Persistence of Activity in Random Boolean Networks Shirshendu Chatterjee and Rick Durrett Cornell University December 20, 2008 Abstract We consider a model for gene regulatory networks that is a modification of Kauffmann s 969 random Boolean networks. There are three parameters: n = the number of nodes, r = the number of inputs to each node, and p = the expected fraction of s in the Boolean functions at each site. Following a standard practice in the physics literature, we use a threshold contact process on a random graph in which each node has in degree r to approximate its dynamics. We show that if r 3 and r 2p p >, then the threshold contact process persists for a long time, which corresponds to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is expcn bp with bp > 0 when r 2p p > and bp = when r 2p p >. Keywords: random graphs, threshold contact process, phase transition, random Boolean networks, gene regulatory networks Both authors were partially supported by NSF grant 0704996 from the probability program at NSF.
Introduction Random Boolean networks were originally developed by Kauffman 969 as an abstraction of genetic regulatory networks. The idea is to identify generic properties and patterns of behavior for the model, then compare them with the behavior of real systems. Protein and RNA concentrations in networks are often modeled by systems of differential equations. However, in large networks the number of parameters such as decay rates, production rates and interaction strengths can become huge. Recent work by Albert and Othmer 2003 on the segment polarity network in Drosophila melanogaster, see also Chaves, Albert and Sontag 2005, has shown that Boolean networks can in some cases outperform differential equation models. Kauffman et al. 2003 used random Boolean networks to model the yeast transcriptional network, and Li et al 2004 have used this approach to model the yeast cell-cycle network. In our random Boolean network model, the state of each node x V n = {, 2,..., n} at time t = 0,, 2,... is η t x {0, }. Each node x has r input nodes y x,..., y r x chosen randomly from the set of all nodes, and we draw oriented edges to each node from its input nodes. So the edge set is E n = {y i x, x : x V n, i r}. Thus the underlying graph G n is a random element chosen from the set of all directed graphs in which the in-degree of any vertex is equal to r. Once chosen the network remains fixed through time. The updating for node x is η t+ x = f x η t y x,..., η t y r x, where the values f x v, x V n, v {0, } r, chosen at the beginning and then fixed for all time, are independent and = with probability p. A number of simulation studies have investigated the behavior of this model. See Kadanoff, Coppersmith, and Aldana 2002 for survey. Flyvberg and Kjaer 988 have studied the degenerate case of r = in detail. Derrida and Pommeau 986 have shown that for r 3 there is a phase transition in the behavior of these networks between rapid convergence to a fixed point and exponentially long persistence of changes, and identified the phase transition curve to be given by the equation r 2p p =. The networks with parameters below the curve have behavior that is ordered and those with parameters above the curve have chaotic behavior. Since chaos is not healthy for a biological network, it should not be surprising that real biological networks avoid this phase. See Kauffman 993, Shmulevich, Kauffman, and Aldana 2005, and Nykter et al. 2008. To explain the intuition behind the result of Derrida and Pomeau 986, we define another process ζ t x for t. The idea is that ζ t x = if and only if η t x η t x. Now if the state of at least one of the inputs y x,..., y r x into node x has changed at time t, then the state of node x at time t + will be computed by looking at a different value of f x. If we ignore the fact that we may have used this entry before, we get the dynamics of the threshold contact process P ζ t+ x = = 2p p if ζ t y x + + ζ t y r x > 0 2
