RESEARCH ARTICLE 2385
Development 137, 2385-2395 (2010) doi:10.1242/dev.048033
© 2010. Published by The Company of Biologists Ltd
Predicting embryonic patterning using mutual entropy
fitness and in silico evolution
Paul François* and Eric D. Siggia
SUMMARY
During vertebrate embryogenesis, the expression of Hox genes that define anterior-posterior identity follows general rules:
temporal colinearity and posterior prevalence. A mathematical measure for the quality or fitness of the embryonic pattern
produced by a gene regulatory network is derived. Using this measure and in silico evolution we derive gene interaction
networks for anterior-posterior (AP) patterning under two developmental paradigms. For patterning during growth (paradigm I),
which is appropriate for vertebrates and short germ-band insects, the algorithm creates gene expression patterns reminiscent of
Hox gene expression. The networks operate through a timer gene, the level of which measures developmental progression (a
candidate is the widely conserved posterior morphogen Caudal). The timer gene provides a simple mechanism to coordinate
patterning with growth rate. The timer, when expressed as a static spatial gradient, functions as a classical morphogen (paradigm
II), providing a natural way to derive the AP patterning, as seen in long germ-band insects that express their Hox genes
simultaneously, from the ancestral short germ-band system. Although the biochemistry of Hox regulation in higher vertebrates is
complex, the actual spatiotemporal expression phenotype is not, and simple activation and repression by Hill functions suffices in
our model. In silico evolution provides a quantitative demonstration that continuous positive selection can generate complex
phenotypes from simple components by incremental evolution, as Darwin proposed.
INTRODUCTION
Numerical simulations of evolution have a long history in ecology
and molecular evolution but are less used in cell and developmental
biology. Evolutionary computations, in the latter context, face the
technical challenges of defining the fitness to be optimized and
inventing plausible mutations. To evolve properties of the cell, the
selection used in ecology, i.e. reproductive success, is too far
removed from the question and it is unclear how to translate
mutation rates at the genome level into morphological change.
In two previous papers we have evolved regulatory networks for
segmentation and adaptation and confronted the problems of what
fitness to optimize and how to chose mutation rates (Francois et al.,
2007; Francois and Siggia, 2008). The fitness has to be specific to
the process, e.g. reward segment number to evolve segmentation,
yet be as general as possible because we can only hope to find
networks common to phyla, not individual species. Detailed
mutation rates became immaterial once it was shown that the
fitness landscape was shaped like a funnel (defining better fitness
to be lower), so that any mutational process that sampled all
directions would move to the bottom of the funnel. This
mathematics recapitulates Darwin’s original insight that small
changes in fitness can rapidly lead to the evolution of complex
structures such as the eye (Nilsson and Pelger, 1994).
Computational evolution functions like a genetic screen in that it
enumerates in an unbiased way all models that can be built from a
predefined set of parts to achieve a certain function. It favors
Center for studies in Physics and Biology, The Rockefeller University, 1230 York
Avenue, 10065 New York, NY, USA.
*Author for correspondence ([email protected])
Accepted 10 May 2010
models that can be built by incremental improvements in fitness,
rather than via multiple neutral steps or transitions through less fit
intermediates. Evolution is rapid when it can march along a fitness
gradient.
The early history of developmental patterning was rich in
concepts, such as morphogen and selector genes, that lacked a
molecular identity, yet informed many experiments (Crick and
Lawrence, 1975), and it is to that level of description that we wish
to return, but in a more quantitative way. There is extensive
literature on homeotic mutants and homeotic genes, starting in the
1960s with the description of bithorax mutants (Lewis, 1963). In
his pioneering work, Lewis proposed that fly segmental identities
were directed by hypothetical bithorax ‘substances’, specific to a
given segment. This idea was generalized to selector genes that had
to be compartment specific, have an instructive role in
development, and function combinatorially and cell-autonomously
(Mann and Morata, 2000). The original selector genes were
engrailed, which defines the anterior-posterior (AP) compartments
within imaginal disks, and the Hox gene Ultrabithorax.
Hox genes are major determinants in patterning the AP axis in
bilaterians. Hox genes have been shown to be the crucial regulators
of segmental identity in the fly (McGinnis and Krumlauf, 1992)
and of vertebrae identity in vertebrates (Burke et al., 1995), such
that Lewis further qualified them as ‘master control genes’ (Lewis,
1992). Other examples of master control genes that direct cellular
fates in different contexts at different stages of development
include MyoD for muscles (Weintraub et al., 1991), Pax6 for eyes
(Kozmik, 2005; Gehring and Ikeo, 1999) and Distal-less for legs
(Gebelein et al., 2002; Pearson et al., 2005) and the so-called
‘terminal selector genes’, such as che-1, that determine ASE
neuronal fates in C. elegans (Hobert, 2008). These master control
genes are often embedded in feedback loops that lock commitment
to a given cellular fate or identity (Weintraub et al., 1991; Zuber et
DEVELOPMENT
KEY WORDS: Hox genes, In silico evolution, Mutual entropy, Systems biology
2386 RESEARCH ARTICLE
patterning, the pattern forms by means of a timer gene that
measures the residence time in the posterior growth zone.
Surprisingly, the evolved networks possess properties that are
qualitatively similar to actual Hox gene networks, such as
anterior homeotic transformations, posterior prevalence and
temporal colinearity. The timer gene also naturally explains how
to interconvert between static fly-like and dynamic morphogens
without gross loss in fitness. Key aspects of Hox regulation
simply follow from the exigencies of morphological evolution,
suggesting that convergent evolution would channel variable
molecular mechanisms towards the same phenotype.
MATERIALS AND METHODS
Mutual entropy as fitness
We simulate networks of interacting genes and proteins as a system of
differential equations, and we model an embryo as a linear array of L cells
sharing the same genetic network. The fitness must be defined for any
network, as we want to begin with networks that pattern poorly or not at
all and evolve something better. We also seek to evolve patterning
networks common to a phyla, so the fitness should be as generic, smooth
and parameter free as possible (Francois et al., 2007; Francois and Siggia,
2008).
We assume that, ultimately, a single gene (which may be the intersection
of several conventionally defined selectors) defines or labels segmental
identity (Lewis, 1963; Lewis, 1992). Since the fitness should be a property
of the pattern of cell types, it will depend only on the spatial distribution
of the label genes, henceforth called realizators (Mann and Morata, 2000).
Computational evolution, as we will explain, has to discover the realizators
as well as the genes and regulatory modules that control them.
Fitness concepts
Fitness should favor: (1) the diversity of genetic expression – the fitness
improves monotonically with the number of realizators; and (2) a unique
cell fate – one cell should express only a single master control gene for a
given segmental identity.
Mathematically, it is natural to measure diversity by entropy as is done
in physics. If a system has N possible states available to it, but only resides
in one of them, the entropy is zero, whereas if it spends equal time in each
state the entropy is maximum and equals log(N). For the embryo, each
realizator gene defines a state and the occupancy of each state is
proportional to the integral of the realizator over the embryo. Thus, we
define an entropy term, H(diversity), in the fitness that varies between zero
when only one realizator is expressed, to log(N) when they are expressed
equally.
