Noisy Attractors and Ergodic Sets in Models

of Genetic Regulatory Networks

Andre S. Ribeiro ∗

Institute for Biocomplexity and Informatics, Univ. of Calgary, Canada

Department of Physics and Astronomy, Univ. of Calgary, Canada

Center for Computational Physics, Univ. of Coimbra, P-3004-516 Coimbra,

Portugal

Stuart A. Kauffman

Institute for Biocomplexity and Informatics, Univ. of Calgary, Canada

Department of Physics and Astronomy, University of Calgary, Canada

Abstract

We investigate the hypothesis that cell types are attractors. This hypothesis was

criticized with the fact that real gene networks are noisy systems and thus, do not

have attractors (Kadanoff et al, 2002). Given the concept of “ergodic set” as a set

of states from which the system, once entering, does not leave when subject to

internal noise, first, using the Boolean network model, we show that if all nodes of

states on attractors are subject to internal state change with a probability p due to

noise, multiple ergodic sets are very unlikely. Thereafter, we show that if a fraction

of those nodes are “locked” (not subject to state fluctuations caused by internal

noise), multiple ergodic sets emerge.

Finally, we present an example of a gene network, modelled with a realistic model

Preprint submitted to Elsevier 13 April 2007

of transcription and translation and gene-gene interaction, driven by a Stochastic

Simulation Algorithm with multiple time-delayed reactions, which has internal noise

and that we also subject to external perturbations. We show that, in this case, two

distinct ergodic sets exist and are stable within a wide range of parameters variations

and, to some extent, to external perturbations.

Key words: Gene Regulatory Networks, Boolean Networks, Stochastic Simulation

Algorithm, Attractor, Cell Type.

PACS: 82.39.k, 87.16.Yc, 82.20.Fd, 05.45.a

1 Introduction

Understanding the integrated behavior of genetic regulatory networks (GRN),

taken in the large sense to comprise genes, RNA, proteins and other molecules

that mutually interact to control the dynamical behavior of the network within

and between cells, has emerged as a fundamental problem in Systems Biol-

ogy. At this stage we only have partial knowledge of the regulatory network

structure and “logic” driving the dynamical behavior of these systems. Nev-

ertheless, we can begin to address questions about these systems using the

known features of these networks, and constructing the family or ensemble,

of all networks consistent with those observations. This “ensemble” approach

(Kauffman, 2004) studies the expected properties of members of the ensemble

and predicts new observables to test against the dynamical behavior of cells

and tissues. It is a further profound issue whether real networks are generic

? The authors would like to thank iCORE, a funding agency of Alberta, Canada.∗ Corresponding author. Phone: + 403 220 24 25.

Email address: ARibeiro@ucalgary.ca (Andre S. Ribeiro).URL: http://www.ucalgary.ca/ aribeiro/index.html (Andre S. Ribeiro).

2

to any ensemble, given 3 billion years of evolution and natural selection.

From the mathematical point of view, there are three broad frameworks in

which to cast the analysis of such networks. At the most precise level one

considers the chemical master equation of the detailed behavior of all compo-

nents in members of some ensemble of networks. Such models are inherently

stochastic. Understanding the consequences of such noise is emerging itself as

a critical problem and is the focus of this article.

At a second level of abstraction, one considers systems of deterministic non-

linear differential equations capturing, in some sense, the mean field behavior

of the real noisy stochastic networks. Since the number of copies of regulatory

molecules in the real system can be one to a few, such deterministic equations

are, at best, an approximation. One approach in this framework is to add

white noise in Langevin equations (Toulouse et al, 2005). However, it remains

to be shown that this modelling strategy captures the real character of cellular

dynamical noise.

At a still higher level of abstraction, one can consider model GRN’s such that

genes states, time and other components are discrete variables. While furthest

from the chemical master equation description, hence the most abstract, such

models allow studying very large networks, with thousands of model genes

or other components. In particular, random Boolean networks (RBN) have

been the subject of considerable analysis (Kauffman, 1993, 1969). The links

between the behaviors of RBN and chemical master equation models of “the

same” network were recently analyzed (Zhu et al, 2006), and some parallels

were found. Also, it is known that several properties of RBNs generalize to

a class of piecewise linear differential equations (Glass, 1975). In particular,

3

a well-established transition from ordered to chaotic dynamics at a critical

phase transition occurs in both systems.

A generic property of many deterministic nonlinear dynamical systems is that

a typical member of the ensemble has a multiplicity of dynamical attractors

such as steady states, limit cycles or strange attractors, each of which drains

a basin of attraction consisting of all states lying on trajectories that flow to

or include that attractor.

At the molecular dynamical level, we do not know what a cell type is. However,

if we consider even the simplest case of a binary valued Boolean network with

30,000 genes, its state space is 230,000. It takes seconds to minutes for genes to

turn on an off. There have been about 1017 seconds since the big bang. Thus,

whatever a cell type may be, it must be a very restricted subset of the states

possible in the GRN.

In the deterministic framework, it is almost an inevitable hypothesis that cell

types correspond to attractors in the dynamics of the network. Such attractors,

in the absence of noise, are the asymptotic behaviors of the network. To be

biologically plausible, such attractors need to be constrained by their dynamics

to very small regions of the state space of the system. Importantly, RBNs in

the ordered regime and their piecewise linear cousins have this property: their

attractors are tiny subsets of state space. In the chaotic regime this is not

true. This suggests that cells may be in the ordered regime.

In this picture, if a cell type is an attractor, then a pathway of differentiation

can be only two things: first, a perturbation from one attractor into a new

basin of attraction from which the cell passes via a trajectory to the new

attractor cell type. Here the perturbation can be a noise fluctuation, or an

4

exogenous signal. Secondly, it is possible that real GRNs have variables that

change dynamically over a wide range of time scales. Then the cell types would

correspond to the attractors of the fast dynamics, and differentiation would

occur due to bifurcations in the fast dynamics as the slow variables change,

such as morphogenesis. These two possibilities do not exclude one another.

However, an important criticism (Kadanoff et al, 2002) with respect to Boolean

networks is that noise may render such attractors a poor model of cell types

since closure of an attractor (a state cycle) in the discrete dynamics is delicate.

This is an important criticism and leads to the question of the effect of noise

on the attractor as the cell type hypothesis.

Some previous work was made in a first attempt to address this problem.

