1609
Stochastic Oscillations in Genetic Regulatory
Networks. Application to Microarray Experiments.
Simon Rosenfeld
National Cancer Institute, Bethesda, MD 20892; sr212a@nih.gov
Summary. We analyze stochastic dynamics of genetic regulatory networks using the system of
nonlinear differential equations. The system of S-functions is applied to capture the role of RNA polymerase in the transcription-translation mechanism. Combining the center manifold theorem of
nonlinear dynamics and probabilistic properties of the chemical rate equations, we derive a system of
stochastic differential equation which is analytically tractable despite high dimension of the regulatory network. Using the stationary solutions of these equations, we explain the apparently paradoxical
results of some recent time-course microarray experiments where mRNA transcription levels are
found to only weakly correlate with the corresponding transcription rates. Combining analytical and
simulation approaches, we determine a set of relations between the size of a regulatory network, its
structural complexity, and chemical variability in the protein-mRNA system.
Key words: Genetic Regulatory Network, Gene Expression, Nonlinear Dynamics, Stochastic Processes
1 Introduction
According to the “central dogma” in molecular biology, the genetic regulatory process involves
two key steps, namely “transcription”, i.e., deciphering the genetic code and creation of the messenger RNA (mRNA), and “translation”, i.e., synthesis of the proteins by ribosomes using mRNAs as
templates. These processes run concurrently for all the genes comprising the genome. Importantly,
each molecular assembly responsible for deciphering the genetic code is itself built from the proteins
produced through transcription and translation of other genes, thus introducing nonlinear interactions
into the regulatory process [Lew04]. In the human genome, for example, typically, from 30 to 100
regulatory proteins are involved in each transcription event in each of about 60,000 genes. This means
that the regulatory network is of a very high dimensionality and very high connectivity simultaneously. Mathematical description of such a network is a challenging task, both conceptually and computationally. Quite paradoxically, however, this seemingly unfavorable combination of two “highs”
opens a new avenue for approximate solutions and understanding global behavior of regulatory systems through the application of asymptotic methods.
1610
In this paper, we pay special attention to quantitative relations between the transcription levels
(TL), i.e., number of mRNA molecules of a certain type per cell, and transcription rates (TR), i.e.,
number of mRNA molecules produced in the cell per unit of time. TLs are the quantities directly derived from microarray experiments, whereas TRs are usually unobservable. Although both of these
quantities seem to be legitimate indicators for characterizing gene activity, yet generally they are different and capture different facets of the regulatory mechanism. Due to the fundamentally nonlinear
nature of gene-to-gene interactions, there cannot be any direct relations between gene-specific TRs
and TLs. Also, due to the inherent instability of high dimensional regulatory systems, nothing like
time-independent “gene activity” may be attributed to a living cell. In our view, these conclusions
may have serious consequences for the interpretation of microarray experiments where the fluctuating
nature of the mRNA levels is frequently ignored, the mRNA abundance is often seen as a direct indicator of the corresponding gene’s activity, and the differential expression (i.e., difference in TLs) is
taken as evidence of differences in the cells themselves.
2 Assumptions and Equations
The system of nonlinear ordinary differential equations (ODE) for description of proteometranscriptome dynamics first appeared in [CHC99].
dr
dt
= F(p) - βr ;
dp
dt
= γr - δp
(1)
p are n -dimensional column vectors of mRNA and protein concentrations measured
in numbers of copies per cell; n is the number of genes in genome; β , γ and δ are the non-
where r and
degenerate diagonal matrices corresponding to the rates of production and degradation in transcription
and translation. The n -dimensional vector-function F is a strongly nonlinear function representing
the mechanism of transcription. [CHC99] linearized the system (1) in the vicinity of a certain hypothesized initial point and formulated general requirements of stability. In what follows, we augment
the system (1) by an explicitly specified model for F(p) and attempt to extract the consequences
from the essentially nonlinear nature of the problem.
