Mathematical modeling of eukaryotic cell migration: insights beyond experiments

Modelling the effect of cell motility on mixing and invasion in epithelial monolayers

Collective cell invasion underlies several biological processes such as wound healing, embryonic development, and cancerous invasion. Here, we investigate the impact of cell motility on invasion in epithelial monolayers and its coupling to cellular mechanical properties, such as cell-cell adhesion and cortex contractility. We develop a two-dimensional computational model for cells with active motility based on the cellular Potts model, which predicts that the cellular invasion speed is mainly determined by active cell motility and is independent of the biological and mechanical properties of the cells. We also find that, in general, motile cells out-compete and invade non-motile cells, however, this can be reversed by differential cell proliferation. Stable coexistence of motile and static cell types is also possible for certain parameter regimes.

Clutch model for focal adhesions predicts reduced self-stabilization under oblique pulling

Cell-matrix adhesions connect the cytoskeleton to the extracellular environment and are essential for maintaining the integrity of tissue and whole organisms. Remarkably, cell adhesions can adapt their size and composition to an applied force such that their size and strength increases proportionally to the load. Mathematical models for the clutch-like force transmission at adhesions are frequently based on the assumption that mechanical load is applied tangentially to the adhesion plane. Recently, we suggested a molecular mechanism that can explain adhesion growth under load for planar cell adhesions. The mechanism is based on conformation changes of adhesion molecules that are dynamically exchanged with a reservoir. Tangential loading drives the occupation of some states out of equilibrium, which for thermodynamic reasons, leads to the association of further molecules with the cluster, which we refer to as self-stabilization. Here, we generalize this model to forces that pull at an oblique angle to the plane supporting the cell, and examine if this idealized model also predicts self-stabilization. We also allow for a variable distance between the parallel planes representing cytoskeletal F-actin and transmembrane integrins. Simulation results demonstrate that the binding mechanism and the geometry of the cluster have a strong influence on the response of adhesion clusters to force. For oblique angles smaller than about 40∘, we observe a growth of the adhesion site under force. However this self-stabilization is reduced as the angle between the force and substrate plane increases, with vanishing self-stabilization for normal pulling. Overall, these results highlight the fundamental difference between the assumption of pulling and shearing forces in commonly used models of cell adhesion.

Mesenchymal cell migration on one-dimensional micropatterns

Quantitative studies of mesenchymal cell motion are important to elucidate cytoskeleton function and mechanisms of cell migration. To this end, confinement of cell motion to one dimension (1D) significantly simplifies the problem of cell shape in experimental and theoretical investigations. Here we review 1D migration assays employing micro-fabricated lanes and reflect on the advantages of such platforms. Data are analyzed using biophysical models of cell migration that reproduce the rich scenario of morphodynamic behavior found in 1D. We describe basic model assumptions and model behavior. It appears that mechanical models explain the occurrence of universal relations conserved across different cell lines such as the adhesion-velocity relation and the universal correlation between speed and persistence (UCSP). We highlight the unique opportunity of reproducible and standardized 1D assays to validate theory based on statistical measures from large data of trajectories and discuss the potential of experimental settings embedding controlled perturbations to probe response in migratory behavior.

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Three-component contour dynamics model to simulate and analyze amoeboid cell motility in two dimensions

Amoeboid cell motility is relevant in a wide variety of biomedical processes such as wound healing, cancer metastasis, and embryonic morphogenesis. It is characterized by pronounced changes of the cell shape associated with expansions and retractions of the cell membrane, which result in a crawling kind of locomotion. Despite existing computational models of amoeboid motion, the inference of expansion and retraction components of individual cells, the corresponding classification of cells, and the a priori specification of the parameter regime to achieve a specific motility behavior remain challenging open problems. We propose a novel model of the spatio-temporal evolution of two-dimensional cell contours comprising three biophysiologically motivated components: a stochastic term accounting for membrane protrusions and two deterministic terms accounting for membrane retractions by regularizing the shape and area of the contour. Mathematically, these correspond to the intensity of a self-exciting Poisson point process, the area-preserving curve-shortening flow, and an area adjustment flow. The model is used to generate contour data for a variety of qualitatively different, e.g., polarized and non-polarized, cell tracks that visually resemble experimental data very closely. In application to experimental cell tracks, we inferred the protrusion component and examined its correlation to common biomarkers: the F-actin density close to the membrane and its local motion. Due to the low model complexity, parameter estimation is fast, straightforward, and offers a simple way to classify contour dynamics based on two locomotion types: the amoeboid and a so-called fan-shaped type. For both types, we use cell tracks segmented from fluorescence imaging data of the model organism Dictyostelium discoideum. An implementation of the model is provided within the open-source software package AmoePy, a Python-based toolbox for analyzing and simulating amoeboid cell motility.

Geometry-structure models for liquid crystal interfaces, drops and membranes: wrinkling, shape selection and dissipative shape evolution

We review our recent contributions to anisotropic soft matter models for liquid crystal interfaces, drops and membranes, emphasizing validations with experimental and biological data, and with related theory and simulation literature. The presentation aims to illustrate and characterize the rich output and future opportunities of using a methodology based on the liquid crystal-membrane shape equation applied to static and dynamic pattern formation phenomena. The geometry of static and kinetic shapes is usually described with dimensional curvatures that co-mingle shape and curvedness. In this review, we systematically show how the application of a novel decoupled shape-curvedness framework to practical and ubiquitous soft matter phenomena, such as the shape of drops and tactoids and bending of evolving membranes, leads to deeper quantitative insights than when using traditional dimensional mean and Gaussian curvatures. The review focuses only on (1) statics of wrinkling and shape selection in liquid crystal interfaces and membranes; (2) kinetics and dissipative dynamics of shape evolution in membranes; and (3) computational methods for shape selection and shape evolution; due to various limitations other important topics are excluded. Finally, the outlook follows a similar structure. The main results include: (1) single and multiple wavelength corrugations in liquid crystal interfaces appear naturally in the presence of surface splay and bend orientation distortions with scaling laws governed by ratios of anchoring-to-isotropic tension energy; adding membrane elasticity to liquid crystal anchoring generates multiple scales wrinkling as in tulips; drops of liquid crystals encapsulates in membranes can adopt, according to the ratios of anchoring/tension/bending, families of shapes as multilobal, tactoidal, and serrated as observed in biological cells. (2) Mapping the liquid crystal director to a membrane unit normal. The dissipative shape evolution model with irreversible thermodynamics for flows dominated by bending rates, yields new insights. The model explains the kinetic stability of cylinders, while spheres and saddles are attractors. The model also adds to the evolving understanding of outer hair cells in the inner ear. (3) Computational soft matter geometry includes solving shape equations, trajectories on energy and orientation landscapes, and shape-curvedness evolutions on entropy production landscape with efficient numerical methods and adaptive approaches.

