^{1}, Stephen Smith

^{1}and Ramon Grima

^{1}

### Abstract

The reaction-diffusion master equation (RDME) is a standard modelling approach for understanding stochastic and spatial chemical kinetics. An inherent assumption is that molecules are point-like. Here, we introduce the excluded volume reaction-diffusion master equation (vRDME) which takes into account volume exclusion effects on stochastic kinetics due to a finite molecular radius. We obtain an exact closed form solution of the RDME and of the vRDME for a general chemical system in equilibrium conditions. The difference between the two solutions increases with the ratio of molecular diameter to the compartment length scale. We show that an increase in the fraction of excluded space can (i) lead to deviations from the classical inverse square root law for the noise-strength, (ii) flip the skewness of the probability distribution from right to left-skewed, (iii) shift the equilibrium of bimolecular reactions so that more product molecules are formed, and (iv) strongly modulate the Fano factors and coefficients of variation. These volume exclusion effects are found to be particularly pronounced for chemical species not involved in chemical conservation laws. Finally, we show that statistics obtained using the vRDME are in good agreement with those obtained from Brownian dynamics with excluded volume interactions.

This work was supported by the BBSRC EASTBIO Ph.D. studentship to S.S. and by a Leverhulme grant award to R.G. (No. RPG-2013-171). C.C. thanks Philipp Thomas for useful discussions.

I. INTRODUCTION II. THE CME, RDME, vRDME, AND vCME III. EXACT SOLUTION OF THE CME, RDME, vRDME, AND vCME IN EQUILIBRIUM CONDITIONS A. Global distribution of molecule numbers assuming point particle interactions B. Local distribution of molecule numbers assuming point particle interactions C. Global distribution of molecule numbers for finite size particle interactions D. Local distribution of molecule numbers for finite size particle interactions IV. RELATIONSHIP OF RATE CONSTANTS IN THE CME AND vCME V. STOCHASTIC DESCRIPTION OF CHEMICAL SYSTEMS WITHOUT CHEMICAL CONSERVATION LAWS A. Derivation of Statement 1 B. Dilute limit C. Statistical measures and physical implications D. Application: Open homodimerisation reaction VI. STOCHASTIC DESCRIPTION OF CHEMICAL SYSTEMS WITH A SPECIAL TYPE OF CHEMICAL CONSERVATION LAWS A. Derivation of Statements 2 and 3 and the dilute limit B. Statistical measures and Physical implications C. Application: Open heterodimerisation reaction VII. STOCHASTIC DESCRIPTION OF CHEMICAL SYSTEMS WITH MORE GENERAL CHEMICAL CONSERVATION LAWS A. Closed dimerisation reaction VIII. COMPARISON OF THE vRDME WITH BROWNIAN DYNAMICS IX. SUMMARY AND CONCLUSION

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### Abstract

The reaction-diffusion master equation (RDME) is a standard modelling approach for understanding stochastic and spatial chemical kinetics. An inherent assumption is that molecules are point-like. Here, we introduce the excluded volume reaction-diffusion master equation (vRDME) which takes into account volume exclusion effects on stochastic kinetics due to a finite molecular radius. We obtain an exact closed form solution of the RDME and of the vRDME for a general chemical system in equilibrium conditions. The difference between the two solutions increases with the ratio of molecular diameter to the compartment length scale. We show that an increase in the fraction of excluded space can (i) lead to deviations from the classical inverse square root law for the noise-strength, (ii) flip the skewness of the probability distribution from right to left-skewed, (iii) shift the equilibrium of bimolecular reactions so that more product molecules are formed, and (iv) strongly modulate the Fano factors and coefficients of variation. These volume exclusion effects are found to be particularly pronounced for chemical species not involved in chemical conservation laws. Finally, we show that statistics obtained using the vRDME are in good agreement with those obtained from Brownian dynamics with excluded volume interactions.

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