Spatial Regime Conversion Method (SRCM)

• Spatial Regime Conversion Method (SRCM) is an adaptive hybrid framework that dynamically switches between stochastic and deterministic regimes based on local concentration thresholds.
• It integrates discrete reaction–diffusion master equations with continuum partial differential equations, ensuring local mass conservation through threshold-driven conversion reactions.
• Empirical analyses of diffusion, morphogen gradients, and FKPP travelling waves demonstrate SRCM’s efficiency and accuracy in capturing both low-copy stochastic dynamics and high-concentration behavior.

The Spatial Regime Conversion Method (SRCM) is an adaptive hybrid modelling framework for simulating reaction–diffusion systems that combines stochastic discrete and deterministic continuum representations within a single spatially distributed model. In its 1D formulation, it represents low-concentration regions by a discrete reaction–diffusion master equation (RDME) and high-concentration regions by reaction–diffusion partial differential equations (PDEs), while allowing both representations to coexist within the same compartment and dynamically switching mass between them through threshold-driven conversion reactions. The method was introduced as a spatial extension of the well-mixed Regime Conversion Method, with explicit emphasis on local mass conservation, mixed discrete–continuous kinetics, and the avoidance of a fixed stochastic–deterministic interface (Cameron et al., 7 Jul 2025).

1. Lineage and defining idea

SRCM extends the earlier well-mixed regime-conversion framework developed for chemical reaction networks, in which each original species is split into a discrete component and a continuous component, with total abundance recovered by summation. In the precursor formulation, the key invariant is

$X_k(t)=C_k(t)+D_k(t),$

where $D_k$ is simulated as a discrete Markov jump process and $C_k$ is evolved by an ODE approximation. Conversion reactions

$C_k \rightleftharpoons D_k$

move mass between the two regimes without changing the total $X_k$, and the extended network is constructed so that, if treated fully stochastically, it is stochastically equivalent to the original well-mixed system (Kynaston et al., 2022).

SRCM carries this molecule-conserving logic into space. It replaces the well-mixed ODE component by a spatial PDE component, replaces the single compartment by a lattice of spatial compartments, and makes regime conversion local to each compartment. The result is essentially “RCM + space + diffusion + PDE solver + spatially local conversion logic”, with the same design objective as the well-mixed method: preserve stochasticity where copy numbers are low, exploit deterministic efficiency where concentrations are high, and move between these descriptions adaptively rather than by a static partition (Cameron et al., 7 Jul 2025).

This distinguishes SRCM from traditional spatial hybrid methods that split the domain into two subdomains with a prespecified interface. In SRCM there is no fixed spatial interface. Any compartment may contain both discrete and continuous mass, and any compartment may switch regime over time. This local rather than interface-based regime logic is one of the method’s defining features.

2. Spatial state space and governing equations

The spatial setting is a 1D domain $\Omega=[0,L]$ containing $M$ species $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$. The domain is partitioned into $K$ compartments $\{\Omega_k\}_{k=1}^K$ of equal length $D_k$0. In the fully stochastic description, $D_k$1 denotes the number of particles of species $D_k$2 in compartment $D_k$3, so the state lies in $D_k$4. The RDME includes both diffusion events between neighbouring compartments and local reaction events within each compartment. Diffusion is parameterized so that

$D_k$5

with $D_k$6 the macroscopic diffusion coefficient and $D_k$7 the per-particle jump rate. Local reactions may be zeroth-, first-, or second-order, with standard compartment-wise stoichiometric updates (Cameron et al., 7 Jul 2025).

The continuum description uses concentrations $D_k$8 governed by reaction–diffusion PDEs of the form

$D_k$9

Numerically, the PDE is solved on a finer grid $C_k$0, $C_k$1, with $C_k$2 for an integer refinement factor $C_k$3, finite differences in space, and 4th-order Runge–Kutta in time. Grid points are mapped to compartments by the left-hand rule: indices $C_k$4 belong to $C_k$5 (Cameron et al., 7 Jul 2025).