In (Klemm and Bornholdt, 2005), Klemm and Bornholdt tested the stability

of attractors with respect to infinitesimal deviations from synchronous up-

date and found that most attractors are artifacts arising from synchronous

clocking. Importantly, the remaining attractors are stable against fluctuating

delays, and its average number grows with the number of nodes, within the

numerically tractable range. A similar scaling law as been observed in a dif-

ferent approach to asynchronous Boolean networks (Greil and Drossel, 2005).

The two works confirm that their models have multiple attractors assuming

asynchronous updating. Yet, these works do not assume or model any prob-

ability of genes “misbehaving”, i.e., act contrary to what inputs states and

Boolean transfer function determines. Only the time at which nodes update

is assumed stochastic.

We here report an analysis of the effects of minimal noise (understood has

genes misbehaving with a certain probability) in the attractors of RBNs with

5

synchronous updating. We also do a similar study using more realistic models

of GRNs.

In the next section, we introduce RBN as models of GRN. The third section

gives our method of analysis of ergodic sets. Next, we introduce a more realistic

model of GRNs recently proposed (Ribeiro et al, 2006), based on the stochastic

simulation algorithm (SSA) (Gillespie, 1977, 1976) that models transcription

and translation as multiple time delayed events (Roussel and Zhu, 2006). Using

this modelling strategy we show an example of such a network that indeed

exhibits two ergodic sets. In the fifth section we present our results followed

by the discussion.

2 Noisy Random Boolean Networks

An RBN consists of N nodes, each representing a gene (Kauffman, 1969) or

other variable, where the nodes can take binary values of “on” and “off”,

and each node receives inputs from among them. Each gene is assigned a

Boolean function from the set of possible Boolean functions of K variables.

Time is discrete and here we consider that all genes update their activities

synchronously. Thus, a state of the network passes to a unique successor state

at each moment. Over time, the system follows a trajectory that ends on a

state cycle attractor. In general, the network has many such attractors.

There are several ways to introduce noise in RBN dynamics. In the Boolean

modelling strategy, we introduce noise by allowing, at each time step t, that

with a small probability, P , any node misbehaves its Boolean rule and assumes

the opposite value to that specified.

6

Although we assume this as “internal noise”, using this noise model, external

and internal noise are not distinguishable since they both consist in “bit flip-

ping” a node state at a given t. Also, there is a very small probability, PN ,

that all nodes misbehave at a given state transition. Given this, the system

is a Markov process with a hierarchy of time scales for single and multiple

gene mistakes. Here, we limit the system to single gene mistakes, which is an

approximation to long-term behavior in the case of very low P. I.e, we assume

that multiple mistakes occur on a very long time scale and can be ignored.

Since we explore all attractors, and misbehaviors on transient states would,

at most, change which attractor the system falls into, we assume the system

falls on a attractor (state cycle) before noise can affect its dynamics. We then

consider all the states of each state cycle of the network. Each of these states

of the state cycle can go to N other states, by doing a single gene state flip.

We start the network on each of the states one can obtain by doing this bit

flip, for each gene of each state of each state cycle, and record numerically the

state cycle to which the system flows.

To analyze such systems, it is essential to explore the entire state space, such

that all state cycle attractors of the network are known. This limits our analysis

to small networks (N < 20).

Sometimes, perturbing a gene in a state of a state cycle leads the system to

fall into another state cycle. That fact made us use the notion of ergodic sets,

which we present in the following section.

7

3 Ergodic Sets

Here we use the standard definition of ergodic set which can be found in

the context of stochastic processes. Define an ergodic set as a closed set of

state cycles such that each state cycle can be reached by a single or more

genes mistakes caused by internal noise from any other state cycle of the

same ergodic set. Also, the ergodic set is such that “state fluctuations due to

internal noise” (here, single bit flips in the Boolean framework and stochastic

fluctuations in the “delayed SSA” framework) are not sufficient to make the

system leave the ergodic set.

Here we assume that, in the Boolean framework, the probability P of a gene

“misbehaving” (do the opposite of what its Boolean rule and inputs genes

states demand) is small enough such that after a noise fluctuation the system

dynamics progresses fast enough for the system to return to a state cycle,

before another noise induced change occurs again.

In the stochastic formulation no such assumption is necessary, because it is

not a synchronous network and the system dynamics is inherently stochastic.

As we show in the results section, the coupled gene network indeed “responds”

to stochastic variations of genes expression levels faster than the time these

need to accumulate a sufficient number of proteins to impose a state change.

Also, in this framework we show examples of, when the system is in an ergodic

set, it also is resistent to some extent to external perturbations, although we

do not impose this as a necessary condition to be considered an ergodic set.

Above we noted that, in the deterministic setting, it was almost an inevitable

8

hypothesis that a cell type corresponds to an attractor. If noise is included,

a first possibility is that a cell type corresponds to a noisy attractor, while a

second possibility is that a cell type corresponds to an ergodic set. One case

does not exclude the other. But, if the second case is true, then in order to

have multiple cell types, the system must have multiple ergodic sets. Here we

examine this possibility. For this purpose we now describe how to determine

ergodic sets in the RBN framework.

We generated RBN of random and scale free topologies (scale free topologies

are generated by the algorithm “scale free 1” in (Airoldi and Carley, 2005),

which creates directed scale free graphs according to the originally proposed

algorithm in (Barabasi, 2002)). Given the inputs distribution of each node,

a random Boolean function is set for all possible input values. A complete

“path of states” matrix is generated, representing to which state any state

will lead, in the system deterministic evolution. Also, initializing the system

in all possible states, a cycle is found for each case. Since it is a deterministic

RBN, all cycles are found using this method.

We now determine which state cycles can reach other state cycles in the limit

of small P , where only a single node can be perturbed at a time. We store

such information in an adjacency matrix of state cycles.

Having the set of state cycles, we perturb each gene of each state belonging

to a cycle, one at a time, and find to which cycle the system dynamics leads.

In general, if perturbing a gene of a state of cycle i, leads the system state to

cycle j, then the position [i,j] of the adjacency matrix of state cycles (initialized

with all zeros) takes the value 1.

Given the adjacency matrix of state cycles, we now check if there is any path,

9

from any state of the merged sets of state cycles, which would allow leaving

to a state not belonging to that set. If this occurs, the set is not an ergodic

set. The remaining sets are the ergodic sets of the system.