The second requirement of our fitness is naturally expressed as a
conditional entropy, H(diversity|position); namely, at a fixed position in
the embryo we want the least diversity possible, i.e. the expression of a
single realizator gene. Thus, our fitness requires us to optimize two
‘contradictory’ constraints on the same function: maximizing entropy at
the global scale and minimizing conditional entropy locally. To define the
trade off between the two desiderata, we assume that duplicating a
realizator gene while keeping its expression domain constant is a neutral
event. Only one combination of our two entropies satisfies this condition
(see Appendix S1 in the supplementary material) and we can define a
unique fitness as:
fitness  –H(diversity) + H(diversity | position) .
(1)
We have chosen signs so that fitness is to be minimized; for N realizators
the optimal fitness is –log(N) when each realizator is restricted to a single
expression domain occupying 1/Nth of the embryo. [Our fitness is also
called ‘mutual information’ (Shannon and Weaver, 1998) and expresses
how well the embryo can define position given concentrations of
realizators.]
The fitness has to be defined for any profile of the realizators and Fig.
1 illustrates several cases decomposed into the two components of the
fitness. The most revealing case is illustrated in Fig. 1D,E. Allowing two
realizators to overlap degrades fitness, although one might think that the
DEVELOPMENT
al., 2003; Hobert, 2008), and transient activation, locked down via
positive-feedback loops, appears to be a rule in developmental gene
networks (Davidson et al., 2002; Oliveri et al., 2008).
In both insects and vertebrates, the 3⬘ to 5⬘ arrangement of Hox
genes on the chromosome matches the order of their anterior
expression boundaries along the AP axis (McGinnis and Krumlauf,
1992), a property termed ‘spatial colinearity’. Furthermore, in
vertebrates and short-germ insects, the temporal order of
expression in the posterior region of the growing embryo follows
the 3⬘ to 5⬘ genomic order (Kmita and Duboule, 2003; Ferrier and
Minguillón, 2003; Wacker et al., 2004; Shippy et al., 2008), a
phenomenon termed ‘temporal colinearity’.
Hox function in the fly follows the posterior prevalence (or
dominance) rule, whereby the most posterior Hox gene imposes its
fate on all the anterior genes (McGinnis and Krumlauf, 1992;
Kmita and Duboule, 2003). Loss-of-function mutations result in
anterior homeotic transformations in which a parasegment(s)
assumes the fate of the immediately anterior expressed Hox gene
(Lewis, 1978). In gain-of-function experiments, a Hox gene that is
ectopically expressed anterior to its normal position results in a
posterior homeotic transformation, whereas expressed posteriorly
it does nothing (Gibson and Gehring, 1988; González-Reyes and
Morata, 1990; McGinnis and Krumlauf, 1992; Morata, 1993).
The situation in vertebrates is similar, but more complex. There
are four paralogous Hox clusters and the clearest phenotypes are
observed when an entire Hox paralog group is removed. There is
definitely combinatorial regulation among the paralog groups, in
that adjacent Hox genes affect overlapping segments (Horan et al.,
1994; Wellik and Capecchi, 2003; McIntyre et al., 2007; Wellik,
2007). However, the number of functional Hox combinations is a
linear function of the number of genes, not an exponential one [e.g.
see figure 2 in Iimura and Pourquie (Iimura and Pourquie, 2007)].
The boundaries between the major types of vertebrae coincide
with Hox domains and are invariant among species, even when the
number of vertebrae in each domain changes (Wellik, 2007).
Posterior homeotic transformations have been observed by ectopic
expression of genes anterior to their normal domains (Kessel et al.,
1990; Lufkin et al., 1992).
Our goal in this article is twofold. We first define quantitatively
a fitness function (technically, the mutual entropy) that rewards
complex patterning. This fitness is an explicit function of the
concentration profiles of specific genes that define cellular
identities (we will call them ‘realizator’ genes) along the AP body
axis. It favors diversity, i.e. there should be many such realizator
genes present, each with a unique stable territory, and knowing the
realizator it should be possible to infer AP position. A standard
mutation-selection evolution algorithm then creates the network
that allows for ordered domains of realizator gene expression.
Mutual entropy (or information) has many mathematical properties
that make it the natural fitness with which to evolve developmental
networks in silico, generalizing its prior use in systems
neuroscience (Rieke et al., 1999).
We use this fitness to evolve networks to pattern the AP axis
under general conditions that span those observed in arthropods
and vertebrates: (1) a global morphogen gradient that disappears
before the end of embryogenesis; and (2) a sliding morphogen
that models patterning during growth (Francois et al., 2007; Peel
et al., 2005; Iimura et al., 2009). Position is defined by exposure
to the morphogen, but the networks are otherwise cellautonomous. We define the types of hierarchical networks that
can be constructed by incremental improvements in fitness. For
the sliding morphogen, which is appropriate to vertebrate Hox
Development 137 (14)
In silico evolution of embryonic pattern
RESEARCH ARTICLE 2387
B
C
D
E
F
overlap region conveys more ‘positional information’. However, we insist
that evolution has to create a gene restricted to the overlap and to designate
it as a realizator before the refined patterning of the axis is recognized by
the fitness.
Mathematical definition
Let cix be the concentration of realizator gene i in cell x. We can define a
conditional probability for cell x to be in the realizator state i as:
pi|x =
cix
∑
ckx
.
(2)
k
Note that:
∑
pi|x = 1 ,
We also define a conditional entropy H(diversity|position) as:
H (diversity | position) = − ∑ pix log pix + ∑ px log px .
i,x
Minimization of H(diversity|position) expresses the realizator gene
hypothesis that only one realizator is expressed at a given position (cell) in
the embryo. Thus, the fitness naturally resolves into the optimization of two
functions: H(diversity) should be large and H(diversity|position) should be
small, a situation we have encountered before (Francois and Siggia, 2008).
If we also impose the condition that duplication of a realizator is a
fitness-neutral event, then only one combination of the two functions is
allowed [see equations (6, 8) in Appendix S1 in the supplementary
material], namely, the mutual entropy between position and realizators for
the probability pix:
i
and pi|x⬇1 if one realizator is much larger than all the others. When only
a single realizator is expressed, we are not concerned about its
concentration as we assume its activity on downstream targets can be
adjusted to ensure their expression.
We define a probability pix normalized over all cells and realizators as:
pix =
cix
L ∑ ckx
,
(3)
k
where L is the number of cells in the embryo. From pix, one can define a
probability pi to find an realizator i anywhere in the embryo as:
pi =
∑
pix ,
(4)
x
and a probability px1/L, which says that each cell x has equal weight. With
these definitions, high diversity equates to large entropy, defined as:
H (diversity) = − ∑ pi log pi .
(5)
i
H(diversity) is the usual Shannon entropy in information theory. This is
maximum for all pi equal to 1/N for N selector genes, and therefore is lower
or equal to its maximum value logN.
(6)
x
F ( p) =
∑
i
pi log pi + ∑ px log px − ∑ pix log pix .
x
(7)
ix
(Smaller fitness is better.)
