Mathematical Models of Cell Motility

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.

Bayesian analysis of coupled cellular and nuclear trajectories for cell migration

Cell migration, the process by which cells move from one location to another, plays crucial roles in many biological events. While much research has been devoted to understand the process, most statistical cell migration models rely on using time-lapse microscopy data from cell trajectories alone. However, the cell and its associated nucleus work together to orchestrate cell movement, which motivates a joint analysis of coupled cell-nucleus trajectories. In this paper, we propose a Bayesian hierarchical model for analyzing cell migration. We incorporate a bivariate angular distribution to handle the coupled cell-nucleus trajectories and introduce latent motility status indicators to model a cell's motility as a time-dependent characteristic. A Markov chain Monte Carlo algorithm is provided for practical implementation of our model, which is used on real experimental data from MDA-MB-231 and NIH 3T3 cells. Through the fitted models, deeper insights into the migratory patterns of these experimental cell populations are gained and their differences are quantified.

Migration of the 3T3 Cell with a Lamellipodium on Various Stiffness Substrates—Tensegrity Model

Changes in mechanical stimuli and the physiological environment are sensed by the cell. Thesechanges influence the cell’s motility patterns. The cell’s directional migration is dependent on the substrate stiffness. To describe such behavior of a cell, a tensegrity model was used. Cells with an extended lamellipodium were modeled. The internal elastic strain energy of a cell attached to the substrates with different stiffnesses was evaluated. The obtained results show that on the stiffer substrate, the elastic strain energy of the cell adherent to this substrate decreases. Therefore, the substrate stiffness is one of the parameters that govern the cell’s directional movement.

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.

Mathematical modelling of cell migration

The complexity of biological systems creates challenges for fully understanding their behaviour. This is particularly true for cell migration which requires the co-ordinated activity of hundreds of individual components within cells. Mathematical modelling can help understand these complex systems by breaking the system into discrete steps which can then be interrogated in silico. In this review, we highlight scenarios in cell migration where mathematical modelling can be applied and discuss what types of modelling are most suited. Almost any aspect of cell migration is amenable to mathematical modelling from the modelling of intracellular processes such as chemokine receptor signalling and actin filament branching to larger scale processes such as the movement of individual cells or populations of cells through their environment. Two common ways of approaching this modelling are the use of models based on differential equations or agent-based modelling. The application of both these approaches to cell migration are discussed with specific examples along with common software tools to facilitate the process for non-mathematicians. We also highlight the challenges of modelling cell migration and the need for rigorous experimental work to effectively parameterise a model.

A DISCRETE MATHEMATICAL MODEL FOR SINGLE AND COLLECTIVE MOVEMENT IN AMOEBOID CELLS

In this paper, we develop a new discrete mathematical model for individual and collective cell motility. We introduce a mechanical model for the movement of a cell on a two-dimensional rigid surface to describe and investigate the cell–cell and cell–substrate interactions. The cell cytoskeleton is modeled as a series of springs and dashpots connected in parallel. The cell–substrate attachments and the cell protrusions are also included. In particular, this model is used to describe the directed movement of endothelial cells on a Matrigel plate. We compare the results from our model with experimental data. We show that cell density and substrate rigidity play an important role in network formation.

Keap1 Inhibits Metastatic Properties of NSCLC Cells by Stabilizing Architectures of F-Actin and Focal Adhesions

