[Paper Review] Parameter Estimation in an SPDE Model for Cell Repolarisation

Randolf Altmeyer, Till Bretschneider+2 more|arXiv (Cornell University)|Oct 13, 2020

Probabilistic and Robust Engineering Design30 references4 citations

TL;DR

This paper proposes a stochastic partial differential equation (SPDE) model based on the Meinhardt activator-inhibitor framework to describe cell repolarisation dynamics under noise, applying parameter estimation techniques from linear SPDEs to a nonlinear, biologically relevant setting. It establishes asymptotic normality of the diffusion coefficient estimator as spatial resolution increases, validated through synthetic and real data simulations.

ABSTRACT

As a concrete setting where stochastic partial differential equations (SPDEs) are able to model real phenomena, we propose a stochastic Meinhardt model for cell repolarisation and study how parameter estimation techniques developed for simple linear SPDE models apply in this situation. We establish the existence of mild SPDE solutions and we investigate the impact of the driving noise process on pattern formation in the solution. We then pursue estimation of the diffusion term and show asymptotic normality for our estimator as the space resolution becomes finer. The finite sample performance is investigated for synthetic and real data.

Motivation & Objective

• To develop a stochastic extension of the Meinhardt model for cell repolarisation that incorporates noise from molecular fluctuations.
• To adapt parameter estimation techniques from linear SPDEs to a nonlinear, biologically realistic SPDE setting.
• To establish theoretical consistency and asymptotic normality of the diffusion coefficient estimator under increasing spatial resolution.
• To evaluate finite-sample performance using both synthetic data and real experimental data from Dictyostelium cells.

Proposed method

• Formulates a coupled system of stochastic reaction-diffusion equations with space-time white noise for activator (A) and inhibitor (I) concentrations.
• Establishes existence of mild solutions to the SPDE using semigroup theory and stochastic integration in Hilbert spaces.
• Derives an estimating equation based on the quadratic variation of the solution, leveraging the noise-induced variability in the data.
• Applies a kernel-based smoothing approach to approximate the infinitesimal generator and construct a contrast function for parameter estimation.
• Uses a finite difference scheme in Julia to simulate the SPDE, ensuring stability via the CFL condition with dt ∝ (dx)^2.
• Performs statistical inference by minimizing a contrast function derived from the empirical quadratic variation, leading to an estimator for the diffusion coefficient.

Experimental results

Research questions

• RQ1 Can parameter estimation techniques developed for linear SPDEs be effectively extended to nonlinear, biologically relevant SPDE models such as the stochastic Meinhardt model?
• RQ2 How does the addition of space-time white noise affect pattern formation and stability in the cell repolarisation process?
• RQ3 What is the asymptotic distribution of the diffusion coefficient estimator as spatial resolution increases?
• RQ4 How well does the proposed estimator perform in finite samples with synthetic and real biological data?
• RQ5 To what extent does the noise level influence the time to repolarisation and the robustness of pattern formation?

Key findings

• The proposed estimator for the diffusion coefficient exhibits asymptotic normality as the spatial resolution δ → 0, with convergence rate δ⁻¹/².
• The variance of the estimator decays as δ⁴ × Var(·) → 0, confirming consistency under increasing spatial sampling.
• Finite-sample simulations show that the estimator performs well on synthetic data, with low bias and good coverage of confidence intervals.
• Real data analysis on 18 Dictyostelium cells confirms the model's ability to capture repolarisation dynamics, with estimated noise levels consistent with biological expectations.
• The model successfully reproduces key features of experimental data, including front repositioning and pattern stability under noise.
• The theoretical framework supports the use of quadratic variation-based estimation in nonlinear SPDEs, extending the applicability of existing statistical methods.