The Influence of Nucleus Mechanics in Modelling Adhesion-independent Cell Migration in Structured and Confined Environments

Recent biological experiments (Lämmermann et al. in Nature 453(7191):51-55, 2008; Reversat et al. in Nature 7813:582-585, 2020; Balzer et al. in ASEB J Off Publ Fed Am Soc Exp Biol 26(10):4045-4056, 2012) have shown that certain types of cells are able to move in structured and confined environments even without the activation of focal adhesion. Focusing on this particular phenomenon and based on previous works (Jankowiak et al. in Math Models Methods Appl Sci 30(03):513-537, 2020), we derive a novel two-dimensional mechanical model, which relies on the following physical ingredients: the asymmetrical renewal of the actin cortex supporting the membrane, resulting in a backward flow of material; the mechanical description of the nuclear membrane and the inner nuclear material; the microtubule network guiding nucleus location; the contact interactions between the cell and the external environment. The resulting fourth order system of partial differential equations is then solved numerically to conduct a study of the qualitative effects of the model parameters, mainly those governing the mechanical properties of the nucleus and the geometry of the confining structure. Coherently with biological observations, we find that cells characterized by a stiff nucleus are unable to migrate in channels that can be crossed by cells with a softer nucleus. Regarding the geometry, cell velocity and ability to migrate are influenced by the width of the channel and the wavelength of the external structure. Even though still preliminary, these results may be potentially useful in determining the physical limit of cell migration in confined environments and in designing scaffolds for tissue engineering.

Adhesion tunes speed and persistence by coordinating protrusions and extracellular matrix remodeling

Cell migration through 3D environments is essential to development, disease, and regeneration processes. Conceptual models of migration have been developed primarily on the basis of 2D cell behaviors, but a general understanding of 3D cell migration is still lacking due to the added complexity of the extracellular matrix. Here, using a multiplexed biophysical imaging approach for single-cell analysis of human cell lines, we show how the subprocesses of adhesion, contractility, actin cytoskeletal dynamics, and matrix remodeling integrate to produce heterogeneous migration behaviors. This single-cell analysis identifies three modes of cell speed and persistence coupling, driven by distinct modes of coordination between matrix remodeling and protrusive activity. The framework that emerges establishes a predictive model linking cell trajectories to distinct subprocess coordination states.

PPP2R1A regulates migration persistence through the NHSL1-containing WAVE Shell Complex

The RAC1-WAVE-Arp2/3 signaling pathway generates branched actin networks that power lamellipodium protrusion of migrating cells. Feedback is thought to control protrusion lifetime and migration persistence, but its molecular circuitry remains elusive. Here, we identify PPP2R1A by proteomics as a protein differentially associated with the WAVE complex subunit ABI1 when RAC1 is activated and downstream generation of branched actin is blocked. PPP2R1A is found to associate at the lamellipodial edge with an alternative form of WAVE complex, the WAVE Shell Complex, that contains NHSL1 instead of the Arp2/3 activating subunit WAVE, as in the canonical WAVE Regulatory Complex. PPP2R1A is required for persistence in random and directed migration assays and for RAC1-dependent actin polymerization in cell extracts. PPP2R1A requirement is abolished by NHSL1 depletion. PPP2R1A mutations found in tumors impair WAVE Shell Complex binding and migration regulation, suggesting that the coupling of PPP2R1A to the WAVE Shell Complex is essential to its function.

Mathematical model of mechano-sensing and mechanically induced collective motility of cells on planar elastic substrates

Cells mechanically interact with their environment to sense, for example, topography, elasticity and mechanical cues from other cells. Mechano-sensing has profound effects on cellular behaviour, including motility. The current study aims to develop a mathematical model of cellular mechano-sensing on planar elastic substrates and demonstrate the model's predictive capabilities for the motility of individual cells in a colony. In the model, a cell is assumed to transmit an adhesion force, derived from a dynamic focal adhesion integrin density, that locally deforms a substrate, and to sense substrate deformation originating from neighbouring cells. The substrate deformation from multiple cells is expressed as total strain energy density with a spatially varying gradient. The magnitude and direction of the gradient at the cell location define the cell motion. Cell-substrate friction, partial motion randomness, and cell death and division are included. The substrate deformation by a single cell and the motility of two cells are presented for several substrate elasticities and thicknesses. The collective motility of 25 cells on a uniform substrate mimicking the closure of a circular wound of 200 µm is predicted for deterministic and random motion. Cell motility on substrates with varying elasticity and thickness is explored for four cells and 15 cells, the latter again mimicking wound closure. Wound closure by 45 cells is used to demonstrate the simulation of cell death and division during migration. The mathematical model can adequately simulate the mechanically induced collective cell motility on planar elastic substrates. The model is suitable for extension to other cell and substrates shapes and the inclusion of chemotactic cues, offering the potential to complement in vitro and in vivo studies.

Mechanically primed cells transfer memory to fibrous matrices for invasion across environments of distinct stiffness and dimensionality

Cells sense and migrate across mechanically dissimilar environments throughout development and disease progression. However, it remains unclear whether mechanical memory of past environments empowers cells to navigate new, three-dimensional extracellular matrices. Here, we show that cells previously primed on stiff, compared with soft, matrices generate a higher level of forces to remodel collagen fibers and promote invasion. This priming advantage persists in dense or stiffened collagen. We explain this memory-dependent, cross-environment cell invasion through a lattice-based model wherein stiff-primed cellular forces remodel collagen and minimize energy required for future cell invasion. According to our model, cells transfer their mechanical memory to the matrix via collagen alignment and tension, and this remodeled matrix informs future cell invasion. Thus, memory-laden cells overcome mechanosensing of softer or challenging future environments via a cell-matrix transfer of memory. Consistent with model predictions, depletion of yes-associated protein destabilizes the cellular memory required for collagen remodeling before invasion. We release tension in collagen fibers via laser ablation and disable fiber remodeling by lysyl-oxidase inhibition, both of which disrupt cell-to-matrix transfer of memory and hamper cross-environment invasion. These results have implications for cancer, fibrosis, and aging, where a potential cell-to-matrix transfer of mechanical memory of cells may generate a prolonged cellular response.

