitness effects of mutations
into main and epistatic effects provides some new insights that we use to characterize
both factors separately and in combination to uncover their effects on evolvability. We
investigate the pleiotropic effects of mutations in 15 drug environments and reveal
how epistasis modifies pleiotropy in these environments. Having fitness effects of
amino acid substitutions might give a much better resolution of mutational variation
than analyzing the effects of gene deletions, RNAi or QTLs. Gene deletion and RNAi
do not work with essential genes, which means that we are missing an important part
of the information. QTL analyses are usually overestimating pleiotropy, because the
identified QTLs often contain several genes. We estimate mutational pleiotropy and
specifically ask how abundant pleiotropy and epistasis are, how they lead to correlations among antiretroviral drugs and whether or not this leads to adaptive constraints.
In summary, our results show that pleiotropy and epistasis are widespread in HIV.
We find evidence that epistasis greatly modifies the pleiotropic degree of mutations
which suggests that the genetic variability also translates into a phenotypic variability
driven by epistasis.
6.2
6.2.1
Materials and methods
The HIV dataset
The data comes from an assay of fitness measurements of over 70’000 HIV-1 sequences
derived from infected patients (Petropoulos et al., 2000). The fitness measure used
here is replicative capacity. To obtain this, the full gene for the protease and most of
the gene for the reverse transcriptase (amino acids 1-305) of the viral sequence were
transferred to an HIV-derived test vector, undergoing routine drug resistance testing.
This vector has been modified such that it can undergo only a single round of replication. The number of infectious progeny virus produced by the patient-derived virus
103
determines the fitness of the viral strain. These measurements have been conducted
in a drug free environment and in presence of 15 different individual antiretroviral drugs (for more details see Petropoulos et al. (2000)). The drugs can be divided
into three classes: protease inhibitors (PIs), nucleoside reverse transcriptase inhibitors
(NRTIs) and non-nucleoside reverse transcriptase inhibitors (NNRTIs). For the full
list of drugs see Appendix 5.6.1.
Hinkley et al. (2011) inferred the fitness effects of single mutations (main effects) and
double mutations (epistatic interactions) from this data. They developed a model to
predict the fitness contribution attributable to main effects of amino acid variants and
epistatic interactions between these variants found in the HIV sequences. The model
was fitted by generalized kernel ridge regression and results in predicted fitness values of 1859 amino acid substitutions (mutations) in 16 conditions (one drug free and
15 drug environments) and of the pairwise epistatic interactions of these 1859 mutations. This data provides us with a differentiation of main effects (hereafter ME)
and pairwise epistatic effects (hereafter EE) of mutations irrespective of the genetic
background.
6.2.2
Pleiotropy measurements
To get the fitness effect of a mutation in a drug environment compared to a drug-free
environment, we subtracted the drug-free fitness values from the values of each drug
condition. This results in the effect a drug has on the replicative capacity of a virus
with a specific point mutation.
We measure pleiotropy by counting the number of drug environments in which a
mutation has a significant fitness effect. The same is done for the double mutations.
The definition of significant fitness effects has always been a difficult task and there is
no straightforward way to do it. The data is affected by different sources of error, e.g.,
in experimental measurement of replicative capacity of viruses and in the computational procedure of predicting the main and epistatic effects of mutations. Therefore
we cannot clearly say whether a small effect is real or not. We define significance
simply be setting a cutoff value of 10−3 , below which we consider mutations to be
neutral. For comparison we also did the analyses with significance cutoffs of 5 ∗ 10−4 ,
10−4 and 5 ∗ 10−5 , which alter the distribution of pleiotropy, but do not qualitatively
change the main results (see Appendix 6.6.2) and therefore show that our findings are
robust to different significance cutoffs.
104
6.3
6.3.1
Results
Pleiotropy and epistasis
Figure 6.2a shows the distribution of pleiotropy among all single mutations. Most
mutations (84.4%) have no significant effect in any drug environment (at the 10−3 significance cutoff). Most effective mutations have effects in only a few environments,
and only some handful of mutations change fitness significantly in many drug environments. The mean pleiotropy is 1.97 (median 2) with a standard deviation of 1.16.
The distribution of pleiotropy is slightly different for the double mutations (Fig. 6.2d).
It first decreases, but has a local maximum at pleiotropy 12 and even many double mutations with effects in all 15 drug environments. However, 44.6% of double mutations
are without significant effect in any environment. The effective double mutations
have a mean pleiotropy of 6.16 (median 5) with a standard deviation of 4.38. These
distributions are significantly different from the random expectation (see Appendix
6.6.1 Fig. 6.11).
We grouped the mutations by their effect signs into only positive, only negative
and antagonistic mutations. Although the group of antagonistic mutations comprises
all mutations with at least one positive and one negative effect, we observe fewer than
expected and their proportion is even decreasing for very high pleiotropy (Fig. 6.2b,e).
The random expectation would show an exponential increase in antagonistic mutations with pleiotropy (Appendix 6.6.1 Fig. 6.12). Instead, there is a large proportion
of mutations with only positive or only negative effects.