As is known from the biology of gene expression, generation of each copy of messenger RNA is preceded by a complex sequence of events in which a large number of proteins bind to the gene's regulatory sites and assemble a reading mechanism known as RNA polymerase [KMP03]. Each act of these
bindings represents a separate bio-chemical reaction involving DNA and proteins, and supported by a
number of smaller molecules. According to the principles of chemical kinetics, the production term
F(p) should have the following general form [DeJ02]
Li
n
k =1
m =1
Fi ( p1 ,..., pn ) = ∑ ωik ∏ pm
rikm
(2)
where Li is the number of concurrent bio-chemical reactions for decoding
rate constants;
rikm
i -th gene; ωik
are the
are the kinetic orders showing how many protein molecules of type m partici-
pate in k -th bio-chemical reaction for the transcription of i -th gene. A detailed account of the assumptions underlying (2) may be found in [SSa87]. Although these assumptions are not free from inevitable simplifications, yet they constitute a reasonably solid basis for studying the dynamics of
genetic regulatory networks because they recognize the central role of RNA polymerase in the
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nonlinear mechanism of gene-to-gene interactions. The rate constants,
ωik , and kinetic orders, rikm ,
are assumed to be positive real and integer numbers, respectively. Due to the very nature of chemical
thermodynamics, each chemical reaction is a trial-and-error process in which fixed values of molecular constants may be attributed only to big ensembles of reactions, but not to the individual ones.
Therefore, we specify
ωik and rikm
as random numbers drawn from the gamma and Poisson popula-
tions, respectively
Pr(ω ik =x)=
x
α −1
exp(- x
Γ (α )θ
θ)
α
λ exp( −λ )
n
; Pr(rikm =n)=
n!
(3)
This choice of probabilistic characterization is a matter of mathematical convenience, and may be
easily replaced by other assumptions compatible with the nature of the problem. Assumptions (3)
provide the basis for ensemble averaging, and, as usual in statistical mechanics, allow for probabilistic statements when deterministic considerations are impossible due to complexity. Although small
networks are conceivable to treat deterministically (e.g. [MSa95]), the probabilistic approach seems
to be the only way of studying the networks with tens of thousands of regulatory units.
Naturally, a great variety of regulatory functions remains outside the scope of this model. In particular, we ignore such important processes as protein post-translational modifications. We also assume
that each protein molecule produced by the ribosome has a specific regulatory function within the cell
of origin and immediately becomes available everywhere within network. Even with these simplifications, mathematical analysis of such systems is not simple, and it does not seem possible to move toward more complex systems without understanding the simpler ones.
3 Outline of the Solution
Following standard procedures in nonlinear dynamics [Car81], we first seek the equilibrium points of
(1-2) and study structure of the solutions in their vicinity. Let P0 be the
n − vector of equilibrium
protein concentrations, and X (t ) be the vector of relative concentrations normalized by the equilibrium values. After some transformation, system (1-2) may be rewritten as
2
d xi
dt
2
+ (βi + δ i )
dxi
dt
Li
∑Ω Y
+ β iδ i xi = β iδ i
ik
ik
(4)
k =1
where Y are the so called S-functions [SVo87] defined as
n
log[Yik (X)]= ∑ rikm log( xm )
(5)
m=1
and
Ω ik = ω ik Yik ( P0 )
n
∑ω Y
ik
ik
( P0 )
m =1
We are looking for a stationary solution of (4-5). According to the center manifold theory [Per01],
this stationary solution may be sought as a linear combination of the eigenvectors corresponding to
the eigenvalues with zero real parts. Because we are dealing with a system containing tens of thousands of regulatory elements, the center manifold may contain up to hundreds of such eigenvalues
forming an essentially continuous spectrum of stationary oscillations. Oscillations with such a spectrum may be reasonably viewed as random processes. In order to find the stochastic structure of these
1612
processes, we apply successive approximation approach. We first notice that the right hand side in (5)
is the sum of identically distributed random functions. Because the first guess trajectories,
xm (t ) ,
correspond to different eigenvalues from the center manifold, we may assume that they are statistically independent, and therefore the random processes
ηik (t )= log(Yik ( X(t ))
are approximately Gaussian. Although difficult to prove theoretically in all its generality, this conjecture has been confirmed by direct numerical simulation under the assumptions of our model. Figure 1
shows an example of such simulation.
