  Biochemical networks play a crucial role in biological systems, implementing
a broad range of vital functions. They normally operate at low copy numbers and
in spatial settings, but this is often ignored and well-stirred conditions are
assumed. Yet, it is increasingly becoming clear that even microscopic spatial
inhomogeneities oftentimes can induce significant differences on the
macroscopic level. Since experimental observation of single-molecule behavior
is extremely challenging, theoretical modeling of biochemical reactions on the
single-particle level is an important tool for understanding spatial effects in
biochemical systems. While purely analytical models quickly become intractable
here, spatial-stochastic simulations can capture a wide range of biochemical
processes with the necessary levels of detail. Here we discuss different
techniques for spatial-stochastic simulation of reaction-diffusion systems, and
explain important precautions 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. We first explain how Brownian Dynamics, a widely used particle-based
diffusion simulation technique with fixed propagation time, can be adapted to
incorporate chemical reactions, and portray a range of schemes that elaborate
on this idea. We then introduce event-driven spatial-stochastic simulation
methods, in which system 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 motion
with large jumps in time and space. We end by briefly presenting recent
developments in the field of spatial-stochastic biochemical simulation.