SRCM superposes these two descriptions. For each species $C_k$6 and compartment $C_k$7, it stores a discrete component

$C_k$8

and a continuous pseudo-particle mass

$C_k$9

where $C_k \rightleftharpoons D_k$0 is the PDE field used internally by SRCM. Total mass in the compartment is then

$C_k \rightleftharpoons D_k$1

The extended state matrix $C_k \rightleftharpoons D_k$2 stacks the discrete and continuous pseudo-particle components. Consistency between the PDE grid and the compartment-scale hybrid state is maintained explicitly: if an SSA event changes $C_k \rightleftharpoons D_k$3 by $C_k \rightleftharpoons D_k$4, then the PDE field is updated in that compartment by

$C_k \rightleftharpoons D_k$5

This local increment or decrement is the basic mass-transfer mechanism linking the SSA state and the PDE state (Cameron et al., 7 Jul 2025).

3. Local regime conversion and hybrid reaction semantics

Regime switching is implemented by formal conversion reactions between continuous and discrete representations. For species $C_k \rightleftharpoons D_k$6 in compartment $C_k \rightleftharpoons D_k$7, continuous-to-discrete conversion has stoichiometric update

$C_k \rightleftharpoons D_k$8

so that

$C_k \rightleftharpoons D_k$9

The associated PDE update is

$X_k$0

If any $X_k$1, the event is aborted to avoid negative PDE concentrations. Discrete-to-continuous conversion is the reverse process,

$X_k$2

with

$X_k$3

and PDE increment

$X_k$4

These updates enforce exact local mass conservation between the two representations (Cameron et al., 7 Jul 2025).

Activation of conversion is threshold-based. Each species $X_k$5 is assigned a concentration threshold $X_k$6, so the corresponding mass threshold in compartment $X_k$7 is $X_k$8. With a global conversion rate $X_k$9, the compartment-wise conversion rates are

$\Omega=[0,L]$0

Thus, if total mass in a compartment falls below threshold, continuous mass is gradually converted to discrete particles; if it rises above threshold, discrete particles are gradually converted to continuous mass. The representation therefore follows local abundance rather than a global fast–slow decomposition (Cameron et al., 7 Jul 2025).

Beyond conversion itself, SRCM specifies hybrid kinetic rules for assigning products to regimes. Rule (C1) states that products of species $\Omega=[0,L]$1 inherit the representation of their reactant(s), with homodimerisation between discrete and continuous forms sending products to the discrete regime. Rule (C2) places any product species not present among the reactants into the discrete regime, in order to preserve stochasticity for newly created species. Rule (C3) states that if all reactants are continuous, then all products are continuous and the reaction is handled by the PDE only. These rules ensure that low-copy interactions remain stochastic, fully continuous channels reduce to mean-field PDE kinetics, and regime switching is not introduced unnecessarily (Cameron et al., 7 Jul 2025).

For second-order chemistry this leads to a characteristic reaction splitting. Each original second-order reaction generates three discrete or mixed subreactions, $\Omega=[0,L]$2, $\Omega=[0,L]$3, and $\Omega=[0,L]$4, plus one continuous $\Omega=[0,L]$5 reaction handled by the PDE. More generally, if the original network contains $\Omega=[0,L]$6 zeroth-order, $\Omega=[0,L]$7 first-order, and $\Omega=[0,L]$8 second-order reactions, then the extended SRCM reaction count is

$\Omega=[0,L]$9

4. Coupled SSA–PDE algorithm

SRCM combines an event-driven SSA for all conversion reactions, discrete diffusion events, and kinetic reactions involving at least one discrete reactant, with a fixed-step PDE solver for continuous-only kinetics and diffusion. For compartment $M$0, the reaction list is

$M$1

where $M$2, the continuous-only kinetic channels, are removed from the SSA and absorbed into the PDE step (Cameron et al., 7 Jul 2025).

Compartment-wise propensities are assembled into

$M$3

with total propensity

$M$4

Mixed second-order propensities are formed directly from the hybrid state. In the FKPP example, for instance,

$M$5

The SSA waiting time is then sampled from an exponential law with rate $M$6, i.e.

$M$7

for $M$8 (Cameron et al., 7 Jul 2025).