-1.0
-0.5
0.0
0.5
1.0
protein oscillations
0
20
40
60
80
100
80
100
time
-1.0
-0.5
0.0
0.5
1.0
transcription rate
0
20
40
60
time
Figure 1. Nonlinear transformation of linear combination of periodic oscillations.
In this Figure, top panel shows 100 separate quasi-periodic oscillations covering wide spectrum of periods. As shown in the bottom panel, corresponding functions Fi (t ) in (2) tend to concentrate around
the same stochastic process whose parameters are determined by the probabilistic properties of
ωik and rikm . This tendency increases with complexity of the network.
In the first guess approximation, all the processes ηik (t ) corresponding to different indexes
may be replaced by a single Ornstein-Ulenbeck process [Gar83] with autocovariance
i
and k
m
Rη (τ ) = ( λ + λ ) ∑ var[ln(σ k )] exp( − τ τ 0 )
2
2
k −1
The correlation radius,
τ0 ,
depends on the width of the center manifold spectrum and becomes
shorter with n increasing. Roughly speaking
τ0
is inversely proportional to the maximum absolute
value of the characteristic roots within the center manifold. Unfortunately, currently known theoretical bounds for eigenvalues of a matrix are too conservative to be used in practice [Bro39]; however,
1613
τ0
can be easily estimated computationally through fitting
η (t )
by the first order (i.e., Markov)
process.
The system (4) is now decoupled on the set of independent equations containing the same "random
force"
d 2 xi
dt
2
+ (βi + δ i )
dxi
dt
+ βiδ i xi = β iδ i eη ( t )
(6)
The process ξ (t ) = exp(η (t )) is log-normally distributed with expectation exp(σ
variance
⎛ τ
2 ) and autoco-
⎞
⎟ (7)
⎝ τ 0 1 − exp( −σ ) ⎠
σ
Rξ (τ )= exp(σ )[exp(σ )-1] exp ⎜ 2
2
2
2
2
where
m
σ = (λ + λ ) ∑ var[ln( xk )]
2
2
(8)
k −1
It is important to note that the correlation radius of ξ (t ) is always smaller than
τ 0 , which means
that ξ (t ) is always closer to a white noise than η (t ) . Applying Fourier transform, equations (6) may
be easily solved, and the solutions are the stochastic processes with expectations
E ( xi ) = β iδ i exp(σ
2
2) ,
variances
var( xi ) =
β iδ i
βi + δi
τ0
[(exp(σ
2
σ
2) -1]
2
2
,
(9)
and autocorrelation function
Rx (τ ) = Ai e
i
−τ τ 0
+ Bi e
− β iτ
+ Δi e
− δ iτ
(explicit expressions for the coefficients are cumbersome and omitted to save page space). These ap-
β iτ 0 1 and δ iτ 0 1 , meaning that the spec-
proximations are valid under the conditions that all
trum of collective random force, ξ (t ) , should be wider than all the individual spectra in (6). We now
can plug the first guess stochastic processes back into expressions (5) and repeat the procedure. It is
worth mentioning that the center manifold spectrum remains the same throughout the iterative process
because it is solely determined by the solution in the vicinity of equilibrium.
