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Computational Cellular Dynamics Based on the Chemical Master Equation: a Challenge for Understanding Complexity

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Computational Cellular Dynamics Based on the Chemical Master Equation: a Challenge for Understanding Complexity Liang J, Qian H. Computational cellular dynamics based on the chemical master equation: A challenge for understanding complexity. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 25(1): 1{ Jan. 2010 Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity Jie Liang1;2 (梁 杰) and Hong Qian3;4 (钱 纮) 1Department of Bioengineering, University of Illinois at Chicago, Chicago, IL 60607, U.S.A. 2Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China 3Department of Applied Mathematics, University of Washington, Seattle, WA 98195, U.S.A. 4Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing 100190, China E-mail: [email protected]; [email protected] Received August 29, 2009; revised November 16, 2009. Abstract Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-¯ve years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the ¯eld that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume (i.e., mesoscopic) in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation (CME). Similar to the wavefunction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved: multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic di®erential equations that o®er alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand \complex behavior" and complexity theory, and from which important biological insight can be gained. Keywords biochemical networks, cellular signaling, epigenetics, master equation, nonlinear reactions, stochastic modeling 1 Introduction The Law of Mass Action and the CME are two parts of a single mathematical theory of chemical reaction Cellular biology has two important foundations: ge- systems, with the latter being fundamental. When the nomics focuses on DNA sequences and their evolu- number of molecules in a reaction system are large, tionary dynamics; and biochemistry studies molecular stochasticity in the CME disappears and the Law of reaction kinetics that involve both small metabolites Mass Action can be shown, mathematically, to arise as and large macromolecules. Computational science has the limit[1-2]. been an essential component of genomics. In recent In this article, we shall introduce the CME approach years, cellular biochemistry is also increasingly relying to biochemical reaction kinetics. We use simply exam- on mathematical models for biochemical reaction net- ples to illustrate some of the salient features of this works. Two approaches have been particularly promi- yet to be fully developed theory. We then discuss the nent: the Law of Mass Action for deterministic non- challenges one faces in applying this theory to computa- linear chemical reactions in terms of the concentrations tional cellular biology. There have been several recent of chemical species, and the Chemical Master Equation texts which cover some of the materials we discuss. See (CME) for stochastic reactions in terms of the numbers [2-3]. of reaction species. Survey This work is supported by US NIH under Grant Nos. GM079804, GM081682, GM086145, GM068610, NSF of USA under Grant Nos. DBI-0646035 and DMS-0800257, and `985' Phase II Grant of Shanghai Jiao Tong University under Grant No. T226208001. ©2010 Springer Science + Business Media, LLC 2 J. Comput. Sci. & Technol., Jan. 2010, Vol.25, No.1 2 A System of Nonlinear Reactions For complex biochemical reactions, master equa- tion like this in general cannot be solved analytically. To illustrate the theory of the CME and the Law Various algorithms exist for simulating its stochastic of Mass Action, let us ¯rst consider a simple system of [8] [4] trajectories . For the above speci¯c example, however, nonlinear chemical reactions ¯rst proposed by SchlÄogl the exact stationary probability distribution to (3), i.e., ® ¯1 after the system reaches stationarity, can be found as A + 2X ­1 3X; B + X ­ C; (1) ®2 ¯2 [9-10]: kY¡1 v in which species A, B and C are at ¯xed concentrations P (k) = C j ; (4) 0 w a, b and c, respectively. The traditional, macroscopic j=0 j kinetics of the system (1), according to the Law of Mass where C0 is a normalization constant such that P1 Action, is described by a deterministic ordinary di®er- k=0 P (k) = 1. The number of X molecules still fluc- ential equation (ODE)[5] tuates in the steady state. We note that for large V , k¡1 k¡1 dx 3 2 X v X v(k=V ) = ¡®2x + ®1ax ¡ ¯1bx + ¯2c; (2) ln P (k) = ln j + C ¼ ln dt w 1 w(k=V ) j=0 j j=0 where x represents the concentration of X. It is ³ ´ Z k=V straightforward to show that (2) exhibits bistabi- 1 v(z) + o + C1 ¼ V ln dz + C1; lity (via the so called pitchfork bifurcation) when V 0 w(z) 2 [4-5] ®2¯1b=(®1a) = 1=3 : that is, the polynomial on in which the right-hand-side switches from having only one pos- itive root to have three positive roots. The system v(z) = z2 + σ; w(z) = z3 + ¹z; also shows another bifurcations when varying another 2 2 3 2 3 ¹ = ®2¯1b=(®1a) , σ = ®2¯2c=(®1a) , and C1 = ln C0. lumped parameter ®2¯2c=(®1a) (this time via the so called saddle-node bifurcation). Therefore in terms of the concentration x = k=V , we We now turn to the CME approach to this reaction have the probability distribution f(x) = VP (V x): system (1). If in a small volume such as that of a cell, 1 1 ln f(x) = ln P (V x) + C the number of X is su±ciently small, its concentration 2V 2V 2 fluctuations become signi¯cant[6]. The dynamics of re- Z x 1 v(z) ^ action (1) then is stochastic, which should be described ¼ ln dz + C (5) 2 0 w(z) in terms of a master equation, also known as a birth- Z 1 x z2 + σ death process in the theory of Markov processes[7]. ^ = ln 3 dz + C: (6) The system is represented by a discrete random vari- 2 0 z + ¹z able nX (t): the number of X at time t (0 6 nX < 1). Therefore, the stationary probability distribution of Let P (k; t) = PrfnX (t) = kg, and we have the concentration of X: dP (k; t) ¡VÁ(x) = v P (k ¡ 1; t) + w P (k + 1; t) f(x) ¼ e ; (7) dt k¡1 k ¡ (v + w )P (k; t); (3) where Z k k¡1 x z2 + σ Á(x) = ¡ ln 3 dz; (8) where 0 z + ¹z ®1ak(k ¡ 1) is independent of V . It is easy to verify that Á(x) is vk = + ¯2c; V 2 at its extrema exactly when the ODE (2) is at its ¯xed and points. The function Á(x) can be thought as a \land- ® (k + 1)k(k ¡ 1) ¯ b(k + 1) scape" for the nonlinear chemical reaction system. w = 2 + 1 : k V 3 V Closed System, Detailed Balance and Chemical Equi- librium. A chemical equilibrium is reached in the reac- Here V is the volume of the reaction system. It is tion system (1) when a very important parameter of the model. The basic 3 rule is still the Law of Mass Action: the rate of one [X] ®1 [C] ¯1 ¯ 1 2 = ; = : (9) step reaction B + X ¡! C, when there are k + 1 num- [A][X] ®2 [B][X] ¯2 ber of X molecules, is ¯ b(k + 1)=V . This gives the 1 This leads to the equilibrium condition that above last term. Similarly, the rate of one step reac- ® 1 ³ ´eq tion A + 2X ¡! 3X, when there are k number of X [C] ®1¯1 2 = : (10) molecules, is ®1ak(k ¡ 1)=V . [A][B] ®2¯2 Jie Liang et al.: Computational Cellular Dynamics Based on the CME 3 In term of the two model parameters ¹ and σ intro- precisely the extrema of Á(x) where duced above, this equilibrium (also called detailed ba- lance) condition is expressed as x2 + σ Á0(x) = ¡ ln = 0: (16) x3 + ¹x 2 3 σ ®2¯2c=(®1a) ®2¯2c = 2 = = 1: (11) Therefore, the nonlinear chemical reaction system is ¹ ®2¯1b=(®1a) ®1¯1ab bistable. The dynamics of the system exhibits multiple This equation has a very strong thermodynamic mean- time scale: the relaxation within each \well" and tran- ing: the term ln(σ=¹) = ¢G=(kBT ) is the chemical sitions between the two wells. The former can be accu- potential di®erence between A + B and C. If one con- rately described by a Gaussian (linear) random process. siders A + B and C as two nodes in a circuit, then ¢G The latter, as two-state transitions, is on a much longer is the potential between them.
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