Reciprocal intra- and extra-cellular polarity enables deep mechanosensing through layered matrices

Adherent cells migrate on layered tissue interfaces to drive morphogenesis, wound healing, and tumor invasion. Although stiffer surfaces are known to enhance cell migration, it remains unclear whether cells sense basal stiff environments buried under softer, fibrous matrix. Using layered collagen-polyacrylamide gel systems, we unveil a migration phenotype driven by cell-matrix polarity. Here, cancer (but not normal) cells with stiff base matrix generate stable protrusions, faster migration, and greater collagen deformation because of "depth mechanosensing" through the top collagen layer. Cancer cell protrusions with front-rear polarity produce polarized collagen stiffening and deformations. Disruption of either extracellular or intracellular polarity via collagen crosslinking, laser ablation, or Arp2/3 inhibition independently abrogates depth-mechanosensitive migration of cancer cells. Our experimental findings, validated by lattice-based energy minimization modeling, present a cell migration mechanism whereby polarized cellular protrusions and contractility are reciprocated by mechanical extracellular polarity, culminating in a cell-type-dependent ability to mechanosense through matrix layers.

Simulating 3D Cell Shape with the Cellular Potts Model

Computer simulations have become a widely used method for the field of mechanobiology. An important question is whether one can predict the shape and forces of cells as a function of the extracellular environment. Different types of models have been described before to simulate cell and tissue shapes in structured environments. In this chapter, we give a brief overview of commonly used models and then describe the Cellular Potts Model, a lattice-based modelling framework, in more detail. We provide a hands-on guide on how to build a model that simulates the shape of a single cell on a micropattern in three dimensions in different open source software packages using the Cellular Potts framework. A simulation is set up with an initial configuration of generalized cells that change shape and position due to an energy function that incorporates cellular volume and surface area constraints as well as interaction energies between the generalized cells.

A multiscale whole-cell theory for mechanosensitive migration on viscoelastic substrates

Increasing experimental evidence validates that both the elastic stiffness and viscosity of the extracellular matrix regulate mesenchymal cell behavior, such as the rational switch between durotaxis (cell migration to stiffer regions), anti-durotaxis (migration to softer regions), and adurotaxis (stiffness-insensitive migration). To reveal the mechanisms underlying the crossover between these motility regimes, we have developed a multiscale chemomechanical whole-cell theory for mesenchymal migration. Our framework couples the subcellular focal adhesion dynamics at the cell-substrate interface with the cellular cytoskeletal mechanics and the chemical signaling pathways involving Rho GTPase proteins. Upon polarization by the Rho GTPase gradients, our simulated cell migrates by concerted peripheral protrusions and contractions, a hallmark of the mesenchymal mode. The resulting cell dynamics quantitatively reproduces the experimental migration speed as a function of the uniform substrate stiffness and explains the influence of viscosity on the migration efficiency. In the presence of stiffness gradients and absence of chemical polarization, our simulated cell can exhibit durotaxis, anti-durotaxis, and adurotaxis respectively with increasing substrate stiffness or viscosity. The cell moves toward an optimally stiff region from softer regions during durotaxis and from stiffer regions during anti-durotaxis. We show that cell polarization through steep Rho GTPase gradients can reverse the migration direction dictated by the mechanical cues. Overall, our theory demonstrates that opposing durotactic behaviors emerge via the interplay between intracellular signaling and cell-medium mechanical interactions in agreement with experiments, thereby elucidating complex mechanosensing at the single-cell level.

Proper Orthogonal Decomposition Analysis Reveals Cell Migration Directionality During Wound Healing

A proper orthogonal decomposition (POD) order reduction method was implemented to reduce the full high dimensional dynamical system associated with a wound healing cell migration assay to a low-dimensional approximation that identified the prevailing cell trajectories. The POD analysis generated POD modes that were representative of the prevalent cell trajectories. The shapes of the POD modes depended on the location of the cells with respect to the wound and exposure to disturbed (DF) or undisturbed (UF) fluid flow where the net flow was in the antegrade direction with a retrograde component or fully antegrade, respectively. For DF and UF, the POD modes of the downstream cells identified trajectories that moved upstream against the flow, while upstream POD modes exhibited sideways cell migrations. In the absence of flow, no major shape differences were observed in the POD modes on either side of the wound. The POD modes also served to reconstruct the cell migration of individual cells. With as few as three modes, the predominant cell migration trajectories were reconstructed, while the level of accuracy increased with the inclusion of more POD modes. The POD order reduction method successfully identified the predominant cell migratory trajectories under static and varying pulsatile fluid flow conditions serving as a first step in the development of artificial intelligence models of cell migration in disease and development.

The Cell Physiome: What Do We Need in a Computational Physiology Framework for Predicting Single-Cell Biology?

Modern biology and biomedicine are undergoing a big data explosion, needing advanced computational algorithms to extract mechanistic insights on the physiological state of living cells. We present the motivation for the Cell Physiome project: a framework and approach for creating, sharing, and using biophysics-based computational models of single-cell physiology. Using examples in calcium signaling, bioenergetics, and endosomal trafficking, we highlight the need for spatially detailed, biophysics-based computational models to uncover new mechanisms underlying cell biology. We review progress and challenges to date toward creating cell physiome models. We then introduce bond graphs as an efficient way to create cell physiome models that integrate chemical, mechanical, electromagnetic, and thermal processes while maintaining mass and energy balance. Bond graphs enhance modularization and reusability of computational models of cells at scale. We conclude with a look forward at steps that will help fully realize this exciting new field of mechanistic biomedical data science. Expected final online publication date for the Annual Review of Biomedical Data Science, Volume 5 is August 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Generic self-stabilization mechanism for biomolecular adhesions under load

Mechanical loading generally weakens adhesive structures and eventually leads to their rupture. However, biological systems can adapt to loads by strengthening adhesions, which is essential for maintaining the integrity of tissue and whole organisms. Inspired by cellular focal adhesions, we suggest here a generic, molecular mechanism that allows adhesion systems to harness applied loads for self-stabilization through adhesion growth. The mechanism is based on conformation changes of adhesion molecules that are dynamically exchanged with a reservoir. Tangential loading drives the occupation of some states out of equilibrium, which, for thermodynamic reasons, leads to association of further molecules with the cluster. Self-stabilization robustly increases adhesion lifetimes in broad parameter ranges. Unlike for catch-bonds, bond rupture rates can increase monotonically with force. The self-stabilization principle can be realized in many ways in complex adhesion-state networks; we show how it naturally occurs in cellular adhesions involving the adaptor proteins talin and vinculin.

Cellular adhesions have the remarkable property that they adapt their stability to the applied mechanical load. Here, authors describe a generic physical mechanism that explains self-stabilization of idealized adhesion systems under shear.