and ζ t+ x = 0 otherwise. Conditional on the state at time t, the decisions on the values of ζ t+ x, x V n are made independently. We content ourselves to work with the threshold contact process since it gives an approximate sense of the original model and we can prove rigorous results about its behavior. To simplify notation and explore the full range of threshold contact processes we let q = 2p p and suppose 0 q. As mentioned above, it is widely accepted that the condition for prolonged persistence of the threshold contact process is qr >. To explain this, we note that vertices in the graph have average out-degree r, so a value of at a vertex will, on the average, produce qr s in the next generation. To state our first result, we will rewrite the threshold contact process as a set valued process ξ t = {x : ζ t x = }. We will refer to the vertices x ξ t as occupied. Let {ξ t } t 0 be the threshold contact process starting from all occupied sites, and ρ be the survival probability of a branching process with offspring distribution p r = q and p 0 = q. By branching process theory ρ = θ, where θ 0, satisfies θ = q + qθ r.. Theorem. Suppose qr > and let δ > 0. There is a positive constant Cδ so that as n ξ inf P t t expcδn n ρ 2δ. To prove this result, we will consider the dual coalescing branching process ˆξ t. In this process if x is occupied at time t, then with probability q all of the sites y x,..., y r x will be occupied at time t+, and birth events from different sites are independent. Writing A and B for the initial sets of occupied sites in the two processes we have the duality relationship: P ξ A t B = P ˆξ B t A, t = 0,, 2,.....2 Taking A = {, 2,..., n} and B = {x} this says P x ξ t = P ˆξ {x} t, or the density of occupied sites in ξt is equal to the probability that ˆξ {x} s survives until time t. Since over small distances our graph looks like a tree in which each vertex has r descendants, the last quantity ρ. From.2 it should be clear that we can prove Theorem by studying the coalescing branching process. The key to this is an isoperimetric inequality. Let Ĝn be the graph obtained from our original graph G n = V n, E n by reversing the edges. That is, Ĝ n = V n, Ên, where Ên = {x, y : y, x E n }. Given a set U V n, let U = {y : x y for some x U},.3 where x y means x, y Ên. Note that U can contain vertices of U. The idea behind this definition is that if U is occupied at time t in the coalescing branching process, then the vertices in U may be occupied at time t +. 3
Theorem 2. Let P m, k be the probability that there is a subset U V n with size U = m so that U k. Given η > 0 there is an ɛ 0 η so that for m ɛ 0 n P m, r ηm exp ηm logn/m/2. In words, the isoperimetric constant for small sets is r. It is this result that forces us to assume qr > in Theorem. To see that Theorem 2 is sharp, define an undirected graph H n on the vertex set V n so that x and y are adjacent if and only if there is a z so that x z and y z in Ĝn. The mean number of neighbors of a vertex in H n is r 2 9, so standard arguments show that there is a c > 0 so that with probability tending to as n there is a connected component K n of H n with K n cn. If U is a connected subset of K n with U = m then by building up U one vertex at a time we see that U + r m. Since the isoperimetric constant is r, it follows that when qr <, there are bad sets A with A nɛ so that E ˆξ A A. Computations from the proof of Theorem 2 suggest that there are a large number of bad sets. We have no idea how to bound the amount of time spent in bad sets, so we have to take a different approach to show persistence when /r < q /r. Theorem 3. Suppose qr >. If δ 0 is small enough, then for any 0 < δ < δ 0, there are constants Cδ > 0 and Bδ = /4 2δ logqr δ/ log r so that as n ξ inf P t t expcδ n Bδ n ρ 3δ. To prove this, we will again investigate persistence of the