Methods
Our network evolution algorithm follows our earlier work (Francois and
Hakim, 2004; Francois et al., 2007; Francois and Siggia, 2008) with some
technical modifications specific to the problems treated here (see Appendix
S1 in the supplementary material). Both the topology of the network [i.e.
the nodes (genes) and the edges (interactions) connecting them] and all the
numerical parameters for the expression rates are mutated and selected. The
tags designating the realizators can move among the available genes and
also be created and destroyed.
Fig. 2 displays a summary of each ‘generation’ of the algorithm. We
evolve a population of networks. In one generation the half that is most fit
is retained unaltered, and a copy of each is mutated and added back to the
population. Therefore, the fitness of the best network can only improve or
remain the same.
When a morphogen is supplied as an input, all interactions are
transcriptional and cell-autonomous. A transcription factor can act as an
activator or repressor, depending on its target. Genes are off, i.e. non-
DEVELOPMENT
A
Fig. 1. Fitness diagram and gene expression profiles
as a function of anterior-posterior (AP) position
from cell 1 to 20 illustrate properties of the fitness.
Only the realizator genes (solid lines) enter the fitness;
other network genes are represented by dashed lines.
(A)The two components of fitness are plotted with
diagonal colored lines showing contours of constant
total fitness (better fitness in red). B-F mark the fitness of
subsequent panels. (B)For three genes ubiquitously
expressed, both H(diversity) and conditional entropy
H(diversity|position) are high and equal, so the actual
fitness is zero. (C)Each cell expresses a single gene
resulting in zero conditional entropy, but gene 1
occupies most of the embryo, lowering the diversity,
giving a fitness of –log1.64. (D)Fitness is defined when
realizators overlap, but neither the diversity nor the
conditional entropy is optimal and the fitness is
–log1.78>–log2. (E)The network in D can be improved
by the addition of a new realizator 4 that accounts for
the overlap of genes 2 and 3, giving a fitness of
–log2.33<–log2. (F)Optimal configuration for three
realizators; diversity is high (log3) and conditional
entropy is zero, so the actual fitness is –log3.
2388 RESEARCH ARTICLE
Development 137 (14)
Fig. 2. Schematic of the
evolutionary algorithm.
Differential equations for each
network are integrated (step 1).
The fitness function is
computed from the steady state
of each network (step 2). The
best half of the networks is
retained (selection), copied
(growth, step 3) and randomly
mutated (mutation, step 4).
Mutations change parameters
(kinetics) or the network itself,
as exemplified here.
practice, gene expression profiles are sharp enough that boundary positions
can be assigned unambiguously, and we include all genes expressed at the
final time in our index count.
We display networks after applying a pruning procedure that eliminates
all genes and interactions, the removal of which does not degrade the
fitness.
RESULTS
We use the fitness function to evolve a patterning network in two
cases that broadly exemplify what is seen in AP patterning in
arthropods and vertebrates: (1) a static morphogen gradient that
disappears before the fitness is measured; and (2) a sliding, or
translating, morphogen, motivated by somitogenesis and Hox
patterning (Francois et al., 2007; Dubrulle et al., 2001; Iimura et
al., 2009).
Since the network is cell-autonomous, AP fate is defined at the
single-cell level, either from the morphogen level (experienced for
a fixed time) or the length of time that the cell ‘feels’ a fixed
morphogen. These two problems become equivalent through the
intermediary of a timer gene. With cell-cell interactions and no
morphogen, our algorithm evolves lateral inhibition (see Appendix
S1 and Figs S13, S14 in the supplementary material).
Evolving with a disappearing morphogen gradient
For each simulation, cells are initialized with an exponential
morphogen gradient that is present for a fixed time and then removed
rapidly (Fig. 3A). (This protocol defines an informative bridge
between a morphogen that never disappears and a sliding morphogen
tied to growth.) The fitness is computed some time after the
morphogen disappears. Fig. 3 provides an example of a network that
evolved under these conditions with our mutual entropy fitness. The
DEVELOPMENT
expressing, unless a transcriptional activator is supplied. A single activator
suffices to turn a gene on, so that positive feedback of a gene on itself can
result in bistability. Activators combine through the equivalent of ‘OR’
functions. Similarly, one repressor is enough to suppress transcription and
repressors work multiplicatively. Gene duplication is permitted and entails
the replication of all interactions impinging on, and emanating from, the
duplicated gene. If a realizator is duplicated, a new realizator tag is created
and assigned to the new gene. The embryo consists of a line of L20 cells,
where cell x1 is anterior.
Although we do not consider molecular noise here, we modestly
randomize the input values and select for an invariant (and static) final gene
profile. When a morphogen is provided, either static or dynamic, to orient
the embryo, we average cix over all initial conditions and then compute pix
and the fitness. When two nearly optimal configurations are mixed in this
way (or the realizator expression domains broken up in multiple ways) the
fitness is clearly degraded.
There are penalties imposed on pathological expression patterns that are
not static at the end of the simulation or in which a cell expresses no
realizators. Their precise form does not matter. For further details of the
methods, see Appendix S1 in the supplementary material.
The morphogen increases from anterior to posterior, in analogy with
Caudal, which is conserved from insects to vertebrates and believed to be an
ancestral morphogen in arthropods (Copf et al., 2004; Olesnicky et al., 2006).
We define an AP order on genes, which potentially overlap in expression
in the following way. A gene ‘A’ is posterior (respectively anterior) to a
gene ‘B’ if the anterior (respectively posterior) boundary of A in space is
posterior (respectively anterior) to that of B. For instance, in Fig. 1E, gene
1 is posterior to genes 2 and 3, whereas gene 2 is anterior to genes 1 and
3. To characterize network-derived patterns, we define a posterior index
(respectively anterior index) by counting the number of repressions in
which one gene is repressed by a more posterior (respectively anterior)
gene. For example, in Fig. 1D, if gene 1 represses gene 2, then this
repression counts for 1 in the posterior index. On the contrary, if gene 2
represses gene 1, then this repression counts for 1 in the anterior index. In
In silico evolution of embryonic pattern
RESEARCH ARTICLE 2389
A
B
→
t < 1500
t= ∞
C
D
Fig. 3. A network evolved under the control of a static morphogen gradient. (A)Dynamics of the morphogen gradient, which is constant
and then disappears. (B)Evolution of the fitness of the best network as a function of generation number. (C)Network topology; the nodes and
edges are defined following the key in Fig. 2. (D)Steady-state profile of genes as a function of their position. Colors and numbers for a given gene
correspond to those in C. Posterior index, 9; anterior index, 0; final fitness, –log7.25.
values of which are 1.1 and 7.2, respectively, for all the networks
in the main text and Appendix S1 that were generated by a
disappearing morphogen gradient.
An extreme case with anterior index 0 is provided by Fig. 3, in
which genes are only repressed by genes posterior to them. As a
consequence, the network itself is very hierarchical, and there is a
correspondence between the position of a gene in this hierarchy
and its domain of expression. For example, consider the
subnetwork made of genes 2, 4, 8 and 1, and order them by the
direction of the repressive links. Gene 2 is at the bottom of the
hierarchy, being repressed by 4, 8 and 1, whereas gene 4 is
repressed by genes 8 and 1, and gene 8 is repressed only by gene
1. Strikingly, this order in network hierarchy (2,4,8,1) is actually
the order of gene expression from anterior to posterior at steady
state, as can be seen in Fig. 3D. This property is general and can
be seen in other evolved examples. So, the bias induced by the
input gradient has a clear signature in the structure of the evolved
network, imposing a hierarchy in which posterior genes are high
and repress anterior genes, which occupy lower positions in the
hierarchy.
