From energy to cellular forces in the Cellular Potts Model: An algorithmic approach

CellularPotts.jl: simulating multiscale cellular models in Julia

SUMMARY

CellularPotts.jl is a software package written in Julia to simulate biological cellular processes such as division, adhesion, and signaling. Accurately modeling and predicting these simple processes is crucial because they facilitate more complex biological phenomena related to important disease states like tumor growth, wound healing, and infection. Here we take advantage of Cellular Potts Modeling to simulate cellular interactions and combine them with differential equations to model dynamic cell signaling patterns. These models are advantageous over other approaches because they retain spatial information about each cell while remaining computationally efficient at larger scales. Users of this package define three key inputs to create valid model definitions: a 2- or 3-dimensional space, a table describing the cells to be positioned in that space, and a list of model penalties that dictate cell behaviors. Models can then be evolved over time to collect statistics, simulated repeatedly to investigate how changing a specific property impacts cellular behavior, and visualized using any of the available plotting libraries in Julia.

AVAILABILITY AND IMPLEMENTATION

The CellularPotts.jl package is released under the MIT license and is available at https://github.com/RobertGregg/CellularPotts.jl. An archived version of the code (v0.3.2) at time of submission can also be found at https://doi.org/10.5281/zenodo.10407783.

Mathematical model for promotion of wound closure with ATP release

To computationally investigate the recent experimental finding such that extracellular ATP release caused by exogeneous mechanical forces promote wound closure, we introduce a mathematical model, the Cellular Potts Model (CPM), which is a popular discretized model on a lattice, where the movement of a "cell" is determined by a Monte Carlo procedure. In the experiment, it was observed that there is mechanosensitive ATP release from the leading cells facing the wound gap and the subsequent extracellular Ca2+ influx. To model these phenomena, the Reaction-Diffusion equations for extracellular ATP and intracellular Ca2+ concentrations are adopted and combined with CPM, where we also add a polarity term because the cell migration is enhanced in the case of ATP release. From the numerical simulations using this hybrid model, we discuss effects of the collective cell migration due to the ATP release and the Ca2+ influx caused by the mechanical forces and the consequent promotion of wound closure.

Excitable dynamics in a molecularly-explicit model of cell motility: Mixed-mode oscillations and beyond

Mesenchymal cell motility is mainly regulated by two members of the Rho-family of GTPases, called Rac and Rho. The mutual inhibition exerted by these two proteins on each other's activation and the promotion of Rac activation by an adaptor protein called paxillin have been implicated in driving cellular polarization comprised of front (high active Rac) and back (high active Rho) during cell migration. Mathematical modeling of this regulatory network has previously shown that bistability is responsible for generating a spatiotemporal pattern underscoring cellular polarity called wave-pinning when diffusion is included. We previously developed a 6V reaction-diffusion model of this network to decipher the role of Rac, Rho and paxillin (along with other auxiliary proteins) in generating wave-pinning. In this study, we simplify this model through a series of steps into an excitable 3V ODE model comprised of one fast variable (the scaled concentration of active Rac), one slow variable (the maximum paxillin phosphorylation rate - turned into a variable) and a very slow variable (a recovery rate - also turned into a variable). We then explore, through slow-fast analysis, how excitability is manifested by showing that the model can exhibit relaxation oscillations (ROs) as well as mixed-mode oscillations (MMOs) whose underlying dynamics are consistent with a delayed Hopf bifurcation with a canard explosion. By reintroducing diffusion and the scaled concentration of inactive Rac into the model, we obtain a 4V PDE model that generates several unique spatiotemporal patterns that are relevant to cell motility. These patterns are then characterized and their impact on cell motility are explored by employing the cellular potts model (CPM). Our results reveal that wave pinning produces purely very directed motion in CPM, while MMOs allow for meandering and non-motile behaviors to occur. This highlights the role of MMOs as a potential mechanism for mesenchymal cell motility.

Computational Modeling to Determine the Effect of Phenotypic Heterogeneity in Tumors on the Collective Tumor-Immune Interactions

Tumor immunotherapy aims to maintain or enhance the killing capability of CD8+ T cells to clear tumor cells. The tumor-immune interactions affect the function of CD8+ T cells. However, the effect of phenotype heterogeneity of a tumor mass on the collective tumor-immune interactions is insufficiently investigated. We developed the cellular-level computational model based on the principle of cellular Potts model to solve the case mentioned above. We considered how asymmetric division and glucose distribution jointly regulated the transient changes in the proportion of proliferating/quiescent tumor cells in a solid tumor mass. The evolution of a tumor mass in contact with T cells was explored and validated by comparing it with previous studies. Our modeling exhibited that proliferating/quiescent tumor cells, exhibiting distinct anti-apoptotic and suppressive behaviors, redistributed within the domain accompanied by the evolution of a tumor mass. Collectively, a tumor mass prone to a quiescent state weakened the collective suppressive functions of a tumor mass on cytotoxic T cells and triggered a decline of apoptosis of tumor cells. Although quiescent tumor cells did not sufficiently do their inhibitory functions, the possibility of long-term survival was improved due to their interior location within a mass. Overall, the proposed model provides a useful framework to investigate collective-targeted strategies for improving the efficiency of immunotherapy.

