Parameter identification problems in the modelling of cell motility

We present a novel parameter identification algorithm for the estimation of parameters in models of cell motility using imaging data of migrating cells. Two alternative formulations of the objective functional that measures the difference between the computed and observed data are proposed and the parameter identification problem is formulated as a minimisation problem of nonlinear least squares type. A Levenberg-Marquardt based optimisation method is applied to the solution of the minimisation problem and the details of the implementation are discussed. A number of numerical experiments are presented which illustrate the robustness of the algorithm to parameter identification in the presence of large deformations and noisy data and parameter identification in three dimensional models of cell motility. An application to experimental data is also presented in which we seek to identify parameters in a model for the monopolar growth of fission yeast cells using experimental imaging data.

💡 Research Summary

The paper introduces a novel algorithm for identifying parameters in mathematical models of cell motility that couple a geometric evolution law for the cell membrane with a surface reaction‑diffusion system. High‑resolution imaging provides time‑series data of membrane positions and concentrations of membrane‑resident species, which are used as observations. Two formulations of the objective functional are proposed. The first, a sharp‑interface functional, measures the Hausdorff distance between observed and simulated membranes and the L²‑norm of concentration differences at nearest‑point correspondences. This formulation is computationally cheap but non‑differentiable. The second, a phase‑field functional, replaces the sharp interface by a smooth indicator φε constructed from signed distance functions with a small interfacial width ε, and extends surface concentrations into a thin diffuse layer. The resulting functional is smooth (C²) and thus amenable to gradient‑based optimization.

The identification problem is cast as a nonlinear least‑squares minimization over admissible parameter sets (e.g., surface tension σ, bending rigidity kb, diffusion coefficients D, reaction parameters). The authors adopt the Levenberg‑Marquardt algorithm, which blends Gauss‑Newton and gradient‑descent steps and is known for robustness when the Jacobian is ill‑conditioned or the problem is highly nonlinear. For the phase‑field functional the Jacobian can be obtained analytically (or via automatic differentiation), while for the sharp‑interface case a numerical approximation is used. Weighting factors wi allow the user to balance the relative importance of positional versus concentration errors, reflecting experimental priorities.

Implementation relies on a parametric surface finite element method previously developed by the authors, avoiding costly embedded methods such as level‑set or phase‑field solvers for the forward problem. The forward problem (membrane evolution and surface reaction‑diffusion) is solved on a moving triangulated surface; the sharp‑interface functional is discretized directly on point‑cloud data using nearest‑neighbor searches, whereas the phase‑field functional is evaluated on a bulk mesh that contains the diffuse interface. The optimization loop proceeds as: solve forward problem → evaluate functional → compute gradient/Jacobian → update parameters via Levenberg‑Marquardt → repeat until convergence criteria (parameter change and functional reduction) are met.

Numerical experiments with synthetic data demonstrate robustness to measurement noise (up to 10 % Gaussian noise) and to variations in ε. Sensitivity analyses show that larger ε blurs the interface, degrading parameter recovery, while too small ε increases computational cost without significant accuracy gain. A three‑dimensional test case involving a highly deforming cell confirms that the method scales to realistic 3D geometries and can recover up to eight parameters simultaneously.

The algorithm is then applied to real experimental data from monopolar growth of fission yeast cells. The authors simplify the model to two parameters (surface tension and a growth‑driving term) and use fluorescence images of the cell membrane and a tagged protein. The identified parameters agree closely with independent experimental estimates, and by varying the weighting wi they illustrate how the fit can be biased toward either membrane shape or concentration profile, offering flexibility for different biological questions.

In conclusion, the study provides a comprehensive framework for parameter identification in coupled geometric‑evolution–reaction‑diffusion models of cell motility. The sharp‑interface functional offers computational efficiency, while the phase‑field functional provides smoothness needed for rigorous optimization. The Levenberg‑Marquardt scheme proves robust across noisy, high‑dimensional, and three‑dimensional scenarios. Limitations include the need to select an appropriate ε for the phase‑field approach and the higher computational load compared to the sharp‑interface version. Future work is suggested on adaptive ε selection, parallel implementation for large 3D datasets, and extension to more complex biochemical networks.