dual. Let d 0 x, y be the length of the shortest oriented path from x to y in Ĝn, let and for any subset A of vertices let dx, y = min z V n [d 0 x, z + d 0 y, z], ma, K = max{ S : dx, y K for x, y S}..4 S A Let R = log n/ log r be the average value of d, y, let a = /4 δ and B = a δ logqr δ/ log r. We will show that if mˆξ s A, 2aR < ρ 2δn B, then with high probability, we will later have mˆξ t A, 2aR ρ 2δn B. To do this we first argue that when we first have mˆξ s A, 2aR < ρ 2δn B, there are at least q δρ 2δn B occupied sites so that the d distance between any two vertices is at least 2aR 2. We run the dual process starting from these vertices until time ar 2, so they are independent. With high probability at least one of these vertices, call it w, has n B descendants. We run the dual processes starting from these n B vertices for another ar units of time, and then pick one offspring from each of the duals which have survived till time ar to have a set that is suitably spread out and has size at least ρ 2δn B. It seems foolish to pick only one vertex w after the first step, but we do not know how to guarantee that the vertices are suitably separated after the second step if we pick more. 4
2 Proof of Theorem We begin with the proof of the isoperimetric inequality, Theorem 2. Proof of Theorem 2. Let P m, k be the probability that there is a set U of vertices in Ĝn of size m with U k. Let pm, k be the probability that there is a set U with U = m and U = k. First we will estimate pm, l where l = r ηm. pm, l = P U = U P U U. {U,U : U =m, U =l} {U,U : U =m, U =l} According to the construction of G n, the other end of each of the r U edges coming out of U is chosen at random from V n so U P U U r U =, n and hence pm, l n m n l l n rm. 2. To bound the right-hand side, we use the trivial bound n nm ne m m m!, 2.2 m where the second inequality follows from e m > m m /m!. Using 2.2 in 2. pm, l ne/m m ne/l l l n rm. Recalling l = r ηm, the last expression becomes = e mr η m/n m[ r η+r] r η r ηm+rm. Letting cη = r η + + η logr η C for η 0, r, we have pm, r ηm exp ηm logn/m + Cm. Summing over integers k = r η m with η η, and noting that there are fewer than rm terms in the sum, we have P m, r ηm exp ηm logn/m + C m. To clean up the result to the one given in Theorem 2, choose ɛ 0 such that η log/ɛ 0 /2 > C. Hence for any m ɛ 0 n, which gives the desired result. η logn/m/2 η log/ɛ 0 /2 > C, 5
To grow the cluster starting at a vertex x in Ĝn, we add the neighbors in a breadth-first search. That is, we add all vertices y for which d 0 x, y = j one at a time, before proceeding to the vertices at distance j +. Lemma 2.. Suppose 0 < δ < /2. Let A x be the event that the cluster starting at x in Ĝn does not intersect itself before reaching size n /2 δ, and let A x,y be the event that the clusters starting from x and y do not intersect before reaching size n /2 δ. Then for all x, y V n P A c x, P A c x,y n 2δ. 2.3 Proof. Let δ = /2 δ. While growing the cluster starting from x up to size n δ, the probability of a self-intersection at any step is n δ /n, so P A x n δ n δ n 2δ = n 2δ. The exact same reasoning applied to the cluster containing x intersecting the cluster of y grown to size n /2 δ. Lemma 2. shows that Ĝn is locally tree-like. In the next lemma, we will use this to get a bound on the survival of the dual process for small times. Let ρ be the branching process survival probability defined in.. Lemma 2.2. If q > /r, δ 0, qr, γ = 20 log r, and b = γ logqr δ then for large n P ˆξ{x} n b ρ δ. 2γ log n Proof. Let A x be the event that the cluster starting at x in Ĝn does not intersect itself before reaching size n /4. On the event A x, for t 2γ log n, ˆξ {x} t = Z t, is a branching process with Z 0 = and offspring distribution p 0 = q and p r = q. Since q > /r, this is a supercritical branching process. Let B x be the event that the branching process survives. If we condition on B x, then, using a large deviation results for branching processes from Athreya 994, Z t+ P qr > δ B x e cδt 2.4 Z t for large enough t. So if F x = {Z t+ qr δz t for γ log n t < 2γ log n}, then P F c x B x 2γ log n t=γ log n e cδt C δ n cδγ 2.5 for large enough n. On the event B x F x, Z 2γ log n qr δ γ log n = n γ logqr δ, since Zγ log n. Combining the error probabilities of 2.3 and 2.5 P ˆξ{x} n γ logqr δ P B x P A c x P Fx B c x P B x δ 2γ log n for large enough n, as P A x n /2 by Lemma 2.. 6
