BIOINFORMATICS DISCOVERY NOTE
Vol. 22 no. 12 2006, pages 1424–1430
doi:10.1093/bioinformatics/btl119
Systems biology
Global dynamics of biological systems from
time-resolved omics experiments
Martin G. Grigorov
Nestlé Research Center, BioAnalytical Science, CH-1000 Lausanne 26, Switzerland
Received on September 13, 2005; revised on March 24, 2006; accepted on March 25, 2006
Advance Access publication April 3, 2006
Associate Editor: Martin Bishop
ABSTRACT
The emergent properties of biological systems, organized around
complex networks of irregularly connected elements, limit the applications of the direct scientific method to their study. The current lack
of knowledge opens new perspectives to the inverse scientific paradigm where observations are accumulated and analysed by advanced
data-mining techniques to enable a better understanding and the formulation of testable hypotheses about the structure and functioning of
these systems. The current technology allows for the wide application of
omics analytical methods in the determination of time-resolved molecular profiles of biological samples. Here it is proposed that the theory of
dynamical systems could be the natural framework for the proper analysis and interpretation of such experiments. A new method is
described, based on the techniques of non-linear time series analysis,
which is providing a global view on the dynamics of biological systems
probed with time-resolved omics experiments.
Contact: [email protected]
INTRODUCTION
A retrospective consideration of the history of science indicates that
new theories come to life in two basic ways (Oldroyd, 1986). Often,
a theory is developed following the direct Baconian approach where
a testable hypothesis is formulated using the currently available
knowledge. The direct approach in the study of the dynamics of
biological networked systems at the molecular level relies on a
number of formalisms such as graphs, Boolean networks, differential equations and stochastic master equations (Smolen et al., 2000;
Hasty et al., 2001; de Jong, 2002; Kappler et al., 2003). These
approaches might be classified in three main groups. A first one
encompasses statistical mechanics ensemble averaging to infer the
dynamical properties of real networks (Kauffman, 2004); a second
one is based on the precise description of the time evolution of the
molecular system by differential equations (de Jong and
Geiselmann, 2005) while a third approach uses stochastic models
of biological networks (Gillespie, 1977). The other major method of
science is the inverse approach (Oldroyd, 1986; Kell and Olliver,
2004). It consists in the accumulation of experimental data,
followed by the inference of a hypothesis and a theory from the
analysis of this data. Statistical data mining and machine learning
techniques, either unsupervised or supervised (Hastie et al., 2001),
enable the inverse approach. This paradigm is gaining importance
in biological research (Kell and Olliver, 2004; Maddox, 1994)
especially after the technological advances achieved during the
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sequencing of the human genome (Hood, 2002). These have triggered the wide application of omics analytical methods (Zhu and
Snyder, 2002) to typically generate several thousand of resolvable
lines that provide quantitative information about the concentrations
of the transcribed genes, the translated proteins, or the synthesized
metabolites (Aggarwal and Lee, 2003; Goodacre et al., 2004;
Lindon, 2004) in a biological sample. The area of research dedicated
to network reconstruction or network reverse engineering is an
example of the successful application of the inverse approach to
the analysis of genomic data (van Someren et al., 2002).
With the increase of throughput of omics technologies time
resolved experiments are now at reach and recent works exemplify
the trend (Fang et al., 2005; Arbeitman et al., 2002). I propose that
the relevant framework to analyse and interpret such data should
be the theory of non-linear dynamical systems (Baker and Gollub,
1996). Within this discipline of theoretical physics, the state of a
dynamical system is represented by a series of n numerical values
that describe its situation in a state space and that are susceptible to
time evolution. A point represents a state with coordinates that
are the n variables. The sets of points that are formed by the
temporal evolution of a deterministic dynamical system, called
the trajectories, converge to specific regions of the state space
that are referred to as attractors. Simple examples are the fixed-point
or limit cycle attractors that are objects of integer topological
dimensions. However, strange attractor objects have been shown
to exist in the state space (Ruelle and Takens, 1971) of deterministic dynamical systems, and to have non-integer topological
dimensions, characteristic of fractals (Mandelbrot, 1977). The
sensitivity to initial conditions is an important property of chaotic
systems. Lyapunov exponents are quantities that are directly
computable from the time-series generated by the dynamic
process, and they are used to measure the divergence of initially
closely situated state space trajectories. When the dynamics occurs
on a chaotic attractor the trajectories diverge, on average, at an
exponential rate characterized by the largest positive Lyapunov
exponent (Eckmann and Ruelle, 1985). An example of a classical
dynamical system is shown in Figure 1, representing the diverging
trajectories generated by the chaotic weather model investigated by
Lorentz in the 1960s, consisting of three coupled differential equations, and shown here for parameter values r ¼ 28, s ¼ 10, b ¼ 8/3.
