Force Estimation during Cell Migration Using Mathematical Modelling.

Cell migration is essential for physiological, pathological and biomedical processes such as, in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion and inflammation. In light of this, quantifying mechanical properties during the process of cell migration is of great interest in experimental sciences, yet few theoretical approaches in this direction have been studied. In this work, we propose a theoretical and computational approach based on the optimal control of geometric partial differential equations to estimate cell membrane forces associated with cell polarisation during migration. Specifically, cell membrane forces are inferred or estimated by fitting a mathematical model to a sequence of images, allowing us to capture dynamics of the cell migration. Our approach offers a robust and accurate framework to compute geometric mechanical membrane forces associated with cell polarisation during migration and also yields geometric information of independent interest, we illustrate one such example that involves quantifying cell proliferation levels which are associated with cell division, cell fusion or cell death.

A sensor kinase controls turgor-driven plant infection by the rice blast fungus.

The blast fungus Magnaporthe oryzae gains entry to its host plant by means of a specialized pressure-generating infection cell called an appressorium, which physically ruptures the leaf cuticle1,2. Turgor is applied as an enormous invasive force by septin-mediated reorganization of the cytoskeleton and actin-dependent protrusion of a rigid penetration hypha3. However, the molecular mechanisms that regulate the generation of turgor pressure during appressorium-mediated infection of plants remain poorly understood. Here we show that a turgor-sensing histidine-aspartate kinase, Sln1, enables the appressorium to sense when a critical turgor threshold has been reached and thereby facilitates host penetration. We found that the Sln1 sensor localizes to the appressorium pore in a pressure-dependent manner, which is consistent with the predictions of a mathematical model for plant infection. A Δsln1 mutant generates excess intracellular appressorium turgor, produces hyper-melanized non-functional appressoria and does not organize the septins and polarity determinants that are required for leaf infection. Sln1 acts in parallel with the protein kinase C cell-integrity pathway as a regulator of cAMP-dependent signalling by protein kinase A. Pkc1 phosphorylates the NADPH oxidase regulator NoxR and, collectively, these signalling pathways modulate appressorium turgor and trigger the generation of invasive force to cause blast disease.

Stability and error analysis for a diffuse interface approach to an advection-diffusion equation on a moving surface.

In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Furthermore, we present test calculations that confirm our analysis.

A computational framework for particle and whole cell tracking applied to a real biological dataset.

Cell tracking is becoming increasingly important in cell biology as it provides a valuable tool for analysing experimental data and hence furthering our understanding of dynamic cellular phenomena. The advent of high-throughput, high-resolution microscopy and imaging techniques means that a wealth of large data is routinely generated in many laboratories. Due to the sheer magnitude of the data involved manual tracking is often cumbersome and the development of computer algorithms for automated cell tracking is thus highly desirable. In this work, we describe two approaches for automated cell tracking. Firstly, we consider particle tracking. We propose a few segmentation techniques for the detection of cells migrating in a non-uniform background, centroids of the segmented cells are then calculated and linked from frame to frame via a nearest-neighbour approach. Secondly, we consider the problem of whole cell tracking in which one wishes to reconstruct in time whole cell morphologies. Our approach is based on fitting a mathematical model to the experimental imaging data with the goal being that the physics encoded in the model is reflected in the reconstructed data. The resulting mathematical problem involves the optimal control of a phase-field formulation of a geometric evolution law. Efficient approximation of this challenging optimal control problem is achieved via advanced numerical methods for the solution of semilinear parabolic partial differential equations (PDEs) coupled with parallelisation and adaptive resolution techniques. Along with a detailed description of our algorithms, a number of simulation results are reported on. We focus on illustrating the effectivity of our approaches by applying the algorithms to the tracking of migrating cells in a dataset which reflects many of the challenges typically encountered in microscopy data.