A hybrid algorithm for coupling partial differential equation and compartment-based dynamics

Abstract

Stochastic simulation methods can be applied successfully to model exact spatio-temporally resolved reaction–diffusion systems. However, in many cases, these methods can quickly become extremely computationally intensive with increasing particle numbers. An alternative description of many of these systems can be derived in the diffusive limit as a deterministic, continuum system of partial differential equations (PDEs). Although the numerical solution of such PDEs is, in general, much more efficient than the full stochastic simulation, the deterministic continuum description is generally not valid when copy numbers are low and stochastic effects dominate. Therefore, to take advantage of the benefits of both of these types of models, each of which may be appropriate in different parts of a spatial domain, we have developed an algorithm that can be used to couple these two types of model together. This hybrid coupling algorithm uses an overlap region between the two modelling regimes. By coupling fluxes at one end of the interface and using a concentration-matching condition at the other end, we ensure that mass is appropriately transferred between PDE- and compartment-based regimes. Our methodology gives notable reductions in simulation time in comparison with using a fully stochastic model, while maintaining the important stochastic features of the system and providing detail in appropriate areas of the domain. We test our hybrid methodology robustly by applying it to several biologically motivated problems including diffusion and morphogen gradient formation. Our analysis shows that the resulting error is small, unbiased and does not grow over time.

Spatially extended hybrid methods: a review

“…All of the macro-meso hybrid papers present novel methods rather than applications of pre-existing methods to real-world systems. paper type system modelled Yates & Flegg [ 41 ] spatially coupled, non-adaptive, non-overlap reaction–diffusion Moro [ 46 ] spatially coupled, non-adaptive, non-overlap reaction–diffusion Spill et al [ 73 ] spatially coupled, adaptive, non-overlap reaction–diffusion Schulze et al [ 74 ] spatially coupled, adaptive, no-overlap epitaxial growth Harrison & Yates [ 75 ] spatially coupled, adaptive, overlap reaction–diffusion Flekkøy et al [ 76 ] spatially coupled, non-adaptive, overlap reaction–diffusion Rossinelli et al [ 77 ] operator splitting reaction–diffusion Lo et al [ 78 ] operator splitting reaction–diffusion Chiam et al [ 79 ] propensity-based spatial splitting reaction–diffusion …”

“…Macro-meso models are used when we want to simulate a region of the domain in which stochastic variation is important but in which the exact locations of every particle are not required, while for the remainder of the domain we have sufficiently high copy numbers to employ the associated continuum model. Typical examples to which these hybrid methods have been applied are the simulation of travelling wave phenomena [ 46 , 75 ]. Behind the wavefront, we have a large number of particles so that the continuum limit is valid, while in front of the wave, fluctuations can play a prominent role in the overall dynamics, including the wave speed.…”

“…Typically, these regions inherit properties from both of the models that are being coupled. Harrison & Yates [ 75 ] use such a region to couple their mesoscopic and macroscopic models of reaction–diffusion. The authors suggest a fixed-time-step, finite-difference scheme for the numerical solution of the macroscopic PDE and use a time-driven algorithm for simulating the stochastic regime (with the same fixed time step as the PDE).…”

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Figure 4. A schematic for the method of Harrison & Yates [ 75 ]. The descriptions for the green line and blue bars are the same as in figure 1 .

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“…Similarly to Harrison & Yates [ 75 ], Flekkøy et al [ 76 ] use an overlap region as part of a non-adaptive algorithm. They introduce a method for coupling a discretized version of the diffusion equation with a discrete-time and -space mesoscopic Markov chain representation of diffusion in which particles can jump to neighbouring voxels in each fixed time step.…”