As an example, suppose a system with state cycles represented by letters A,

B, C, D and E, and that according to the adjacency matrix of state cycles,

there is a pathway from A to B, from B to C, from C to B, from C to D, and

from D to C. Additionally, each state cycle has a pathway leaving from it onto

itself.

We represent this in the form of a directed graph in Fig. 1, from where one sees

that the possible pathways between state cycles, due to small perturbations,

result in a graph such that the system always end up on ergodic sets (B, C,

D) or (E), and once entering them, cannot leave.

In the stochastic modelling strategy (described in the following section), the

analog is defined with the use of the K-means clustering algorithm (MacQueen,

1967), which we use to classify the products of gene expression quantity over

time (proteins time series), as “0” or “1”.

The algorithm works as follows. Let Kcluster be the number of clusters one

wants to cluster the data points into: i) Place Kcluster points into the space

represented by the objects that are being clustered. These points are used as

initial group centroids, ii) assign each object to the group that has the closest

centroid, iii) when all objects have been assigned, recalculate the positions

of the Kcluster centroids, iv) repeat steps 2 and 3 until the centroids do not

change. This separates the objects into groups from which the metric to be

minimized can be calculated. The function to be minimized here is (eq. 1):

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Fig. 1. A directed graph representing the possible pathways between state cycles

A, B, C, D and E of an imaginary RBN with 2 ergodic sets. In this RBN, once

entering the ergodic set of state cycles (B,C,D) or the ergodic set of state cycle E,

the system cannot leave them.

J =Kcluster∑

j = 1

n∑

i=1

∥∥∥x(j)i − cj

∥∥∥2

(1)

The quantity∥∥∥x(j)

i − cj

∥∥∥ is the distance measure between each of the n data

points, x(j)i , and their respective cluster centers value, cj, at each step of the

algorithm. Since we intend to binarize the time series of the number of proteins

of each gene, we set Kcluster = 2, while n equals the number of data points

of the proteins time series, and is equal to the ratio between the time series

total length and the sampling period. The variables x(j)i are the values of the

data points which in our case is the number of proteins in each sample of the

system state, and the variables cj are the data points chosen as centroids at

each step of the algorithm. We note that the temporal order of the proteins

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levels is not taken in account for the clustering procedure.

Because we here want to binarize the proteins expression level, choosing how

many clusters one wants is not a problem (generally, this choice might be prob-

lematic when using K-means). However, the results of K-means also depend on

the initial choice for centroids. We used to standard approach to minimize this

problem. Since we choose the initial centroids values randomly from the set

of data points, we make 1000 independent runs of the K-means algorithm to

each time series and choose the solution that minimizes the distance measure.

Given the time series of the number of the proteins (data points) of a gene

of the GRN, the algorithm begins by choosing randomly two data points,

and uses them as initial centroids. Then, it computes the distance between

each data point and the centroids (using eq. 1), and places the data point

(independently of the moment in time to which the data point corresponds

to) into the cluster that has the smaller distance between the data point value

and the centroid value.

After the first clustering step, the algorithm computes the mean value of the

data points in each cluster. These mean values are used as centroids in the

next iteration. Once the new centroids values are computed, again all data

points are placed in one of the two clusters, and again new centroids are

computed. When the centroids value is constant from one step to the next of

the algorithm, the process stops MacQueen (1967).

Using this clustering algorithm is adequate in our case since the GRN in the

stochastic framework, used as example, is such that the proteins levels of all

genes have two very distinct levels (“high” and “low”), and when in each of

these states, only vary due to stochastic fluctuations. Therefore, setting Kcluster

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to 2, one can distinguish clearly if a gene is “on” or “off” at a given moment.

E.g., if one had a gene whose protein quantity could be in three distinct levels,

Kcluster should be set at 3, and binarizing this time series would not provide

clear results. In general, the time series characteristics determine what is the

most appropriate clustering method to use in each case.

Using the K-means algorithm also allows determining in our examples, what

should be considered a “perturbation”. Namely, here we consider that a per-

turbation in a protein concentration must be such that, given the threshold

that defines if a state of a gene is “on” or “off” determined by the K-means

algorithm, the variation of the number of proteins externally imposed must

be sufficient to cross that threshold.

4 A model of a GRN dynamically driven by a multiple-delays SSA

A gene network modelling strategy was proposed (Ribeiro et al, 2006) that

models GRNs by coupling genes via protein-protein interactions and protein-

operator sites interactions. Also, it models transcription and translation as

multiple time delayed events. Its dynamics is driven by the “delay SSA”

(Roussel and Zhu, 2006), that consists of a modification of the original SSA

(Gillespie, 1977, 1976) and uses a waiting list to store delayed output events.

The waitlist consists of a list of elements (e.g., proteins being produced and

occupied promoter regions), each to be released after a certain time interval

(such time duration is also stored on the waitlist). The algorithm proceeds as

follows (Roussel and Zhu, 2006):

1) Set t ← 0, tstop ← stop time, read initial number of molecules and reactions,

13

create empty waiting list L for delayed generating events.

2) Do an SSA step for reacting events to get next reacting event R1 and

corresponding occurrence time t + t1.

3) Compare t1 with least time in L, tmin. If t1 < tmin or L is empty, set

t ← t + t1. Update number of molecules by performing R1, adding delayed

products (if existing) and the time delay they have to stay from appropriate

distribution into L, as necessary.

4) If L is not empty and if t1 ≥ tmin, set t ← t + tmin. Update number of

molecules and L, by releasing the first element in L, else go to step 5.

5) If t < tstop, go to step 2, else stop.

In these networks we represent a gene by its promoter (Proi) occupancy state

(available to transcribe or occupied due to binding to some molecule). The sys-

tem “state” is defined at a certain moment t by the number of the proteins (pi)

of each gene present in the system. In general, the level of expression is a func-

tion of number of RNA polymerase (RNAp ) available, promoter time delay,

protein production time delay, and rate constant of transcription/translation

reaction. The way to use this modelling strategy to generate ensembles of net-

works was proposed in (Ribeiro et al, 2006). Here, for simplicity, we model

transcription and translation as single step multi-delayed reactions (Roussel

and Zhu, 2006; Ribeiro et al, 2006).