Lemma 2.2 shows that the dual process starting from one vertex will with probability ρ δ survive until there are n b particles. The next lemma will show that if the dual starts with n b particles, then it can reach ɛn particle with high probability. Lemma 2.3. If qr >, then there exists ɛ 0 = ɛ 0 q > 0 such that for any A with A = n b the dual process {ˆξ t A } t 0 satisfies P max ˆξ A t ɛ 0 n n b t < ɛ 0 n exp n b/4. Proof. Choose η > 0 such that q ηr η > and let ɛ 0 η be the constant in Theorem 2. Let τ = min{t : ˆξ A t ɛ 0 n}. For t < τ, let F t = { ˆξ A t ˆξ A t + }. Then for t 0 P F t+ P B t C t, where B t = {at least q η ˆξ A t particles of ˆξ A t give birth}, C t = { U t r η U t }, where U t = {x ˆξ A t : x gives birth}. Using the binomial large deviations, see Lemma 2.3.3 on page 40 in Durrett 2007 P B t exp γq η/qq ˆξ A t, 2.6 where γx = x log x x + > 0 for x. On G t = t s=f s, ˆξ A t n b and so P B t G t exp γq η/qqn b. Since for t < τ, we have ˆξ t A < ɛ 0 n, we can use Theorem 2 and the fact, that U t q ηn b n b /r on G t B t, to get P C t G t B t exp η n b nr log. 2 r n b Combining these two bounds we get P B t C t G t exp n b/2 for large n. Since τ ɛ 0 n n b on G ɛ0 n n b, P τ > ɛ 0 n n b P ɛn nb t= F c t ɛ 0 n n b exp n b/2 exp n b/4 for large n and we get the result. The next result shows that if there are ɛn particles at some time, then the dual survives for time expcn. Lemma 2.4. If qr >, then there exists constants c, ɛ 0 > 0 such that for T = expcn and any A with A ɛ 0 n, P inf ˆξ t A < ɛ 0 n 2 exp cn. t T 7
Proof. Choose η > 0 so that q ηr η >, and then choose ɛ 0 η > 0 as in Theorem 2. For any A with A ɛ 0 n, let U t = {x ˆξ A t : x gives birth}. If U t ɛ 0 n, then take U t = U t. If U t > ɛ 0 n, we have too many vertices to use Theorem 2, so we let U t be the subset of U t consisting of the ɛ 0 n vertices with smallest indices. Let F t = { ˆξ t A ɛ 0 n} G t = t s=0f s B t = {at least q η ˆξ t A particles of ˆξ t A C t = { Ut r η U t }. give birth} Now for t 0, F t+ G t B t C t G t. Using our binomial large deviations result 2.6 again, P B t exp γq η/qq ˆξ t A. On the event F t, ˆξ t A ɛ 0 n, and so P B t G t exp γq η/qqɛ 0 n. Since U t ɛ 0 n, and on the event G t B t U t q ηɛ 0 n ɛ 0 n/r, using Theorem 2 we have P C t G t B t exp η ɛ 0 n 2 r log r. ɛ 0 Combining these two bounds P F t+ G t 2 exp 2cηn, where cη = q η {γ 2 min qɛ 0, η ɛ 0 q 2 r log r }. ɛ 0 Hence for T = expcηn P inf t T ˆξ A t T < ɛ 0 n P T t=ft c t=0 P F c t+ G t 2 exp cηn. Lemma 2.4 confirms prolonged persistence for the dual. We will now give the Proof of Theorem. Choose δ 0, qr and γ = 20 log r. Define the random variables Y x, x n, as Y x = if the dual process {ˆξ {x} t } t 0 starting at x satisfies ˆξ {x} 2γ log n nb for b = γ logqr δ and Y x = 0 otherwise. By Lemma 2.2, if n is large then EY x ρ δ for any x. If we grow the cluster starting from x in a breadth-first search then all the vertices at distance 2γ log n from x are in the cluster of size n 2γ log r n /0. So if A x,z is the event that the clusters of size n /0 starting from x and z do not intersect, then P Y x =, Y z = P Y x = P Y z = P A c x,z n 4/5 by Lemma 2.. Using this bound, n var Y x n + nn n 4/5, 8