4 Interrelations between nonlinearity, instability and complexity
Parameter λ in the Poisson distribution (3) is a natural measure of complexity of the system because
the quantity λ n is an average number of the proteins participating in the transcription. We now formally introduce the "index of complexity", I c =(λ + λ ) n . If this index was small then the vast ma2
jority of characteristic roots would be stable (i.e., have negative real parts) and the center manifold
spectrum would be narrow. Obviously, this is not the case in reality with usually I c ≈ 30 - 100
1614
[Lew04]. In the systems of such a great complexity, a substantial number of characteristic roots will
reside in the right half of the complex plane thus signifying a greater instability. It is seen in Figures
2-3 where we show two examples of the distribution of characteristic roots over the complex plane for
small and big I c .
n=300; poisson lambda=0.05; spectral width=2.21
complexity index=15.75; stability index=3.47
6
4
imaginary parts
2
0
-2
-4
-6
-6
-4
-2
0
2
4
6
real parts
Figure 2 Positions of characteristic roots in case of low complexity
0
-6
-4
-2
imaginary parts
2
4
6
n=300; poisson lambda=0.5; spectral width=4.15
complexity index=225; stability index=1.93
-6
-4
-2
0
2
4
6
real parts
Figure 3 Positions of characteristic roots in case of high complexity
We also define the "index of stability",
I s , as a ratio of number of roots with negative real parts to
that with positive ones. With complexity increasing, the stability decreasing, spectral width of the
central manifold is increasing, thus making the correlation radius,
τ 0 , smaller and the spectrum of
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collective "random force", ξ (t ) , "whiter." Effectively, that means that the more complex the system
the more favorable are the conditions for applying the proposed approach. Figure 4, left panel, shows
the dependence of stability on complexity. The right panel illustrates the fact that the correlation radius of ξ (t ) (open circles) is always substantially smaller than that of η (t ) (solid circles) and both
Ic
drastically decrease with
increasing.
correlation radii
12
4
0.6
6
0.8
8
10
corr.rad
1.2
1.0
log(stbl.ind)
1.4
14
1.6
stability vs complexity
0
20
40
cmplx.ind
60
0
20
40
60
cmplx.ind
Figure 4 Stability and correlation radii vs. complexity of network
5 Interrelations between transcription levels and transcription rates
In the model adopted here, the entire gene expression mechanism is seen as driven by a collective
random force which in turn is generated by all the individual transcription-translation events. This
kind of "self-consistent" or "average field" approach is widely employed in physics, with such notable
examples as the Thomas-Fermi equation in atomic physics [PYa89] and Landau-Vlasov equations in
physics of plasma [Che84], to name just a few. Transcript levels (TL) and transcription rates (TR) are
presented by the quantities
ri
and Fi in (1), respectively. Since Fi are the stochastic processes gen-
erated, generally speaking, by the entire network, there is no noticeable correlation between them and
ri . Therefore, one cannot expect any substantial similarity between the temporal behavior of TR and
TL. This conclusion is of fundamental importance for the interpretation of microarray experiments.
Also, despite the fact that in our model each mRNA molecule entering the ribosome translates into
exactly one protein, there is no similarity between temporal behavior of protein and mRNA concentrations. The dissimilarities increase with the regulatory network complexity because of a longer
chain of intermediate events being involved in each act of gene expression. Figure 5 depicts the median correlation coefficient as a function of complexity. As seen from this Figure, in the case of high
complexity half of all the protein-mRNA pairs are correlated at the level below 0.5. This level of correlation is close to that observed in the [GAP04] experiment where about half of
~5500 TLs in the yeast genome are found not correlated with corresponding TRs. Based on this comparison, we may conclude that complexity of the yeast genome is about 45 ÷ 60
0.7
0.6
0.5
median cor.coef
0.8
1616
0
20
40
60
complexity index
Figure 5. Median correlation coefficients vs complexity
6. Interrelations between limits of complexity and total variability
Equation (9) expresses the individual variances of protein oscillations through the individual degradation rates and total variance, which in turn, due to (8), is a linear combination of variances of