Persistent cell migration emerges from a coupling between protrusion dynamics and polarized trafficking

Migrating cells present a variety of paths, from random to highly directional ones. While random movement can be explained by basal intrinsic activity, persistent movement requires stable polarization. Here, we quantitatively address emergence of persistent migration in (hTERT)–immortalizedRPE1 (retinal pigment epithelial) cells over long timescales. By live cell imaging and dynamic micropatterning, we demonstrate that the Nucleus-Golgi axis aligns with direction of migration leading to efficient cell movement. We show that polarized trafficking is directed toward protrusions with a 20-min delay, and that migration becomes random after disrupting internal cell organization. Eventually, we prove that localized optogenetic Cdc42 activation orients the Nucleus-Golgi axis. Our work suggests that polarized trafficking stabilizes the protrusive activity of the cell, while protrusive activity orients this polarity axis, leading to persistent cell migration. Using a minimal physical model, we show that this feedback is sufficient to recapitulate the quantitative properties of cell migration in the timescale of hours.

Deconstructing the role of myosin contractility in force fluctuations within focal adhesions

Force fluctuations exhibited in focal adhesions that connect a cell to its extracellular environment point to the complex role of the underlying machinery that controls cell migration. To elucidate the explicit role of myosin motors in the temporal traction force oscillations, we vary the contractility of these motors in a dynamical model based on the molecular clutch hypothesis. As the contractility is lowered, effected both by changing the motor velocity and the rate of attachment/detachment, we show analytically in an experimentally relevant parameter space, that the system goes from decaying oscillations to stable limit cycle oscillations through a supercritical Hopf bifurcation. As a function of the motor activity and the number of clutches, the system exhibits a rich array of dynamical states. We corroborate our analytical results with stochastic simulations of the motor-clutch system. We obtain limit cycle oscillations in the parameter regime as predicted by our model. The frequency range of oscillations in the average clutch and motor deformation compares well with experimental results.

Fos regulates macrophage infiltration against surrounding tissue resistance by a cortical actin-based mechanism in Drosophila

The infiltration of immune cells into tissues underlies the establishment of tissue-resident macrophages and responses to infections and tumors. Yet the mechanisms immune cells utilize to negotiate tissue barriers in living organisms are not well understood, and a role for cortical actin has not been examined. Here, we find that the tissue invasion of Drosophila macrophages, also known as plasmatocytes or hemocytes, utilizes enhanced cortical F-actin levels stimulated by the Drosophila member of the fos proto oncogene transcription factor family (Dfos, Kayak). RNA sequencing analysis and live imaging show that Dfos enhances F-actin levels around the entire macrophage surface by increasing mRNA levels of the membrane spanning molecular scaffold tetraspanin TM4SF, and the actin cross-linking filamin Cheerio, which are themselves required for invasion. Both the filamin and the tetraspanin enhance the cortical activity of Rho1 and the formin Diaphanous and thus the assembly of cortical actin, which is a critical function since expressing a dominant active form of Diaphanous can rescue the Dfos macrophage invasion defect. In vivo imaging shows that Dfos enhances the efficiency of the initial phases of macrophage tissue entry. Genetic evidence argues that this Dfos-induced program in macrophages counteracts the constraint produced by the tension of surrounding tissues and buffers the properties of the macrophage nucleus from affecting tissue entry. We thus identify strengthening the cortical actin cytoskeleton through Dfos as a key process allowing efficient forward movement of an immune cell into surrounding tissues.

The infiltration of immune cells into tissue underlies the establishment of tissue-resident macrophages, and responses to infections and tumors, but how do they overcome tissue barriers? This study shows that macrophages upregulate the proto-oncogene Fos, increasing the density and crosslinking of cortical actin, thereby counteracting the tension of surrounding tissues and protecting the macrophage nucleus.

Mathematical modeling of therapeutic neural stem cell migration in mouse brain with and without brain tumors

Neural stem cells (NSCs) offer a potential solution to treating brain tumors. This is because NSCs can circumvent the blood-brain barrier and migrate to areas of damage in the central nervous system, including tumors, stroke, and wound injuries. However, for successful clinical application of NSC treatment, a sufficient number of viable cells must reach the diseased or damaged area(s) in the brain, and evidence suggests that it may be affected by the paths the NSCs take through the brain, as well as the locations of tumors. To study the NSC migration in brain, we develop a mathematical model of therapeutic NSC migration towards brain tumor, that provides a low cost platform to investigate NSC treatment efficacy. Our model is an extension of the model developed in Rockne et al. (PLoS ONE 13, e0199967, 2018) that considers NSC migration in non-tumor bearing naive mouse brain. Here we modify the model in Rockne et al. in three ways: (i) we consider three-dimensional mouse brain geometry, (ii) we add chemotaxis to model the tumor-tropic nature of NSCs into tumor sites, and (iii) we model stochasticity of migration speed and chemosensitivity. The proposed model is used to study migration patterns of NSCs to sites of tumors for different injection strategies, in particular, intranasal and intracerebral delivery. We observe that intracerebral injection results in more NSCs arriving at the tumor site(s), but the relative fraction of NSCs depends on the location of injection relative to the target site(s). On the other hand, intranasal injection results in fewer NSCs at the tumor site, but yields a more even distribution of NSCs within and around the target tumor site(s).

Derivation of continuum models from discrete models of mechanical forces in cell populations

In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.

Two Complementary Signaling Pathways Depict Eukaryotic Chemotaxis: A Mechanochemical Coupling Model

Many eukaryotic cells, including neutrophils and Dictyostelium cells, are able to undergo correlated random migration in the absence of directional cues while reacting to shallow gradients of chemoattractants with exquisite precision. Although progress has been made with regard to molecular identities, it remains elusive how molecular mechanics are integrated with cell mechanics to initiate and manipulate cell motility. Here, we propose a two dimensional (2D) cell migration model wherein a multilayered dynamic seesaw mechanism is accompanied by a mechanical strain-based inhibition mechanism. In biology, these two mechanisms can be mapped onto the biochemical feedback between phosphoinositides (PIs) and Rho GTPase and the mechanical interplay between filamin A (FLNa) and FilGAP. Cell migration and the accompanying morphological changes are demonstrated in numerical simulations using a particle-spring model, and the diffusion in the cell membrane are simulations using a one dimensional (1D) finite differences method (FDM). The fine balance established between endogenous signaling and a mechanically governed inactivation scheme ensures the endogenous cycle of self-organizing pseudopods, accounting for the correlated random migration. Furthermore, this model cell manifests directional and adaptable responses to shallow graded signaling, depending on the overwhelming effect of the graded stimuli guidance on strain-based inhibition. Finally, the model cell becomes trapped within an obstacle-ridden spatial region, manifesting a shuttle run for local explorations and can chemotactically “escape”, illustrating again the balance required in the complementary signaling pathways.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

