A Generating Function Approach To Biochemical Reactions With Complex Distribution Profiles

Systems biology has recently emerged as a new discipline to understand how reactions between species lead to specific cellular behavior.At the molec-ular level,chemical reactions between molecules are the most basic form of expression,A variety of biochemical reaction networks can be formed in cells.We can describe the biochemical reactions in the networks by establishing math-ematical models.When the stochasticity of system is neglected,we can use deterministic reaction rate equation to describe the dynamic behavior of each molecule.For most systems,the stochasticity has a significant impact on the change of system state,and deterministic rate equations are no longer applicable.We can choose the master equation to describe the probability density function of system state,but in most cases,the master equation can not be solved analytically.So Gille-spie proposed a stochastic simulation algorithm,which can simulate each possi-ble reaction in turn to generate a reaction trajectory.The probability distribution of each species can be obtained accurately by simulating the reaction trajecto-ries in large quantities.Because Gillespie algorithm needs to simulate every reaction occurring,its simulation efficiency becomes very low.Gillespie then proposed the tau-leaping algorithm.Later,many researchers improved the tau-leaping algorithm.However,the numerical simulation algorithm can not avoid simulating the sample orbits many times,so the computational efficiency with various molecules and more reactions will be significantly reduced.Based on master equation,we can transform a large number of ODEs in the master equation into a small number of ODEs by generating function method and variational method,and obtain a probability density function describing the system state by solving these few ODEs.We need to choose the appropriate left basis function and right basis function to solve ODEs.We need to choose appropriate right and left basis functions.Through the conservation between molecules in the system,we can determine the minimum number of molecules that can describe the system.Then the right basis func-tion is obtained by using various distribution forms and the correlation among molecules in different reaction types.The left basis function can be represented by a linear combination of differential operators.There are two aspects to con-sider in determining this right basis function:1.Determine the distribution form that each type of molecule satisfies,such as Binomial distribution,Poisson dis-tribution,etc.2.Consider the interaction between different type of molecules in the biochemical system.We give several forms of approximate solution for 2-step,3-step,3-step with feedback enzymatic cascade reactions.In addition,we present approxi-mate solutions for three other different types of biochemical reaction models.The approximate solution is in good agreement with the Gillespie algorithm,and the simulation speed of the approximate solution is not affected by the complexity of the model.It can also visually describe the interaction between molecules and the effect of molecules on the whole system.