Gene duplication and mutational events
To understand why there is such a bias in network structure, we
looked at a typical evolutionary pathway that creates a new
expression domain (Fig. 4). Gene duplication is central to the
events pictured, but after all interactions are replicated for the new
gene, several (e.g. the mutual activation of genes 1 and 3 in Fig.
4B) are superfluous and disappear.
Then, while the expression domain of one copy is first kept
essentially unchanged (gene 3 in Fig. 4C), the domain of
expression of the other copy starts drifting to another position
(improving the fitness) and becomes locked more posteriorly, new
regulations being evolved to ensure that the domains do not overlap
(Fig. 4D). Single neutral mutations and fitness-improving
parameter changes suffice to traverse the entire pathway. A less
DEVELOPMENT
fitness clearly decreases with generation number, ending up close to
–log7.25, with eight realizators expressed at steady state defining
eight slightly uneven domains. Movie 1 in the supplementary
material shows the time course of embryonic development. For other
examples of networks evolved under identical conditions, see
Appendix S1 and Figs S1-S8 in the supplementary material.
Despite some network-to-network variability, common structural
features can be observed in the network topologies. All the
realizator genes are self-sustaining in the final state (the morphogen
is gone) and many are individually bistable. Bistability also plays
a crucial role in pattern establishment because transient activation
by upstream genes can push the realizator above the threshold that
leads to a self-maintained high-activity state. This simple process
has been described as the basic mechanism for stabilization of
regulatory states in the sea urchin embryo (Davidson et al., 2002;
Oliveri et al., 2008). Bistability also implies that gene repression
can occur by transiently forcing a realizator below its autoactivating threshold, provided that no other activators are present.
When a set of realizators are individually bistable, then
supplying only a single repressive interaction between each pair of
realizators is sufficient to stabilize just those configurations with at
most one realizator active. This is most evident in the Boolean
limit. Any configuration with two realizators on, would activate the
repression between them and eliminate one. The sets of genes
(8,4,2) and (1,8,4) in Fig. 3 are examples of this widely used
arrangement.
Another prominent feature of the evolved networks is the
asymmetry in how the anterior versus posterior limits of the
realizator domains are established due to the morphogen. Most
posterior boundaries are defined by repression from genes
expressed posterior to the one in question, rather than by the loss
of an anteriorly expressed activator. Anterior boundaries are
generally defined by activation thresholds, often in response to the
morphogen. To quantify this asymmetry, we defined (see Materials
and methods) network anterior/posterior indexes, the average
2390 RESEARCH ARTICLE
Development 137 (14)
A
B
C
D
Fig. 4. Example of a typical pathway going from fitness –log2 to –log3. (A-D)Starting with a network with two realizators and two domains
of equal size, fitness is –log2 (A); gene 1 is duplicated to create gene 3, with all incoming and outgoing interactions copied and no change in
fitness (B). Parameter selection shifts the relative domains of the duplicated realizators, with fitness –log2.1 (C). Eventually, the most posterior gene
represses the anterior ones, confining gene 3 to the middle, each step improving fitness until finally, fitness is –log3 (D).
frequent pathway leading to an equivalent final state is shown in
Fig. S15 in the supplementary material. If we freeze the topology
of Fig. 4D, randomize the parameters and re-optimize with our
fitness, the same hierarchical expression pattern re-emerges (see
Fig. S16 in the supplementary material). Other frequent
evolutionary pathways following gene duplication are shown in
Fig. S17 in the supplementary material, leading to the same
topology as in Fig. 4.
boundaries, however, it is not possible just to read a morphogen
level, but rather the gene has to measure the time that it is exposed
to input. In several cases (e.g. see Fig. S11 in the supplementary
material) this is accomplished by early-activated genes repressing
those that appear later, i.e. setting their anterior boundaries. In
general, it was harder with a sliding gradient than a static one to
initiate multi-domain patterns starting from nothing.
To accelerate the evolution, we began with the small network
shown in Fig. 5B. It is not unreasonable, biologically, to bias
evolution in this way because the sliding morphogen really only
patterns the trunk and posterior, not the head. Our initial network
is common in developmental biology [e.g. the gsc/bra system
(Green, 2002)] and was the generic two-domain network we found
for a static input. One activator controls two genes with different
thresholds, and the high-threshold gene represses the lowthreshold gene to make their expression domains disjoint. We
generalize it slightly by adding a potential timer protein 3, which
is activated by the input, to delay the activation of gene 1 by the
morphogen. The subsequent evolution of this small network in
shown in Fig. 5, along with the expression profile it generates.
Another example of such evolution is shown in Fig. S12 in the
supplementary material.
The initial network bias and timer gene 3 facilitate the evolution
of simple networks with many states that exhibit topological
properties that are similar to those produced with a static input.
Again, posterior index is high while anterior index is low. As with
the static gradient, posterior boundaries are set up by repression by
posterior genes. However, because there is no AP gradient, the
mechanism of establishment of anterior boundaries is different, and
relies heavily on the timer gene. In Fig. 5, gene 3 accumulates
slowly and by virtue of graded activation thresholds, gene 3
directly activates genes 4 then 5 then 6, and indirectly controls the
remaining realizators other than gene 2 (which is activated directly
by the input and is repressed by gene 3 to restrict it to the anterior).
DEVELOPMENT
Using a timer gene
Evolving with a sliding morphogen
There is good evidence from somitogenesis in vertebrates that
patterning occurs as cells emerge from a posterior growth zone and
transit to lower levels of the posterior morphogens FGF and Wnt
(Aulehla and Pourquié, 2010). Hox patterns develop
contemporaneously (Aulehla and Pourquié, 2010; Iimura and
Pourquie, 2006) with somitogenesis. In Xenopus, there are
particularly clear data showing that Hox genes are expressed in a
3⬘ to 5⬘ temporal progression in the non-organizer mesoderm and
acquire a fixed position when the cells converge into the organizer
and then extend to create the AP axis (Wacker et al., 2004; Durston
et al., 2010). Thus, to model Hox patterning coupled to growth, we
assume a step-like morphogen profile that translates down a line of
cells. High morphogen defines the posterior growth zone, and the
step itself corresponds to the organizer, where the morphogen is
withdrawn and cells assume a fixed fate. Thus, cells sense their
distance from the rostral pole only once they are exposed to
morphogen. Cell fate can be controlled by the exposure time to
morphogen in the context of digit formation (Nelson et al., 1996),
and possibly also during axis formation (Aulehla and Pourquié,
2010), although the process by which time is measured is
completely unknown.
Typical networks derived with these input dynamics are
presented in Figs S9-S11 in the supplementary material. As for
static gradients, realizators tend to auto-activate and be bistable so
as to persist after the input disappears. Posterior boundaries are also
controlled by repression from posterior genes. For the anterior
RESEARCH ARTICLE 2391
A
C
B
D
E
F
As a consequence of these successive activations in response to
the accumulating timer and in spite of the rather complex network
topology, there is an almost perfect correspondence between the
temporal order of expression in the posterior cells and the spatial
order (Fig. 5C; see Movie 2 in the supplementary material). This
property can be related to the so-called temporal colinearity of Hox
genes and is therefore a direct consequence of evolution under the
control of a timer gene.