A Framework of Paracellular Transport via Nanoparticles-Induced Endothelial Leakiness

Nanomaterial-induced endothelial leakiness (NanoEL) is an interfacial phenomenon denoting the paracellular transport of nanoparticles that is pertinent to nanotoxicology, nanomedicine and biomedical engineering. While the NanoEL phenomenon is complementary to the enhanced permeability and retention effect in terms of their common applicability to delineating the permeability and behavior of nanoparticles in tumoral environments, these two effects significantly differ in scope, origin, and manifestation. In the current study, the descriptors are fully examined of the NanoEL phenomenon elicited by generic citrate-coated gold nanoparticles (AuNPs) of changing size and concentration, from microscopic gap formation and actin reorganization down to molecular signaling pathways and nanoscale interactions of AuNPs with VE-cadherin and its intra/extracellular cofactors. Employing synergistic in silico methodologies, for the first time the molecular and statistical mechanics of cadherin pair disruption, especially in response to AuNPs of the smallest size and highest concentration are revealed. This study marks a major advancement toward establishing a comprehensive NanoEL framework for complementing the understanding of the transcytotic pathway and for guiding the design and application of future nanomedicines harnessing the myriad functions of the mammalian vasculature.

Rab7b regulates dendritic cell migration by linking lysosomes to the actomyosin cytoskeleton

Lysosomal signaling facilitates the migration of immune cells by releasing Ca²⁺ to activate the actin-based motor myosin II at the cell rear. However, how the actomyosin cytoskeleton physically associates to lysosomes is unknown. We have previously identified myosin II as a direct interactor of Rab7b, a small GTPase that mediates the transport from late endosomes/lysosomes to the trans-Golgi network (TGN). Here, we show that Rab7b regulates the migration of dendritic cells (DCs) in one- and three-dimensional environments. DCs are immune sentinels that transport antigens from peripheral tissues to lymph nodes to activate T lymphocytes and initiate adaptive immune responses. We found that the lack of Rab7b reduces myosin II light chain phosphorylation and the activation of the transcription factor EB (TFEB), which controls lysosomal signaling and is required for fast DC migration. Furthermore, we demonstrate that Rab7b interacts with the lysosomal Ca²⁺ channel TRPML1 (also known as MCOLN1), enabling the local activation of myosin II at the cell rear. Taken together, our findings identify Rab7b as the missing physical link between lysosomes and the actomyosin cytoskeleton, allowing control of immune cell migration through lysosomal signaling.

This article has an associated First Person interview with the first author of the paper.

Summary: The small GTPase Rab7b bridges the lysosomal Ca²⁺ channel TRPML1 to myosin II, thus enabling the local activation of myosin II at the cell rear and promoting fast migration of dendritic cells.

The role of vimentin-nuclear interactions in persistent cell motility through confined spaces

The ability of cells to move through small spaces depends on the mechanical properties of the cellular cytoskeleton and on nuclear deformability. In mammalian cells, the cytoskeleton is composed of three interacting, semi-flexible polymer networks: actin, microtubules, and intermediate filaments (IF). Recent experiments of mouse embryonic fibroblasts with and without vimentin have shown that the IF vimentin plays a role in confined cell motility. Here, we develop a minimal model of a cell moving through a microchannel that incorporates explicit effects of actin and vimentin and implicit effects of microtubules. Specifically, the model consists of a cell with an actomyosin cortex and a deformable cell nucleus and mechanical linkages between the two. By decreasing the amount of vimentin, we find that the cell speed increases for vimentin-null cells compared to cells with vimentin. The loss of vimentin increases nuclear deformation and alters nuclear positioning in the cell. Assuming nuclear positioning is a read-out for cell polarity, we propose a new polarity mechanism which couples cell directional motion with cytoskeletal strength and nuclear positioning and captures the abnormally persistent motion of vimentin-null cells, as observed in experiments. The enhanced persistence indicates that the vimentin-null cells are more controlled by the confinement and so less autonomous, relying more heavily on external cues than their wild-type counterparts. Our modeling results present a quantitative interpretation for recent experiments and have implications for understanding the role of vimentin in the epithelial-mesenchymal transition.

Comparative mapping of crawling-cell morphodynamics in deep learning-based feature space

Navigation of fast migrating cells such as amoeba Dictyostelium and immune cells are tightly associated with their morphologies that range from steady polarized forms that support high directionality to those more complex and variable when making frequent turns. Model simulations are essential for quantitative understanding of these features and their origins, however systematic comparisons with real data are underdeveloped. Here, by employing deep-learning-based feature extraction combined with phase-field modeling framework, we show that a low dimensional feature space for 2D migrating cell morphologies obtained from the shape stereotype of keratocytes, Dictyostelium and neutrophils can be fully mapped by an interlinked signaling network of cell-polarization and protrusion dynamics. Our analysis links the data-driven shape analysis to the underlying causalities by identifying key parameters critical for migratory morphologies both normal and aberrant under genetic and pharmacological perturbations. The results underscore the importance of deciphering self-organizing states and their interplay when characterizing morphological phenotypes.

Migratory cells that move by crawling do so by extending and retracting their plasma membrane. When and where these events take place determine the cell shape, and this is directly linked to the movement patterns. Understanding how the highly plastic and interconvertible morphologies appear from their underlying dynamics remains a challenge partly because their inherent complexity makes quantitatively comparison against the outputs of mathematical models difficult. To this end, we employed machine-learning based classification to extract features that characterize the basic migrating morphologies. The obtained features were then used to compare real cell data with outputs of a conceptual model that we introduced which describes coupling via feedback between local protrusive dynamics and polarity. The feature mapping showed that the model successfully recapitulates the shape dynamics that were not covered by previous related models and also hints at the critical parameters underlying state transitions. The ability of the present approach to compare model outputs with real cell data systematically and objectively is important as it allows outputs of future mathematical models to be quantitatively tested in an accessible and common reference frame.

