  Open biochemical systems of interacting molecules are ubiquitous in
life-related processes. However, established computational methodologies, like
molecular dynamics, are still mostly constrained to closed systems and
timescales too small to be relevant for life processes. Alternatively,
particle-based reaction-diffusion models are currently the most accurate and
computationally feasible approach at these scales. Their efficiency lies in
modeling entire molecules as particles that can diffuse and interact with each
other. In this work, we develop modeling and numerical schemes for
particle-based reaction-diffusion in an open setting, where the reservoirs are
mediated by reaction-diffusion PDEs. We derive two important theoretical
results. The first one is the mean-field for open systems of diffusing
particles; the second one is the mean-field for a particle-based
reaction-diffusion system with second-order reactions. We employ these two
results to develop a numerical scheme that consistently couples particle-based
reaction-diffusion processes with reaction-diffusion PDEs. This allows modeling
open biochemical systems in contact with reservoirs that are time-dependent and
spatially inhomogeneous, as in many relevant real-world applications.