Furthermore, we also find a positive correlation between pleiotropy and mutational effect size for main effects (Spearman’s ρ = 0.3990, P = 1.7 ∗ 10−12 ) as well as
epistatic effects (Spearman’s ρ = 0.7844, P < 2.2 ∗ 10−16 ) (Fig. 6.2c,f). Mutational effect
size is measured as the absolute per-trait effect of a mutation (in Manhattan distance).
This means the effect that a mutation has in each environment is increasing with its
pleiotropy. Intuitively one would expect a negative or no correlation. However, a
randomization of fitness effects of the ME data resulted in a significant Spearman’s
correlation in 78.4% of the 10,000 runs. For the EE data a randomization even led
to a significant correlation between pleiotropy and relative effect size in all 10,000
runs. On the other hand, the real correlation was larger than the correlation from the
randomized data in all runs.
To investigate how main effects and epistatic effects interact, we combined the two
datasets by comparing the sets of traits (drug environments) affected by single and
double mutations. There is no clear relationship between pleiotropy as single mutation (ME pleiotropy) and pleiotropy in double mutations (EE pleiotropy) (Fig. 6.3a).
There is a lot of variation with respect to the number of drug environments in which
mutations affect viral fitness and there are large differences between pleiotropy of
main and epistatic effects. Taking their main effects, many mutations do not affect
105
Pleiotropy of single mutations
Effect signs of single mutations
anta
4 58
neg
Fractions of effect signs
3
pos
120
100
80
60
2
0
20
40
Frequency
1
1
2
3
4
5
6
7
8
Pleiotropy
Pleiotropy
(a)
(b)
Pleiotropy of double mutations
2
3 4 5 6 7 8 9101112131415
neg
Fractions of effect signs
60000
40000
1
0
pos
80000
anta
Effect signs of double mutations
20000
Frequency
(c)
1
2
3
4
5
6
7
8
9
10
Pleiotropy
(d)
11
12
13
14
15
Pleiotropy
(e)
(f)
Figure 6.2. Analysis of fitness effects of single (upper row) and double mutations (lower
row). First column: distribution of pleiotropy. Second column: proportion of mutational
classes (antagonistic, only negative, only positive) for each pleiotropic degree. Bar widths
proportional to the number of mutations in each pleiotropy class. Third column: mutational
per-trait effect for each pleiotropic degree. The red lines and values for R2 result from linear
regression models.
fitness in any drug environment (84.4%), but in combination with other mutations on
the same viral sequence they can have effects in many drug environments (even in
all 15). The plot suggests that the data is divided into two classes, mutations which
have an effect in many drug environments through epistatic interactions and mutations having effects in only a few drug environments through epistatic interactions.
We find a strong positive correlation between the mean pleiotropy in epistatic interactions and the number of epistatic interactions of a single mutation (Fig. 6.3b), which
suggests that the number of epistatic interactions of a mutation drives the pattern
seen in Figure 6.3a. Therefore, we divided the data into highly interactive and lowly
interactive mutations by choosing the mean number of interactions (654) as a cutoff
point and did a regression analysis on these two sets separately. As a result, we find
106
8
6
2
4
Mean epistatic pleiotropy
8
6
4
2
●
0
Mean pleiotropy as double mutation
Pleiotropy: single vs. double
0
1
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4
5
6
7
Pleiotropy as single mutation
(a)
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2.0
2.5
3.0
log10(No. epistatic interactions)
(b)
Figure 6.3. Difference between pleiotropy of main and epistatic effects. In (a) each dot stands
for one of the 1859 mutations. Depicted on the x-axis is the pleiotropic degree of a single
mutation, on the y-axis the average pleiotropic degree that a mutation has occurring together
with any other mutation it interacts with. Red lines result from linear regression models
for mutations with less than 654 interactions (lower) and for mutations with more than 654
interactions (upper). (b) shows the relationship between the number of epistatic interactions
of a mutation and the mean pleiotropy in these interactions. Colors indicate neutral mutations
(pleiotropy = 0), pleiotropic mutations and highly interactive mutations.
that in lowly interactive mutations there is no significant relationship (Spearman’s
ρ = 0.023, P = 0.4287), whereas in highly interactive mutations mean pleiotropy in
double mutations is increasing with the pleiotropy of its main effects (Spearman’s
ρ = 0.240, P = 8.1 ∗ 10−11 ). The question is whether epistasis rather increases or decreases pleiotropy. To answer this, we calculated the number of traits that are added
to the trait repertoire of a mutation by epistatic interactions. That means, how many
additional traits does a mutation affect through epistatic interactions (that are not affected by the main effects alone)? We find that epistasis greatly increases the trait
repertoire of a single mutation (Fig. 6.4a), even if we do not consider mutations with
no significant main effects (Fig. 6.4d). Note here, that these analyses include all observed epistatic interactions, which do not exist on a single viral genome of course. It
however shows the huge potential of epistasis for changing the pleiotropic degree of
mutations and therefore broadening their adaptive potential.
We did a similar analysis for double mutations by counting how many traits the
double mutation adds to the union of the trait repertoires of both single mutations.