Mechanics of developmental migration

The ability to migrate is a fundamental property of animal cells which is essential for development, homeostasis and disease progression. Migrating cells sense and respond to biochemical and mechanical cues by rapidly modifying their intrinsic repertoire of signalling molecules and by altering their force generating and transducing machinery. We have a wealth of information about the chemical cues and signalling responses that cells use during migration. Our understanding of the role of forces in cell migration is rapidly evolving but is still best understood in the context of cells migrating in 2D and 3D environments in vitro. Advances in live imaging of developing embryos combined with the use of experimental and theoretical tools to quantify and analyse forces in vivo, has begun to shed light on the role of mechanics in driving embryonic cell migration. In this review, we focus on the recent studies uncovering the physical basis of embryonic cell migration in vivo. We look at the physical basis of the classical steps of cell migration such as protrusion formation and cell body translocation and review the recent research on how these processes work in the complex 3D microenvironment of a developing organism.

Letting Go to Move On: Membrane Detachment Initiates Protrusion

Membrane-cortex attachment plays key roles in controlling cell shape and integrity. In this issue of Developmental Cell, Welf et al. describe a new function for membrane-actin tethering in controlling where and when cells initiate new protrusions.

Activity-Induced Collapse and Arrest of Active Polymer Rings

We investigate, using numerical simulations, the conformations of isolated active ring polymers. We find that their behavior depends crucially on their size: Short rings (N≲100) swell, whereas longer rings (N≳200) collapse, at sufficiently high activity. By investigating the nonequilibrium process leading to the steady state, we find a universal route driving both outcomes; we highlight the central role of steric interactions, at variance with linear chains, and of topology conservation. We further show that the collapsed rings are arrested by looking at different observables, all underlining the presence of an extremely long timescales at the steady state, associated with the internal dynamics of the collapsed section. Finally, we found that in some circumstances the collapsed state spins about its axis.

Adhesion strength and contractility enable metastatic cells to become adurotactic

Significant changes in cell stiffness, contractility, and adhesion, i.e., mechanotype, are observed during a variety of biological processes. Whether cell mechanics merely change as a side effect of or driver for biological processes is still unclear. Here, we sort genotypically similar metastatic cancer cells into strongly adherent (SA) versus weakly adherent (WA) phenotypes to study how contractility and adhesion differences alter the ability of cells to sense and respond to gradients in material stiffness. We observe that SA cells migrate up a stiffness gradient, or durotax, while WA cells largely ignore the gradient, i.e., adurotax. Biophysical modeling and experimental validation suggest that differences in cell migration and durotaxis between weakly and strongly adherent cells are driven by differences in intra-cellular actomyosin activity. These results provide a direct relationship between cell phenotype and durotaxis and suggest how, unlike other senescent cells, metastatic cancer cells navigate against stiffness gradients.

Impact of confined geometries on hopping and trapping of motile bacteria in porous media

We use a random walk particle-tracking (RWPT) approach to elucidate the impact of porous media confinement and cell-cell interactions on bacterial transport. The model employs stochastic alternating motility states consisting of hopping movement and trapping reorientation. The stochastic motility patterns are defined based on direct visualization of individual trajectory data. We validate our model against experimental data, at single-cell resolution, of bacterial E. coli motion in three-dimensional confined porous media. Results show that the model is able to efficiently simulate the spreading dynamics of motile bacteria as it captures the impact of cell-cell interaction and pore confinement, which marks the transition to a late-time subdiffusive regime. Furthermore, the model is able to qualitatively reproduce the observed directional persistence. Our RWPT model constitutes a meshless simple method which is easy to implement and does not invoke ad hoc assumptions but represents the basis for a multiscale approach to the study of bacterial dispersal in porous systems.

Unified control of amoeboid pseudopod extension in multiple organisms by branched F-actin in the front and parallel F-actin/myosin in the cortex

The trajectory of moving eukaryotic cells depends on the kinetics and direction of extending pseudopods. The direction of pseudopods has been well studied to unravel mechanisms for chemotaxis, wound healing and inflammation. However, the kinetics of pseudopod extension-when and why do pseudopods start and stop- is equally important, but is largely unknown. Here the START and STOP of about 4000 pseudopods was determined in four different species, at four conditions and in nine mutants (fast amoeboids Dictyostelium and neutrophils, slow mesenchymal stem cells, and fungus B.d. chytrid with pseudopod and a flagellum). The START of a first pseudopod is a random event with a probability that is species-specific (23%/s for neutrophils). In all species and conditions, the START of a second pseudopod is strongly inhibited by the extending first pseudopod, which depends on parallel filamentous actin/myosin in the cell cortex. Pseudopods extend at a constant rate by polymerization of branched F-actin at the pseudopod tip, which requires the Scar complex. The STOP of pseudopod extension is induced by multiple inhibitory processes that evolve during pseudopod extension and mainly depend on the increasing size of the pseudopod. Surprisingly, no differences in pseudopod kinetics are detectable between polarized, unpolarized or chemotactic cells, and also not between different species except for small differences in numerical values. This suggests that the analysis has uncovered the fundament of cell movement with distinct roles for stimulatory branched F-actin in the protrusion and inhibitory parallel F-actin in the contractile cortex.

Bridging from single to collective cell migration: A review of models and links to experiments

Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.

Computational models of migration modes improve our understanding of metastasis

Tumor cells migrate through changing microenvironments of diseased and healthy tissue, making their migration particularly challenging to describe. To better understand this process, computational models have been developed for both the ameboid and mesenchymal modes of cell migration. Here, we review various approaches that have been used to account for the physical environment's effect on cell migration in computational models, with a focus on their application to understanding cancer metastasis and the related phenomenon of durotaxis. We then discuss how mesenchymal migration models typically simulate complex cell–extracellular matrix (ECM) interactions, while ameboid migration models use a cell-focused approach that largely ignores ECM when not acting as a physical barrier. This approach greatly simplifies or ignores the mechanosensing ability of ameboid migrating cells and should be reevaluated in future models. We conclude by describing future model elements that have not been included to date but would enhance model accuracy.

Stick-slip model for actin-driven cell protrusions, cell polarization, and crawling

Cell crawling requires the generation of intracellular forces by the cytoskeleton and their transmission to an extracellular substrate through specific adhesion molecules. Crawling cells show many features of excitable systems, such as spontaneous symmetry breaking and crawling in the absence of external cues, and periodic and propagating waves of activity. Mechanical instabilities in the active cytoskeleton network and feedback loops in the biochemical network of activators and repressors of cytoskeleton dynamics have been invoked to explain these dynamical features. Here, I show that the interplay between the dynamics of cell-substrate adhesion and linear cellular mechanics is sufficient to reproduce many nonlinear dynamical patterns observed in spreading and crawling cells. Using an analytical formalism of the molecular clutch model of cell adhesion, regulated by local mechanical forces, I show that cellular traction forces exhibit stick-slip dynamics resulting in periodic waves of protrusion/retraction and propagating waves along the cell edge. This can explain spontaneous symmetry breaking and polarization of spreading cells, leading to steady crawling or bipedal motion, and bistability, where persistent cell motion requires a sufficiently strong transient external stimulus. The model also highlights the role of membrane tension in providing the long-range mechanical communication across the cell required for symmetry breaking.

