Spatial-Stochastic Simulation of Reaction-Diffusion Systems

In biological systems, biochemical networks play a crucial role, implementing a broad range of vital functions from regulation and communication to resource transport and shape alteration. While biochemical networks naturally occur at low copy numbers and in a spatial setting, this fact often is ignored and well-stirred conditions are assumed for simplicity. Yet, it is now increasingly becoming clear that even microscopic spatial inhomogeneities can profoundly influence reaction mechanisms and equilibria, oftentimes leading to apparent differences on the macroscopic level. Since experimental observations of spatial effects on the single-particle scale are extremely challenging under in vivo conditions, theoretical modeling of biochemical reactions on the single-particle level is an important tool for understanding spatial effects in biochemical systems. While the combined requirement of incorporating space and stochasticity quickly limits the tractability of purely analytical models, spatialstochastic simulations can capture a wide range of biochemical processes with the necessary minimal levels of detail and complexity. In this chapter we discuss different simulation techniques for spatial-stochastic modeling of reaction-diffusion systems, and explain important working steps required to make them biochemically accurate and efficient. We illustrate non-negligible accuracy issues arising even in the most simple approaches to biochemical simulation, and present methods to deal with them. In the first part of the chapter we explain how Brownian Dynamics, a widely used particle-based diffusion simulation technique with a fixed propagation time, can be adapted to simulate chemical reactions as well, and portray a range of simulation schemes that elaborate on this idea. In the second part, we introduce event-driven spatial-stochastic simulation methods, in which simulation updates are performed asynchronously with situation-dependent, varying time steps; here we particularly focus on eGFRD, a computationally efficient particle-based algorithm that makes use of analytical functions to accurately sample interparticle reactions and diffusive movements with large jumps in time and space. We end by briefly presenting recent developments in the field of spatial-stochastic biochemical simulation. ∗to appear in: B. Munsky, W.S. Hlavacek, and L. Tsimring (editors), “Quantitative Biology: Theory, Computational Methods and Examples of Models”, MIT Press, Cambridge, MA, U. S. A. (expected in early 2018) †e-mail: [email protected] 1 ar X iv :1 70 5. 08 66 9v 1 [ qbi o. M N ] 2 4 M ay 2 01 7