Physical confinement and cell proximity increase cell migration rates and invasiveness: A mathematical model of cancer cell invasion through flexible channels

Cancer cell migration between different body parts is the driving force behind cancer metastasis, which is the main cause of mortality of patients. Migration of cancer cells often proceeds by penetration through narrow cavities in locally stiff, yet flexible tissues. In our previous work, we developed a model for cell geometry evolution during invasion, which we extend here to investigate whether leader and follower (cancer) cells that only interact mechanically can benefit from sequential transmigration through narrow micro-channels and cavities. We consider two cases of cells sequentially migrating through a flexible channel: leader and follower cells being closely adjacent or distant. Using Wilcoxon's signed-rank test on the data collected from Monte Carlo simulations, we conclude that the modelled transmigration speed for the follower cell is significantly larger than for the leader cell when cells are distant, i.e. follower cells transmigrate after the leader has completed the crossing. Furthermore, it appears that there exists an optimum with respect to the width of the channel such that cell moves fastest. On the other hand, in the case of closely adjacent cells, effectively performing collective migration, the leader cell moves 12% faster since the follower cell pushes it. This work shows that mechanical interactions between cells can increase the net transmigration speed of cancer cells, resulting in increased invasiveness. In other words, interaction between cancer cells can accelerate metastatic invasion.

Polarity and mixed-mode oscillations may underlie different patterns of cellular migration

In mesenchymal cell motility, several migration patterns have been observed, including directional, exploratory and stationary. Two key members of the Rho-family of GTPases, Rac and Rho, along with an adaptor protein called paxillin, have been particularly implicated in the formation of such migration patterns and in regulating adhesion dynamics. Together, they form a key regulatory network that involves the mutual inhibition exerted by Rac and Rho on each other and the promotion of Rac activation by phosphorylated paxillin. Although this interaction is sufficient in generating wave-pinning that underscores cellular polarization comprised of cellular front (high active Rac) and back (high active Rho), it remains unclear how they interact collectively to induce other modes of migration detected in Chinese hamster Ovary (CHO-K1) cells. We previously developed a six-variable (6V) reaction-diffusion model describing the interactions of these three proteins (in their active/phosphorylated and inactive/unphosphorylated forms) along with other auxiliary proteins, to decipher their role in generating wave-pinning. In this study, we explored, through computational modeling and image analysis, how differences in timescales within this molecular network can potentially produce the migration patterns in CHO-K1 cells and how switching between migration modes could occur. To do so, the 6V model was reduced to an excitable 4V spatiotemporal model possessing three different timescales. The model produced not only wave-pinning in the presence of diffusion, but also mixed-mode oscillations (MMOs) and relaxation oscillations (ROs). Implementing the model using the Cellular Potts Model (CPM) produced outcomes in which protrusions in the cell membrane changed Rac-Rho localization, resulting in membrane oscillations and fast directionality variations similar to those observed experimentally in CHO-K1 cells. The latter was assessed by comparing the migration patterns of experimental with CPM cells using four metrics: instantaneous cell speed, exponent of mean-square displacement ([Formula: see text]-value), directionality ratio and protrusion rate. Variations in migration patterns induced by mutating paxillin's serine 273 residue were also captured by the model and detected by a machine classifier, revealing that this mutation alters the dynamics of the system from MMOs to ROs or nonoscillatory behaviour through variation in the scaled concentration of an active form of an adhesion protein called p21-Activated Kinase 1 (PAK). These results thus suggest that MMOs and adhesion dynamics are the key mechanisms regulating CHO-K1 cell motility.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Biophysical models of early mammalian embryogenesis

Embryo development is a critical and fascinating stage in the life cycle of many organisms. Despite decades of research, the earliest stages of mammalian embryogenesis are still poorly understood, caused by a scarcity of high-resolution spatial and temporal data, the use of only a few model organisms, and a paucity of truly multidisciplinary approaches that combine biological research with biophysical modeling and computational simulation. Here, we explain the theoretical frameworks and biophysical processes that are best suited to modeling the early mammalian embryo, review a comprehensive list of previous models, and discuss the most promising avenues for future work.

