Use of the parameterised finite element method to robustly and efficiently evolve the edge of a moving cell

Computational framework for single-cell spatiotemporal dynamics of optogenetic membrane recruitment

We describe a modular computational framework for analyzing cell-wide spatiotemporal signaling dynamics in single-cell microscopy experiments that accounts for the experiment-specific geometric and diffractive complexities that arise from heterogeneous cell morphologies and optical instrumentation. Inputs are unique cell geometries and protein concentrations derived from confocal stacks and spatiotemporally varying environmental stimuli. After simulating the system with a model of choice, the output is convolved with the microscope point-spread function for direct comparison with the observable image. We experimentally validate this approach in single cells with BcLOV4, an optogenetic membrane recruitment system for versatile control over cell signaling, using a three-dimensional non-linear finite element model with all parameters experimentally derived. The simulations recapitulate observed subcellular and cell-to-cell variability in BcLOV4 signaling, allowing for inter-experimental differences of cellular and instrumentation origins to be elucidated and resolved for improved interpretive robustness. This single-cell approach will enhance optogenetics and spatiotemporally resolved signaling studies.

Selection of established tumour cells through narrow diameter micropores enriches for elevated Ras/Raf/MEK/ERK MAPK signalling and enhanced tumour growth

As normal cells become cancer cells, and progress towards malignancy, they become progressively softer. Advantages of this change are that tumour cells become more deformable, and better able to move through narrow constraints. We designed a positive selection strategy that enriched for cells which could move through narrow diameter micropores to identify cell phenotypes that enabled constrained migration. Using human MDA MB 231 breast cancer and MDA MB 435 melanoma cancer cells, we found that micropore selection favoured cells with relatively higher Ras/Raf/MEK/ERK mitogen-activated protein kinase (MAPK) signalling, which affected actin cytoskeleton organization, focal adhesion density and cell elasticity. In this follow-up study, we provide further evidence that selection through micropores enriched for cells with altered cell morphology and adhesion. Additional analysis of RNA sequencing data revealed a set of transcripts associated with small cell size that was independent of constrained migration. Gene set enrichment analysis identified the 'matrisome' as the most significantly altered gene set linked with small size. When grown as orthotopic xenograft tumours in immunocompromised mice, micropore selected cells grew significantly faster than Parent or Flow-Sorted cells. Using mathematical modelling, we determined that there is an interaction between 1) the cell to gap size ratio; 2) the bending rigidity of the cell, which enable movement through narrow gaps. These results extend our previous conclusion that Ras/Raf/MEK/ERK MAPK signalling has a significant role in regulating cell biomechanics by showing that the selective pressure of movement through narrow gaps also enriches for increased tumour growth in vivo.

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction-diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

Tools for computational analysis of moving boundary problems in cellular mechanobiology

A cell's ability to change shape is one of the most fundamental biological processes and is essential for maintaining healthy organisms. When the ability to control shape goes awry, it often results in a diseased system. As such, it is important to understand the mechanisms that allow a cell to sense and respond to its environment so as to maintain cellular shape homeostasis. Because of the inherent complexity of the system, computational models that are based on sound theoretical understanding of the biochemistry and biomechanics and that use experimentally measured parameters are an essential tool. These models involve an inherent feedback, whereby shape is determined by the action of regulatory signals whose spatial distribution depends on the shape. To carry out computational simulations of these moving boundary problems requires special computational techniques. A variety of alternative approaches, depending on the type and scale of question being asked, have been used to simulate various biological processes, including cell motility, division, mechanosensation, and cell engulfment. In general, these models consider the forces that act on the system (both internally generated, or externally imposed) and the mechanical properties of the cell that resist these forces. Moving forward, making these techniques more accessible to the non-expert will help improve interdisciplinary research thereby providing new insight into important biological processes that affect human health. This article is categorized under: Cancer > Cancer>Computational Models Cancer > Cancer>Molecular and Cellular Physiology.

An Adaptive Moving Mesh Method for Forced Curve Shortening Flow

We propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation. Control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle, where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve. Central finite differences are used to discretize in space the governing evolution equation for the position vector, and a second-order implicit scheme is used for the temporal integration. Simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second-order accuracy. Furthermore, the new method delivers improved solution accuracy compared to the use of uniform arc-length meshes.

Fam49/CYRI interacts with Rac1 and locally suppresses protrusions

Actin-based protrusions are reinforced through positive feedback, but it is unclear what restricts their size, or limits positive signals when they retract or split. We identify an evolutionarily conserved regulator of actin-based protrusion: CYRI (CYFIP-related Rac interactor) also known as Fam49 (family of unknown function 49). CYRI binds activated Rac1 via a domain of unknown function (DUF1394) shared with CYFIP, defining DUF1394 as a Rac1-binding module. CYRI-depleted cells have broad lamellipodia enriched in Scar/WAVE, but reduced protrusion-retraction dynamics. Pseudopods induced by optogenetic Rac1 activation in CYRI-depleted cells are larger and longer lived. Conversely, CYRI overexpression suppresses recruitment of active Scar/WAVE to the cell edge, resulting in short-lived, unproductive protrusions. CYRI thus focuses protrusion signals and regulates pseudopod complexity by inhibiting Scar/WAVE-induced actin polymerization. It thus behaves like a 'local inhibitor' as predicted in widely accepted mathematical models, but not previously identified in cells. CYRI therefore regulates chemotaxis, cell migration and epithelial polarization by controlling the polarity and plasticity of protrusions.

Actin-Based Cell Protrusion in a 3D Matrix

Cell migration controls developmental processes (gastrulation and tissue patterning), tissue homeostasis (wound repair and inflammatory responses), and the pathobiology of diseases (cancer metastasis and inflammation). Understanding how cells move in physiologically relevant environments is of major importance, and the molecular machinery behind cell movement has been well studied on 2D substrates, beginning over half a century ago. Studies over the past decade have begun to reveal the mechanisms that control cell motility within 3D microenvironments – some similar to, and some highly divergent from those found in 2D. In this review we focus on migration and invasion of cells powered by actin, including formation of actin-rich protrusions at the leading edge, and the mechanisms that control nuclear movement in cells moving in a 3D matrix.

Cell migration has been well studied in 2D, but how this relates to movement in physiological 3D tissues and matrix is not clear, particularly in vertebrate interstitial matrix.

In 3D matrix cells actin polymerisation directly contributes to the formation of lamellipodia to facilitate migration and invasion (mesenchymal movement), analogous to 2D migration; actomyosin contractility promotes bleb formation to indirectly promote protrusion (amoeboid movement).

Mesenchymal migration can be characterised by polymerisation of actin to form filopodial protrusions, in the absence of lamellipodia.

Translocation of the nucleus is emerging as a critical step due to the constrictive environment of 3D matrices, and the mechanisms that transmit force to the nucleus and allow movement are beginning to be uncovered.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

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. Our numerical tests allow us to compare the method with the two different formulations of the objective functional and we conclude that the results with both objective functionals seem to agree.

Modelling cell motility and chemotaxis with evolving surface finite elements

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/∼maskae/CV_Warwick/Chemotaxis.html.