Half of double mutations do not further increase the trait repertoire of their two single
mutations, but there is a broad spectrum of cases in which the epistatic interaction
107
1400
800
1000
1200
p=8
p=7
p=6
p=5
p=4
p=3
p=2
p=1
p=0
400
200
1e+05
4
5
6
7
8
9
10 11 12 13 14 15
0
1
2
3
4
5
(a)
7
8
9
10 11 12 13 14 15
0
1
2
3
4
5
120000
80000
60000
8
9
10 11 12 13 14 15
Net pleiotropy (ME + EE)
p=8
p=7
p=6
p=5
p=4
p=3
p=2
p=1
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
No. traits added to repertoire
(d)
0
0
0
20000
50
Frequency
100000
p = 13
p = 12
p = 11
p = 10
p=9
p=8
p=7
p=6
p=5
p=4
p=3
p=2
p=1
40000
40
20
7
(c)
Change in trait repertoire of double mutations
p=8
p=7
p=6
p=5
p=4
p=3
p=2
p=1
6
Pleiotropy
(b)
Change in trait repertoire of single mutations
60
6
No. traits added to repertoire
150
3
100
2
No. traits added to repertoire
Frequency
1
0
0e+00
200
0
0
Frequency
Net pleiotropy (ME + EE)
600
3e+05
Frequency
4e+05
5e+05
p = 13
p = 12
p = 11
p = 10
p=9
p=8
p=7
p=6
p=5
p=4
p=3
p=2
p=1
p=0
2e+05
600
800
Change in trait repertoire of double mutations
400
Frequency
1000
1200
p=8
p=7
p=6
p=5
p=4
p=3
p=2
p=1
p=0
Frequency
Change in trait repertoire of single mutations
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
No. traits added to repertoire
(e)
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Pleiotropy
(f)
Figure 6.4. The number of traits that epistatic interactions add to the trait repertoire of mutations. The trait repertoire contains the traits that are affected by main effects only. The bars
indicate the number of mutations for which epistatic interactions add traits to that repertoire.
The colors indicate different pleiotropy levels of main effects. Upper row: including neutral
mutations. Lower row: excluding neutral mutations.
increases the repertoire of its two interacting mutations up to the maximum of all 15
traits (Fig. 6.4b,e). This modification of pleiotropy by epistasis leads to a higher net
pleiotropy of all mutations with most of them affecting even all traits in some way
(Fig. 6.4c,f).
The crucial question is: does epistasis rather lead to adaptive constraints or is it
enhancing adaptation? The data does not provide a clear answer to this question.
There is a strong positive correlation between fitness of main effects and fitness of
epistatic effects (Spearman’s ρ = 0.402, P < 2.2 ∗ 10−16 , Fig. 6.5). This does not
support that epistasis largely compensated deleterious mutations. We however find
37 mutations for which this might be the case, that is, which show negative main
effects and positive epistatic effects.
4
2
0
-4
-2
EE fitness (sum)
6
8
108
-0.015
-0.010
-0.005
0.000
0.005
0.010
ME fitness (sum)
Figure 6.5. Fitness effects of main effects (ME) versus fitness effects of epistatic effects (EE)
for all mutations. Each dot is represented by the sum of all main effects and the sum of all
epistatic effects across the 15 drug environments.
Mutation classes – neutral, pleiotropic, highly interactive
Noticeable in Fig. 6.3b are the different bands. Our analyses revealed that these bands
reflect different classes of mutations that follow slightly different dynamics. First,
there are those mutations that have no significant effects at the threshold of 10−3 ,
which we call neutral (colored red in the figure). They cover the whole range of
the number of epistatic interactions with a maximum around 250. Second, there are
mutations that have effects in at least one drug environment, which we call pleiotropic
(colored in blue in the figure). These mutations have a bimodal distribution of the
number of epistatic interactions with a high maximum at 500 and a small maximum at
1750. Pleiotropic mutations interact with at least 381 other mutations, which is exactly
the same number of mutations that the third class comprises. This class contains all
mutations that have the maximum number of epistatic interactions (1858) at the 10−5
threshold, which we call highly interactive (colored in green in the figure). In this
mutation class we find all levels of pleiotropy and also neutral mutations and the
distribution of the number of interactions is bimodal with a large peak around 600
and a smaller peak around 1800. It appears however that the equality of the minimum
number of interactions for pleiotropic mutations and the number of mutations in the
highly interactive class is just a coincidence. The overlap is very high, but pleiotropic
mutations do not always interact with all of the highly interactive mutations. The
mutations in each of these classes are spread over the whole genome and there are no
109
highinteract
pleiotropic
neutral
Mutation class
visible clusters that would indicate a cumulation of mutations of one class in a certain
genomic region (Fig. 6.6).
1
81.6
162.2
242.8
323.4
404
Genome position
Figure 6.6. Distribution of mutations of three identified mutation classes across the HIV
genome. The first 99 positions belong to the protease (PT) and the other 305 positions belong
to the reverse transcriptase (RT). Color code: white – no mutation, yellow – few mutations,
red – many mutations.
Modifiers of pleiotropy
Pavlicev and colleagues suggest the existence of relationship loci that modify the
pleiotropic degree of other loci (Pavlicev et al., 2008, 2011; Pavlicev and Wagner,
2012). We test this hypothesis on our data, first by investigating the role of neutral mutations, and second by identifying mutations that modify pleiotropy in HIV.