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.

Computational approaches for simulating luminogenesis

Lumens, liquid-filled cavities surrounded by polarized tissue cells, are elementary units involved in the morphogenesis of organs. Theoretical modeling and computations, which can integrate various factors involved in biophysics of morphogenesis of cell assembly and lumens, may play significant roles to elucidate the mechanisms in formation of such complex tissue with lumens. However, up to present, it has not been documented well what computational approaches or frameworks can be applied for this purpose and how we can choose the appropriate approach for each problem. In this review, we report some typical lumen morphologies and basic mechanisms for the development of lumens, focusing on three keywords - mechanics, hydraulics and geometry - while outlining pros and cons of the current main computational strategies. We also describe brief guidance of readouts, i.e., what we should measure in experiments to make the comparison with the model's assumptions and predictions.

A Review of Mathematical and Computational Methods in Cancer Dynamics

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Density-Dependent Migration Characteristics of Cancer Cells Driven by Pseudopod Interaction

The ability of cancer cells to invade neighboring tissue from primary tumors is an important determinant of metastatic behavior. Quantification of cell migration characteristics such as migration speed and persistence helps to understand the requirements for such invasiveness. One factor that may influence invasion is how local tumor cell density shapes cell migration characteristics, which we here investigate with a combined experimental and computational modeling approach. First, we generated and analyzed time-lapse imaging data on two aggressive Triple-Negative Breast Cancer (TNBC) cell lines, HCC38 and Hs578T, during 2D migration assays at various cell densities. HCC38 cells exhibited a counter-intuitive increase in speed and persistence with increasing density, whereas Hs578T did not exhibit such an increase. Moreover, HCC38 cells exhibited strong cluster formation with active pseudopod-driven migration, especially at low densities, whereas Hs578T cells maintained a dispersed positioning. In order to obtain a mechanistic understanding of the density-dependent cell migration characteristics and cluster formation, we developed realistic spatial simulations using a Cellular Potts Model (CPM) with an explicit description of pseudopod dynamics. Model analysis demonstrated that pseudopods exerting a pulling force on the cell and interacting via increased adhesion at pseudopod tips could explain the experimentally observed increase in speed and persistence with increasing density in HCC38 cells. Thus, the density-dependent migratory behavior could be an emergent property of single-cell characteristics without the need for additional mechanisms. This implies that pseudopod dynamics and interaction may play a role in the aggressive nature of cancers through mediating dispersal.

Computational modelling of cell motility modes emerging from cell-matrix adhesion dynamics

Lymphocytes have been described to perform different motility patterns such as Brownian random walks, persistent random walks, and Lévy walks. Depending on the conditions, such as confinement or the distribution of target cells, either Brownian or Lévy walks lead to more efficient interaction with the targets. The diversity of these motility patterns may be explained by an adaptive response to the surrounding extracellular matrix (ECM). Indeed, depending on the ECM composition, lymphocytes either display a floating motility without attaching to the ECM, or sliding and stepping motility with respectively continuous or discontinuous attachment to the ECM, or pivoting behaviour with sustained attachment to the ECM. Moreover, on the long term, lymphocytes either perform a persistent random walk or a Brownian-like movement depending on the ECM composition. How the ECM affects cell motility is still incompletely understood. Here, we integrate essential mechanistic details of the lymphocyte-matrix adhesions and lymphocyte intrinsic cytoskeletal induced cell propulsion into a Cellular Potts model (CPM). We show that the combination of de novo cell-matrix adhesion formation, adhesion growth and shrinkage, adhesion rupture, and feedback of adhesions onto cell propulsion recapitulates multiple lymphocyte behaviours, for different lymphocyte subsets and various substrates. With an increasing attachment area and increased adhesion strength, the cells’ speed and persistence decreases. Additionally, the model predicts random walks with short-term persistent but long-term subdiffusive properties resulting in a pivoting type of motility. For small adhesion areas, the spatial distribution of adhesions emerges as a key factor influencing cell motility. Small adhesions at the front allow for more persistent motility than larger clusters at the back, despite a similar total adhesion area. In conclusion, we present an integrated framework to simulate the effects of ECM proteins on cell-matrix adhesion dynamics. The model reveals a sufficient set of principles explaining the plasticity of lymphocyte motility.