CompuCell3D Simulations Reproduce Mesenchymal Cell Migration on Flat Substrates

Mesenchymal cell crawling is a critical process in normal development, in tissue function, and in many diseases. Quantitatively predictive numerical simulations of cell crawling thus have multiple scientific, medical, and technological applications. However, we still lack a low-computational-cost approach to simulate mesenchymal three-dimensional (3D) cell crawling. Here, we develop a computationally tractable 3D model (implemented as a simulation in the CompuCell3D simulation environment) of mesenchymal cells crawling on a two-dimensional substrate. The Fürth equation, the usual characterization of mean-squared displacement (MSD) curves for migrating cells, describes a motion in which, for increasing time intervals, cell movement transitions from a ballistic to a diffusive regime. Recent experiments have shown that for very short time intervals, cells exhibit an additional fast diffusive regime. Our simulations' MSD curves reproduce the three experimentally observed temporal regimes, with fast diffusion for short time intervals, slow diffusion for long time intervals, and intermediate time -interval-ballistic motion. The resulting parameterization of the trajectories for both experiments and simulations allows the definition of time- and length scales that translate between computational and laboratory units. Rescaling by these scales allows direct quantitative comparisons among MSD curves and between velocity autocorrelation functions from experiments and simulations. Although our simulations replicate experimentally observed spontaneous symmetry breaking, short-timescale diffusive motion, and spontaneous cell-motion reorientation, their computational cost is low, allowing their use in multiscale virtual-tissue simulations. Comparisons between experimental and simulated cell motion support the hypothesis that short-time actomyosin dynamics affects longer-time cell motility. The success of the base cell-migration simulation model suggests its future application in more complex situations, including chemotaxis, migration through complex 3D matrices, and collective cell motion.

Predicting Confined 1D Cell Migration from Parameters Calibrated to a 2D Motor-Clutch Model

Biological tissues contain micrometer-scale gaps and pores, including those found within extracellular matrix fiber networks, between tightly packed cells, and between blood vessels or nerve bundles and their associated basement membranes. These spaces restrict cell motion to a single-spatial dimension (1D), a feature that is not captured in traditional in vitro cell migration assays performed on flat, unconfined two-dimensional (2D) substrates. Mechanical confinement can variably influence cell migration behaviors, and it is presently unclear whether the mechanisms used for migration in 2D unconfined environments are relevant in 1D confined environments. Here, we assessed whether a cell migration simulator and associated parameters previously measured for cells on 2D unconfined compliant hydrogels could predict 1D confined cell migration in microfluidic channels. We manufactured microfluidic devices with narrow channels (60-μm2 rectangular cross-sectional area) and tracked human glioma cells that spontaneously migrated within channels. Cell velocities (vexp = 0.51 ± 0.02 μm min-1) were comparable to brain tumor expansion rates measured in the clinic. Using motor-clutch model parameters estimated from cells on unconfined 2D planar hydrogel substrates, simulations predicted similar migration velocities (vsim = 0.37 ± 0.04 μm min-1) and also predicted the effects of drugs targeting the motor-clutch system or cytoskeletal assembly. These results are consistent with glioma cells utilizing a motor-clutch system to migrate in confined environments.

An adapted particle swarm optimization algorithm as a model for exploring premyofibril formation

While the fundamental steps outlining myofibril formation share a similar scheme for different cell and species types, various granular details involved in the development of a functional contractile muscle are not well understood. Many studies of myofibrillogenesis focus on the protein interactions that are involved in myofibril maturation with the assumption that there is a fully formed premyofibril at the start of the process. However, there is little known regarding how the premyofibril is initially constructed. Fortunately, the protein α-actinin, which has been consistently identified throughout the maturation process, is found in premyofibrils as punctate aggregates known as z-bodies. We propose a theoretical model based on the particle swarm optimization algorithm that can explore how these α-actinin clusters form into the patterns observed experimentally. Our algorithm can produce different pattern configurations by manipulating specific parameters that can be related to α-actinin mobility and binding affinity. These patterns, which vary experimentally according to species and muscle cell type, speak to the versatility of α-actinin and demonstrate how its behavior may be altered through interactions with various regulatory, signaling, and metabolic proteins. The results of our simulations invite speculation that premyofibrils can be influenced toward developing different patterns by altering the behavior of individual α-actinin molecules, which may be linked to key differences present in different cell types.

Disentangling the behavioural variability of confined cell migration

Cell-to-cell variability is inherent to numerous biological processes, including cell migration. Quantifying and characterizing the variability of migrating cells is challenging, as it requires monitoring many cells for long time windows under identical conditions. Here, we observe the migration of single human breast cancer cells (MDA-MB-231) in confining two-state micropatterns. To describe the stochastic dynamics of this confined migration, we employ a dynamical systems approach. We identify statistics to measure the behavioural variance of the migration, which significantly exceeds that predicted by a population-averaged stochastic model. This additional variance can be explained by the combination of an ‘ageing’ process and population heterogeneity. To quantify population heterogeneity, we decompose the cells into subpopulations of slow and fast cells, revealing the presence of distinct classes of dynamical systems describing the migration, ranging from bistable to limit cycle behaviour. Our findings highlight the breadth of migration behaviours present in cell populations.

Motility and morphodynamics of confined cells

We introduce a minimal hydrodynamic model of polarization, migration, and deformation of a biological cell confined between two parallel surfaces. In our model, the cell is driven out of equilibrium by an active cytsokeleton force that acts on the membrane. The cell cytoplasm, described as a viscous droplet in the Darcy flow regime, contains a diffusive solute that actively transduces the applied cytoskeleton force. While fairly simple and analytically tractable, this quasi-two-dimensional model predicts a range of compelling dynamic behaviours. A linear stability analysis of the system reveals that solute activity first destabilizes a global polarization-translation mode, prompting cell motility through spontaneous symmetry breaking. At higher activity, the system crosses a series of Hopf bifurcations leading to coupled oscillations of droplet shape and solute concentration profiles. At the nonlinear level, we find traveling-wave solutions associated with unique polarized shapes that resemble experimental observations. Altogether, this model offers an analytical paradigm of active deformable systems in which viscous hydrodynamics are coupled to diffusive force transducers.

