Biyokimyasal Reaksiyon Sistemlerinin Modellenmesi için Deterministik ve Stokastik Yaklaşım

Biyokimyasal süreçler, birbirleriyle, farklı reaksiyon kanallarıyla etkileşime giren türleri içeren reaksiyon ağları olarak düşünülebilirler. Deterministik yaklaşım ve stokastik yaklaşım bu sistemlerin dinamiklerini modelleyen iki temel yaklaşımdır. Deterministik yaklaşım geleneksel olandır ve bu tip sistemleri modellemek için Reaksiyon Oran Denklemleri (ROD) adı verilen Adi Diferansiyel Denklemleri (ADD) kullanır. Bu yaklaşıma göre sistem dinamikleri sürekli ve deterministiktir. Diğer taraftan, stokastik yaklaşım sistem dinamiklerinin stokastik ve kesikli olduğunu düşünür. Bu yaklaşımda, sistem dinamiklerini modelleyen olasılık fonksiyonunun zamana göre türevi ünlü Temel Kimyasal Denklemini (TKD) sağlar. Stokastik Simülasyon Algoritmaları (SSAs), TKD’nin davranışlarını tam olarak yansıtan bilgisayar tabanlı algoritmalardır. SSA’nın doğrudan ve ilk reaksiyon metodu olmak üzere iki farklı versiyonu vardır. Bu çalışmada, deterministik ve stokastik yaklaşımın temellerini ve birbirleriyle olan ilişkilerini açıkladık. Farklı boyutlardaki sistemlerin doğrudan metot ve ROD algoritmalarını R programlama dili ile yazdık ve kodlarımız ile birlikte simülasyon sonuçlarımızı sunduk.

Anahtar Kelimeler:

deterministik yaklaşım, stokastik yaklaşım, reaksiyon oran denklemleri, temel kimyasal denklemi, stokastik simülasyon algoritmaları

Deterministic and Stochastic Approach for Modelling Biochemical Reaction Systems

Biochemical processes can be thought as a reaction network containing species interacting with each other via different reaction channels. Deterministic approach, stochastic approach are two fundamental approaches modelling the dynamics of these systems. Deterministic approach is the traditional one and it uses Ordinary Differential Equations (ODEs), namely, Reaction Rate Equations (RREs) to model these kind of systems. According to this approach, the system dynamics are continuous and deterministic. On the other hand, stochastic approach assumes that the system dynamics are stochastic adn deterministic. In this approach, the time derivative of the probability function representing the dynamics of the system satisfies the celebrated Chemical Master Equation (CME). Stochastic Simulation Algorithms (SSAs) are computer based algorithms which generate exact realizations of the given CME. There are two versions of SSAs which are direct method and first reaction method. In this study, we explain the bases of deterministic approach, stochastic approach and their relations with each other. We have written SSA direct and RRE algorithms of systems in different sizes by using R programming language and presented our simulation results together with our codes.

Keywords:

Deterministic approach, Stochastic approach, Reaction rate equations, Chemical master equation, Stochastic simulation algorithms,

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Reference 1: Arkin, A., Ross, J., McAdams, H. H., Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells, Genetics, 149(4), 1633–1648 (1998).
Reference 1-2: Altıntan, D., Koeppl, H. Hybrid master equation for jump diffusion approximation of biomolecular reaction networks,BIT Numerical Mathematics, vol. 60,no. 2, pp. 261. 294, (2020) .
Reference 3: Anderson, D. F., Kurtz, T. G. Continuous time Markov chain models for chemical reaction networks”. Design and analysis of biomolecular circuits. Editörler: Koeppl, H., Setti,G., Bernardo, M. d., Densmore D., New York: Springer-Verlag, (2011).
Reference 4: Arkin, A., Ross, J., McAdams, H. H., Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells, Genetics, 149(4), 1633–1648 (1998).
Reference 5: Cao, Y., Li, H., Petzold, L., Efficient formulation of the stochastic simula- tion algorithm for chemically reacting systems, Journal of Chemical Physics, 121(9), 4059–4067 (2004).
Reference 6: Cao, Y. , Gillespie, D. T. and Petzold, L. R. , The slow-scale stochastic simulation algorithm, J. Chem. Phys., 122 ,014116 (2005).
Reference 7: Cao, Y. , Gillespie, D. T. and Petzold, L. R. , Efficient step size selection for the tau-leaping simulation method,The Journal of Chemical Physics, vol. 124, p.044109, (2006).
Reference 8: Crudu A, Debussche A and Radulescu, Hybrid stochastic simplifications for multiscale gene networks BMC Systems of Biology 3, 89, (2009).
Reference 9: Ganguly, A., Altıntan, D. and H. Koeppl, Jump-diffusion approximation of stochastic reaction dynamics: Error bounds and algorithms,Multiscale Model. Simul., vol. 13, no. 4, pp. 1390-1419, (2015).
Reference 10: Gibson, M. A. and Bruck, J., Efficient exact stochastic simulation of chemical systems with many species and many channels, Journal of Physical Chemistry, 104, 1876-1889 (2000).
Reference 11: Gillespie, D. T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, Journal of Computational Physics, 22(4), 403–434 (1976).
Reference 12: Gillespie, D. T., Exact stochastic simulation of coupled chemical reactions, Journal of Physical Chemistry, 81(25), 2340–2361 (1977).
Reference 13: Gillespie, D. T., A rigorous derivation of the chemical master equation, Physica A, 188(1–3), 404–425 (1992).
Reference 14: Gillespie, D. T., Approximate accelerated stochastic simulation of chemically reacting systems, Journal of Chemical Physics, 115(4), 1716–1733 (2001).
Reference 15: Gillespie, D. T., Stochastic simulation of chemical kinetics, Annual Review of Physical Chemistry, 58, 35–55 (2007).