During immunosurveillance, lymphocytes patrol through tissues to interact with cancer cells, other immune cells, and pathogens. The efficiency of this process depends on the kinds of trajectories taken, ranging from simple Brownian walks to Lévy walks. The composition of the extracellular matrix (ECM), a network of macromolecules, affects the formation of cell-matrix adhesions, thus strongly influencing the way lymphocytes move. Here, we present a model of lymphocyte motility driven by adhesions that grow, shrink and rupture in response to the ECM and cellular forces. Compared to other models, our model is computationally light making it suitable for generating long term cell track data, while still capturing actin dynamics and adhesion turnover. Our model suggests that cell motility is affected by the force required to break adhesions and the rate at which new adhesions form. Adhesions can promote cell protrusion by inhibiting retrograde actin flow. After introducing this effect into the model, we found that it reduces the cellular diffusivity and that it promotes stick-slip behaviour. Furthermore, location and size of adhesion clusters determined cell persistence. Overall, our model explains the plasticity of lymphocyte behaviour in response to the ECM.

Supracellular organization confers directionality and mechanical potency to migrating pairs of cardiopharyngeal progenitor cells

Physiological and pathological morphogenetic events involve a wide array of collective movements, suggesting that multicellular arrangements confer biochemical and biomechanical properties contributing to tissue-scale organization. The Ciona cardiopharyngeal progenitors provide the simplest model of collective cell migration, with cohesive bilateral cell pairs polarized along the leader-trailer migration path while moving between the ventral epidermis and trunk endoderm. We use the Cellular Potts Model to computationally probe the distributions of forces consistent with shapes and collective polarity of migrating cell pairs. Combining computational modeling, confocal microscopy, and molecular perturbations, we identify cardiopharyngeal progenitors as the simplest cell collective maintaining supracellular polarity with differential distributions of protrusive forces, cell-matrix adhesion, and myosin-based retraction forces along the leader-trailer axis. 4D simulations and experimental observations suggest that cell-cell communication helps establish a hierarchy to align collective polarity with the direction of migration, as observed with three or more cells in silico and in vivo. Our approach reveals emerging properties of the migrating collective: cell pairs are more persistent, migrating longer distances, and presumably with higher accuracy. Simulations suggest that cell pairs can overcome mechanical resistance of the trunk endoderm more effectively when they are polarized collectively. We propose that polarized supracellular organization of cardiopharyngeal progenitors confers emergent physical properties that determine mechanical interactions with their environment during morphogenesis.

Cellular Tango: how extracellular matrix adhesion choreographs Rac-Rho signaling and cell movement

The small GTPases Rac and Rho are known to regulate eukaryotic cell shape, promoting front protrusion (Rac) or rear retraction (Rho) of the cell edge. Such cell deformation changes the contact and adhesion of cell to the extracellular matrix (ECM), while ECM signaling through integrin receptors also affects GTPase activity. We develop and investigate a model for this three-way feedback loop in 1D and 2D spatial domains, as well as in a fully deforming 2D cell shapes with detailed adhesion-bond biophysics. The model consists of reaction-diffusion equations solved numerically with open-source software, Morpheus, and with custom-built cellular Potts model simulations. We find a variety of patterns and cell behaviors, including persistent polarity, flipped front-back cell polarity oscillations, spiral waves, and random protrusion-retraction. We show that the observed spatial patterns depend on the cell shape, and vice versa.

The mechanics and dynamics of cancer cells sensing noisy 3D contact guidance

Contact guidance is a major physical cue that modulates cancer cell morphology and motility, and is directly linked to the prognosis of cancer patients. Under physiological conditions, particularly in the three-dimensional (3D) extracellular matrix (ECM), the disordered assembly of fibers presents a complex directional bias to the cells. It is unclear how cancer cells respond to these noncoherent contact guidance cues. Here we combine quantitative experiments, theoretical analysis, and computational modeling to study the morphological and migrational responses of breast cancer cells to 3D collagen ECM with varying degrees of fiber alignment. We quantify the strength of contact guidance using directional coherence of ECM fibers, and find that stronger contact guidance causes cells to polarize more strongly along the principal direction of the fibers. Interestingly, sensitivity to contact guidance is positively correlated with cell aspect ratio, with elongated cells responding more strongly to ECM alignment than rounded cells. Both experiments and simulations show that cell-ECM adhesions and actomyosin contractility modulate cell responses to contact guidance by inducing a population shift between rounded and elongated cells. We also find that cells rapidly change their morphology when navigating the ECM, and that ECM fiber coherence modulates cell transition rates between different morphological phenotypes. Taken together, we find that subcellular processes that integrate conflicting mechanical cues determine cell morphology, which predicts the polarization and migration dynamics of cancer cells in 3D ECM.