Using this model we here build a specific gene circuit that exhibits two ergodic

sets that can be reached from the same initial condition. Once reached they

are stable to all sorts of internal noise (and external perturbations of single

genes states).

14

The gene circuit here used was built with the intent of showing that a gene

network with a realistic model of noise can in fact exhibit multiple ergodic

sets, via the coupling of two circuits which, if isolated, would toggle.

One sub-circuit is a toggle switch (Gardner et al, 2000) without cooperative

binding (a modified version of the model presented in (Lipshtat et al, 2006)

that here includes time delays from transcription and translation). With in-

ternal noise present, i.e., if the dynamics is driven by the delayed SSA, the

system can toggle between two states (either gene A is on and B is off, or the

opposite).

The other sub-circuit is a 4 genes repressilator. In a deterministic framework,

a 4 genes repressilator exhibits, like the toggle switch, two “attractors” (odd

genes on and even genes off, or the opposite). Again, using our stochastic

modelling strategy due to the values chosen for the several parameters, these

are not stable although toggles are less frequent than in the toggle switch.

The reason to choose these two bistable circuits is that it has long been posed

the hypothesis that differentiation is based on bistable mechanisms (Monod

and Jacob, 1961), and recent experiments support this hypothesis (Chang et

al, 2006).

The toggle switch is the best know example of a bistable circuit. Also, its

dynamical behavior is well known from experimental measurements (Gardner

et al, 2000). The 4 genes repressilator is similar to the toggle switch, toggling

between two states but at different average rates. Our goal is to show how one

can attain stability simply by coupling the two known sub-circuits which, by

themselves, do not settle on a single attractor.

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Due to their relevance, these bistable sub-circuits’ dynamical behavior (if not

coupled to one another) have been studied in the literature (mostly using the

continuous o.d.e.’s modelling strategy). See, e.g. (Muller et al, 2006; Li et al,

2006; Smith, 1987; Allwright, 1977; MacDonald, 1977; Zhu et al, 2006).

As shown in the next section, the stability of the system, i.e., the resistance to

internal noise and external perturbations arises from the coupling of the two

circuits. The 4 genes repressilator can be described by the following reactions:

RNAp + Proi0.001−→Proi(1) + RNAp(20) + pi(100) (2)

Proj+pi

0.1

À0.01

Projpi (3)

pi0.002−→ ∅ (4)

Proipj0.002−→ Proi (5)

In these reactions, N = 4, i = 1, ..., N and j = i + 1, except for i = N , where

j = 1. In reaction 2, a time delay τ is associated to each product X of the

reaction representing gene expression, using the notation: X(τ). Reaction 3

represents two independent reactions: binding and unbinding of the repressor

to the promoter. The rate constants of these two reactions, represented in

the numbers associated to the arrows, are not equal. The unbinding reaction

allows the repressor to disassociate from the promoter. The repressor can also

decay while on the promoter via reaction 5. This reaction is needed to allow

the protein to decay when bound to the promoter at the same rate as if not

bound. If this reaction was absent, binding to the promoter would act as a

“protection” against decay.

The toggle switch is described by the following reactions:

16

RNAp + ProA0.05−→ProA(2) + RNAp(20) + A(100) (6)

RNAp + ProB0.05−→ProB(2) + RNAp(20) + B(100) (7)

A0.001−→ ∅, B

0.001−→ ∅, P roBA0.001−→ ProB, P roAB

0.001−→ ProA (8)

ProB + A0.1

À0.001

ProBA (9)

ProA + B0.1

À0.001

ProAB (10)

Reactions 8 and 9 are representing two reactions each: the binding and unbind-

ing of the repressors to the genes promoter regions. We use the same values

for time delays in both sub circuits’ gene expression reactions. Different time

delays in the two sub-circuits would affect the transient to reach stability

(here used in a loose sense because its a noisy system), but have no effect

on long term stability. The circuits are then coupled. The coupling reactions

were made assuming that genes proA and pro4, and genes proB and pro3, have

very similar consensus sequences and therefore, share the same inputs. Thus,

the 4 genes repressilator proteins react with the toggle switch genes promoter

regions as follows:

ProA + p3

1

À0.001

ProAp3 (11)

ProB + p4

1

À0.001

ProBp4 (12)

ProBp40.001−→ ProB, P roAp3

0.001−→ ProA (13)

Reciprocally, the toggle switch proteins react with the repressilator genes pro-

moter regions as follows:

Pro3 + A1

À0.001

Pro3A (14)

17

Pro4 + B1

À0.001

Pro4B (15)

Pro3A0.001−→ Pro3, P ro4B

0.001−→ Pro4 (16)

A detailed justification of the values chosen for delays and number of RNAp

’s can be found in (Roussel and Zhu, 2006). The reactions rate constants were

tuned to attain toggling behavior for each network when uncoupled (toggle

switch and 4 genes repressilator), and stability when coupled (with 2 possible

ergodic sets). In the next section we present our results.

5 Results

We divide this section in two subsections: first, the results using the Boolean

networks modelling strategy, and afterward, using the gene networks driven

by the multi-delayed SSA.

5.1 Boolean networks

As described above, we test on synchronous Boolean networks, when subject

to noise, if these networks have 1 or more ergodic sets. We considered networks

with N = 6, 10 and 14 nodes.

For the RBNs we were able to examine exhaustively, if all genes are subject

to perturbations, generically only a single ergodic set exists. We ran 100.000

independent experiments for a 10 nodes, random topology with average con-

nectivity of 2 and with unbiased random Boolean functions. Not a single case of

multiple ergodic sets was found. We also ran, for 6, 10 and 14 genes networks,

for k equal to 1,2,3,4, doing 1000 runs for each case (1000 distinct randomly

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generated networks). Again, in these 12000 runs, no case of multiple ergodic

sets was found. We tested the same in scale free inputs, outputs and inputs-

outputs distributions with γ = −2,−2.5, generating randomly 1000 networks

for each case. Again, only single ergodic sets were found.

Nevertheless, in figures 2 and 3, we show two Boolean networks that do have

multiple ergodic sets even when subject to all possible 1 bit perturbations once

in the state cycles, hence showing that such systems exist. Notice, in Fig. 3,

that one of the ergodic sets consists of two states.