Low expression of the tumor suppressor Kelch-like ECH-associated protein 1 (KEAP1) in non-small cell lung cancer (NSCLC) often results in higher malignant biological behavior and poor prognosis; however, the underlying mechanism remains unclear. The present study demonstrates that overexpression of Keap1 significantly suppresses migration and invasion of three different lung cancer cells (A549, H460, and H1299). Highly expressed Keap1, compared with the control, promotes formation of multiple stress fibers with larger mature focal adhesion complexes in the cytoplasm where only fine focal adhesions were observed in the membrane under control conditions. RhoA activity significantly increased when Keap1 was overexpressed, whereas Myosin 9b expression was reduced but could be rescued by proteasome inhibition. Noticeably, mouse tumor xenografts with Keap1 overexpression were smaller in size and less metastatic relative to the control group. Taken together, these results demonstrate that Keap1 stabilizes F-actin cytoskeleton structures and inhibits focal adhesion turnover, thereby restraining the migration and invasion of NSCLC. Therefore, increasing Keap1 or targeting its downstream molecules might provide potential therapeutic benefits for the treatment of patients with NSCLC.Implications: This study provides mechanistic insight on the metastatic process in NSCLC and suggests that Keap1 and its downstream molecules may be valuable drug targets for NSCLC patients. Mol Cancer Res; 16(3); 508-16. ©2018 AACR.

How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time

We provide a tool for data-driven modeling of motility, data being time-lapse recorded trajectories. Several mathematical properties of a model to be found can be gleaned from appropriate model-independent experimental statistics, if one understands how such statistics are distorted by the finite sampling frequency of time-lapse recording, by experimental errors on recorded positions, and by conditional averaging. We give exact analytical expressions for these effects in the simplest possible model for persistent random motion, the Ornstein-Uhlenbeck process. Then we describe those aspects of these effects that are valid for any reasonable model for persistent random motion. Our findings are illustrated with experimental data and Monte Carlo simulations.

Stochastic model explains formation of cell arrays on H/O-diamond patterns

Cell migration plays an important role in many biological systems. A relatively simple stochastic model is developed and used to describe cell behavior on chemically patterned substrates. The model is based on three parameters: the speed of cell movement (own and external), the probability of cell adhesion, and the probability of cell division on the substrate. The model is calibrated and validated by experimental data obtained on hydrogen- and oxygen-terminated patterns on diamond. Thereby, the simulations reveal that: (1) the difference in the cell movement speed on these surfaces (about 1.5×) is the key factor behind the formation of cell arrays on the patterns, (2) this difference is provided by the presence of fetal bovine serum (validated by experiments), and (3) the directional cell flow promotes the array formation. The model also predicts that the array formation requires mean distance of cell travel at least 10% of intended stripe width. The model is generally applicable for biosensors using diverse cells, materials, and structures.

Unidirectional cell crawling model guided by extracellular cues

Cell migration is a highly regulated and complex cellular process to maintain proper homeostasis for various biological processes. Extracellular environment was identified as the main affecting factors determining the direction of cell crawling. It was observed experimentally that the cell prefers migrating to the area with denser or stiffer array of microposts. In this article, an integrated unidirectional cell crawling model was developed to investigate the spatiotemporal dynamics of unidirectional cell migration, which incorporates the dominating intracellular biochemical processes, biomechanical processes and the properties of extracellular micropost arrays. The interpost spacing and the stiffness of microposts are taken into account, respectively, to study the mechanism of unidirectional cell locomotion and the guidance of extracellular influence cues on the direction of unidirectional cell crawling. The model can explain adequately the unidirectional crawling phenomena observed in experiments such as "spatiotaxis" and "durotaxis," which allows us to obtain further insights into cell migration.

On the mechanical interplay between intra- and inter-synchronization during collective cell migration: a numerical investigation

Collective cell migration is a fundamental process that takes place during several biological phenomena such as embryogenesis, immunity response, and tumorogenesis, but the mechanisms that regulate it are still unclear. Similarly to collective animal behavior, cells receive feedbacks in space and time, which control the direction of the migration and the synergy between the cells of the population, respectively. While in single cell migration intra-synchronization (i.e. the synchronization between the protrusion-contraction movement of the cell and the adhesion forces exerted by the cell to move forward) is a sufficient condition for an efficient migration, in collective cell migration the cells must communicate and coordinate their movement between each other in order to be as efficient as possible (i.e. inter-synchronization). Here, we propose a 2D mechanical model of a cell population, which is described as a continuum with embedded discrete cells with or without motility phenotype. The decomposition of the deformation gradient is employed to reproduce the cyclic active strains of each single cell (i.e. protrusion and contraction). We explore different modes of collective migration to investigate the mechanical interplay between intra- and inter-synchronization. The main objective of the paper is to evaluate the efficiency of the cell population in terms of covered distance and how the stress distribution inside the cohort and the single cells may in turn provide insights regarding such efficiency.