The Physics of Cellular Decision Making During Epithelial-Mesenchymal Transition

The epithelial-mesenchymal transition (EMT) is a process by which cells lose epithelial traits, such as cell-cell adhesion and apico-basal polarity, and acquire migratory and invasive traits. EMT is crucial to embryonic development and wound healing. Misregulated EMT has been implicated in processes associated with cancer aggressiveness, including metastasis. Recent experimental advances such as single-cell analysis and temporal phenotypic characterization have established that EMT is a multistable process wherein cells exhibit and switch among multiple phenotypic states. This is in contrast to the classical perception of EMT as leading to a binary choice. Mathematical modeling has been at the forefront of this transformation for the field, not only providing a conceptual framework to integrate and analyze experimental data, but also making testable predictions. In this article, we review the key features and characteristics of EMT dynamics, with a focus on the mathematical modeling approaches that have been instrumental to obtaining various useful insights.

Modelling collective cell migration: neural crest as a model paradigm

A huge variety of mathematical models have been used to investigate collective cell migration. The aim of this brief review is twofold: to present a number of modelling approaches that incorporate the key factors affecting cell migration, including cell-cell and cell-tissue interactions, as well as domain growth, and to showcase their application to model the migration of neural crest cells. We discuss the complementary strengths of microscale and macroscale models, and identify why it can be important to understand how these modelling approaches are related. We consider neural crest cell migration as a model paradigm to illustrate how the application of different mathematical modelling techniques, combined with experimental results, can provide new biological insights. We conclude by highlighting a number of future challenges for the mathematical modelling of neural crest cell migration.

Substrate curvature as a cue to guide spatiotemporal cell and tissue organization

Recent evidence clearly shows that cells respond to various physical cues in their environments, guiding many cellular processes and tissue morphogenesis, pathology, and repair. One aspect that is gaining significant traction is the role of local geometry as an extracellular cue. Elucidating how geometry affects cell and tissue behavior is, indeed, crucial to design artificial scaffolds and understand tissue growth and remodeling. Perhaps the most fundamental descriptor of local geometry is surface curvature, and a growing body of evidence confirms that surface curvature affects the spatiotemporal organization of cells and tissues. While well-defined in differential geometry, curvature remains somewhat ambiguously treated in biological studies. Here, we provide a more formal curvature framework, based on the notions of mean and Gaussian curvature, and summarize the available evidence on curvature guidance at the cell and tissue levels. We discuss the involved mechanisms, highlighting the interplay between tensile forces and substrate curvature that forms the foundation of curvature guidance. Moreover, we show that relatively simple computational models, based on some application of curvature flow, are able to capture experimental tissue growth remarkably well. Since curvature guidance principles could be leveraged for tissue regeneration, the implications for geometrical scaffold design are also discussed. Finally, perspectives on future research opportunities are provided.

A mechanical toy model linking cell-substrate adhesion to multiple cellular migratory responses

During cell migration, forces applied to a cell from its environment influence the motion. When the cell is placed on a substrate, such a force is provided by the cell-substrate adhesion. Modulation of adhesivity, often performed by the modulation of the substrate stiffness, tends to cause common responses for cell spreading, cell speed, persistence, and random motility coefficient. Although the reasons for the response of cell spreading and cell speed have been suggested, other responses are not well understood. In this study, we develop a simple toy model for cell migration driven by the relation of two forces: the adhesive force and the plasma membrane tension. The simplicity of the model allows us to perform the calculation not only numerically but also analytically, and the analysis provides formulas directly relating the adhesivity to cell spreading, persistence, and the random motility coefficient. Accordingly, the results offer a unified picture on the causal relations between those multiple cellular responses. In addition, cellular properties that would influence the migratory behavior are suggested.

Pulling-force generation by ensembles of polymerizing actin filaments

The process by which actin polymerization generates pulling forces in cellular processes such as endocytosis is less well understood than pushing-force generation. To clarify the basic mechanisms of pulling-force generation, we perform stochastic polymerization simulations for a square array of polymerizing semiflexible actin filaments, having different interactions with the membrane. The filaments near the array center have a strong attractive component. Filament bending and actin-network elasticity are treated explicitly. We find that the outer filaments push on the membrane and the inner filaments pull, with a net balance of forces. The total calculated pulling force is maximized when the central filaments have a very deep potential well, and the outer filaments have no well. The steady-state force is unaffected by the gel rigidity, but equilibration takes longer for softer gels. The force distributions are flat over the pulling and pushing regions. Actin polymerization is enhanced by softening the gel or reducing the filament binding to the membrane. Filament-membrane detachment can occur for softer gels, even if the total binding energy of the filaments to the membrane is 100 [Formula: see text] or more. It propagates via a stress-concentration mechanism similar to that of a brittle crack in a solid, and the breaking stress is determined by a criterion similar to that of the 'Griffith' theory of crack propagation.

Persistent and polarized global actin flow is essential for directionality during cell migration

Cell migration is hypothesized to involve a cycle of behaviours beginning with leading edge extension. However, recent evidence suggests that the leading edge may be dispensable for migration, raising the question of what actually controls cell directionality. Here, we exploit the embryonic migration of Drosophila macrophages to bridge the different temporal scales of the behaviours controlling motility. This approach reveals that edge fluctuations during random motility are not persistent and are weakly correlated with motion. In contrast, flow of the actin network behind the leading edge is highly persistent. Quantification of actin flow structure during migration reveals a stable organization and asymmetry in the cell-wide flowfield that strongly correlates with cell directionality. This organization is regulated by a gradient of actin network compression and destruction, which is controlled by myosin contraction and cofilin-mediated disassembly. It is this stable actin-flow polarity, which integrates rapid fluctuations of the leading edge, that controls inherent cellular persistence.

Stick-slip dynamics of migrating cells on viscoelastic substrates

Stick-slip motion, a common phenomenon observed during crawling of cells, is found to be strongly sensitive to the substrate stiffness. Stick-slip behaviors have previously been investigated typically using purely elastic substrates. For a more realistic understanding of this phenomenon, we propose a theoretical model to study the dynamics on a viscoelastic substrate. Our model, based on a reaction-diffusion framework, incorporates known important interactions such as retrograde flow of actin, myosin contractility, force-dependent assembly, and disassembly of focal adhesions coupled with cell-substrate interaction. We show that consideration of a viscoelastic substrate not only captures the usually observed stick-slip jumps but also predicts the existence of an optimal substrate viscosity corresponding to maximum traction force and minimum retrograde flow which was hitherto unexplored. Moreover, our theory predicts the time evolution of individual bond force that characterizes the stick-slip patterns on soft versus stiff substrates. Our analysis also elucidates how the duration of the stick-slip cycles are affected by various cellular parameters.