Pavlicev and Wagner (2012) describe relationship loci as ’private’ genes that interact
with pleiotropic loci and being themselves not pleiotropic for the focal traits. We test
mutations in HIV for these characteristics and specifically ask whether neutral mutations serve as modifiers of pleiotropy. If this was the case, one would expect neutral
mutations to mostly interact with pleiotropic mutations. This however does not seem
to be the case. We even find that neutral mutations have less epistatic interactions
with pleiotropic mutations than expected by chance (Fig. 6.7).
In a next step, we identified mutations that modify pleiotropy of other mutations.
Apparently, all mutations function as modifiers somehow. The minimum is 45 times
and the maximum 1813 times. Again, the neutral mutations do not seem to be the
ones doing most modifications, modifying on average 517 mutations (pleiotropic: 680,
highly interactive: 909). There is a positive correlation between the number of mutations modified and the mean epistatic pleiotropy (Fig. 6.8a), similar to Figure 6.3b.
Note here, that if we pick mutations with the same number of modified mutations, the
neutral mutations are the ones with the highest epistatic pleiotropy, which means they
have effects on many traits and therefore the potential to do modifications in more
110
100
Pleiotropic mutations
100
Neutral mutations
80
theory
observed
80
theory
observed
30.6
60
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60
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70.9
10.3
9.3
0
0
7.2
neut−neut
neut−plei
(a)
neut−highint
plei−plei
plei−neut
plei−highint
(b)
Figure 6.7. Epistatic interactions of neutral and pleiotropic mutations. Comparison between
the expected and observed number of interactions within and between mutation classes for
neutral (a) and pleiotropic mutations (b). The expected number of interactions (theory) is
based on the number of mutations in each class. neut = neutral mutations, plei = pleiotropic
mutations, highint = highly interactive mutations.
traits than other mutations. Additionally, we find a positive correlation between the
number of modified mutations and the epistatic fitness effects (Spearman’s ρ = 0.1716,
P = 9.4 ∗ 10−14 , Fig. 6.8b), that is, modifying more mutations is associated with having larger positive epistasis. Finally, there is evidence that neutral mutations can be
important for epistatic compensation. This class of mutations has significantly higher
epistatic fitness effects than all mutations together (Wilcoxon-Test: W = 1060408,
P = 0.0007), whereas highly interactive mutations have significantly smaller effects
(Wilcoxon-Test: W = 420602.5, P = 7.5 ∗ 10−9 ).
6.3.2
Drug class effects
By dividing the data into the two main drug classes, protease inhibitors (PIs) and
reverse transcriptase inhibitors (RTIs), we test for class specific effects. We specifically
ask whether PIs select for more mutations in the protease (PT) gene and RTIs select
for more mutations in the reverse transcriptase (RT) gene. We find no sign that mutations in the protease region have greater impact in PI environments than mutations in
the reverse transcriptase region, and vice versa. Pleiotropy of protease mutations and
reverse transcriptase mutations is not significantly different in the RTI environments
and in the PI environments, respectively (Tab. 6.1). Additionally, the fitness effects
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111
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Modified mutations
(a)
(b)
Figure 6.8. Mutations that modify many other mutations have high epistatic pleiotropy (b) and
also larger positive epistasis (c). Colors indicate neutral mutations (pleiotropy = 0), pleiotropic
mutations and highly interactive mutations.
of PT mutations are not significantly greater than those of RT mutations in PI environments and fitness effects of RT mutations are not significantly greater than those
of PT mutations in RTI environments. Hence, mutations in other regions than those
targeted by a drug might be as important as mutations in the target region.
Table 6.1. Comparison between mutations in the two proteins protease (PT) and reverse
transcriptase (RT) and between the two main drug classes protease inhibitors (PIs) and reverse
transcriptase inhibitors (RTIs).
Mean pleiotropy
PIs
RTIs
PT mutations
RT mutations
Wilcoxon-test
2.15
1.97
1.85
1.85
Mean fitness
PIs
RTIs
-0.000433
-0.000222
-0.000442
-0.000277
W = 581
W = 4097 W = 330258 W = 333771
P = 0.3096 P = 0.3713 P = 0.9147
P = 0.1515
Looking at all three drug classes (PIs, NRTIs, NNRTIs), we find a modular distribution of mutational effects among the drug classes. Of the 290 mutations with
significant main effects in at least one drug environment, 256 (88.3%) affect only a
single drug class, 33 (11.4%) affect two drug classes and only 1 mutation (0.3%) has
fitness effects in all 3 drug classes. Note that 134 mutations affect only a single trait
112
and can thus only belong to a single drug class.
Furthermore, epistatic interactions of each mutation are spread among the whole
genome and are not limited to a local area around the mutation or the same gene
(Fig. 6.9). However, we still find a significant positive correlation of the median position of interacting mutations with the position of a mutation in the genome (Spearman’s ρ = 0.1606, P = 3.3 ∗ 10−12 ).
RT
↑
PT
↓
PT
← →RT
Figure 6.9. Mutations have epistatic interactions across the whole genome. For visibility only
the medians of the positions of interacting mutations are plotted. The red line results from a
linear regression model and indicates a slightly positive correlation between the position of
a mutation on the genome and the position of interacting mutations. The first 99 positions
belong to the protease (PT) and the other 305 positions belong to the reverse transcriptase
(RT).
