Double Truncation Method for Simulation of Stochastic Chemical Reaction Networks

The Chemical Master Equation (CME) accurately describes the probability distribution of the system under the markov assumption. SSA lifts the curse of dimensionality, but it requires too many realizations, which makes it less practical, in cases of the system consisting of large numbers of molecules or very different time scale reactions. We want to keep track of the probability distribution of the states in a chemical reaction network.

The probability of a state is determined based on the law of mass action, which is an underlying hypothesis from chemical kinetics.[2] It states that the speed of a chemical reaction is proportional to the amount of reactants. For a uni-molecular reaction, the probability is assumed to be proportional to the number of reactants. It is therefore assumed that the probability is proportional to the number of cases encountered.[3].

However, it is very difficult to solve the CME analytically or numerically due to the curse of dimensionality. Due to the lack of boundary conditions, we cannot use conventional schemes as the finite difference method. Since the coefficient implies the probability of the state by the definition of PGF, the sum of the small coefficient can be seen as a weight for the system.

Chemical master equation (CME) is the governing equation for the probability distribution of the states in a chemical reaction system.

Backgrounds

The probability that reaction j occurs in the infinitesimal time dt is given by aj(n)dt. In 1864 Peter Waage and Cato Guldberg discovered the law of mass kinetics that the speed of a chemical reaction depends on the numbers of the reactants. The propensity function for uni-molecular reaction is therefore proportional to the number of reactants.

The reaction constant is an inherent value and means that every encounter does not lead to a reaction.

Chemical Master Equation

Stochastic Simulations

Stochastic Simulation Algorithm

Generate two random numbers to determine the next reaction j and the time until the next reaction τ.

Tau-leaping Method

Although SSA is accurate, but in case the system consists of very different timescale reactions or a large number of molecules, it converges too slowly. The Probability Generating Function (PGF) method gains the upper hand for a high-dimensional calculation compared to CME. The PGF is structurally simple because the probability distribution is nothing more than the coefficients of the terms.

Probability Generating Function (PGF) is an efficient function to describe the probability distribution for a discrete state space. Forz= (z1,· · ·, zA) and n= (n1,· · ·, nA), let's note.

PGF-PDE

Probability Generating Function Method

This boundary condition is trivial and does not give us any clue to solve the given PDE. By taking a semi-analytical approach based on the power series and the Pade approximation, we can handle it. From the definition of PGF and the fact that all events are independent, we can calculate the time-dependent probability.

By putting the initial conditionf0(z) =zn0 in the PGF-PDE, we can derive fn recursively. But it takes some time to approximate Paths and the error cannot be strictly predicted. We propose Double Truncation Method (DTM) for an efficient calculation by taking advantage of the PGF method.

The first trimming for order N in time dt means the maximum number of reactions isN for time. The second trimming for small coefficients means the elimination of rare events in the system. It works well for the first few expressions as long as the time step is infinitesimally small.

Second Truncation

Therefore, we can reasonably conclude the error for the second truncation as above, as long as the first truncation does not have a decisive effect on the system.

Enzyme Kinetics

G 2 /M Transition Model

Brusselator Model

In this work, we propose an efficient method to simulate the chemical reaction network using Probability Generating Function (PGF) method. From conventional methods, such as Chemical Master Equation(CME), Stochastic Simulation Algorithms(SSA) to recently developed PGF method, those methods are less practical in some cases. Although SSA solves the dimensional problem, it is not suitable for the stiff system and too slow.

2004).Biochemical network stochastic simulator (BioNetS): software for stochastic modeling of biochemical networks.BMC bioinformatics 5(1): 24. A general method for numerical simulation of the stochastic time evolution of coupled chemical reactions.Journal of computational physics. Petzold (2007). Stochastic simulation of biochemical systems on the graphics processing unit.Santa Barbara: Department of Computer Science, University of California. 1920). Analytical note on certain rhythmic relations in organic systems. Proceedings of the National Academy of Sciences of the United States of America 6(7): 410.

확률적 화학 동역학 및 준정상 상태 가정: Gillespie 알고리즘에 적용. 화학 물리학 저널. 확률론적 화학 반응 시스템의 강성: 암시적 타우 점프 방법. 화학 물리학 저널. 먼저, 본 논문을 작성하는데 많은 도움을 주신 김필원 교수님께 감사의 말씀을 전하고 싶습니다.

기초과목인 선형대수학 공부부터 연구주제 선정, 논문 작성까지 지도해주시고, 힘들 때 격려해주셔서 감사합니다. 앞으로 누군가를 가르칠 수 있는 상황이 된다면 교수님을 생각하며 가르치고 싶습니다. 제가 질문할 때마다 최선을 다해 답변해 주시고 좋은 질문도 주시는 권봉석 교수님 감사드립니다.

그리고 논문심사위원으로서 많은 조언을 해주신 이창형 교수님, 서병기 교수님에게도 감사의 말씀을 전하고 싶습니다. 그리운 신상묵 교수님에게도 감사의 말씀을 전하고 싶습니다. 그리고 연구실 동료인 선이 박사님에게도 감사의 말씀을 전하고 싶습니다. 문성환, 룸메이트 비누셀비, 그리고 지금은 헤어진 절친한 친구들. 항상 배려심 많고 관대하며 나와 가장 오랜 시간 함께 살면서 고난과 어려움을 함께 해 준 친구들.

마지막으로 늘 응원해주시는 가족들에게 감사 인사를 전하고 싶습니다. 자랑스러운 부모님과 누나, 남동생에게 감사와 사랑을 전하고 싶습니다.