The common topological features of networks evolved under the
control of static versus sliding morphogens allow an easy translation
between the two cases. Since our networks act cell-autonomously,
we can literally replace the static morphogen as in Fig. 3D with a
timer gene that grows without saturation while exposed to
morphogen, and then decays exponentially after the morphogen
disappears from the cell in question. When this is done for the
network in Fig. 3, the realizator expression order from anterior to
posterior does not change (see Figs S19, S20 and Movie 3 in the
supplementary material). Similarly, if we turn the ‘timer’ gene into
a static morphogen, the network of Fig. 5 patterns an embryo with
the exact same anterior to posterior order of gene expression (see
Fig. S21 and Movie 4 in the supplementary material). Such a
change in morphogen dynamics could explain the evolution from
short to long germ-band insects (Liu and Kaufman, 2005).
Lastly, the existence of a timer gene suggests an easy way to
couple growth of the embryo to the dynamics of patterning, a
property necessary to explain the fact that embryos can develop
normally at different rates, such as happens in amphibians at reduced
temperature. If the dynamics of the timer gene scale with the growth
rate (simulated by the speed of the sliding input), the embryo patterns
properly, as shown in Movie 5 in the supplementary material.
Fig. 5. Network evolved with a sliding
morphogen beginning from a twodomain network. (A)The concentration of
gene 0 recedes from anterior to posterior to
model the coupling of patterning to growth.
(B)Initial network topology and the evolved
network topology after 5000 generations.
(C)Steady-state profile for the evolved
network. Posterior index, 5; anterior index,
0; fitness, –log4.95. (D)Gene expression as a
function of time in the posterior-most cell
follows the AP order in C, with the
exception of the one realizator (gene 8) that
is repressed by the ’input’. (E,F)We created
two genes (shown by a common color) from
each realizator as explained in the text and
in Fig. S22 in the supplementary material.
The dashed lines display the Hox-like genes
of the pair with nested expression profiles,
and the solid lines show the new realizator
genes with non-overlapping expression
domains. (E)Final pattern. (F)For a smoothed
morphogen step, most of the genes display
anterior spreading: they are first expressed
posteriorly and then move anteriorly (see
also Movie 6 in the supplementary material).
Connecting realizators to nested Hox pattern
It is well known that Hox expression domains overlap considerably
and are often nested, whereas their ‘master gene’ activity is
segmentally restricted. Unknown post-transcriptional events ensure
that the activity of anterior Hox genes is repressed by posterior Hox
genes, a property called phenotypic repression (or posterior
dominance) in fly and posterior prevalence in vertebrates (Duboule
and Morata, 1994; Kmita and Duboule, 2003).
The network in Fig. 5 has manifest temporal colinearity in its
realizators, but the remaining upstream genes 1, 6 and 9 are not
nested in the typical Hox pattern. This is because we have made all
of the genes multifunctional, i.e. simultaneously repressors and
activators, so that we do not make any distinction between
transcriptional and post-transcriptional activities.
However, it is nevertheless possible to formally disentangle these
two activities to retrieve a more usual Hox gene transcriptional
pattern. For each realizator, we create a new upstream gene that
captures all the activating inputs (including from itself). This new
gene activates a realizator, which retains all the repressive
connections of the original, and accounts for the actual ‘master
gene’ activity. Then, the bistable upstream genes will have Hoxlike nested expression domains, and the new realizators will retain
the localized expression of the original ones (Fig. 5E). Fig. S24 in
the supplementary material shows the complete network derived
from Fig. 5B by this transformation; for further details, see
Appendix S1 and Figs S22-S24 in the supplementary material.
If we smooth our sliding input step to make the transition from
off to on gradual, then our model can readily capture another
feature of Hox patterning termed ‘anterior spreading’ (Kmita and
Duboule, 2003; Iimura et al., 2009). It is most obvious in chick,
DEVELOPMENT
In silico evolution of embryonic pattern
A
Development 137 (14)
B
and as the name implies, Hox genes first turn on in the posterior
and are then expressed more anteriorly. When we simulate the
network in Fig. S24 in the supplementary material and Fig. 5E, we
see very evident anterior spreading (Fig. 5F; for the full time course
for this network, see Movie 6 in the supplementary material).
Simulation of anterior homeotic transformations
The high posterior index that typifies hierarchical networks has a
characteristic signature in a loss-of-function experiment when one
gene is set to zero. In analogy with the anterior homeotic
transformation seen with Hox gene loss of function (Horan et al.,
1994; Wellik and Capecchi, 2003; McIntyre et al., 2007), the
expression domain of the immediate anterior gene expands
posteriorly to fill the hole left by the deactivated gene (Fig. 6),
irrespective of whether the network was evolved under a static or
sliding input.
DISCUSSION
Network hierarchy and posterior prevalence
In silico evolution generates gene regulatory networks with
properties similar to the Hox gene networks that pattern the AP
body axis. Orientation is derived solely from the morphogen, be it
static or sliding (our surrogate for patterning during growth); the
gene networks are entirely cell-autonomous, which is not
unreasonable biologically, as noted in the Introduction. Since the
morphogen disappears before we assess the expression pattern of
the realizator genes that define the fitness, the gene networks are
inherently multi-stable and are directed towards the appropriate fate
by the morphogen.
Our networks were all hierarchical, as abstracted in the module
in Fig. 4. The morphogen defines the anterior gene expression
boundaries, whereas the posterior boundaries are defined by
repression from other genes. The two-gene limit is well known
(Green, 2002). We quantified the excess of repression from posterior
versus anterior genes by the respective indices (as defined in the
Materials and methods) and found, for the 11 networks described in
the text and in Appendix S1 in the supplementary material that were
evolved under the control of gradient or timer, averages of 6.9 and
1.1. The phenomenon of posterior prevalence is a direct
consequence of repression from the posterior gene (Fig. 6).
We obtained gene networks with the phenotypic properties of
Hox genes and do not explicitly distinguish transcriptional and
post-transcriptional regulation. The ‘genes’ in the simulation may
be composites of real genes. The classic nested Hox expression
territories occurred when we segregated activator and repressor
activities into separate genes (Fig. 5E).
Our conclusions are largely insensitive to mutation parameters
in the biologically plausible limit in which the rate of addition of
new links/nodes in the network is much slower than parameter
Fig. 6. Change in expression domains
when a single gene 8 is forced to zero.
(A)See Fig. 3C; (B) see Fig. 5C. In both cases,
the gene directly anterior to 8 extends
posteriorly, until it is repressed by the next
posterior gene.
mutation and link/node removal. Repeated simulations for the fate
of duplicated genes or parameter reoptimization with fixed network
topologies (see Figs S16, S17 in the supplementary material)
generally yielded the same outcome (Fig. 4).