This leads to the conclusion that RBNs with multiple ergodic sets, even for

this low level of noise, appear to be improbable. Yet, notice that, e.g., for a

10 nodes RBN of average connectivity 2, there are 2210distinct networks as

for the wiring diagram and the Boolean functions distributions, thus, it is not

possible from our set of experiments to conclude that the conditions for the

existence of multiple ergodic sets are very rare, since we only explored a very

small sample of the state space, and networks with multiple ergodic sets are

not necessarily homogeneously spread in that state space.

We then asked ourselves the following question: Could the network have sev-

eral ergodic sets if a few genes were not subject to noise? Cells, do in fact

possess several mechanisms by which they control gene expression, usually by

“shutting off” genes. For example, its is known that in eucaryotes, DNA forms

chromatin which serves as a mechanism to control gene expression (Holde,

1989).

We tested whether the mean number of ergodic sets was greater than one

when only a single gene is perturbed for random k = 1, 2, 3, 4 topologies.

We determined the average number of ergodic sets, for random and scale

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free topologies, by perturbing the activities from 1 to N genes (the ones to

be perturbed are, in all cases, randomly chosen). We tested this in random

topologies with k equal to 1, 2,3 and 4, and in all cases, if a small fraction of

the genes could not be perturbed, we found networks that possessed multiple

ergodic sets. As an example, in Fig. 4 we plot the average number of ergodic

sets, over 1000 networks, of networks with random topologies and k = 2.

The number of ergodic sets falls as the number of genes able to flip increases

toward N. Very similar results were obtained using scale free topologies of

inputs, outputs and both, for γ equal to -2.0 and -2.5 and N = 6, 10, 14. In

Fig. 5, we show the results for 14 genes networks and the 3 kinds of scale free

topologies with -2 slopes.

Given the results above, we went to search, using more realistic modelling

strategies of GRNs if, given internal noise (stochastic dynamics) and external

perturbations, a simple genetic network could be built, robust to both noise

sources.

5.2 A gene network driven by the multi-delayed SSA

A gene network was built, according to the reactions 2 to 16. The dynamics is

simulated according to the algorithm from (Roussel and Zhu, 2006) and here

described. In all cases, we started with the following initial concentrations:

RNAp = 1000, all promoters free and no proteins. Also, the results are all

plotted in graphs of number of molecules versus time in seconds.

The gene network connections diagram can be seen in Fig. 6. The lines ending

in small perpendicular lines represent repression reactions on that gene, by the

20

Fig. 2. Inputs diagram (top left) of a 3 genes Boolean network, Boolean functions

table (bottom left), and the resulting directed graph of all state transitions (top

right). The Boolean rules were chosen to obtain a system with two ergodic sets.

The state cycles are {(0,0,0)} and {(1,1,1)}. Perturbing, by bit flipping, any single

gene when in one of these 2 states, leads to a state of the same state cycle the system

was in. Thus, state cycles {(0,0,0)} and {(1,1,1)} are ergodic sets. The boxes are

drawn such that include only the states the system can go to, due to single bit

perturbations of the states of the ergodic sets.

gene from where the line starts. The toggle switch and the 4 genes repressilator

are driven by similar reactions apart from the different number of genes, rate

constants and delays values, as seen when comparing reactions 2 to 5 and 6

to 10.

Notice one cannot define precisely an attractor in a noisy system. We consider

21

Fig. 3. Inputs diagram (top left) of a 4 genes Boolean network, Boolean functions

table (bottom left), and the resulting directed graph of all state transitions (top

right). The Boolean rules were chosen to obtain a system with two ergodic sets. The

state cycles are {(0,0,0,0),(0,1,0,0)} and {(1,1,1,1)}. Perturbing, by bit flipping, any

single gene of all the states of both state cycle, leads to a state of the same state

cycle where the system was. Thus, state cycles {(0,0,0,0),(0,1,0,0)} and {(1,1,1,1)}are ergodic sets. The boxes are drawn so that they include only the states the system

can go to, due to single bit perturbations of the states of the ergodic sets.

here an attractor as a region of the state space where the system can stay for

a long time. The concept of ergodic set remains as previously stated: a region

of the state space that once entered, the system cannot leave if not affected

by external perturbations.

22

Fig. 4. Average number of ergodic sets, for k = 2 random topologies for 6,10 and

14 genes networks, allowing from 1 to N genes to flip. Each point represents the

average of 1000 networks.

We begin by observing the temporal behavior of the toggle switch and the 4

genes repressilator, when independent from one another (not coupled). Both

the toggle switch and the 4 genes repressilator have two “unstable attractors”

(regions of the state space where they stay most of the time) and both toggle

from one unstable attractor to the other due to stochastic fluctuations (Gard-

ner et al, 2000; Ribeiro et al, 2006). Therefore, in both cases, only one ergodic

set exists (Figs 7 and 8).

Comparing the two figures (Figs 7 and 8), its observable that the toggle switch

is more subject to noise, i.e., it toggles more times in the same time interval

between the two unstable attractors than the 4 genes repressilator.

The reason for the 4 genes repressilator to toggle less is that it requires sto-

23

Fig. 5. Average number of ergodic sets for 14 genes networks, scale free inputs,

outputs, and inputs and outputs topologies with γ = −2. Each point represents the

average of 1000 networks.

chastic fluctuations such that, at the same time both proteins levels of the

genes “on” are low enough so that the other two genes (previously “off”) can

now start expressing and repressing the two genes previously “on”. This event

is less likely to occur than the stochastic fluctuation of the protein level of the

only gene “on” in the toggle switch, resulting in less frequent toggling in the

same time interval.

Adding more genes to the repressilator unidirectional chain (keeping the to-

tal number an even quantity) would diminish the toggling between the two

alternative states even more.

Given the toggle switch time series (Fig. 7) and using K-means to binarize the

24

Fig. 6. A 2 genes (A and B) toggle switch and a 4 genes (1, 2, 3 and 4) repressilator

coupled by reciprocal repression reactions between genes A and 3 and between genes

B and 4. The arrows ending in perpendicular lines represent repression interactions

of one gene protein in the other gene promoter.

levels of proteins A and B, one observes from the binarized time series that

the system, after an initial transient of approximately 2000 seconds, has two

unstable “single state” attractors. One is (A,B) = (0,1) and the other is (A,B)

= (1,0). These two states are by far the most represented ones in the binarized

time series, out of the possible 4 binary states. We note that sometimes one

detects also the states (0,0) and (1,1), if the sampling occurred during the

transitions between the two unstable attractors. These transitions have very

short time duration in comparison with the total time that the system remains

on one of the two unstable attractors.

