Ordered and disordered dynamics in random networks

Transcription

1 EUROPHYSICS LETTERS 5 March 998 Eurohys. Lett., 4 (6), (998) Ordered and disordered dynamics in random networks L. Glass ( )andc. Hill 2,3 Deartment of Physiology, McGill University 3655 Drummond Street, Montreal, Quebec, Canada H3G Y6 2 Deartment of Physics, McGill University - Montreal, Quebec, Canada 3 Deartment of Physics, Cornell University - Ithaca, NY, USA (received 20 October 997; acceted in final form 29 January 998) PACS b Theory and models of chaotic systems. PACS q Lattice theory and statistics; Ising roblems. PACS Jb Muscle contraction, nerve conduction, synatic transmission, memorization, and other neurohysiological rocesses (excluding ercetion rocesses and seech). Abstract. Random Boolean networks that model genetic networks show transitions between ordered and disordered dynamics as a function of the number of inuts er element, K, and the robability,, that the truth table for a given element will have a bias for being, in the limit as the number of elements N. We analyze transitions between ordered and disordered dynamics in randomly constructed ordinary differential equation analogues of the random Boolean networks. These networks show a transition from order to chaos for finite N. Qualitative features of the dynamics in a given network can be redicted based on the comutation of the mean dimension of the subsace admitting outflows during the integration of the equations. Genetic networks have been modeled by random Boolean networks in which time is discrete and each element comutes a Boolean function based on the values of inuts to that element []. Since the number of human genes is of the order of 00,000, and each gene is idealized as either on () or off (0), the state sace for the human gene activity is huge. An order-disorder transition has been described for random Boolean networks in the limit that the number of variables, N, as a function of the number of inuts er variable, K, and the robability,, that the truth table for a given element will have a bias for being [2]-[8]. The order-disorder boundary is given by K c = 2 c ( c ), () where K c and c reresent the values of K and on the boundary [2], [4], [6]. Kauffman has argued that for a network to be biologically meaningful, it should have relatively few attractors, ( ) glass@cnd.mcgill.ca c EDP Sciences

2 600 EUROPHYSICS LETTERS and the cycle length of attractors should be comaratively short []. In real biological systems there are not clocking devices to generate synchronous udating and theoretical models are more aroriately formulated as continuous differential equations [9]-[4]. Here we resent numerical evidence for an order-disorder transition in differential equation analogues of the discrete switching networks. We also resent a robabilistic model of the dynamics, in which we show that () also alies to the continuous equations. First consider a logical network consisting of N binary variables, X i =0,, i =,...,N. Since we consider the networks as models of genes, X i reresents the activity of gene i. In other contexts, logical variables may reresents sins or voltages. The network is udated by means of the dynamical equation X i (j +)=Λ i (X i (j),x i2 (j),...,x ik (j)), i =,...,N, (2) where Λ i (X i (j),x i2 (j),...,x ik (j)) {0,}and K is the number of inuts. More comactly, we have X(j +)=Λ(X(j)), i =,...,N. (3) Thus, for any state X(j), Λ is a truth table determining X(j +). The logical structure of eq. (2) can be catured by a differential equation [9]. To a continuous variable x i (t), we associate a discrete variable X i (t), X i (t) =0 if x i (t)<0; otherwise X i (t) =. (4) For any logical network, we define an analogous differential equation, dx i dt = x i + λ i (X i (j),x i2 (j),...,x ik (j)), i =,...,N, (5) where λ i (X i (j),x i2 (j),...,x ik (j)) is a scalar whose sign is negative (ositive) if the corresonding logical variable Λ i (X i (j),x i2 (j),...,x ik (j)) is 0 (). For each variable, the temoral evolution is governed by a first-order iecewise linear differential equation. Let {t,t 2,...,t k } denote the switch times when any variable of the network crosses 0. The solution of eq. (5) for each variable x i for t j <t<t j+ is x i (t) =x i (t j )e (t tj) +λ i (X i (j),x i2 (j),...,x ik (j))( e (t tj) ). (6) Deending on the articular network, eq. (5) can dislay steady states, limit cycles, quasieriodicity or chaos [9]-[3]. Although the origin of chaos in one articular fourdimensional network has been analyzed [3], no general methods have yet been develoed to determine whether chaotic dynamics exists in any given equation without integrating it. Chaos is the usual behavior in these networks for N = 64, 9 K 25, =[4]. In carrying out the comutations, we have made several additional assumtions concerning the structure of the equations. i) There is no self-inut or recirocal inut. This means that i cannot be an inut to itself, and if i is an inut to k, k is not also an inut to i. These assumtions eliminate the ossibility of asymtotic aroach to stable steady states in the neighborhood of threshold axes x i = 0 [9], []. If these assumtions are not made, the techniques for analysis of the dynamics based on the symbolic dynamics described below fail. ii) We set λ i in eq. (5) to be in the range ± or ±. iii) In the truth tables we assume that for a given value of, half the variables of the network have entries that are biased towards and half are biased towards 0. Although the iecewise-linear nature of eq. (5) facilitates the seed and accuracy of numerical integration over other integration techniques, numerical studies nevertheless grow