The figure depicts the evolution of the dynamical system using
the so-called Poincaré space section. This is a way of presenting a trajectory in an n-dimensional state space within an
(n 1)-dimensional space. The Poincaré space section is obtained
by picking one state space variable constant while plotting the
Ó The Author 2006. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
Global dynamics of biological systems
Fig. 1. The Lorentz chaotic attractor. The sensitivity to initial conditions is a characteristic property of chaotic systems. Here the property is illustrated by the
divergence of two initially closely situated trajectories generated by the chaotic weather model investigated by Lorentz in the 1960s, consisting of three coupled
differential equations, and shown here for parameter values r ¼ 28, s ¼ 10, b ¼ 8/3. Samples over the two initially nearby trajectories are highlighted at exactly
the same time points in green and red color, respectively.
values of the other variables each time the selected variable has the
desired value.
In the early years of the theory of dynamical systems its practical
applications were rather limited as it was suitable only for the study
of systems for which the state space variables and the governing
differential equations were known. A breakthrough came with the
theorem of Takens (Eckmann and Ruelle, 1985; Takens, 1981)
stating that a map exists between the original state space and a
reconstructed state space that is an embedding. The theorem assures
that one does not have to measure all the state space variables of the
system as in fact almost any one of the variables will be sufficient to
reconstruct the dynamics. The dynamical properties of the system in
the true state space are preserved under the embedding transformation. The relevance of a dynamical system state space reconstruction
from the time series of a single observable was first demonstrated
numerically by Packard et al. (1980) with the method of derivative
coordinates. This opened new avenues to practical applications in
the area of economics, physiology, physics and chemical kinetics, to
cite a few (Weigend and Gershenfled, 1993).
The investigation of these systems led to the discovery of strange
attractors with a limited number of degrees of freedom and one
positive Lyapunov exponent, characteristic of low-dimensional
chaos, such as the Lorentz attractor (Fig. 1). However, the study
of biological systems, such as species populations, biological
rhythms or neuronal firings in the brain, demonstrated the existence
of dynamical systems with more than one positive Lyapunov exponent, termed as high-dimensional chaotic systems. The limited
interest towards high-dimensional systems was attributed to the
fact that the existing methods for non-linear systems analysis
experienced difficulties with describing high-dimensional systems.
However, some researchers were able to outline solid mathematical
background and numerical methods for high-dimensional dynamical systems using chaotic itinerancy (Kaneko and Tsuda, 2003),
attractor ruins (Kaneko and Tsuda, 2000) and Milnor attractors
(Milnor, 1985).
MATERIALS AND METHODS
Drosophila melanogaster developmental gene
expression timecourse
The biological data used in this study were obtained from the original work
of Arbeitman et al. (2002) who used cDNA microarrays to follow the fate
of over 4000 genes at 66 sequential time points during the fly life cycle.
The data were downloaded from the LifeCycle website made available by
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M.G.Grigorov
Dr Kevin White (http://genome.med.yale.edu/Lifecycle/Data_download/
wholeanimaldata.zip).
Data pre-processing
The original study was carried by Arbeitman et al. (2002) who used cDNA
microarrays to follow the fate of over 4000 genes at 66 sequential time points
during the fruit fly life cycle. Experimental data were not available at every
single time point for a number of genes. The original publication of
Arbeitman et al (2002) does not specify whether the missing values were
owing to experimental errors or to the fact that the particular genes were only
expressed in a specific developmental stage. In the lack of this information
these highly interesting genes were removed from all other stages of the
analysis, although several methods were described in the literature for the
estimation of missing values in microarray experiments. However, these
methods were specifically developed to support single time point experiments and it was estimated that they would be of limited reliability in the
context of the current investigation. The filtering was therefore undertaken as
a necessary step that enabled the current analysis. A second limitation
inherent to the used methodology consisted in removing from consequent
analysis the time points corresponding to either the male or the female phases
of adult development. Indeed, in its current form singular spectrum analysis
does not support bifurcating time series. In the present analysis I took in
account the male phase of adult development. The remaining 3104 genes
expressed at the 66 sequential time points were ordered according to the
modulations that they exhibited in the very early developmental stage of
unfertilized eggs, by putting the down-regulated genes first. This formed a
data matrix with a column containing a given gene modulation at the 66
different time points, and a row containing the modulations of all the 3104
different genes at this specific time point. Here and forth I use the term
modulation to name the level of differential expression of a gene as compared with its expression in all stages. In this I follow Arbeitman et al. (2002)
who used a pool of RNA from all developmental stages as a reference. The
modulations of the different genes appear on the y-axis in Figure 2a.