The 4 genes repressilator behaves similar to the toggle switch. Using K-means

to binarize the time series of proteins p1, p2, p3, and p4, in Fig. 8, shows that the

system, after an initial transient of approximately 2500 seconds, has two un-

stable attractors. One is (p1,p2,p3,p4) = (1,0,1,0) and the other is (p1,p2,p3,p4)

25

Fig. 7. A toggle switch proteins quantities time series. The system toggles from one

“unstable attractor” to the other.

= (0,1,0,1). These two states (each a single state unstable attractor) are the

most observed states in the binarized time series.

Next, the two systems were coupled according to equations 11 to 16. We

assume similar consensus sequences between some of the genes of the two sub

circuits, which therefore share the same inputs.

When coupling the two systems, two ergodic sets emerge, as opposed to the

unstable attractors seen before in each of the systems. In Fig. 9, the pro-

teins expression level corresponding to one of the two possible ergodic sets is

observed. Their levels are all approximately constant, aside small stochastic

fluctuations, as opposed to what is observed in Figs. 7 and 8, where there is

a constant toggling of proteins levels, corresponding to the toggling between

the two unstable attractors of the systems. Additionally, the proteins levels of

26

Fig. 8. A 4 genes repressilator proteins quantities time series. The system toggles

from one “unstable attractor” to the other. Only p1 and p2 time series are shown.

p3 follows the same trajectory as p1, and p4 as p2.

the genes in the “on” state fluctuate far less in the coupled system.

Applying the K-means algorithm to one time series of a single simulation

of this gene network, represented in Fig. 6), will result in, depending on

the ergodic set reached by the system in that run, either (p1,p2,p3,p4,A,B)

= (1,0,1,0,0,1) or (p1,p2,p3,p4,A,B) = (0,1,0,1,1,0). In Fig. 9, the ergodic set

reached by the system after the initial transient was (1,0,1,0,0,1).

We have not observed in all our experiments (above 10.000 independent exper-

iments) the coupled system leaving any of the two ergodic sets once reaching

them, in the parameters range of values here used. Depending on the rates

constants, e.g., the ratio between gene expression and decay, the two subsys-

tems can be made more or less stable. For example, if the transcription rate

27

constants are set to very small values or decays to very high, the system will

never be able to reach any of the ergodic sets and will indefinitely remain on a

long transient with very small total number of proteins. In this extreme regime,

stochastic fluctuations of the proteins time series would be the predominant

dynamical feature and no ergodic set would be observed.

It is interesting to notice that, if an analogy is to be made with the Boolean

networks dynamics, that is, convert the time series to a binary series, a “zero”

corresponds to null or very near null concentrations. Higher decay would move

the “threshold” between 0’s and 1’s to a higher value (relatively to the maxi-

mum concentrations of proteins observed).

Fig. 9. Coupled repressilator and toggle switch. No toggling due to internal noise is

observed. Only p1 and B proteins are shown. Proteins p2, p4 and A quantities are

null after a short transient, while protein p3 has the same level as p1.

For example, the binarization using the K-means method of the system state

28

in Fig. 9 after the initial transient, would assign the following Boolean states:

p1, p3 and B = 1, p2, p4 and A = 0. These values do not change over time,

since the proteins concentrations are fairly stable aside stochastic fluctuations.

Figure 9 shows that this system multiple ergodic sets are resistant to the

internal noise due to stochastic fluctuations.

Notice that in this more realistic model of GRNs, because it is based on the

multiple delayed SSA, the gene expression levels, when observed in detail, show

that proteins are produced by bursts (for a very detailed study on these GRNs

models dynamics at the single gene level see (Zhu et al, 2006) and for studies

with experimental measurements see (Chubb et al, 2006; Golding et al, 2005;

Blake et al, 2006)). Additionally, decay (which can happen to proteins even

when bound to the promoter) is also a stochastic process and so promoters’

states are inherently stochastic.

The gene network here modelled was always resistant to all these forms of

internal noise, i.e., once the ergodic set was reached, a transition to the other

possible ergodic set was never observed in very long time intervals (107s).

Thus, it is an example of a GRN driven by the delayed SSA with multiple

ergodic sets.

Since the system ergodic sets proved to be resistant to any internal fluctua-

tions (i.e. stochastic fluctuations in promoter states and proteins decay, among

others), and because cells can also be affected by external perturbations, we

wanted to see how our gene network reacted to external perturbations.

We observed the system response to external perturbations that consist in the

addition or removal of large amounts of proteins. For all proteins, we changed

29

its concentration from “high” to “low” or vice versa, one at a time (corre-

sponding to flipping the protein level from 0 to 1 or vice versa after binarized

via K-means). Additionally, these perturbations are similar to testing what

would happen if such very large (extremely unlikely) internal fluctuations of

proteins concentrations occurred. Because we never observed these rare events

in the cases where no external perturbation is made, we imposed them to test

if they would be sufficient to make the system leave the ergodic set.

Here we present in figures 10 and 11 a perturbation to the inactive gene

of the toggle switch, and another to an inactive gene of the repressilator,

as examples. In all cases, the coupling was strong enough to maintain the

steady state that the ergodic set consists of. In figure 10, a perturbation was

introduced externally to the toggle switch, by adding 500 A molecules to the

system at each 100 000 seconds. As seen, due to the coupling, the perturbation

was unable to remove the system from its stable state. We also tried adding

at each 20 000 seconds, with the same results (data not shown). In figure 11, a

perturbation was introduced externally to the repressilator, by adding 500 p2

molecules to the system at each 100 000 seconds. As seen, due to the coupling

to the toggle switch, the perturbation was unable to remove the system from

its stable state. Notice that sometimes, protein B and p1 went down due to

the perturbation but, due to coupling they “recovered” the previous levels,

after all proteins externally introduced, decayed.

Also, we tried adding all other proteins, one at a time, and again, due to the

stabilization given by the coupling, the system did not changed state (data not

shown). In all cases, 100 experiments were made, all analyzed independently

and with similar results, but we plot only the results of single experiments as

an illustration, since plots of average behavior would not allow seeing pertur-

30

bations effects.

Fig. 10. Perturbation of the toggle switch by adding A’s at each 100 000 seconds.