logarithms of individual conentrations. It allows for the derivation of a nonlinear algebraic equation
for so far unknown variance σ
2
σ = Ic
2
1
n
n
∑
⎧
ln ⎨1 + 2τ 0
⎩
i =1
β iδ i cosh σ 2 − 1 ⎫
βi + δ i
σ
⎬ (10)
⎭
2
In a sense, solution of the original strongly nonlinear problem is now reduced to the solution of
this equation. Simple analysis shows that the solution does exist and is unique if
T0 >I cτ 0
;
−1
T0 =
⎡ 1 n β iδ i ⎤
⎢n ∑ β +δ ⎥
⎣ i =1 i i ⎦
(11)
−1
where parameter T0 characterizes a network-wide average degradation rate. The inequality above,
rewritten as I c <T0
τ 0 , tells us that in a regulatory network with n units there exists an upper limit
of complexity determined by two global parameters, i.e., by the overall rate of degradation and spectral width of the center manifold. If all the parameters reside within the limits required by (11), then
equation (10) may be easily solved numerically using a well known delta-method [Oeh92]. Here,
again, we take advantage of the system being asymptotically large. Namely, suppose that we evaluate
the S N = N
−1
∑
N
i =1
f ( zi ) , where { zi } are iid random values drawn from the population with PDF
P ( z ) . Using the delta-method, we obtain E(SN ) = f ( E ( z )) + 0.5σ z f ( E ( z )) . Assuming, in ac2
//
cordance with (10), that z= βδ ( β + δ ) , and, in accordance with (3), that β and
δ are gamma-
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distributed, we easily find that E(z)= θα
2
(2α + 1) . Substituting this expression into (10) we end
up with a simple transcendental algebraic equation with respect to σ
2
(not shown here.)
The total variance σ is the global measure of intensity of fluctuations within the regulatory network and is somewhat analogous to temperature in statistical mechanics. This “network temperature”
2
is a function of three global parameters, i.e.,
I c , τ 0 , and T0 . It is of great interest to establish asymp-
totic behavior of the network temperature as a function of the size of network. Preliminary analysis
shows that depending on the structure of the center manifold generated by the matrix Ωik in (4), the
network temperature may increase, may remain relatively constant, or may even decrease. The main
obstacle to better understanding the asymptotic behavior of σ is the lack of realistic knowledge regarding the bounds of characteristic roots of matrices, in general, and spectral radii of center mani2
folds (and therefore of
τ 0 ). The last of the three scenarios mentioned above , if takes place in reality,
would help to understand how a very complex network, although being inherently unstable, is nevertheless able to maintain the regulatory regime with comparatively low variability. This kind of behavior could provide some insight into the nature of “functional determinism” of a high-dimensional dynamical system arising from nearly chaotic behavior of all its regulatory subunits. At this point, this
intriguing question is open for further investigation.
7 Cautionary Notes Regarding Microarray Data Interpretation
There exist two sets of legitimate quantitative parameters which characterize “gene activity”, i.e.,
transcription level and transcription rate. Microarray experiments provide us with mRNA abundances,
i.e., transcription levels. What we would rather like to know are the mRNA transcription rates, or
number of mRNA copies produced per unit of time. This quantity, if available, would be a more direct measure of gene activity. The distinction between the TL and TR has been repeatedly highlighted
in literature [WCB02], however it seems to remain largely ignored by microarray community. As
shown above, in a complex regulatory network, transcription level, generally speaking, is a poor predictor for transcription rates. It is often tacitly assumed in the interpretation of microarray data that
there exists some kind of equilibrium between production and degradation of mRNA for each gene
separately, in which case a direct proportionality would exist between TLs and TRs. Global equilibrium, however, is not possible in a complex network of interacting units with many channels of coregulation. In order to judge which TRs and TLs are in equilibrium and which are not, a detailed information about the timing of corresponding bio-chemical reactions would be required. This kind of
information is usually unavailable from the microarray experiments, unless it is a time-course experiment with sufficiently high rate of sampling.