3 l. glass et al.: ordered and disordered dynamics etc. 60 (a) (b) (c) x(t) 0 x(t) 0 x(t) t t t Fig.. Time series for a tyical variable in a network with N = 50, K =8. (a)=, (b) =0.8, (c) =0.85. In anels (a) and (b), the time series satisfy our tests for chaos and in (c) the dynamics are eriodic. large raidly. Consequently, we focus on the order-disorder transition for limited regions of arameter sace. For each network, we investigated one initial condition and classified the dynamics after a transient of switch times. The dynamics can be classified by using symbolic dynamics, keeing track of the variable that switches as a function of time. Provided the network does not reach a steady state during the course of the integration, we generate a sequence of integers denoting the label of the variable that switches at each consecutive switch time. We search for eriodicities in this sequence of integers, searching for eriodicities u to length Given the restrictions on the equations mentioned above, if the symbolic sequence is eriodic, in the differential equation there is a stable limit cycle oscillation [9]. If the sequence is not eriodic, then in the differential equation, there can either be quasieriodic dynamics or chaotic dynamics. To identify quasieriodic dynamics, we need to kee track of the exact switching times of all the variables of the network. However, ilot investigations of several hundred randomly constructed networks failed to identify quasieriodic dynamics. Consequently, to simlify the analysis we lum quasieriodic rhythms together with chaotic networks, where we exect that the incidence of quasieriodic dynamics is negligible (less than %). Figure illustrates tyical dynamics for a single variable in a network with N = 50, K =8. The same connection matrix was assumed for three values, =,0.8, In fig. (a) and (b), the symbolic transition sequences corresonding to the time series do not show eriodicities. The Lyaunov number, which can be evaluated numerically using techniques in Ott [4], [5], is ositive for anels (a) and (b) and negative for anel (c). Based on the above, the dynamics are deterministic chaos in anels (a) and (b) and eriodic in anel (c). Figure 2 shows the numbers of networks dislaying steady states, limit cycles, and chaos for K =8andN= 50 anel (a), N = 00 anel (b), and N = 200 anel (c), for 50 different randomly generated networks for [,.0] incremented in stes of 2. As increases the number of chaotic networks decreases and the number of networks dislaying steady states increases. The values of associated with limit cycles are centered in an increasingly narrow range as N increases. Since the total number of orthants of hase sace that are visited during the simulations is a tiny fraction of the 2 N orthants of hase sace, we cannot exclude the ossibility that networks classified as chaotic will eventually reach a stable steady state or limit cycle if the integration times are extended. We now characterize the dynamics in eq. (5). Let j designate the time interval (t j,t j+ ), in eq. (6). At any given time in the interval j, the state in the continuous equation is maed to the logical state X(j). The distance between two logical states is the number of variables in which the activities differ. We call the distance between X(j) andx(j+ ), determined

4 602 EUROPHYSICS LETTERS from eq. (2), the outflow dimension, h(j), of X(j). The outflow dimension is a measure of the number of different variables that have the otential to cross 0 at the next switch time of the network [9]. If h(j) = 0, then the system will aroach a steady state. Using methods similar to those develoed in earlier work by Derrida and others [2], [5], [8], we comute the mean value of h. Denote the number of variables in logical state α, and with truth table entry β in time interval j by N αβ (j), where α, β {0,}.IfN 0 (j) =N 0 (j) =0, there is a fixed oint. If this is not the case there will eventually be a transition to a new state. The robability that there will be a transition of a variable from state 0 to state is P 0 = N 0 (j)/(n 0 (j)+n 0 (j)), and the robability that there will be a transition of a variable from state to state 0 is P 0 = N 0 (j)/(n 0 (j)+n 0 (j)). The exected number of inuts from the switching variable to variables in state (α, β) is ρn αβ,whereρ=k/n. Now assume that there is a transition of a variable from 0 at switch time t j+. We adot a robabilistic aroach to determine N αβ (j + ). Consider first the value of N 00 (j + ). This may be different from N 00 (j) if inuts from the variable that changed its state have inuts to variables in N 00 that leads to a change to N 0, or inuts to variables in N 0 that leads to changes to state N 00. Thus, we find that N 00 (j +)=N 00 (j) ρ( )N 00 (j)+ρn 0 (j). If there is a transition of a variable from 0attimet j, then the above exression would be changed to N 00 (j +)=N 00 (j)+ ρ( )N 00 (j)+ρn 0 (j). In similar fashion, by weighting the resective transition robabilities, we are led to the following system of equations for N αβ (j +): N 00 (j +)=N 00 (j)+p 0 ρ( )N 00 (j)+ρn 0 (j), N 0 (j +)=N 0 (j) P 0 + ρ( )N 00 (j) ρn 0 (j), N 0 (j +)=N 0 (j) P 0 ρ( )N 0 (j)+ρn (j), (7) N (j +)=N (j)+p 0 +ρ( )N 0 (j) ρn (j). At steady state, we have N αβ (j +)=N αβ (j) =Nαβ, where the asterisk reresents the steady-state value. Substituting this relation in eq. (7), we find that N0 = N 0. Substituting fraction of networks.0 (a) (b) (c) Fig. 2. Variation in the number of networks dislaying steady states (circles), limit cycles (squares), and chaos (diamonds) for K = 8, as a function of, (a)n= 50, (b) N = 00, (c) N = 200. A single initial condition was selected for each of 50 different networks for each value of.