The algorithm compares the tentative event time $M$9 with the next PDE update time $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$0. If $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$1, an SSA event occurs first. A compartment $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$2 is chosen according to $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$3, then a reaction $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$4 within that compartment according to $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$5. If the stoichiometric vector $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$6 changes any continuous pseudo-particle component $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$7, the PDE grid in $$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$8 is updated immediately by the corresponding mass increment or decrement. The extended state is then advanced by

$$\mathbf{A}=[\mathrm{A}^{(1)},\dots,\mathrm{A}^{(M)}]$$9

If instead $K$0, the PDE is advanced from $K$1 to $K$2 by finite differences in space and 4th-order Runge–Kutta in time, yielding updated grid values $K$3. The compartment-scale continuous pseudo-particle masses are then recomputed by numerical quadrature,

$K$4

the extended state is refreshed, and the next PDE time is set to $K$5. In this way, the method interleaves variable-time stochastic events with fixed-time deterministic updates. Stability constraints for the PDE solver remain standard, with $K$6 chosen according to diffusion-dominated finite-difference restrictions (Cameron et al., 7 Jul 2025).

5. Validation systems and empirical behaviour

The method was validated on four one-dimensional systems: diffusion from a concentrated region, alternating growth and decay, morphogen gradient formation, and FKPP travelling-wave propagation. Across these tests, SRCM was compared against canonical fully stochastic RDME simulations and pure PDE solutions using mean profiles, normalized histogram distances, and, for travelling waves, wave-speed estimates (Cameron et al., 7 Jul 2025).

For pure diffusion from a top-hat initial condition, all mass is initially discrete in two adjacent compartments. As the profile smooths, compartments with total mass above threshold convert to continuous representation, while low-density tails remain discrete. At later times, when concentration falls below threshold everywhere, mass converts back to discrete form. The combined SRCM profile $K$7 closely matches both the ensemble-averaged SSA and the PDE mean. Error is quantified by normalized histogram distance, with hybrid-versus-SSA error $K$8, SSA self-error $K$9, and PDE-versus-SSA error $\{\Omega_k\}_{k=1}^K$0. In this case, $\{\Omega_k\}_{k=1}^K$1 is slightly higher at early times because conversion events are frequent, but later fluctuates around or below $\{\Omega_k\}_{k=1}^K$2, while $\{\Omega_k\}_{k=1}^K$3 remains small.

In the alternating growth-and-decay system, the reactions switch between

$\{\Omega_k\}_{k=1}^K$4

A key subtlety is that zeroth-order production is implemented only in the discrete regime, following rule (C2), to avoid double-counting. Consequently, the SRCM PDE contains only diffusion and decay, while production enters through SSA and may later be converted to continuous mass. For spatially uniform initial data, the ensemble means remain uniform, so the comparison reduces to total mass. SRCM total mass closely tracks both SSA and PDE, and the hybrid error remains comparable to the SSA self-error.

For morphogen gradient formation, production occurs only at the left boundary, degradation is spatially uniform, and diffusion spreads mass across the domain. The mean-field PDE is

$\{\Omega_k\}_{k=1}^K$5

with a production flux boundary condition at $\{\Omega_k\}_{k=1}^K$6. In SRCM, the PDE component uses zero-flux boundaries and production is inserted only through the discrete reaction channels. The resulting hybrid solution shows a high-concentration left region dominated by the continuous regime, a mixed middle region near threshold, and a low-copy right tail dominated by the discrete regime. The leftmost compartment exhibits elevated discrete mass because production is created stochastically and then converted. Hybrid error is slightly larger than, but of the same order as, the SSA self-error.

The FKPP travelling-wave example is the most diagnostically important test because the wave front is pulled and its speed is controlled by low-density leading-edge dynamics. The underlying reactions are

$\{\Omega_k\}_{k=1}^K$7

with PDE

$\{\Omega_k\}_{k=1}^K$8

Under a naive SRCM implementation, first-order production is allowed in both regimes when local mass is above threshold. This yields a wave that initially tracks SSA but gradually accelerates toward the PDE wave speed. The mechanism identified in the analysis is that PDE mass diffuses into the leading edge and is then amplified by PDE growth, making the front dynamics too deterministic. Since FKPP fronts are pulled, even a small PDE contribution at the front changes the macroscopic wave speed.

