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Approximating particle-based clustering dynamics by stochastic PDEs

Nathalie Wehlitz, Mohsen Sadeghi, Alberto Montefusco, Christof Schütte, Grigorios A. Pavliotis, Stefanie Winkelmann

TL;DR

The paper tackles the challenge of efficiently reproducing particle-based clustering dynamics on membranes by using a regularized Dean–Kawasaki SPDE to model diffusive, pairwise-interacting particles with a Morse potential. It shows that the SPDE captures both the initial cluster formation and long-term merging behavior, matching key statistics of the full particle-based model where mean-field PDE alone fails to account for finite-N fluctuations. By leveraging SPDE simulations, the authors estimate long-time cluster-count statistics and construct a reduced Markov jump process for the number of clusters, obtaining good agreement with the spatially resolved models while achieving substantial computational savings. The work suggests broad applicability for SPDE-based modeling of clustering in biological membranes and provides a pathway to parameter estimation and multi-scale hybrid modeling, with future extensions to higher dimensions and multi-body interactions.

Abstract

This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane-mediated receptor dynamics on cell surfaces, we formulate a stochastic particle-based model for diffusion and pairwise interaction of particles, leading to intriguing clustering phenomena. Employing numerical simulation and cluster detection methods, we explore the approximation of the particle-based clustering dynamics through mean-field approaches. We find that SPDEs successfully reproduce spatiotemporal clustering dynamics, not only in the initial cluster formation period, but also on longer time scales where the successive merging of clusters cannot be tracked by deterministic mean-field models. The computational efficiency of the SPDE approach allows us to generate extensive statistical data for parameter estimation in a simpler model that uses a Markov jump process to capture the temporal evolution of the cluster number.

Approximating particle-based clustering dynamics by stochastic PDEs

TL;DR

The paper tackles the challenge of efficiently reproducing particle-based clustering dynamics on membranes by using a regularized Dean–Kawasaki SPDE to model diffusive, pairwise-interacting particles with a Morse potential. It shows that the SPDE captures both the initial cluster formation and long-term merging behavior, matching key statistics of the full particle-based model where mean-field PDE alone fails to account for finite-N fluctuations. By leveraging SPDE simulations, the authors estimate long-time cluster-count statistics and construct a reduced Markov jump process for the number of clusters, obtaining good agreement with the spatially resolved models while achieving substantial computational savings. The work suggests broad applicability for SPDE-based modeling of clustering in biological membranes and provides a pathway to parameter estimation and multi-scale hybrid modeling, with future extensions to higher dimensions and multi-body interactions.

Abstract

This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane-mediated receptor dynamics on cell surfaces, we formulate a stochastic particle-based model for diffusion and pairwise interaction of particles, leading to intriguing clustering phenomena. Employing numerical simulation and cluster detection methods, we explore the approximation of the particle-based clustering dynamics through mean-field approaches. We find that SPDEs successfully reproduce spatiotemporal clustering dynamics, not only in the initial cluster formation period, but also on longer time scales where the successive merging of clusters cannot be tracked by deterministic mean-field models. The computational efficiency of the SPDE approach allows us to generate extensive statistical data for parameter estimation in a simpler model that uses a Markov jump process to capture the temporal evolution of the cluster number.
Paper Structure (16 sections, 19 equations, 9 figures)

This paper contains 16 sections, 19 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model setup
  3. Stochastic particle-based model
  4. Mean-field approximations
  5. Exemplary numerical simulation
  6. Cluster formation for classical mean-field approximation
  7. Stochastic clustering dynamics
  8. Qualitative spatial comparison
  9. Cluster detection based on relative local maxima
  10. Statistics of cluster counts
  11. Markov jump process for the number of clusters
  12. Discussion and conclusion
  13. Appendix
  14. Numerical discretization of the SPDE
  15. Stability of cluster detection based on relative local maxima
  16. ...and 1 more sections

Figures (9)

  • Figure 1: Individual numerical simulations of initial cluster formation period. Stochastic dynamics of $N=10^3$ particles on the torus of length $L=5$ until time $t=10$, after starting at time $t=0$ with a uniform distribution. (a) spatiotemporal trajectories of the particle-based dynamics \ref{['eq:PBD']}; (b) empirical distribution for particle-based dynamics; (c) simulation of the Dean--Kawasaki equation \ref{['eq:Dean-Kawasaki']}. For clarity, (a) shows only $10^2$ of the $10^3$ trajectories, while (b) is the empirical distribution of all trajectories. Due to the logarithmic scale, values smaller than $10^{-3}$ (especially negative values for the SPDE) are displayed in white in (b) and (c). Please note: The two simulations, particle-based and Dean-Kawasaki-based, are subject to different realizations of the respective noise process and thus cannot be compared directly.
  • Figure 2: Individual long-term simulations. Stochastic dynamics of $N=10^3$ particles until time $t=2 \cdot 10^3$, after starting at time $t=0$ with a uniform distribution. (a) Empirical distribution for particle-based dynamics \ref{['eq:PBD']} and (b) simulation of the Dean--Kawasaki equation \ref{['eq:Dean-Kawasaki']}. These are the same realizations as in Figure \ref{['fig:PBD_compared_SPDE']}, but for a longer simulation period and on a logarithmic timescale. Please note: The two simulations, particle-based and Dean-Kawasaki-based, are subject to different realizations of the respective noise process and thus cannot be compared directly.
  • Figure 3: Statistics of cluster counts over time. (a) Relative frequencies of cluster counts and (b) associated mean and standard deviation depending on time. These results are based on $S=10^3$ independent particle-based simulations (solid lines) and $S=10^3$ independent SPDE-simulations (dashed lines), respectively, each with a population size of $N=10^3$ and starting at time $t=0$ with a uniform distribution. See Figure \ref{['fig:number_cluster_average_methods']} for an analysis of the same realizations but with HDBSCAN.
  • Figure 4: Long-term statistics for SPDE-simulations. Mean and standard deviation (std) of cluster counts for $S=10^3$ independent SPDE-simulations until time $t=12\cdot 10^3$ for $N=10^3$ in (a) linear scale and (b) linear-log scale after starting with a uniform distribution. (c) Standard deviation in linear-log scale.
  • Figure 5: Distribution of waiting times for cluster merge. Relative frequencies of waiting times $\tau_{i,i-1}$ to jump from $i$ to $i-1$ clusters, see \ref{['tau']}, computed from $S=10^3$ independent SPDE-simulations with $N=10^3$ until time $t=12 \cdot 10^3$ after starting in $t=0$ with a uniform distribution, compared to best approximation by exponential distributions (black lines) with mean $\Bar{\tau}_{i,i-1}$.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 1: Application of linear stability analysis