Although the final reordering of the columns within the matrix might
seem arbitrary at first sight, it had no effect on the analysis that was further
performed, and this because the eigenvalue decomposition of a matrix is
invariant under reordering of its columns.
Data analysis
The time resolved microarray dataset was processed with the Singular
Spectrum Analysis toolkit (Ghil et al., 2002). This non-parametric data
analysis method overcomes the problems of finite sample length and noisiness of time series by using a data-adaptive basis set, instead of fitting the
signal using some predefined basis functions. The method uses a principal
component analysis performed on the auto-covariance data matrix to decompose a time series into a succession of signals of decreasing variance that are
referred to as the empirical orthogonal eigenfunctions (EOFs). In this work
the Burg method was used to derive the auto-covariance matrix of the time
series under scrutiny. The formal similarity between classical laggedcovariance analysis and the method of delays was demonstrated, and it
was found that pairs of SSA eigenmodes corresponding to nearly equal
eigenvalues and the associated EOFs that are nearly in quadrature could
represent a non-linear, anharmonic oscillation (Vautard et al., 1992). The
figures were produced with the Xgobi data visualization software (http://
www.research.att.com/areas/stat/xgobi/) (Swayne et al., 1998).
RESULTS AND DISCUSSION
Here I show that time-resolved omics data could be scrutinized by
the methods of non-linear time-series analysis much in the same
way as this is done for some time now with physiological rhythms
such as electrocardiograms and electroencephalograms (Glass
and Kaplan, 1993). After thorough investigation of these physiological time series it was concluded that they are generated by
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low-dimensional chaotic dynamics. For this, classical methods
were employed such as the determination of a proper embedding
dimension, calculation of the correlation dimension, or the identification of a positive Lyapunov exponent. However, it became clear
that it would be a true challenge to operate a simple translation of
these methods to the investigation at the molecular level of the
dynamical behavior of the complex and complicated living organisms. A fundamental reason for this is that gene expression and
regulation is an intrinsically noisy process (McAdams and Arkin,
1999), whereas a technical reason is that currently the recorded time
series in -omics experiments are still short. This does not allow for
the rigorous application of methods that were originally developed
for the study of time series of several thousands of points, recorded
on mechanical systems. For example in the present case the calculation of the Lyapunov spectrum turned out to be impossible owing
to the insufficient amount of data. In this situation the emerging
interest in the study of biological systems by the methods of the
theory of dynamical systems is pursued by the reconstruction of the
phase space of the generating dynamical process. For example
Rifkin et al. (2000) applied the methodology to the analysis of
microarray experiments of recurrent patterns occurring within the
Saccharomyces cerevisae cell cycle (Rifkin et al., 2000; Rifkin and
Kim, 2002). In this work I am using a similar methodology to
analyse one of the most extensive time-resolved microarray experiments available to date, aimed at investigating the dynamics of
D.melanogaster development. The processing of omics experimental data that I propose consists in assuming that the dynamical
system (here the fruit fly) ‘walks’during its development over a
trajectory in a state space that is spanned by the gene expression
levels, measured by the respective concentrations of mRNA, taken
as state space variables.
In this way a developmental time series was generated by the
time-ordered concatenation of the levels of gene expression at every
developmental phase that is by concatenating the rows of the data
matrix described in the previous section, starting with the first row
(Fig. 2a). Often, time-series analysis is aimed at searching for oscillating signal components, responsible for the recurrent behaviour of
the trajectories in the state space. To this end, I have used singular
spectrum analysis to identify several complex cyclical patterns in
the developmental time series of the fruit fly. Singular spectrum
analysis is essentially a principal components analysis but carried
out in the time domain. The method was demonstrated to be able to
separate short and noisy time series into statistically independent
components that can be classified as trends, deterministic oscillations, or noise (Vautard et al., 1992), and therefore to overcome the
difficulties in studying high-dimensional non-linear systems just
discussed. To carry such an analysis Vautard, You and Ghil proposed to construct from the standardized time series and a maximum
lag M (or window size) the corresponding Toeplitz lagged correlation matrix. Consequently, the matrix is subjected to eigenanalysis
to determine the eigenvectors, or EOFs, sorted in descending order
of the corresponding eigenvalues. The Toeplitz matrix generated
from the time series shown in Figure 2a was therefore constructed
and diagonalized, with a time lag M ¼ 3104 corresponding to the
number of genes used to define a system’s state. Clear analogies
exist between singular spectrum analysis and Fourier analysis as
both methods decompose the original signal in a set of basis functions. But in contrast to Fourier analysis that is using a set of
predetermined harmonics, singular spectrum analysis is not making
Global dynamics of biological systems
Fig. 2. State space reconstruction of the developmental dynamics of D.melanogaster. The reconstruction is based on data from microarray experiments that
quantifies the levels of gene expression during embryonic (green), larval (cyan), pupal (orange) and adult male (red) phases. (a) Developmental time-series
obtained by concatenation of state space variables represented by the levels of gene expression. The x-axis represents the time arrow, while the modulations of the
different genes appear on the y-axis. (b) Vase-shaped funnel representation of the four major phases in the life of the fruit fly in the space spanned by the first three
EOFs. The first point of the ‘developmental’ time series is labelled as ‘1’. (c) The complexity of the reconstructed attractor appears when taking other projections.