6.3.3
Mutation frequencies
The 70’000 HIV sequences provide information about the frequency of the mutations
considered in the above analyses. These frequencies also play a role in the prediction algorithm for the single and double mutation effects by assigning frequencydependent weights to the mutations. Thus the unequal distribution of frequencies
might introduce a bias in the estimation of fitness effects towards those mutations
with high abundance. However, we do not find a correlation between mutation frequency and main fitness effects. Only when dividing the mutations into two classes
with a cutoff at 50’000 (or higher), the high frequency mutations have a significantly
113
higher fitness effect than the low frequency mutations. There is also no relationship
between the frequency and the pleiotropic degree of a mutation.
Pleiotropy of single mutations
0
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(c)
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4
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10 11 12 13 14 15
No. traits added to repertoire
(d)
Figure 6.10. Distribution of pleiotropy (upper row) and change in trait repertoire (lower row)
for mutations with low frequency (left) and high frequency (right).
Since the distribution of mutation frequencies is bimodal with a large peak at low
frequencies, a smaller peak at high frequencies and very few mutations with intermediate frequencies, low and high frequency mutations might differ in other characteristics than fitness and pleiotropy of main effects. Therefore we decided to define those
mutations that appear in less than 10% of the sequences (frequency < 7’000) as low
frequency mutations (1411) and all the others as high frequency mutations (448) to
analyze these two classes separately. High and low frequency mutations do not differ
114
much in their main effects, but for the epistatic effects we observe higher pleiotropy,
a larger proportion of effective double mutations and a larger fraction of antagonistic double mutations in the high frequency mutation class. Note here, that by this
separation we only consider within-class epistatic interactions and completely disregard epistatic effects caused by the interaction of low with high frequency mutations.
Therefore, the high frequency mutations have a large number of epistatic interactions
whereas the low frequency mutations on average only interact with few other mutations. As a consequence, there is a larger increase in trait repertoire of single and
double mutations by epistatic interactions in the high frequency class, compared to
the low frequency class. This makes almost all mutations with high frequency fully
pleiotropic.
6.4
Discussion
Understanding mutational patterns brings us further to understanding the evolution of HIV and the disease it causes. The present study deals with the patterns
of pleiotropy and epistasis in fitness effects of mutations in patient-derived HIV-1
sequences in different drug environments.
The distribution of pleiotropy we find for HIV is in good agreement with previous
studies that investigated pleiotropy in other organisms (e.g., mouse, yeast, nematodes,
E. coli) and with different techniques (e.g., QTL, gene deletion, RNAi) (Dudley et al.,
2005; Ohya et al., 2005; Ostrowski et al., 2005; Sonnichsen et al., 2005; Ericson et al.,
2006; Wang et al., 2010). Thus, this L-shaped distribution (Fig. 6.2a) seems to be
very conserved among species and across scales with many mutations having low
pleiotropy, but there are still more highly pleiotropic mutations than expected if we
randomly assign effects to mutations.
This pattern is only observed when the significance threshold for ’real’ effects is
high, i.e., only when we include just the largest fitness effects of mutations in our
analysis. As soon as we start to decrease this threshold the distribution of pleiotropy
starts to change and shifts to larger values. In essence, our data is able to support
different hypotheses on the nature of pleiotropy. A recent review by Paaby and Rockman (2012) describes three main hypotheses of pleiotropy: the Strong Hypothesis of
Universal Pleiotropy (SHUP), the Weak Hypothesis of Universal Pleiotropy (WHUP)
and the Hypothesis of Modular Pleiotropy (HMP). By considering all fitness effects as
’real’, each mutation affects all traits, which exactly meets the criterium of the SHUP.
But even if we apply a small threshold and take the net pleiotropy (of main and
epistatic effects together) as a measure, we end up at full pleiotropy for all mutations.
If we choose a slightly higher threshold for the significance of fitness effects, each mutation still affects at least one of the traits and thus supports the WHUP which states
that mutations do not need to have an impact on all traits, but we observe a certain
115
degree of pleiotropy for each mutation. Finally, when the threshold is high (to make
sure only taking real effects), many mutations do not affect any of the measured traits.
On the other hand, the effects of pleiotropic mutations are largely clustered among related traits, which supports the modular pleiotropy hypothesis (HMP). In a nutshell,
depending on the kind of data and how one attempts to analyze it, it is easy to argue
for and against all of these hypotheses on the nature of pleiotropy.
The observation that the mutational effect size per trait is positively correlated
with pleiotropy seems to be the rule rather than the exception (Wagner et al., 2008;
Wang et al., 2010). The reason for this remains unclear and it is also possible that this
relationship is an artifact of measurement. However, Wagner et al. (2008) have shown
that it is not due to the detection limit for small mutational effects or an overestimation of the degree of pleiotropy. One possible explanation for the positive correlation
between pleiotropy and mutational effect size might be an adaptive advantage opposing the cost of complexity. Another one is simply that mutations altering many
functions are most important for overall fitness and thus have a larger impact on
each function. For instance, Guillaume and Otto (2012) show that pleiotropy is linked
to the robustness of phenotypes and organismal fitness to variation in expression of
the underlying genes. They suggest that pleiotropy and expression may coevolve, i.e.,
more pleiotropic genes evolve higher expression levels to maintain functionality at the
traits they affect. Consequently, mutations in these genes may have a greater per-trait
effect.
