Deterministic and Stochastic Approach for Modelling Biochemical Reaction Systems

Abstract

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

Biyokimyasal Reaksiyon Sistemlerinin Modellenmesi için Deterministik ve Stokastik Yaklaşım

Abstract

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.

Keywords

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

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