Changing mutation rates influences the speed of evolution, not
its qualitative outcome, in agreement with previous findings
(Francois et al., 2007). Only neutral or fitness-improving
mutations survive selection and the fitness consists of plateaux
separated by jumps when the topology changes and parameters are
optimized in a few generations. The ease of evolving elaborate
patterning networks suggests the fitness landscape is funnel-like,
providing another argument for the parameter independence of our
results.
Morphogen, timer and temporal colinearity
The correlation between the spatial, genomic, and (for vertebrates)
temporal order of Hox expression across many species has
motivated experiments for mechanistic explanations. Genomeordered Hox arrays are not required for proper AP expression (Seo
et al., 2004); they may facilitate temporally ordered expression
(Duboule, 1995), but genome order is not part of our phenotypic
model. For mouse, experiments that disrupt one of four Hox
clusters alter the temporal expression but do not disrupt the final
spatial order (reviewed by Tschopp et al., 2009).
Our model for vertebrates assumes that axial patterning occurs
progressively as cells exit a region of high posterior morphogen
(e.g. Wnt). In amniotes, this is coupled to axial growth from longterm progenitors in the tail bud (Tzouanacou et al., 2009). In
amphibians, Hox patterning occurs as cells converge from the
ventral-posterior mesoderm and pass through the organizer
(Wacker et al., 2004; Durston et al., 2010). Both situations are
modeled as a morphogen step that slides (translates) down a line of
cells (there is no need to explicitly model cell proliferation or
movements). The step inflection is analogous to the organizer. Our
networks are cell-autonomous, so the only cue as to AP position
comes from the residence time in the high-morphogen posterior
region.
In silico evolution connects time to AP position by creating a
timer gene that builds up monotonically while the cells experience
the morphogen and dies after they exit the morphogen. Cell fate
records, through multi-stability, the highest value of the timer gene
seen by that cell. Temporal colinearity of Hox expression in the
undifferentiated posterior cells is one consequence of the timer
gene. Temporal order does not itself generate spatial order, and
morphogen versus temporal control is not an issue [except when
we compare long and short germ-band insects (see below)]. Other
biological properties, such as anterior spreading (Gaunt and
Strachan, 1994), can be easily explained by translating a smoothed
step.
DEVELOPMENT
2392 RESEARCH ARTICLE
A timer gene is a very natural way to coordinate growth and
patterning, which is particularly important in amphibians, the
embryonic development of which can vary by a factor of two with
temperature. The vertebrate homolog of the posterior insect
morphogen Caudal, Cdx (Duprey et al., 1988), has been shown to
posteriorize Xenopus embryos in a dose-dependent manner
(Pownall et al., 1996; Isaacs et al., 1998). Overexpression or
downregulation of Cdx genes shifts multiple Hox domains in a
graded way (van den Akker et al., 2002; Gaunt et al., 2008),
indicating an upstream position in the global control of Hox genes
that is suggestive of a role as a potential timer. Recently,
overexpression of Hox genes has been shown to influence axial
growth in mice (Young et al., 2009), which is beyond the scope of
our model. However, the concept of a timer gene as a mediator
between growth and axial patterning has not been proposed
previously and we hope that our ‘timer and wavefront’ model
functions in an analogous role to the ‘clock and wavefront’ model
for somitogenesis.
Transition from short to long germ-band insects
There is now abundant evidence from the arthropod phylogeny that
long germ-band segmental patterning arose from the ancestral short
germ-band mode multiple times (Peel et al., 2005), and recent
reviews focus mostly on segmentation (Peel, 2008; Rosenberg et
al., 2009). Much less is known about Hox patterning in short as
opposed to long germ-band insects, so we assume that its dynamics
are qualitatively similar to those of vertebrates. Thus, we model the
short to long germ-band transition as the transition from a sliding
to static gradient.
The timer gene in the short germ-band ancestor that converts
residence time in the growth zone to a protein level simply
becomes the new static morphogen (Fig. 7; see Fig. S21 and Movie
4 in the supplementary material), which could happen gradually as
more nuclei become available in the blastula. What the timer gene
concept accomplishes is to preserve unchanged the entire
downstream gene hierarchy, which is necessary for a viable
embryo. The fate of the Hox genes is seldom addressed in this
radical developmental transition, although it has been proposed that
the ancestral function of gap genes was to pattern Hox genes
(reviewed by Peel, 2008). Interestingly, in view of vertebrate
analogies, it has been suggested that the master morphogen
controlling segmentation in the long germ-band insect Nasonia is
Caudal in conjunction with Otd (Olesnicky et al., 2006; Lynch et
al., 2006), and Caudal might be the timer in short germ-band
insects.
The temporal colinearity of expression that we observed for the
sliding gradient is readily lost after the transition to static gradient
patterning (see Movie 1 in the supplementary material). There is
pressure (in both the simulation and the embryo) to make the Hox
pattern rapidly, so that when a static morphogen is supplied the
posterior genes turn on early and repress the initiation of the
anterior genes in posterior territories, thus generating simultaneous
temporal induction.
Mutual entropy as a fitness and the role of gene
duplications
We used the mutual entropy between realizator genes and AP
position to define a fitness for embryonic patterning. The same
expression has been used to quantify information transmission in
the presence of molecular noise for small genetic networks (Ziv et
al., 2007; Tkacik et al., 2008; Tkacik et al., 2009). The measure
used in those studies implies that combinatorial expression conveys
RESEARCH ARTICLE 2393
Fig. 7. Model for AP patterning in vertebrates and short germband insects and the transition to a long germ-band insect.
(Left) AP patterning in vertebrates and short germ-band insects.
(Right) The transition to a long germ-band insect. The timer (green)
accumulates while cells are exposed to the sliding posterior
morphogen (blue) and then decays. Its maximum value in a cell
defines the cell fate. The timer, when applied statically as a
morphogen gradient (right), creates the same pattern, now
simultaneously rather than sequentially.
more information about position, whereas our fitness penalizes two
realizators that overlap and it is only when a third realizator is
created and confined to the overlap region that the fitness will
improve.
Our mutual entropy fitness offers a mathematical framework in
which complex evolutionary events at the network level can be
observed and quantified. Lewis proposed that Hox genes evolved
by duplication, in which one copy keeps the ‘original’ function
while the other one is able to assume a new segmental identity
(Lewis, 1992). However, we find a different scenario that is more
symmetric between the two duplicated genes. They are expressed
contiguously and perturb the expression domains of their
immediate neighbors (Fig. 4). Some phylogenetic data on
homeobox genes seem to support the idea that duplicated genes
remain adjacent [for example, see figure 2 in Ryan et al. (Ryan et
al., 2007), where Hox1/2, Hox6-8 and Hox9-14 respectively cluster
together].
Incremental evolution and embryonic patterning
Although a connection between posterior prevalence, temporal
colinearity and anterior spreading has long been part of the Hox
phenomenology, this is not equivalent to creating a differential
equation model that expresses it. Nor was it obvious that the natural
mathematical measure of diversity of gene expression along an axis
could serve as the fitness, for which the properties of temporal
colinearity and posterior prevalence could arise by incremental
evolution. Duplication is the obvious route to multiple Hox genes,
but we furnished a detailed path for how this can occur
incrementally.