In this case the object is represented in the space spanned by the second, third and fourth EOFs. (d) Same representation but in the space defined by first, third and
fifth EOFs. (e) Same representation but in the space defined by first, fourth and sixth EOFs.
any a priori assumption about the form of the basis functions.
Rather these emerge within a data-adaptive algorithm under the
form of EOFs. The resulting reconstruction of the state space of
the fly’s lifespan was carried out in the space spanned by the first
10 significant EOFs that accounted for 27% of the variance of the
original dataset. In our case the null hypothesis of a generating noise
process was rejected at the 99% confidence level for the first 5 pairs
of eigenvalues (Fig. 3). The captured variance reached 43% when
the first 32 components taken into account were found to
be significant at the 90% (data not shown). The flattening of the
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M.G.Grigorov
Fig. 3. Singular spectrum of the D.melanogaster developmental time series. The whole spectrum is depicted with the lower (red) and upper (green) 99%
confidence limits. In the upper right corner appears a magnification of the dominant singular values, where deviations from the statistical limits derived by Monte
Carlo numerical sumilations are clearly identifiable for the first 10 eigenpairs. In the far upper right corner are provided the shapes of the first 2 EOFs. All graphs
use the same x-axis, representing the eigenvalue ranks. On the y-axis appear the respective magnitudes of the eigenvalues and eigenfunctions, going to the upper
right corner.
significant part of the eigenvalue spectrum and the spread of the
information among a number of different EOFs confirms the
hypothesis about an underlying high-dimensional chaotic generating process. The presence of irregular oscillations in the original
dataset is evidenced by the pairs of nearly equal eigenvalues that
appear in Figure 3, with the associated eigenvectors and principal
components being in quadrature (Vautard et al., 1992). The statistical significance of each of the EOFs was estimated using a special
extension of the SSA method, known as Monte Carlo SSA, first
proposed by Allen and Smith (1996). Monte Carlo SSA can be used
to establish whether a given time series is distinguishable from any
well-defined process, including the output of a deterministic chaotic
system. In this work the focus was on testing against linear stochastic processes that are usually considered as noise. The method
consists in measuring the resemblance of a given surrogate randomized time-series with the original time-series. This resemblance is
quantified by taking as reference the eigenspectrum of a noise
process that is known to have a particular shape. If the data spectrum
lies above this theoretical noise spectrum it is generally considered
as significant. However a single realization of the noise process is
not sufficient here—it is only the average of such sample spectra
over many realizations that will tend to the theoretical eigenspectrum of the ideal noise process. The statistical distribution of
the eigenvalues, determined from the ensemble of Monte Carlo
simulations, gives confidence intervals outside which a time series
can be considered to be significantly different from a generic noise
simulation. For instance, if an eigenvalue lies outside a 95% noise
percentile, then the noise null hypothesis for the associated EOF can
be rejected with this confidence.
A complex attractor was identified in the reconstructed space that
appeared as a vase-shaped funnel when the initial developmental
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time series was projected in the space defined by the first three EOFs
(Fig. 2b). This ‘vase of life’ turned out to be an object with complex
topology as it can be seen when a projection is taken in the space of
the second, third and fourth EOFs (Fig. 2c). However, the topological structures of the attractor projections are similar when taken over
the axes of highest information content, with some non-linear deformations appearing on a finer scale (Fig. 2d and e). The width of a
cyclic pattern in the state space could be related to the number of
different states visited by the dynamical process. It is therefore
representative of the entropic content or the degree of order within
the system during its evolution. The visual inspection of Figure 2
provides arguments to speculate that during the early developmental
stages of D.melanogaster, gene expression activity is disordered in
time, and has a high entropic content—the trajectories explore
distant regions of the state space. During the evolution of the organism to its well-structured mature state, entropy decreases significantly. The system evolution is characterized by a confined bundle
of trajectories exploring only a very limited region of the state space
that corresponds to a well-defined pattern of gene activity. The view
that an organism cell’s differentiation states are determined by a
pre-existing program, an attractor, and that only few external
instructional inputs are required for the system to reach this attractor
was already formulated by Kaufmann (2004), based on his extensive investigation of the dynamics of Boolean networks, and more
recently by Huang et al. (2005), relying on a microarray-based
investigation of the mechanism of stem cells differentiation.