and Chebyshev s inequality shows that as n n P Y x EY x nδ n + nn n 4/5 0. n 2 δ 2 Since EY x ρ δ, this implies lim P n n Y x nρ 2δ =. 2.7 Choose η > 0 so that q ηr η > 0. Let ɛ 0 and cη be the constants in Lemma 2.4. Now if Y x =, Lemma 2.2 shows that ˆξ {x} T n b for T = 2γ log n. Combining the error probabilities of Lemmas 2.3 and 2.4, shows that within the next T 2 = expcηn + ɛ 0 n n b units of time the dual process contains at least ɛ 0 n many particles with probability 2 exp n b/4 for large n. Using the duality property of the threshold contact process we see that for any subset B of vertices ˆξ{x} P ξt +T 2 B = P T +T 2 x B P ˆξ {x} T +T 2 ɛ 0 n x B P Y x = x B 2 B exp n b/4 P Y x = x B 2 exp n b/8, as B n. Hence for T = T + T 2 using the attractiveness property of the threshold contact process and combining the last calculation with 2.7 we conclude that as n ξ inf P t ξ t T n > ρ 2δ = P T n > ρ 2δ n P ξt {x : Y x = }, Y x nρ 2δ. This completes the proof of Theorem. 3 Proof of Theorem 3 Recall the definition of ma, K given in.4. Let R = log n/ log r, a = /4 δ and let ρ be the branching process survival probability defined in.. Lemma 3.. If qr > and δ 0 is small enough, then for any 0 < δ < δ 0 there are constants Cδ > 0, Bδ = /4 2δ logqr δ/ log r and a stopping time T satisfying P T < 2 exp Cδn Bδ 3 exp Cδn Bδ such that for any A with ma, 2aR ρ 2δn Bδ, ˆξ T A ρ 2δn Bδ. 9
Proof. Let m t = mˆξ A t, 2aR. We define the stopping times σ i and τ i as follows. σ 0 = 0, and for i 0 τ i+ = min{t > σ i : m t < ρ 2δn B }, σ i+ = min{t > τ i+ : m t ρ 2δn B }. First we estimate the probability of the event E i = {mˆξ A τ i, 2aR 2 q δρ 2δn B }. Since τ i > σ i for i, m τi ρ 2δn B, and hence there is a set S of size ρ 2δn B such that any two vertices in S are 2aR apart. Using the binomial large deviation estimate 2.6, at least q δρ 2δn B vertices of S will give birth at time τ i with probability exp γq δ/qqρ 2δn B. If we choose one offspring from the vertices of S which give birth, then we have at least q δρ 2δn B vertices at time τ i so that they are separated by distance 2aR 2 from each other. So P E c i exp c δn B, 3. where c δ = γq δ/qqρ 2δ. On the event E i, ˆξ τ A i ζ i where ζ i q δρ 2δn B and all vertices of ζ i are distance 2aR 2 apart from each other. Let A x be the event that the cluster starting at x in Ĝn does not intersect itself before reaching size n 2a.On the event A x, as shown in Lemma 2.2, ˆξ {x} t = Zt x for t 2aR, where Zt x is a supercritical branching process with mean offspring number qr. Let B x be the event of survival for Zt x, P B x = ρ > 0 and F x = ar 3 t=δr 2 {Zx t+ qr δzt x }. Using the error probability of 2.4 P F c x B x ar 3 t=δr 2 e c δt C δ e c δδ log n/log r = C δ n c δδ/log r. On the event B x F x Z x ar 2 qr δa δr = n a δ logqr δ/ log r = n B. Hence for G x = A x { ˆξ {x} ar 2 nb }, P G x P A x B x F x P B x P A c x P F c x B x P B x δ = ρ δ 3.2 for large enough n, as P A x n 4δ using Lemma 2.. Since each vertex of ζ i is 2aR 2 apart from the other vertices of ζ i, ˆξ ζ i t is a disjoint {x} union of ˆξ t over x ζ i for t ar 2. Let H i be the event that there is at least one x ζ i for which G x occurs. Then P H c i E i ρ + δ ζ i ρ + δ q δρ 2δnB = exp c 2 δn B, 3.3 where c 2 δ = q δρ 2δ log/ ρ + δ. If H i E i occurs, choose any vertex w i ζ i such that G wi occurs and let S i = ˆξ {w i} ar 2 = S i. By the choice of w i, S i n B and the clusters of size n a starting from any two vertices of 0
ˆξ {x} S i do not intersect. So ˆξ S i t is a disjoint union of P ˆξ {x} {x} ar ρ δ. We choose one vertex from ˆξ ar have a set W i ˆξ S i ar. t over x S i for t ar. Using 3.2 for each x S i for which to Let J i be the event that there are at least ρ 2δn B {x} vertices in S i such that ˆξ ar. Using our binomial large deviation estimate 2.6 ρ 2δ P Ji c exp γ ρ δn B = exp c 3 δn B. 3.4 ρ δ So on the intersection E i H i J i there is a set W i ˆξ {w i} 2aR 2 with W i ρ 2δn B, and each vertex of W i is separated by a distance 2aR from other vertices of W i. Hence using monotonicity of the threshold contact process σ i τ i + 2aR on this intersection. So P σ i > τ i + 2aR P E c i + P H c i E i + P