Another important implication of nonlinearity and complexity of regulatory network is that a living cell cannot reside in a global state of equilibrium, simply because such state cannot be stable. Stochastic oscillatory behavior is in the very nature of the regulatory process. Figuratively speaking, the
cell should continuously depart from the point of equilibrium in order to activate the mechanism of
returning back. A usual way of thinking in microarray data interpretation is to attribute the differences
in mRNA abundances to the cells themselves. However, depending on the frequency of sampling and
duration of the sample preparation, the cell can be arrested in different phases of its oscillatory cycle,
thus mimicking the differential expression. That means that covariances of expression profiles may be
quite different in different time scales. These covariances, usually obtained through cluster analysis or
1618
classification, are often used as a basis for the pathway analysis. However, if the temporal dynamics
of the regulatory processes is ignored, this analysis may produce misleading results.
Many statistical procedures in microarray data analysis, especially in the context of disease biomarker discovery, include the notion that only a small subset of all genes participate in the disease
process and therefore are actually differentially expressed, while a vast majority of genes are not involved in this process and "do business as usual." Contrary to this notion, it is quite possible that rapidly fluctuating components of the regulatory network are the integral parts of the process as a whole,
and their high-frequency variations manifest the preparatory work of supplying the mRNAs for
slower processes with bigger amplitudes.
8 Discussion
The model formalized in equations (1-3) possesses rich gamut of features capable of simulating
the properties of living cells. We briefly discuss some of them here.
Formally speaking, equations (1-3) are written for the entire genome, and therefore, as shown in
[Lws91], there is only one global point of equilibrium. However, if random sets of
rikm
and
ωik
are
clustered into a number of comparatively independent subsets, then the entire system (1) is also decomposed into comparatively independent subsystems, with their own equilibria. In this case, it
would be reasonable to expect that the system may switch between different equilibria and produce
different oscillatory repertoires. The concept of differentiation, i.e., the ability of living cells to perform different functions despite the fact that they have basically identical molecular structures, has
been extensively discussed within a number of previously proposed regulatory models [DeJ02]. The
model proposed here has the capability of mimicking the cell differentiation as well. Results of extensive simulation of “tunneling” between different oscillatory repertoires will be published elsewhere.
Regulatory mechanisms in living systems are highly redundant and able to maintain their functionality even when a number of regulatory elements are “knocked out.” In the model proposed
herein, all the individual transcription-translation subunits are driven by the “collective” random force
whose stochastic structure is basically determined by the spectrum of center manifold. Because this
spectrum is generated by a large number of individual processes, it follows that if a certain number of
genes is “knocked out”, then the majority of the remaining genes will not generally change their behavior. Due to the same reasons, the model suggested here has wide basins of attractions [Wue99],
i.e., low sensitivity to initial conditions. This property is considered desirable for any formal scheme
in models of living systems.
In this work, the S-system has been selected to represent nonlinear interactions within genetic
regulatory networks for two reasons. First, the S-system originates from (and adequately represents
the dynamics of) bio-chemical reactions, a material basis of all the intra-cellular processes. Second,
the S-system is known to be a universal approximator, i.e., to have the capability of representing a
wide range of nonlinear functions under mild restrictions on their regularity and differentiability
[Voi98]. However, the S-approximation is in no way unique in this sense. Sometimes it would seem
desirable to maintain a more general view on the nonlinear structure, such as provided by the artificial
neural networks (ANN), for example. Our numerical experiments show that a properly constructed
ANN retains many of the same features as the S-functions. In fact, the only requirement which is necessary to fulfill when selecting a nonlinear model is that it has the “mixing” capability, i.e., provides
strong interaction between normal oscillatory modes resulting in stochastic-like behavior of F(p) .
It is not our goal in this paper to present a sort of “theory of everything”, however, it is worth noting that the system described by equations (1-3) allows for many extension and generalizations, retaining an advantage of being compact and mathematically tractable. In particular, any bio-chemical
1619
network, alone or in conjunction with the genetic regulatory network, may be considered in a similar
manner.
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