Spatially extended hybrid methods: a review

Abstract

Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as partial differential equations are typically fast to simulate, they lack the individual-level detail that may be required in regions of low concentration or small spatial scale. However, to simulate at such an individual level throughout a domain and in regions where concentrations are high can be computationally expensive. Spatially coupled hybrid methods provide a bridge, allowing for multiple representations of the same species in one spatial domain by partitioning space into distinct modelling subdomains. Over the past 20 years, such hybrid methods have risen to prominence, leading to what is now a very active research area across multiple disciplines including chemistry, physics and mathematics. There are three main motivations for undertaking this review. Firstly, we have collated a large number of spatially extended hybrid methods and presented them in a single coherent document, while comparing and contrasting them, so that anyone who requires a multiscale hybrid method will be able to find the most appropriate one for their need. Secondly, we have provided canonical examples with algorithms and accompanying code, serving to demonstrate how these types of methods work in practice. Finally, we have presented papers that employ these methods on real biological and physical problems, demonstrating their utility. We also consider some open research questions in the area of hybrid method development and the future directions for the field.

Keywords: hybrid modelling; modelling; multiscale; reaction–diffusion.

Figure 1.

A schematic for the PCM [41]. The green line represents the PDE solution, while the blue boxes represent particles within each compartment. The red line denotes the interface between the two subdomains. The green boxes residing in the pseudo-compartment represent the number of pseudo-particles within the pseudo-compartment, calculated by direct integration of the solution over that region. The arrows in the centre represent the movement of pseudo-particles over the interface between the pseudo-compartment and the first compartment of the mesoscopic domain. (Online version in colour.)

Figure 2.

A replication of results from Yates & Flegg [41] using the PCM. The green line corresponds to the PDE part of the hybrid solution, the red vertical line at x = 0 is the interface and the blue bars are the compartment-based part of the hybrid solution. The dashed black line is the analytical solution of the mean-field PDE model (the diffusion equation) across the entire domain. Parameter values are as in the text. (Online version in colour.)

Figure 3.

A schematic for the method from Spill et al. [73]. The green line and blue boxes are as in figure 1, while the red boxes denote an extra compartment between the PDE and compartment subdomains. The coloured double-headed arrows denote how the flux over each of the two red interfaces are calculated. (Online version in colour.)

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. The overlap region is denoted by the red region. The width of the overlap region can be any integer number of compartment widths (here, for simplicity, we have chosen a two-compartment-width overlap region). In the overlap region, the sum of the densities of the two methods gives the overall solution. (Online version in colour.)

Figure 5.

Schematic for the GCM [43]. The blue boxes represent particles within each compartment and the yellow dots represent individual particles. These particles are shown with a volume, but in the simulations do not have a mass or volume. The particles reside on the one-dimensional line, but have been illustrated in the plane in order to show the directions and magnitudes of their next movement clearly (black arrows). The yellow boxes within the ghost cell correspond to the number of Brownian particles which reside within it. The coloured arrows in the centre are similar to those in figure 1. (Online version in colour.)

Figure 6.

A replication of results from the GMC [43]. Descriptions are as in figure 2, with the addition that yellow bars denote the ‘binned’ solution of the individual-based simulation in the hybrid method. Parameter values are as in the text. (Online version in colour.)

Figure 7.

(a) Schematic for the TRM [42]. The blue blocks and yellow dots are as described in figure 5. The arrow from left to right over the interface denotes the jump in this direction, with the specified altered jump rate. In this jump rate, D is the macroscopic diffusion coefficient, h_c is the width of a compartment and Δt is the time step used to evolve the particles in the Brownian-based subdomain. The other cross interface arrow represents jumps in the other direction. The yellow rectangle and blue particle near the interface represent particles converted from one modelling regime to the other upon crossing the interface in either direction according to the method described. (b) Schematic for the application of the TRM to the problem of calcium-induced calcium release [1]. The blue outlined box denotes the outer boundaries of the compartment-based subdomain. All boundaries are absorbing, apart from the grey one (bottom), which is reflective. The yellow box in the centre of the lower face is the microscopic subdomain, containing nine ion channels (yellow circles). For simplicity, no particles or compartments are displayed in this schematic. (Online version in colour.)

Figure 8.

Schematic for the ARM [44]. The green line and yellow dots represent the same phenomena as in figures 1 and 5, respectively. The auxiliary regions on either side of the interface are highlighted in red. The green and yellow boxes within auxiliary regions represent compartment-based particle numbers in the PDE and Brownian auxiliary regions, respectively. The coloured arrows in the centre represent the conversion of particles between the mesoscopic and microscopic auxiliary regions, similar to those in figure 1. (Online version in colour.)

Figure 9.

Replication of results from the ARM [44]. Descriptions for the PDE and Brownian domains are as in figures 2 and 6, respectively, with parameter values given in the text. (Online version in colour.)

Figure 10.

Schematic for the method by Franz et al. [89] (without overlap region). The green line and yellow dots represent the same quantities as in figure 8. The orange mass labelled α is the amount of mass that flows over the interface in a small time interval (comprising several PDE updates). Its total mass is used to find the probability of a particle being initialised in the microscopic subdomain, and its profile acts as a scaled probability density function for the position of the new molecule. The spike in the PDE solution is representative of a Dirac delta function which is added to the PDE at the location that a Brownian particle has jumped to from the Brownian subdomain. (Online version in colour.)

Figure 11.

(a) Schematic for the method presented by Geyer et al. [88]. The green lines and yellow dots represent the same phenomena as in figure 8. The additional green line which resides in the microscopic subdomain is the mass which flows over the interface after a given time, where ρ₀ is the density at the PDE mesh point adjacent to the interface and is the average Brownian step size during a time interval of length Δt. (b) Schematic for the application presented by Gorba et al. [87]. The yellow dots are the same as in figure 8, while the green region is a constant density heat-bath. There are reflective boundary conditions on all sides of the computational domain, with the exception of the lower boundary, denoted in orange. This is a repulsive boundary caused by the van der Waals forces, representing the charged boundary. (Online version in colour.)

Figure 12.

Schematic for the method by Alexander et al. [90]. The green line and yellow dots represent the same phenomena as in figure 8. The green dots residing within the PDE subdomain are particles initialized at the beginning of a time step (the numbers of particles within the corresponding region obtained by direct integration of the PDE solution). Black arrows show the directions and magnitudes of next movement of all particles. (Online version in colour.)

Figure 13.

A schematic for the method presented by Erban [45]. The large yellow circle is an individual particle (protein molecule) with mass, volume and velocity. The small purple dots represent the molecular dynamics particles (air/water molecules) and also have a mass, volume and velocity. (Online version in colour.)