Very unexpected is the high number of mutations that only have negative (main)
effects. These should be eliminated by selection and not be observed. A reason
for their persistence might be positive epistasis (Bonhoeffer et al., 2004). We find
some mutations for which this is certainly true, but the majority of deleterious mutations also shows negative epistasis and it is not clear how they are maintained or
whether this is an artifact of the fitness prediction model. Since most mutations are of
low frequency, the potentially deleterious mutations might just be a snapshot of the
mutation-selection balance. For those mutations with high frequency we observe an
increased proportion with negative main fitness effects and positive epistatic fitness
effects. This is even more pronounced when we consider only large fitness effects
and suggests that negative main effects get compensated by positive epistasis. On
the other hand, the large number of mutations with only positive effects hints at
many resistance mutations with cross-resistance. This has to be proofed by specifically investigating the effects of known resistance mutations. However, the pattern of
widespread consistency of pleiotropic effects among traits has been observed before
by Wolf et al. (2005, 2006) in mouse mandible QTLs. They call loci with only positive
or only negative fitness effects positive pleiotropic whereas loci exhibiting both, positive and negative effects are negative pleiotropic. Thus, our results are consistent with
their studies. The evolutionary implication of positive pleiotropy is that it contributes
to integration of traits, which enhances modularity.
116
The low number of antagonistic mutations lets us conclude that most mutations
have similar effects in different drug environments, i.e., only increase or only decrease
fitness. This makes sense for mutations with low pleiotropy where traits usually fall
within the same drug class. However, for highly pleiotropic mutations this finding
is very surprising. We want to point out of course, that the viruses used in this
study never experienced all 15 drugs at the same time. Fitness was measured in one
drug at a time, but since they are patient-derived, the viruses might have experienced
up to three drugs during treatment. Hence, in reality it might be much easier for
compensatory mutations or epistasis to keep the deleterious mutations at a low level.
There is little overlap between the traits that a mutation affects alone and those that
are affected when it occurs together with other mutations in the same viral particle.
Epistasis increases pleiotropy and leads to a broader range of mutational effects across
different drug environments. The mutations that do not have significant main effects,
but a large number of epistatic effects, might be equivalent to rQTLs, as suggested
by Pavlicev and Wagner (2012). These mutations interact with other mutations and
change their pleiotropy.
The way mutations affect fitness traits and how mutations interact is crucial for
evolution. We have shown that pleiotropy and epistasis are very abundant in HIV.
Main and epistatic effects can change very different aspects of organismal functioning. In HIV, the interplay between pleiotropy and epistasis often leads to extended
possibilities of changing the viral performance. This is of particular importance in the
context of adapting to a new (drug) environment and developing resistance. The data
suggests that epistasis is enhancing evolvability and even is the driving force in the
adaptation to drug-enriched environments.
6.5
Conclusions
Our work shows that epistasis modifies pleiotropic effects of mutations which leads
to a variation of the pleiotropic degree. This variation is essential for adaptation to
new environmental conditions, like different drug encounters for a viral population.
Depending on the genetic background (i.e., other mutations present in the genome),
covariances between traits may evolve or change. In our study system HIV, this has
implications for the emergence of multi-drug resistance mutations.
In summary, we find that epistasis is the major player in mutational variation
in HIV. There is a high variability, not only in the size of mutational effects, but
also with respect to the pleiotropic degree. The interplay between main effects and
epistatic interactions is one of the reasons why HIV is so versatile. Our results support
the hypothesis that modular pleiotropy together with larger fitness effects in highly
pleiotropic mutations enhance evolvability.
117
What we do not know yet is, how approved resistance mutations fit into this picture. We would expect that they have predominantly positive fitness effects. But do
they have low or high pleiotropy, i.e., are they mostly drug specific or is there a lot of
cross-resistance? We would also like to find out whether it is the main effects or the
epistatic interactions that make them resistant. To investigate this we have to combine
this data with information of known resistance mutations.
Furthermore, it is important that more data of this kind becomes available, especially with new drugs, since some of the drugs in this study are not in clinical use
anymore (Sarafianos et al., 2009). Also, research on other human pathogens could
greatly benefit from such kind of data and analyses. As Tyler et al. (2009) points out:
epistasis and pleiotropy are ubiquitous and important in human diseases!
118
6.6
6.6.1
Appendix
Random expectations
Pleiotropy of double mutations
data
random expectation
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Figure 6.11. Random expectations for the distribution of pleiotropy. Effects have been randomly assigned to mutation-drug pairs and pleiotropy was calculated based on this shuffled
matrix. Grey lines indicate standard deviations from 10’000 runs.
Random expectation − double mutations
anta
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Random expectation − single mutations
Pleiotropy
Pleiotropy
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Figure 6.12. Random expectations for the mutational classes. Each real effect was assigned a
random value from the set of fitness values in the data. Plotted are only the results of a single
run of this process for the ME (a) and the EE (b) data respectively, since results of replicates
were very similar.
119
Results for different significance levels
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Figure 6.13.
Distribution of
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Figure 6.14.
Proportions of mutational classes for different significance levels.