Hox regulation during growth might be achieved by different
combinations of molecular pathways in different species in analogy
with the somite clock, i.e. the Wnt, Notch and FGF pathways may
have different oscillatory properties in different species (Goldbeter
and Pourquié, 2008; Giudicelli and Lewis, 2004), yet the
phenomenon itself is invariant, and its ease of evolution by
continuous improvements in fitness suggests an explanation for
this.
DEVELOPMENT
In silico evolution of embryonic pattern
Acknowledgements
We thank John Guckenheimer and Bill Bialek for discussions. Support was
provided by the NSF under grant number DMR0804721 to E.D.S.
Competing interests statement
The authors declare no competing financial interests.
Supplementary material
Supplementary material for this article is available at
http://dev.biologists.org/lookup/suppl/doi:10.1242/dev.048033/-/DC1
References
Aulehla, A. and Pourquié, O. (2010). Signaling gradients during paraxial
mesoderm development. Cold Spring Harbor Perspect. Biol. 2, a000869.
Burke, A. C., Nelson, C. E., Morgan, B. A. and Tabin, C. (1995). Hox genes and
the evolution of vertebrate axial morphology. Development 121, 333-346.
Copf, T., Schröder, R. and Averof, M. (2004). Ancestral role of caudal genes in
axis elongation and segmentation. Proc. Natl. Acad. Sci. USA 101, 1771117715.
Crick, F. H. and Lawrence, P. A. (1975). Compartments and polyclones in insect
development. Science 189, 340-347.
Davidson, E. H., Rast, J. P., Oliveri, P., Ransick, A., Calestani, C., Yuh, C.-H.,
Minokawa, T., Amore, G., Hinman, V., Arenas-Mena, C. et al. (2002). A
genomic regulatory network for development. Science 295, 1669-1678.
Duboule, D. (1995). Vertebrate hox genes and proliferation: an alternative
pathway to homeosis? Curr. Opin. Genet. Dev. 5, 525-528.
Duboule, D. and Morata, G. (1994). Colinearity and functional hierarchy among
genes of the homeotic complexes. Trends Genet. 10, 358-364.
Dubrulle, J., McGrew, M. J. and Pourquie, O. (2001). Fgf signaling controls
somite boundary position and regulates segmentation clock control of
spatiotemporal hox gene activation. Cell 106, 219-232.
Duprey, P., Chowdhury, K., Dressler, G. R., Balling, R., Simon, D., Guenet, J.
L. and Gruss, P. (1988). A mouse gene homologous to the Drosophila gene
caudal is expressed in epithelial cells from the embryonic intestine. Genes Dev. 2,
1647-1654.
Durston, A. J., Jansen, H. J. and Wacker, S. A. (2010). Time-space translation
regulates trunk axial patterning in the early vertebrate embryo. Genomics 95,
250-255.
Ferrier, D. E. K. and Minguillón, C. (2003). Evolution of the hox/parahox gene
clusters. Int. J. Dev. Biol. 47, 605-611.
Francois, P. and Hakim, V. (2004). Design of genetic networks with specified
functions by evolution in silico. Proc. Natl. Acad. Sci. USA 101, 580-585.
Francois, P. and Siggia, E. D. (2008). A case study of evolutionary computation of
biochemical adaptation. Phys. Biol. 5, 26009.
Francois, P., Hakim, V. and Siggia, E. D. (2007). Deriving structure from
evolution: metazoan segmentation. Mol. Syst. Biol. 3, 9.
Gaunt, S. J. and Strachan, L. (1994). Forward spreading in the establishment of a
vertebrate hox expression boundary: the expression domain separates into
anterior and posterior zones, and the spread occurs across implanted glass
barriers. Dev. Dyn. 199, 229-240.
Gaunt, S. J., Drage, D. and Trubshaw, R. C. (2008). Increased cdx protein dose
effects upon axial patterning in transgenic lines of mice. Development 135,
2511-2520.
Gebelein, B., Culi, J., Ryoo, H. D., Zhang, W. and Mann, R. S. (2002).
Specificity of distalless repression and limb primordia development by abdominal
hox proteins. Dev. Cell 3, 487-498.
Gehring, W. J. and Ikeo, K. (1999). Pax 6, mastering eye morphogenesis and eye
evolution. Trends Genet. 15, 371-377.
Gibson, G. and Gehring, W. (1988). Head and thoracic transformations caused
by ectopic expression of antennapedia during Drosophila development.
Development 102, 657.
Giudicelli, F. and Lewis, J. (2004). The vertebrate segmentation clock. Curr. Opin.
Genet. Dev. 14, 407-414.
Goldbeter, A. and Pourquié, O. (2008). Modeling the segmentation clock as a
network of coupled oscillations in the notch, wnt and fgf signaling pathways. J.
Theor. Biol. 252, 574-585.
González-Reyes, A. and Morata, G. (1990). The developmental effect of
overexpressing a ubx product in Drosophila embryos is dependent on its
interactions with other homeotic products. Cell 61, 515-522.
Green, J. (2002). Morphogen gradients, positional information, and Xenopus:
Interplay of theory and experiment. Dev. Dyn. 225, 392-408.
Hobert, O. (2008). Regulatory logic of neuronal diversity: terminal selector genes
and selector motifs. Proc. Natl. Acad. Sci. USA 105, 20067-20071.
Horan, G. S., Wu, K., Wolgemuth, D. J. and Behringer, R. R. (1994). Homeotic
transformation of cervical vertebrae in hoxa-4 mutant mice. Proc. Natl. Acad.
Sci. USA 91, 12644-12648.
Iimura, T. and Pourquie, O. (2006). Collinear activation of hoxb genes during
gastrulation is linked to mesoderm cell ingression. Nature 442, 568-571.
Development 137 (14)
Iimura, T. and Pourquie, O. (2007). Hox genes in time and space during
vertebrate body formation. Dev. Growth Differ. 49, 265-275.
Iimura, T., Denans, N. and Pourquié, O. (2009). Establishment of hox vertebral
identities in the embryonic spine precursors. Curr. Top. Dev. Biol. 88, 201-234.
Isaacs, H. V., Pownall, M. E. and Slack, J. M. (1998). Regulation of hox gene
expression and posterior development by the Xenopus caudal homologue
xcad3. EMBO J. 17, 3413-3427.
Kessel, M., Balling, R. and Gruss, P. (1990). Variations of cervical vertebrae after
expression of a hox-1.1 transgene in mice. Cell 61, 301-308.
Kmita, M. and Duboule, D. (2003). Organizing axes in time and space; 25 years
of colinear tinkering. Science 301, 331-333.
Kozmik, Z. (2005). Pax genes in eye development and evolution. Curr. Opin.
Genet. Dev. 15, 430-438.
Lewis, E. B. (1963). Genes and developmental pathways. Am. Zool. 3, 33-56.
Lewis, E. B. (1978). A gene complex controlling segmentation in Drosophila.
Nature 276, 565-570.
Lewis, E. B. (1992). The 1991 Albert Lasker medical awards. Clusters of master
control genes regulate the development of higher organisms. JAMA 267, 15241531.
Liu, P. Z. and Kaufman, T. C. (2005). Short and long germ segmentation:
unanswered questions in the evolution of a developmental mode. Evol. Dev. 7,
629-646.