5 l. glass et al.: ordered and disordered dynamics etc. 603 h*/n (a) fraction of networks.0 (b) 2 3 h* Fig. 3. (a) Normalized mean outflow dimension, h /N, as a function of for N = 50 (triangles), N = 00 (asterisks), N = 200 (lus signs) comared with the theoretical result from eq. (9). For each network, the value of h was averaged over 3000 switch times following a transient of switch times. (b) Fraction of chaotic networks as a function of h for N = 50 (triangles), N = 00 (asterisks), N = 200 (lus signs). this result in eq. (7), we find 2 = ρ( )N 00 + ρn0, 2 = ρ( )N 0 ρn. (8) Using the conservation condition, N00 +2N 0 +N = N, we have three simultaneous equations. Solving these equations and recalling that at steady state the mean outflow dimension, h,is given by N0 + N 0 =2N 0, we comute h N = K ( +2K 2K2 ). (9) In fig. 3(a), we comare the theoretical estimates from eq. (9) (solid curve) with the numerical comutations for N = 50, 00, 200. The theoretical estimate lies consistently above the numerically comuted values, but shows a similar functional deendence on. Whenh =0, we recover eq. (). In fig. 3(b), we lot the fraction of chaotic networks as a function of h. The transition occurs aroximately in the range 0 <h <20 for all values of N considered. The results in fig. 3 rovide a challenge for further theoretical analysis. These results have connections with the extensive studies carried out on the discrete time and discrete state sace switching networks [2]-[8]. In contrast to the earlier work, in which all finite networks must eventually cycle in the limit t and which do not therefore admit deterministic chaos, in eq. (5) deterministic chaos is ossible [3], [4]. At the moment, there are no general techniques to assert deterministic chaos in any given network, and it is ossible that eventually networks identified as chaotic here will reach limit cycles or steady states. Nevertheless, the extremely long transient behavior would aear to render these architectures imrobable for the highly constrained dynamics in real biological systems. In the continuous equations, the critical line () defines the line at which h = 0 and almost all networks with c dislay steady states. Thus, the continuous equations show transitions in dynamics even for finite N. Asthevalueofis varied from, there is a transition from chaotic dynamics to steady states, with an intervening zone of eriodic dynamics that becomes increasingly narrow as N increases.

6 604 EUROPHYSICS LETTERS *** This research has been suorted by grants from NSERC and FCAR. We thank T. Mestl, B. Sawhill, and S. Kauffman for helful conversations. The authors thank the Santa Fe Institute for hositality and for the use of comutational facilities. REFERENCES [] Kauffman S. A., J. Theor. Biol., 22 (969) 437; Kauffman S. A., Origins of Order: Self- Organization and Selection in Evolution (Oxford University Press, Oxford) 993. [2] Derrida B. and Pomeau Y., Eurohys. Lett.,, (986) 45; Derrida B. and Pomeau Y., Eurohys. Lett., 2 (986) 739. [3] Derrida B., Philos. Mag., 56 (987) 97. [4] Flyvbjerg H., J. Phys. A, 2 (988) L955. [5] Weisbuch G., Comlex Systems Dynamics (Addison-Wesley, Redwood City) 99. [6] Bastolla U. and Parisi G., Physica D, 98 (996) ; J. Theor. Biol., 87 (997) 7. [7] Bhattacharjya A. and Liang S., Phys. Rev. Lett., 77 (996) 644; Physica D, 95 (997) 29. [8] Luque B. and Solé R.V.,Phys. Rev. E, 55 (997) 257. [9] Glass L., J. Chem. Phys., 63 (975) 325; Glass L. and Pasternack J., J. Math. Biol., 6 (978) 207. [0] Thomas R. and Àri R. D., Biological Feedback (CRC Press, Boca Raton) 990. [] Mestl T., Plahte E. and Omholt S. W., J. Theor. Biol., 76 (995) 29; Mestl T., Plahte E. and Omholt S. W., Dyn. Stab. Systems, 2 (995) 79. [2] Bagley R. J. and Glass L., J. Theor. Biol., 83 (996) 269. [3] Mestl T., Lemay C. and Glass L., Physica D, 98 (996) 35. [4] Mestl T., Bagley R. J. and Glass L., Phys. Rev. Lett., 79 (997) 653. [5] The Lyaunov number can be exlicitly evaluated numerically using techniques in Ott E., Chaos in Dynamical Systems (Cambridge) 993, Chat. 4.