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Pleiotropy
Figure 6.15. Relative mutational effect size for different significance levels.
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120
7 | Final discussion and outlook
7.1
The nature of pleiotropy
In Part I of the thesis I used data from gene-deletion experiments in yeast to test
the hypothesis that modular pleiotropy explains more of the variation in the rate of
molecular evolution of genes than the pure measure of pleiotropy. Gene-based modularity measures were used to integrate structural information of the GP map. The
hope was to improve the coefficient of determination to a great extend, but unfortunately the results show that in general, these measures only marginally increase the
association between pleiotropy and evolutionary rates of genes. Weighting the effect
of the genes on their traits by the level of gene expression greatly increases this association, because gene expression itself is a good explanatory factor for variation in
the rate of evolution. However, what can be learned from this study, is, that a species
does not reveal its secrets so easily. The results of the studies I used, were all based on
experiments that, in different ways, try to find genes involved in the expression of phenotypic traits. Using this data in a different context, like describing the distribution of
pleiotropy in our case, the analysis of only one of these datasets will most likely only
give an incomplete picture of the underlying process and could consequently lead
to wrong conclusions. One issue we observed, is that pleiotropy substantially differs
across experiments. For those studies, where I did not find any significant correlation between pleiotropy and evolutionary rate, there is complete lack of correlation of
pleiotropic degrees with other datasets. I therefore think that the observed traits are
not appropriate to study pleiotropy and that they will be confounding factors when
studying the relationship between pleiotropy and other evolutionary factors. At least,
all datasets are consistent in their prediction for the distribution of pleiotropic degrees
among genes. This confirms and strengthens the hypothesis that the distribution is
really L-shaped with many low pleiotropic and few high pleiotropic genes.
121
122
7.2
The modular genetic architecture
The second part deals with the transition of modular pleiotropy on the level of the GP
map to modularity in the genetic covariance matrix G, and its role for the evolvability
of a population. First, in contrast to previous studies, I was able to demonstrate that,
under reasonable assumptions about the genetic architecture, it is possible to use the
G-matrix for predicting the structure of the GP map. The required assumptions are
modular pleiotropy, mutational correlations of pleiotropic effects and heterogeneity in
module sizes. However, it is true that the G-matrices of two populations with similar
GP maps can be very divergent, when these assumptions are not fulfilled. Therefore,
there is need for more empirical studies to validate these assumptions and to verify
the results presented in this work.
Based on the conclusions from above, I further investigated the role of modularity
in the G-matrix. I find that modularity has positive effects on the adaptive potential
of a population when selection favors trait combinations with low genetic variation.
Hence, an important result of this work is, that modularity should be beneficial in
frequently changing environments. In this chapter I also cover the issue of measuring
genetic constraints. Many methods exist for analyzing the G-matrix in this light, but
they all tackle different aspects of evolution. The results from my comparative study
propose to apply both static and dynamic analysis approaches to the G-matrix to describe the adaptive potential of a population. Doing so, one can get an idea about
the genetic variation of traits and predict how correlations among them confine or
support the response to any selection gradient. The approach could be approved further, if there was more knowledge available about the real strength and direction of
selection a population encounters, which at the moment we cannot take under consideration. Another not yet covered extension to my approach is, applying selection over
several generations, because G might change over time. Jones et al. (2003) performed
a simulation study on the stability of a 2-dimensional G-matrix over 2000 generations
and found that correlational selection, large effective population size and mutational
correlation of pleiotropic effects greatly enhance stability of G. This suggests that not
only the G-matrix, but also the genetic architecture can be considered relatively stable
under these assumptions. Applying the above to a higher-dimensional modular GP
map, would probably result in stable modules, but there should still be potential for
evolutionary change when key factors like selection or population size change.
7.3
Evolutionary lessons from HIV
In Part III of the thesis I investigated the role of pleiotropy and epistasis in influencing evolutionary dynamics of the human immunodeficiency virus (HIV). Through
the available data on HIV fitness measurements in several drug environments, it was
123
possible to analyze the effects of single mutations (main effects) and double mutations (epistatic effects) independently. My analyses of the data reveal that main and
epistatic effects alone resemble the information from the complete sequences quite
well. Three drugs of the NRTI class provide the lowest potential for resistance evolution, whereas the NNRTI drugs induce the highest mutational variation and therefore
bear the greatest potential for viral evolvability under drug treatment. Of course,
these results are based on this specific dataset of sampled viruses from several infected patients. This does not necessarily imply that exactly these patterns will be
found in other viral populations and therefore this analysis cannot provide a general
suggestion for future drug treatment strategies.
In concurrence with previous studies (Martins et al., 2010; Kouyos et al., 2012), the
high correlations between drug environments hint at large adaptive constraints for the
virus, because fitness effects are very similar across environments. In the second chapter of this part, I show that epistasis can substantially increase the pleiotropic degree
of mutations and thereby broaden the range of mutational effects across drugs. Epistasis is non-additive and even non-pleiotropic genes can have effects in many drug
environments by epistatic interactions. As suggested before (Hinkley et al., 2011), this
indicates that epistasis is a very important factor for the genetic variability of HIV.
