  Molecular Dynamics - Green's Functions Reaction Dynamics (MD-GFRD) is a
multiscale simulation method for particle dynamics or particle-based
reaction-diffusion dynamics that is suited for systems involving low particle
densities. Particles in a low-density region are just diffusing and not
interacting. In this case one can avoid the costly integration of microscopic
equations of motion, such as molecular dynamics (MD), and instead turn to an
event-based scheme in which the times to the next particle interaction and the
new particle positions at that time can be sampled. At high (local)
concentrations, however, e.g. when particles are interacting in a nontrivial
way, particle positions must still be updated with small time steps of the
microscopic dynamical equations. The efficiency of a multi-scale simulation
that uses these two schemes largely depends on the coupling between them and
the decisions when to switch between the two scales. Here we present an
efficient scheme for multi-scale MD-GFRD simulations. It has been shown that
MD-GFRD schemes are more efficient than brute-force molecular dynamics
simulations up to a molar concentration of $10^{2}\mu M$. In this paper, we
show that the choice of the propagation domains has a relevant impact on the
computational performance. Domains are constructed using a local optimization
of their sizes and a minimal domain size is proposed. The algorithm is shown to
be more efficient than brute-force Brownian dynamics simulations up to a molar
concentration of $10^{3}\mu M$ and is up to an order of magnitude more
efficient compared with previous MD-GFRD schemes.