To correct this, the production channel is modified by an adaptive switching rule based on

$\{\Omega_k\}_{k=1}^K$9

Production is handled by the SSA when $D_k$00 and by the PDE when $D_k$01. The PDE becomes

$D_k$02

where

$D_k$03

With this adaptive production rule, the smoothed SRCM wave speed nearly coincides with the SSA wave speed and remains below the PDE speed. The paper estimates wave speed from total mass by

$D_k$04

following Robinson et al. 2014. In computational terms, the adaptive FKPP study also shows the characteristic efficiency crossover: below threshold, SRCM may be more expensive per run than SSA because the PDE and conversion machinery add overhead; near threshold, runtime peaks because of frequent back-and-forth conversions; at high concentrations, however, SSA runtime grows steeply while SRCM runtime grows much more slowly and becomes significantly more efficient. This is the clearest empirical demonstration of the claim that SRCM offers substantial gains in computational efficiency over fully stochastic models (Cameron et al., 7 Jul 2025).

6. Relation to other hybrid methods, limitations, and scope

SRCM belongs to the broader class of spatial hybrid stochastic–deterministic methods, but its structure is distinct from interface-based domain-decomposition schemes such as the pseudo-compartment method, TRM, and auxiliary-region methods. Those methods partition the domain into stochastic and deterministic subregions and impose special coupling conditions across a sharp interface. SRCM instead permits every compartment to be mixed and to switch regime locally. This eliminates the need to prescribe or track a single moving interface and allows multiple disjoint stochastic patches to arise naturally. In that sense, SRCM inherits from the well-mixed regime-conversion paradigm rather than from interface-tracking PDE/SSA coupling (Kynaston et al., 2022).

Its nearest conceptual relatives are the well-mixed Regime Conversion Method and other hybrid frameworks such as Germano et al.’s JSF framework. The difference is that RCM is non-spatial and JSF uses species-wise switching at the system level, whereas SRCM uses local thresholds per compartment. The method also differs from microscopic–mesoscopic hybrids such as CPM and GCM, whose purpose is to couple Brownian dynamics and compartment-based descriptions rather than discrete and continuum representations of the same compartment. A common misconception is therefore to treat SRCM as merely another moving-interface PDE/SSA scheme; its defining property is instead local mixed representation with ordinary SSA conversion events (Cameron et al., 7 Jul 2025).

Several formal properties delimit the method’s scope. Every conversion event conserves mass exactly. When reactants and products are both discrete, SRCM reduces to the RDME description; when all reactants are continuous, it reduces to the PDE description. In the limit $D_k$05, switching becomes rapid once thresholds are crossed; in the limit of extreme thresholds, SRCM reduces to pure SSA for very high $D_k$06 or pure PDE for very low $D_k$07. These limits make the framework analytically interpretable as a continuously tunable interpolation between fully stochastic and fully deterministic spatial models.

The limitations are also explicit. The current implementation is for 1D domains; extension to 2D and 3D is described as conceptually straightforward but numerically more complex. A single threshold $D_k$08 can induce oscillatory conversions when mass fluctuates around the switching point, so the authors identify two-threshold hysteresis or smooth sigmoidal switching functions as natural future modifications. Problems with pulled fronts, exemplified by FKPP, require care because naive production in both regimes can distort key macroscopic observables such as wave speed. Boundary conditions may differ slightly between the SRCM PDE component and the corresponding pure PDE, as in the morphogen example, producing small systematic discrepancies. Finally, stiff kinetics may necessitate implicit or adaptive PDE solvers rather than explicit Runge–Kutta stepping. These caveats do not alter the central result: SRCM provides a mass-conserving, locally adaptive hybridization of RDME and PDE dynamics that is especially useful when stochasticity is essential only in spatially localized, moving, or transient low-copy regions (Cameron et al., 7 Jul 2025).