Cell migration with multiple pseudopodia: temporal and spatial sensing models

Cell migration triggered by pseudopodia (or "false feet") is the most used method of locomotion. A 3D finite element model of a cell migrating over a 2D substrate is proposed, with a particular focus on the mechanical aspects of the biological phenomenon. The decomposition of the deformation gradient is used to reproduce the cyclic phases of protrusion and contraction of the cell, which are tightly synchronized with the adhesion forces at the back and at the front of the cell, respectively. First, a steady active deformation is considered to show the ability of the cell to simultaneously initiate multiple pseudopodia. Here, randomness is considered as a key aspect, which controls both the direction and the amplitude of the false feet. Second, the migration process is described through two different strategies: the temporal and the spatial sensing models. In the temporal model, the cell "sniffs" the surroundings by extending several pseudopodia and only the one that receives a positive input will become the new leading edge, while the others retract. In the spatial model instead, the cell senses the external sources at different spots of the membrane and only protrudes one pseudopod in the direction of the most attractive one.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

Continuum modeling of a neuronal cell under blast loading

Traumatic brain injuries have recently been put under the spotlight as one of the most important causes of accidental brain dysfunctions. Significant experimental and modeling efforts are thus underway to study the associated biological, mechanical and physical mechanisms. In the field of cell mechanics, progress is also being made at the experimental and modeling levels to better characterize many of the cell functions, including differentiation, growth, migration and death. The work presented here aims to bridge both efforts by proposing a continuum model of a neuronal cell submitted to blast loading. In this approach, the cytoplasm, nucleus and membrane (plus cortex) are differentiated in a representative cell geometry, and different suitable material constitutive models are chosen for each one. The material parameters are calibrated against published experimental work on cell nanoindentation at multiple rates. The final cell model is ultimately subjected to blast loading within a complete computational framework of fluid-structure interaction. The results are compared to the nanoindentation simulation, and the specific effects of the blast wave on the pressure and shear levels at the interfaces are identified. As a conclusion, the presented model successfully captures some of the intrinsic intracellular phenomena occurring during the cellular deformation under blast loading that potentially lead to cell damage. It suggests, more particularly, that the localization of damage at the nucleus membrane is similar to what has already been observed at the overall cell membrane. This degree of damage is additionally predicted to be worsened by a longer blast positive phase duration. In conclusion, the proposed model ultimately provides a new three-dimensional computational tool to evaluate intracellular damage during blast loading.

'Run-and-tumble' or 'look-and-run'? A mechanical model to explore the behavior of a migrating amoeboid cell

Single cell migration constitutes a fundamental phenomenon involved in many biological events. Amoeboid cells are single cell organisms that migrate in a cyclic manner like worms. In this paper, we propose a 3D finite element model of an amoeboid cell migrating over a 2D surface. In particular, we focus on the mechanical aspect of the problem. The cell is able to generate cyclic active deformations, such as protrusion and contraction, in any direction. The progression of the cell is governed by a tight synchronization between the adhesion forces, which are alternatively applied at the front and at the rear edges of the cell, and the protrusion-contraction phases of the cell body. Finally, two important aspects have been taken into account: (1) the external stimuli in response to which the cell migrates (e.g. need to feed, morphogenetic events, normal or abnormal environment cues), (2) the heterogeneity of the 2D substrate (e.g. obstacles, rugosity, slippy regions) for which two distinct approaches have been evaluated: the 'run-and-tumble' strategy and the 'look-and-run' strategy. Overall, the results show a good agreement with respect to the experimental observations and the data from the literature (e.g. velocity and strains). Therefore, the present model helps, on one hand, to better understand the intimate relationship between the deformation modes of a cell and the adhesion strength that is required by the cell to crawl over a substrate, and, on the other hand, to put in evidence the crucial role played by mechanics during the migration process.