J c i 3 exp 2Cδn B, where Cδ = min{c δ, c 2 δ, c 3 δ}/2. Let K = inf{i : σ i > τ i + 2aR}. Then P K > expcδn B [ 3 exp 2Cδn B ] expcδn B 3 exp Cδn B. Since σ i > τ i > σ i, σ K 2K. As ˆξ A σ K ρ 2δn B, we get our result if we take T = σ K. As in the proof of Theorem, survival of the dual process gives persistence of the threshold contact process. Proof of Theorem 3. Let 0 < δ < δ 0, ρ, a = /4 δ and B = /4 2δ logqr δ/ log r be the constants from the previous proof. Define the random variables Y x, x n, as {x} Y x = if the dual process ˆξ t starting at x satisfies mˆξ {x} 2aR 2, 2aR ρ 2δnB and Y x = 0 otherwise. Consider the event G x as defined in Lemma 3.. For large n, P G x P B x δ = ρ δ. If G x occurs, then ˆξ {x} ar 2 nb {x} and clusters starting from any two vertices of ˆξ ar do not intersect until their size become n a. Using arguments that led to 3.4 P mˆξ {x} 2aR 2, 2aR < ρ 2δnB G x exp cn B, ˆξ {x} ar which implies that if n is large P Y x = ρ 2δ. If we grow the cluster starting from x using a breadth-first search, then all the vertices at distance 2aR from x are within the cluster of size n 2a. So if A x,z be the event that the clusters of size n 2a starting from x and z do not intersect, then P Y x =, Y z = P Y x = P Y z = P A c x,z n 4a = n 4δ
by Lemma 2.. Using the bound on the covariances, n var Y x n + nn n 4δ, and Chebyshev s inequality gives that as n n P Y x EY x nδ n + nn n 4δ n 2 δ 2 0. Since EY x ρ 2δ, this implies lim P n n Y x nρ 3δ =. 3.5 If Y x =, mˆξ {x} T, 2aR ρ 2δn B, for T = 2aR 2. Using Lemma 3., after an additional T 2 2 expcδn B units of time, the dual process contains at least ρ 2δn B many particles with probability 3 exp Cδn B. Using duality of the threshold contact process, for any subset S of vertices ˆξ{x} P ξt +T 2 S = P T +T 2 x S P ˆξ {x} T +T 2 ρ 2δn B x S P Y x = x S 3 S exp Cδn B P Y x = x S exp n B/2, since S n. Hence for T = T + T 2 using the attractiveness property of the threshold contact process, and combining the last calculation with 3.5 we conclude that as n ξ inf P t ξ t T n > ρ 3δ = P T n > ρ 3δ n P ξt {x : Y x = }, Y x nρ 3δ, which completes the proof of Theorem 3. References Albert, R. and Othmer, H. G. 2003 The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. Journal of Theoretical Biology, 223, 8 2
Athreya, K.B. 994 Large deviations for branching processes, I. Single type case. Ann. Appl. Prob. 4, 779 790 Chaves, M., Albert, R., and Sontag, E.D. 2005 Robustness and fragility of Boolean models for genetic regulatory networks. J. Theor. Biol. 235, 43 449 Derrida, B. and Pomeau, Y. 986 Random networks of automata: a simplified annealed approximation. Europhysics Letters,, 45 49 Durrett, R. 2006 Random Graph Dynamics. Cambridge University Press. Flyvbjerg, H. and Kjaer, N. J. 988 Exact solution of Kaufmann s model with connectivity one. Journal of Physics A, 2, 695 78 Kadanoff, L.P., Coppersmith, S., and Aldana, M. 2002 Boolean dynamics with random couplings. arxiv:nlin.ao/0204062 Kauffman, S. A. 969 Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22, 437 467 Kauffman, S. A. 993 Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press. Kauffman, S.A., Peterson, C., Samuelsson, B., and Troein, C. 2003 Random Boolean models and the yeast transcriptional network. Proceedings of the National Academy of Sciences 0, 4796 4799 Li, F., Long, T., Lu Y., Ouyang Q., Tang C. 2004 The yeast cell-cycle is robustly designed. Proceedings of the National Academy of Sciences, 0, 478 4786. Liggett, T. M.999 Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes. Springer. Nyter, M., Price, N.D., Aldana, M., Ramsey. S.A., Kauffman, S.A., Hood, L.E., Yli-Harja, O., and Shmuelivich, I. 2008 Proceedings of the National Academy of Sciences. 05, 897 900 Shmulevih, I., Kauffmann, S.A., and Aldana, M. 2005 Eukaryotic cells are dynamically ordered or critical but not chaotic. Proceedings of the National Academy of Sciences. 02, 3439 3444 3