With the present work, I was able to provide a concrete reason for this importance,
namely increasing the trait repertoire of mutations. Something that should be kept in
mind is, that epistasis is usually more limited than considered here, because not all
the mutations are always present. Therefore its influence might be weaker than the
results demonstrate.
7.4
Outlook
Whenever possible, I tried to back up my findings with results from other studies.
For example, the analyses of phenotypic covariance matrices of mammalian skulls by
Marroig et al. (2009) and Porto et al. (2009) support the hypothesis that modularity
positively correlates with autonomy and conditional evolvability. And the conclusion
that the HIV drugs of the NNRTI class have the highest potential to provoke resistance
evolution is supported by a study on transmission networks of resistant strains from
San Francisco (Smith et al., 2010). However, some of the hypotheses in this thesis
are purely based on theoretical models and have been tested only on synthetic data
or on publicly available data that was not designed for answering my questions. An
important extension of the current work would be to specifically design biological
experiments to test the hypotheses.
Another addition to this project is to show more clearly that the epistatic effects
are not an additive interaction of the main effects. That means the traits affected by
an epistatic interaction of two mutations are not the union of the traits affected by
124
the two mutations alone. This difference could be quantified by comparing it with
expected values obtained from randomization of the data.
Also, the HIV data analysis could be further improved in several aspects. For
filtering the significant effects, only confidence intervals from bootstraps of the drugfree environment were used. As preliminary analyses indicate, there is quite some
variation in the probability that a mutation has a significant effect across drug environments. This seems to quantitatively change some of the results, but not the
conclusions. Another possibility to overcome the issue of significance of fitness effects, is to re-run the regression algorithm of Hinkley et al. (2011) with additional
neutral mutations, to test how large the effects of neutral mutations are and which
real mutation effects can be considered significant. I also think that this dataset contains much more interesting information that we were not able to capture so far. It
would be interesting to combine this data with a model of within-host evolution. It
is known that a viral population consists of several ’quasi-species’ that compete with
each other (Eigen et al., 1989). Modeling this viral population structure, based on
the sequences of the data I used, and performing the analyses on this altered population could provide a more realistic view on the within-host dynamics of resistance
evolution.
With this work I took a start in modeling modular pleiotropy in a more realistic
way, and if experimental studies are conducted to precisely validate my results, I am
confident that we will be able to get a much more profound understanding about the
interplay of genes and traits and the consequences for adaptive evolution in real-life
systems.
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Acknowledgments
There are a couple of people that accompanied me during my time as a PhD student
that I would like to mention here. Most of all I’d like to thank Frédéric Guillaume for
giving me the opportunity to do a PhD with him. Fred guided me through my doctorate, provided ideas, help and useful comments, and gave me a kick in the ass when
I needed it. I wish him all the best for his Professorship position at the University
Zürich. Another big thanks goes to Sebastian Bonhoeffer for providing this excellent
research environment. I really appreciate being part of his group. I thank everyone
in the Theoretical Biology group for discussions and helpful feedback. Special thanks
go to João Martins, Rafał Mostowy, Pia Abel zur Wiesch, Carsten Magnus, Helen
Alexander and Dorita Avila for being not only work mates but also good friends. Especially my former office mates João and Rafał made commuting between Basel and
Zürich less annoying, because I knew there was someone waiting for me in Zürich! I
also thank the awesome Mensa group for sharing the Mensa experience with me, including fruitful discussions about the food, weather, football, movies, politics, history,
group gossip and other irrelevant things. The frequent Mensa eaters are/were: João,
Rafał, Carsten, Gabriel Cisarovsky, Frederik Graw, Victor Garcia, Roland Regoes, Jan
Engelstaedter, Danesh Moradi, Dominique Cadosch, Alexey Mikaberidze and Ramith
Nair. I thank Tanja Stadler and Dominik Refardt for always having good advice and
Nicco Yu for sharing her life experiences. Not to be forgotten shall be Paul SchmidHempel, Martin Ackermann, Greg Velicer and their groups who are all part of this
nice working environment. I also want to acknowledge Jérôme Goudet for being the
external examiner of my thesis. Finally, I’d like to thank my parents and family and
the biggest thanks and kisses go to my girlfriend, fiancé and wife Juliane for her
endless love and support.
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test
Curriculum Vitae
The author of this thesis, Robert Polster, was born on the 28th of January 1984 in
Neubrandenburg (Germany). He attended the secondary school at Alexander-vonHumboldt Gymnasium in Greifswald and graduated in 2003 (Abitur). From October
2003 onwards, he studied Biomathematics at the Ernst-Moritz-Arndt University in
Greifswald, focusing on discrete mathematics and molecular genetics. During his
studies, he did a semester abroad at Massey University in Palmerston North (New
Zealand) in 2006, with an emphasis on computational biology and phylogenetics.
Back at the University of Greifswald, Robert wrote his diploma thesis in 2008 on phylogenetic analyses of rabies live vaccines under supervision of Prof. Dietmar Cieslik and in collaboration with Dr. Lutz Geue from the Friedrich-Loeffler-Institute in
Wusterhausen. In 2010, he joined the Theoretical Biology group of Prof. Sebastian
Bonhoeffer at ETH Zurich (Switzerland), where he worked with Dr. Frédéric Guillaume on evolutionary aspects of pleiotropy and modularity. The results of this work
are described in this thesis.