Cell-biomaterial mechanical interaction in the framework of tissue engineering: insights, computational modeling and perspectives

Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.

Effects of compressive residual stress on the morphologic changes of fibroblasts

Recently, the term tensotaxis was proposed to describe the phenomenon that tensile stress or strain affects cell migration. Even so, less attention has been paid to the effects of compressive stress on cell behavior. In this study, by using an injection-molded method combined with photoelastic technology, we developed residual stress gradient-controlled poly-L-lactide discs. After culturing NIH-3T3 fibroblasts on the stress gradient substrate, the cell distributions for high- and low-stress regions were measured and compared. Our results showed that there were significantly more cells in the low-compressive stress region relative to their high-stress analogs (p < 0.05). In addition, NIH-3T3 fibroblasts in the low-compressive stress region expressed more abundant extensive filopodia. These findings provide greater insight into the interaction between cells and substrates, and could serve as a useful reference for connective tissue development and repair.

A numerical model for durotaxis

Cell migration is a phenomenon that is involved in several physiological processes. In the absence of external guiding factors it shares analogies with Brownian motion. The presence of biochemical or biophysical cues, on the other hand, can influence cell migration transforming it in a biased random movement. Recent studies have shown that different cell types are able to recognise the mechanical properties of the substratum over which they move and that these properties direct the motion through a process called durotaxis. In this work a 2D mathematical model for the description of this phenomenon is presented. The model is based on the Langevin equation that has been modified to take into account the local mechanical properties of the substratum perceived by the cells. Numerical simulations of the model provide individual cell tracks, whose characteristics can be compared with experimental observations directly. The present model is solved for two important cases: an isotropic substratum, to check that random motility is recovered as a subcase, and a biphasic substratum, to investigate durotaxis. The degree of agreement is satisfactory in both cases. The model can be a useful tool for quantifying relevant parameters of cell migration as a function of the substratum mechanical properties.

A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION

Much is known about the biophysical mechanisms involved in cell crawling, but how these processes are coordinated to produce directed motion is not well understood. Here, we propose a new hypothesis whereby local cytoskeletal contraction generates fluid flow through the lamellipodium, with the pressure at the front of the cell facilitating actin polymerization which pushes the leading edge forward. The contraction, in turn, is regulated by stress in the cytoskeleton. To test this hypothesis, finite element models for a crawling cell are presented. These models are based on nonlinear poroelasticity theory, modified to include the effects of active contraction and growth, which are regulated by mechanical feedback laws. Results from the models agree reasonably well with published experimental data for cell speed, actin flow, and cytoskeletal deformation in migrating fish epidermal keratocytes. The models also suggest that oscillations can occur for certain ranges of parameter values.

A sub-cellular viscoelastic model for cell population mechanics

Understanding the biomechanical properties and the effect of biomechanical force on epithelial cells is key to understanding how epithelial cells form uniquely shaped structures in two or three-dimensional space. Nevertheless, with the limitations and challenges posed by biological experiments at this scale, it becomes advantageous to use mathematical and 'in silico' (computational) models as an alternate solution. This paper introduces a single-cell-based model representing the cross section of a typical tissue. Each cell in this model is an individual unit containing several sub-cellular elements, such as the elastic plasma membrane, enclosed viscoelastic elements that play the role of cytoskeleton, and the viscoelastic elements of the cell nucleus. The cell membrane is divided into segments where each segment (or point) incorporates the cell's interaction and communication with other cells and its environment. The model is capable of simulating how cells cooperate and contribute to the overall structure and function of a particular tissue; it mimics many aspects of cellular behavior such as cell growth, division, apoptosis and polarization. The model allows for investigation of the biomechanical properties of cells, cell-cell interactions, effect of environment on cellular clusters, and how individual cells work together and contribute to the structure and function of a particular tissue. To evaluate the current approach in modeling different topologies of growing tissues in distinct biochemical conditions of the surrounding media, we model several key cellular phenomena, namely monolayer cell culture, effects of adhesion intensity, growth of epithelial cell through interaction with extra-cellular matrix (ECM), effects of a gap in the ECM, tensegrity and tissue morphogenesis and formation of hollow epithelial acini. The proposed computational model enables one to isolate the effects of biomechanical properties of individual cells and the communication between cells and their microenvironment while simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue.