The system remains stable.

Also interesting to notice is that the time series of both repressilator and

toggle switch in Figs. 10 and 11 are much less noisy than their time series

when uncoupled (Figs 7 and 8).

As observed from the previous cases, this GRN requires more than 1 pertur-

bation at a time (more than one protein level significantly changed), due to

the coupling between the toggle switch and the 4 genes repressilator.

The ergodic sets are stable to all possible stochastic fluctuations, e.g., on

proteins quantities and promoter states. Only fluctuations that persist for a

long time in “one direction” and happening to at least three of the six proteins

at the same time, would make the system change ergodic set.

31

Fig. 11. Perturbation of the repressilator by adding p2’s at each 100 000 seconds.

The system remains stable.

We tested the system response to simultaneous perturbations, i.e., adding

more than 1 kind of proteins at the same time. Our simulations results are

that (data not shown), only when introducing the three kinds of proteins not

present in the system at the moment the perturbation is imposed, can the

system be removed from the ergodic set he is in. Another possibility would

be removing at a given moment, all proteins present in the system, included

those bound to promoters, thus resetting the system state. For example, if

the levels of proteins p1, p3 and B are high, only adding to the system, at the

same time, many proteins p2, p4 and A, will the system, in some cases, leave

the ergodic set he is in.

Notice that the time series of the proteins in the coupled system (Fig. 9)

are much less “noisier” than in the two uncoupled systems (Figs. 7 and 8).

32

For that reason, perturbing the levels of two of the three proteins “holding

state”, is not sufficient to remove the system from its ergodic set, while, e.g.,

perturbing the level of 1 of the 2 proteins holding state in the uncoupled 4

genes repressilator, is sufficient to change its state.

Given a detailed analysis of the set of reactions defining the coupled system,

one identifies, at least, two parameters whose values must be set within a cer-

tain interval of possible values, for the system to behave as described. Also,

these parameters need to be considered together (varying one may be “com-

pensated” with the opposite variation of another): i) the slower is the proteins

rate constant of decay, the more likely is one perturbation to affect the sys-

tem state since the externally added proteins will remain in the system for

a longer period, ii) the larger is the time delay of promoters release in the

transcription/translation reaction, the less likely is that a perturbation causes

any effect on the system dynamics since the promoters will be, on average,

a longer time unavailable for reactions, resulting in more time for externally

added proteins to decay without affecting the system state. Other parameters,

such as the rate constants of binding and unbinding of repressors to promoters,

also play an important role.

In the coupled system, given our parameters values, fluctuations resulting in

the events able to force the system to leave its ergodic set are very unlikely and

can be made even less likely by the coupling of more switches (or rate constants

tuning, e.g.). The probability of the necessary consecutive events described to

occur is thereby extremely remote and can be considered non-existent in a

realistic time scale. For example, never do any proteins concentrations reach,

due to stochastic fluctuations, levels as those imposed by us when adding

external perturbations.

33

Symmetrical experiments were made when the system chose the other ergodic

set with the same results. In all cases, the system never went from one ergodic

set to another, thereby we conclude that this system has two ergodic sets which

can be reached from the same initial condition. Starting from a different initial

condition, for example, non null proteins concentrations biased towards one

of the ergodic sets, or one of the genes repressed, is equivalent to start from

an initial state, in the Boolean framework, belong to one of ergodic sets basin

of attraction.

The results obtained in this section are robust to some variations of the pa-

rameters values. Changing any of the rate constants and time delays by a

factor of 10 did not change the system dynamics significantly, and its only

consequence is varying the proteins levels at “equilibrium” that depend on

the relationships between rate constants of production and decay, and time

delays of the transcription/translation reactions.

6 Discussion

We tested whether in noisy Boolean networks, in the regime of low noise (slow

dynamics versus the system fast dynamics), multiple ergodic sets exist.

If all genes can be perturbed, the results on Boolean networks we report here

show that, although possible, (Fig 2), the property of having multiple ergodic

sets is not generic in the ensembles examined.

Yet, if a fraction of nodes are “protected” from perturbations, for which we

presented concrete possible mechanisms, multiple ergodic sets do exist, even in

the small region of the network space we searched and for very small networks

34

(and thus with very small number of attractors even without the presence of

noise).

Namely, we observed in the Boolean framework that, for RBNs of 6 nodes,

random topology and average connectivity of 2, one of the six genes had to be

“protected from noise fluctuations” to attain a few networks (out of a set of

1000 randomly generated) with more than 1 ergodic set. For RBNs of 10 nodes,

2 genes needed to be protected and, for RBNs of 14 nodes, 3 nodes had to be

protected. Unfortunately, due to the simulations computational complexity,

we do not know if, as the number of nodes increases, it will require a larger or

smaller fraction of nodes to be protected against noise for multiple ergodic sets

to appear. It is likely however, that some networks will not require protection

against noise of any node because even for small networks there are such cases.

In this framework, as more nodes are “protected” against noise, the number

of randomly generated networks that possess multiple ergodic sets grows sig-

nificantly as our results show, but perhaps more relevant is to point out that

RBNs with multiple ergodic sets do exist even when no node is “protected

from noise” and the fact that only a few exist not necessarily makes them

impossible to attain. For example, it could be the case that they can be se-

lected for, rather than randomly attained. One could, for example, imagine

algorithms able to find these networks in some faster way other than the ran-

dom search method here used. As said, we do not know if RBNs with multiple

ergodic sets are homogeneously distributed in the state space. If they are not,

imposing some simple conditions in the generation of RBNs to restrict them

to the phase space regions where they are more likely to occur would make

the search for RBNs with multiple ergodic sets easier.

35

Thereafter, we constructed a small stochastic genetic circuit by coupling two

genetic circuits which are known for being able to toggle between “attractors”.

We showed that using a very simple scheme of coupling, they gain enough

stability that removes the ability of toggling both due to internal noise and

to any “single gene state” external perturbation. That is, this GRN with

stochastic dynamics and a realistic model of gene expression does indeed hold

state once it settles on one of the ergodic sets.

Our initial work hypothesis in the Boolean framework, of considering that

gene state flipping due to internal noise is a slow event in comparison to the

dynamics time scale (where by dynamics we mean state transitions) appears to

be a limitation. In this framework, if many genes could flip randomly at each

time step, most likely only one ergodic set would exist. That is, if noise can

occur more often than here assumed, multiple ergodic sets would be extremely

rare.