The results presented here suggest that the stable evolution of
the biological system on the attractor is limited in time. At the
later stages of the mature state, gene activity seems to reach
again a disordered regime, the trajectories start to explore broader
regions of the state space, and as a result entropy increases. It is
Global dynamics of biological systems
tempting to speculate that the analysis of microarray experiments
that I propose here provides some arguments to what could be an
intuitively appealing interpretation of life—it is a complex process
driving biological forms of matter from a highly disordered, entropically rich states through a set of entropically poor, highly structured
and uniform states for a limited amount of time, returning back to an
entropically rich, unstructured states at the latest phases of development of an organism. Unfortunately, the developmental microarray experiment was acquired over a limited time, and data was not
available for the latest stages of D.melanogaster lifespan to
strengthen this conclusion. Another point to question this conclusion is the high noise content present within the dataset as it turned
out that the strictly statistically significant irregular oscillations
spanning the attractor accounted for only 27% of the temporal
variance. These are questions that remain open for the moment,
but with the advent of inexpensive microarray technologies to be
applied to capture the dynamical genetic behaviour of biological
systems they will certainly find soon the necessary answers.
The EOFs in a singular spectrum analysis are the data-adaptive
basis functions or modes on which the complex dynamics of the
investigated system is decomposed. The shape of an EOF is determined by the contributions of each of the 3104 genes included in the
analysis. The shape of the first pair of EOFs is shown in Figure 3.
The EOFs or dynamical modes represent natural selections of genes
in direct relation with the time evolution of the gene transcription
activity in the biological system that was investigated. For example,
the shape of EOF1 indicates that approximately half of the genes are
down-regulated in the early phases of life, while the other half is
up-regulated in the more advanced phases of live, and vice versa. It
was interesting to project the genes associated to the first EOF on
gene pathways, such as the ones made available by the Ingenuity
Pathway Analyzer (http://www.ingenuity.com), and to find out that
the gene components that were up-regulated in the early phase of
D.melanogaster life were related to molecular functions such as
‘Cellular Assembly and Organization’ and ‘Amino acid metabolism’ for instance. In the later phases of the fly’s life other molecular
processes were identified as being dominant such as ‘Small
Molecule Biochemistry’ or ‘Vitamin and Mineral Metabolism’.
In this respect the discussed approach seem to hold promises for
a new way for interpreting time-resolved functional genomics
experimental data and these aspects will be investigated in forthcoming works.
CONCLUSIONS
The importance of the proposed analysis consists in providing a
global view on the dynamics of the biological process that is probed
with a time-resolved omics experiment, a view that is not obtainable
by other means. Indeed, by applying the new method the information about the gene activity across the lifespan of D.melanogaster
became accessible in a glimpse, such as the increased and divergent
gene activity during the embryonic phase of life, evidenced by
widely circling trajectories. In difference, the larval and the
early pupal phases had a very restricted gene activity, while in
the adulthood gene activity was again relatively elevated. The finding reported by Arbeitman et al. (2002) that ‘about half the genes
changed during embryogenesis, whereas only 118 changed during
adult life’ confirmed the evidence for switching of a multitude of
genes in the very early phases of the life of the fruit fly, easily
identifiable by visual inspection of Figure 2.
In this work the application of the full set of tools used to
study low-dimensional chaotic systems was limited by the
high-dimensional non-linear time evolution of biological systems,
unravelled by several other similar studies, and confirmed by the
present one. An intrinsic limitation to the application of these
methods is related to the high amount of noise in gene expression
and regulation events, whereas a current technical drawback certainly consists in the relatively short records over time that are at
hand. However, important theoretical developments are appearing
that will provide the necessary computational tools to study the
complexity of biological systems and their evolution in time.
This will allow answering important questions such as the relations
in time between gene expression, protein synthesis and metabolite
turnover.
ACKNOWLEDGEMENTS
The author would like to thank Prof. Dr Jeremy Nicholson,
Dr Laurent-Bernard Fay and Dr Sunil Kochhar for the discussions
we had on some aspects of the subject. The author would like to
adress his thanks to the reviewers who by suggestions and advises
contributed to the significant improvement of this work.
Conflict of Interest: none declared.
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