Lufkin, T., Mark, M., Hart, C. P., Dollé, P., LeMeur, M. and Chambon, P. (1992).
Homeotic transformation of the occipital bones of the skull by ectopic
expression of a homeobox gene. Nature 359, 835-841.
Lynch, J. A., Brent, A. E., Leaf, D. S., Pultz, M. A. and Desplan, C. (2006).
Localized maternal orthodenticle patterns anterior and posterior in the long
germ wasp Nasonia. Nature 439, 728-732.
Mann, R. S. and Morata, G. (2000). The developmental and molecular biology of
genes that subdivide the body of Drosophila. Annu. Rev. Cell Dev. Biol. 16, 243271.
McGinnis, W. and Krumlauf, R. (1992). Homeobox genes and axial patterning.
Cell 68, 283-302.
McIntyre, D. C., Rakshit, S., Yallowitz, A. R., Loken, L., Jeannotte, L.,
Capecchi, M. R. and Wellik, D. M. (2007). Hox patterning of the vertebrate rib
cage. Development 134, 2981-2989.
Morata, G. (1993). Homeotic genes of Drosophila. Curr. Opin. Genet. Dev. 3, 606614.
Nelson, C. E., Morgan, B. A., Burke, A. C., Laufer, E., DiMambro, E.,
Murtaugh, L. C., Gonzales, E., Tessarollo, L., Parada, L. F. and Tabin, C.
(1996). Analysis of hox gene expression in the chick limb bud. Development
122, 1449-1466.
Nilsson, D. E. and Pelger, S. (1994). A pessimistic estimate of the time required
for an eye to evolve. Proc. Biol. Sci. 256, 53-58.
Olesnicky, E. C., Brent, A. E., Tonnes, L., Walker, M., Pultz, M. A., Leaf, D.
and Desplan, C. (2006). A caudal mRNA gradient controls posterior
development in the wasp Nasonia. Development 133, 3973-3982.
Oliveri, P., Tu, Q. and Davidson, E. H. (2008). Global regulatory logic for
specification of an embryonic cell lineage. Proc. Natl. Acad. Sci. USA 105, 59555962.
Pearson, J. C., Lemons, D. and McGinnis, W. (2005). Modulating hox gene
functions during animal body patterning. Nat. Rev. Genet. 6, 893-904.
Peel, A. D. (2008). The evolution of developmental gene networks: lessons from
comparative studies on holometabolous insects. Philos. Trans. R. Soc. Lond. B
Biol. Sci. 363, 1539-1547.
Peel, A. D., Chipman, A. D. and Akam, M. (2005). Arthropod segmentation:
beyond the Drosophila paradigm. Nat. Rev. Genet. 6, 905-916.
Pownall, M. E., Tucker, A. S., Slack, J. M. and Isaacs, H. V. (1996). efgf, xcad3
and hox genes form a molecular pathway that establishes the anteroposterior
axis in Xenopus. Development 122, 3881-3892.
Rieke, F., Warland, D., de Ruyter van Steveninck, R. and Bialek, W. (1999).
Spikes: Exploring the Neural Code. Cambridge, MA: The MIT Press.
Rosenberg, M. I., Lynch, J. A. and Desplan, C. (2009). Heads and tails:
evolution of antero-posterior patterning in insects. Biochim. Biophys. Acta 1789,
333-342.
Ryan, J. F., Mazza, M. E., Pang, K., Matus, D. Q., Baxevanis, A. D.,
Martindale, M. Q., Finnerty, J. R. and Fay, J. (2007). Pre-bilaterian origins of
the hox cluster and the hox code: evidence from the sea anemone, Nematostella
vectensis. PLoS ONE 2, e153.
Seo, H.-C., Edvardsen, R. B., Maeland, A. D., Bjordal, M., Jensen, M. F.,
Hansen, A., Flaat, M., Weissenbach, J., Lehrach, H., Wincker, P. et al.
(2004). Hox cluster disintegration with persistent anteroposterior order of
expression in Oikopleura dioica. Nature 431, 67-71.
Shannon, C. E. and Weaver, W. (1998). The Mathematical Theory of
Communication. Urbana, IL: University of Illinois Press.
Shippy, T. D., Ronshaugen, M., Cande, J., He, J., Beeman, R. W., Levine, M.,
Brown, S. J. and Denell, R. E. (2008). Analysis of the tribolium homeotic
complex: insights into mechanisms constraining insect hox clusters. Dev. Genes
Evol. 218, 127-139.
DEVELOPMENT
2394 RESEARCH ARTICLE
Tkacik, G., Callan, C. G. and Bialek, W. (2008). Information flow and
optimization in transcriptional regulation. Proc. Natl. Acad. Sci. USA 105,
12265-12270.
Tkacik, G., Walczak, A. M. and Bialek, W. (2009). Optimizing information flow in
small genetic networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80, 031920.
Tschopp, P., Tarchini, B., Spitz, F., Zakany, J. and Duboule, D. (2009).
Uncoupling time and space in the collinear regulation of hox genes. PLoS Genet.
5, e1000398.
Tzouanacou, E., Wegener, A., Wymeersch, F. J., Wilson, V. and Nicolas, J.-F.
(2009). Redefining the progression of lineage segregations during mammalian
embryogenesis by clonal analysis. Dev. Cell 17, 365-376.
van den Akker, E., Forlani, S., Chawengsaksophak, K., de Graaff, W., Beck,
F., Meyer, B. I. and Deschamps, J. (2002). Cdx1 and cdx2 have overlapping
functions in anteroposterior patterning and posterior axis elongation.
Development 129, 2181-2193.
Wacker, S. A., McNulty, C. L. and Durston, A. J. (2004). The initiation of hox
gene expression in Xenopus laevis is controlled by brachyury and bmp-4. Dev.
Biol. 266, 123-137.
RESEARCH ARTICLE 2395
Weintraub, H., Davis, R., Tapscott, S., Thayer, M., Krause, M., Benezra, R.,
Blackwell, T. K., Turner, D., Rupp, R. and Hollenberg, S. (1991). The myod
gene family: nodal point during specification of the muscle cell lineage. Science
251, 761-766.
Wellik, D. M. (2007). Hox patterning of the vertebrate axial skeleton. Dev. Dyn.
236, 2454-2463.
Wellik, D. M. and Capecchi, M. R. (2003). Hox10 and hox11 genes are
required to globally pattern the mammalian skeleton. Science 301, 363367.
Young, T., Rowland, J. E., van de Ven, C., Bialecka, M., Novoa, A., Carapuco,
M., van Nes, J., de Graaff, W., Duluc, I., Freund, J.-N. et al. (2009). Cdx and
hox genes differentially regulate posterior axial growth in mammalian embryos.
Dev. Cell 17, 516-526.
Ziv, E., Nemenman, I. and Wiggins, C. H. (2007). Optimal signal processing in
small stochastic biochemical networks. PLoS ONE 2, e1077.
Zuber, M. E., Gestri, G., Viczian, A. S., Barsacchi, G. and Harris, W. A. (2003).
Specification of the vertebrate eye by a network of eye field transcription
factors. Development 130, 5155-5167.
DEVELOPMENT
In silico evolution of embryonic pattern