Additionally, there are evidences that real genes often have bursts in produc-

tion (see e.g. (Zhu et al, 2006)), which could be seen as corresponding to “fast

bit flipping” noise of gene states.

We stress that, in agreement with those experimental observations, our sto-

chastic modeling strategy has gene expression bursts and all other kinds of

stochastic fluctuations (a very detailed description of this can be found in

(Zhu et al, 2006)).

Yet, as observed in Fig. 9, this system ergodic sets are stable once reached.

That is, they are robust to the genes expression bursts (these are not easily

visible in the figure due to their short time duration) and all the other kinds

of intrinsic noise due to stochastic fluctuations.

36

The reason for this is that burst of gene expression and other “noise caused”

fluctuations in the stochastic framework do not correspond to the “state flip-

ping” in the Boolean framework. This state flipping of a node in the Boolean

model actually corresponds to very large proteins quantities variations in

the stochastic framework which we never observed, since its occurrence is

very sparse, if not impossible. The stochastic fluctuations are indeed “small”

enough for the system to “recover”.

For example, in the toggle switch (when not coupled), the toggling is due to

stochastic fluctuations of the proteins levels. Notice in Fig 7 that the number of

toggles is extremely small compared to the number of proteins concentrations

fluctuations.

A set of stochastic events is needed to cause the system to toggle in this case.

Also, these events must occur at approximately the same time. For example,

if the repressed gene produces proteins in a stochastic burst, unless these

proteins can repress the other gene (while the promoter is free) they will

decay relatively fast and not cause a toggling. The time delay of the promoter

region acts as a protection against this “noise”. Additionally, even if these

two events happen, for some time the two genes will be competing (like in

the initial transient) to decide who represses who. At this point there is a

50% chance that a toggling will occur. The 4 genes repressilator as a similar

behavior.

When coupling the toggle switch to the 4 genes repressilator, the conditions

for toggling become virtually impossible within a reasonable time scale, as

seen. Namely, in this case, at least 4 independent events must occur at ap-

proximately the same time and last for a significant time interval: at least two

37

of the three active genes must stochastically be “down” simultaneously and,

at the same time, one previously repressed gene must be persistently being

allowed to express and maintain a high protein level (at least high enough for

a K-means analysis to detect a “bit flip”).

By comparing the number of these bursts with the toggling frequency of a

single toggle switch one concludes that they rarely cause toggling due to the

reasons stated.

We observed that in the case of the 4 genes repressilator the toggling is even

less frequent, and finally, in the case of the coupled toggle switch and 4 genes

repressilator, stochastic fluctuations never caused a toggling.

If one converts the proteins time series into a binary string via K-means, the

resulting bit string of state transitions shows that the stochastic fluctuations,

unless when causing a toggling, do not cause a bit flip.

Thus, the assumption of “low noise frequency” in the Boolean framework is

in agreement with what was observed in the dynamics of the more realistic

modeling strategy, the multiple-delayed SSA.

Taking the above in consideration, regarding the external perturbations analy-

sis, since the network here modeled has only 6 genes, we only perturbed one

gene protein level at a time. Perturbing all genes at the same time for a sig-

nificant time interval definitely will cause a state change (this is equivalent to

“resetting the network state”), but this is certainly a very rare event to observe

in large scale networks, if not virtually impossible (and we never observed it

even in our 6 genes network).

The study (Figs. 10 and 11) of the effect of external perturbations (where

38

many proteins were added and removed artificially) in the stochastic model

was made to be the most similar possible to the “bit flipping” in the Boolean

framework. Binarizing the time series via K-means will show bit flipping at

the moments a perturbation is applied. The GRN ergodic sets were always

robust to these external perturbations.

7 Evidence that Cell Types are Ergodic Sets.

At least two experimental results and one mathematical observation suggest

that, at present, the most plausible hypothesis is that cell types correspond

to ergodic sets, which may occasionally be unstable to external perturbations.

Neubauer and Calef (Neubauerz and Calef, 1970), constructed N-P-lysogens of

phage lambda in E. coli. Two genes, C1 and Cro, repress one another (toggle

switch). This molecular circuit would be expected to have two steady states,

C1 on and Cro off, called the immunity + state, and C1 off, Cro on, called the

immunity - state. The authors found that E. coli could be cultured indefinitely

in the im+ or im- state, but that, at rare intervals a cell would jump to the

other state. That new state could then be reliably cultured indefinitely. This

behavior appears to be consistent with a ergodic set hypothesis.

Similarly, Gardner et al, (Gardner et al, 2000) made a small genetic circuit

of two genes that repress one another, cloned these into E. coli, and observed

the analogue of the im+ and im- states, each stably heritable for hundreds of

generations without jumps.

One possible interpretation of these results, for real genes in real cells, is that

ergodic sets can be stable against cellular noise fluctuations at least for long

39

periods, if not indefinitely. Nevertheless, they are not infinitely robust and

some specific external perturbations can make them jump between ergodic

sets.

Recently, Suel et al (Suel et al , 2006), showed that some types of cellular

differentiation are probabilistic and transient. They also show that the most

correct modeling strategy of gene networks is the stochastic modeling ap-

proach. One possible interpretation of these results, in light of the present

work, is that in such case, one is observing the going back and forth between

two attractors of the same ergodic set. It is likely that these two noisy attrac-

tors consist of states which differ significantly and thus, two “transient cell

types” would be identified in this case. However, many differentiation path-

ways do not have this feature, namely, they are irreversible and most known

cases of reversibility are caused by specific external perturbations and even

in this case only some cells are affected. For example, recent work (Chang et

al, 2006) examining differentiation of HL60 cells induced to differentiate into

polymorphoneuterophils has yielded the first evidence that cell types are high

dimensional (many variable) ergodic sets. Also, their results suggest bistabil-

ity in the gene regulatory mechanism underlying the dynamics (Chang et al,

2006).

As noted, our simple results are the first look at transitions between noisy

attractors in idealized Boolean models and in a more realistic model of GRNs.

Many questions remain unanswered and require further study. It may turn

out, and would have very deep medical implications if true, that there is but

one or a few ergodic sets for modest perturbations to real genetic regulatory

networks. If so, the potential for tissue engineering of such epigenetic plasticity

will merit even more attention.

40

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