A model of fibroblast motility on substrates with different rigidities

To function efficiently in the body, the biological cells must have the ability to sense the external environment. Mechanosensitivity toward the extracellular matrix was identified as one of the sensing mechanisms affecting cell behavior. It was shown experimentally that a fibroblast cell prefers locomoting over the stiffer substrate when given a choice between a softer and a stiffer substrate. In this article, we develop a discrete model of fibroblast motility with substrate-rigidity sensing. Our model allows us to understand the interplay between the cell-substrate sensing and the cell biomechanics. The model cell exhibits experimentally observed substrate rigidity sensing, which allows us to gain additional insights into the cell mechanosensitivity.

Analysis and modelling of motility of cell populations with MotoCell

Background

Cell motility plays a central role in development, wound-healing and tumour invasion. Cultures of eucariotic cells are a complex system where most cells move according to 'random' patterns, but may also be induced to a more coordinate migration by means of specific stimuli, such as the presence of chemical attractants or the introduction of a mechanical stimulus. Various tools have been developed that work by keeping track of the paths followed by specific objects and by performing statistical analysis on the recorded path data. The available tools include desktop applications or macros running within a commercial package, which address specific aspects of the process.

Results

An online application, MotoCell, was developed to evaluate the motility of cell populations maintained in various experimental conditions. Statistical analysis of cell behaviour consists of the evaluation of descriptive parameters such as average speed and angle, directional persistence, path vector length, calculated for the whole population as well as for each cell and for each step of the migration; in this way the behaviour of a whole cell population may be assessed as a whole or as a sum of individual entities. The directional movement of objects may be studied by eliminating the modulo effect in circular statistics analysis, able to evaluate linear dispersion coefficient (R) and angular dispersion (S) values together with average angles. A case study is provided where the system is used to characterize motility of RasV12 transformed NIH3T3 fibroblasts.

Conclusion

Here we describe a comprehensive tool which takes care of all steps in cell motility analysis, including interactive cell tracking, path editing and statistical analysis of cell movement, all within a freely available online service. Although based on a standard web interface, the program is very fast and interactive and is immediately available to a large number of users, while exploiting the web approach in a very effective way. The ability to evaluate the behaviour of single cells allows to draw the attention on specific correlations, such as linearity of movement and deviation from the expected direction. In addition to population statistics, the analysis of single cells allows to group the cells into subpopulations, or even to evaluate the behaviour of each cell with respect to a variable reference, such as the direction of a wound or the position of the closest cell.

Multi-scale models of cell and tissue dynamics

Cell and tissue movement are essential processes at various stages in the life cycle of most organisms. The early development of multi-cellular organisms involves individual and collective cell movement; leukocytes must migrate towards sites of infection as part of the immune response; and in cancer, directed movement is involved in invasion and metastasis. The forces needed to drive movement arise from actin polymerization, molecular motors and other processes, but understanding the cell- or tissue-level organization of these processes that is needed to produce the forces necessary for directed movement at the appropriate point in the cell or tissue is a major challenge. In this paper, we present three models that deal with the mechanics of cells and tissues: a model of an arbitrarily deformable single cell, a discrete model of the onset of tumour growth in which each cell is treated individually, and a hybrid continuum-discrete model of the later stages of tumour growth. While the models are different in scope, their underlying mechanical and mathematical principles are similar and can be applied to a variety of biological systems.