Theory and simulation for equilibrium glassy dynamics in cellular Potts model of confluent biological tissue

Glassy dynamics in a confluent monolayer is indispensable in morphogenesis, wound healing, bronchial asthma, and many others; a detailed theoretical framework for such a system is, therefore, important. Vertex-model (VM) simulations have provided crucial insights into the dynamics of such systems, but their nonequilibrium nature makes theoretical development difficult. The cellular Potts model (CPM) of confluent monolayers provides an alternative model for such systems with a well-defined equilibrium limit. We combine numerical simulations of the CPM and an analytical study based on one of the most successful theories of equilibrium glass, the random first-order transition theory, and develop a comprehensive theoretical framework for a confluent glassy system. We find that the glassy dynamics within the CPM is qualitatively similar to that in the VM. Our study elucidates the crucial role of geometric constraints in bringing about two distinct regimes in the dynamics, as the target perimeter P_{0} is varied. The unusual sub-Arrhenius relaxation results from the distinctive interaction potential arising from the perimeter constraint in such systems. The fragility of the system decreases with increasing P_{0} in the low-P_{0} regime, whereas the dynamics is independent of P_{0} in the other regime. The rigidity transition, found in the VM, is absent within the CPM; this difference seems to come from the nonequilibrium nature of the former. We show that the CPM captures the basic phenomenology of glassy dynamics in a confluent biological system via comparison of our numerical results with existing experiments on different systems.

Artistoo, a library to build, share, and explore simulations of cells and tissues in the web browser

The cellular Potts model (CPM) is a powerful in silico method for simulating biological processes at tissue scale. Their inherently graphical nature makes CPMs very accessible in theory, but in practice, they are mostly implemented in specialised frameworks users need to master before they can run simulations. We here present Artistoo (Artificial Tissue Toolbox), a JavaScript library for building ‘explorable’ CPM simulations where viewers can change parameters interactively, exploring their effects in real time. Simulations run directly in the web browser and do not require third-party software, plugins, or back-end servers. The JavaScript implementation imposes no major performance loss compared to frameworks written in C++; Artistoo remains sufficiently fast for interactive, real-time simulations. Artistoo provides an opportunity to unlock CPM models for a broader audience: interactive simulations can be shared via a URL in a zero-install setting. We discuss applications in CPM research, science dissemination, open science, and education.

A Cellular Potts Model for Analyzing Cell Migration across Constraining Pillar Arrays

Cell migration in highly constrained environments is fundamental in a wide variety of physiological and pathological phenomena. In particular, it has been experimentally shown that the migratory capacity of most cell lines depends on their ability to transmigrate through narrow constrictions, which in turn relies on their deformation capacity. In this respect, the nucleus, which occupies a large fraction of the cell volume and is substantially stiffer than the surrounding cytoplasm, imposes a major obstacle. This aspect has also been investigated with the use of microfluidic devices formed by dozens of arrays of aligned polymeric pillars that limit the available space for cell movement. Such experimental systems, in particular, in the designs developed by the groups of Denais and of Davidson, were here reproduced with a tailored version of the Cellular Potts model, a grid-based stochastic approach where cell dynamics are established by a Metropolis algorithm for energy minimization. The proposed model allowed quantitatively analyzing selected cell migratory determinants (e.g., the cell and nuclear speed and deformation, and forces acting at the nuclear membrane) in the case of different experimental setups. Most of the numerical results show a remarkable agreement with the corresponding empirical data.

Tools for computational analysis of moving boundary problems in cellular mechanobiology

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

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.

Self-organized multicellular structures from simple cell signaling: a computational model

Recent synthetic biology experiments reveal that signaling modules designed to target cell-cell adhesion enable self-organization of multicellular structures Toda et al (2018 Science 361 156-162). Changes in homotypic adhesion that arise through contact-dependent signaling networks result in sorting of an aggregate into two- or three-layered structures. Here we investigate the formation, maintenance, and robustness of such self-organization in the context of a computational model. To do so, we use an established model for Notch/ligand signaling within cells to set up differential E-cadherin expression. This signaling model is integrated with the cellular Potts model to track state changes, adhesion, and cell sorting in a group of cells. The resulting multicellular structures are in accordance with those observed in the experimental reference. In addition to reproducing these experimental results, we track the dynamics of the evolving structures and cell states to understand how such morphologies are dynamically maintained. This appears to be an important developmental principle that was not emphasized in previous models. Our computational model facilitates more detailed understanding of the link between intra- and intercellular signaling, spatio-temporal rearrangement, and emergent behavior at the scale of hundred(s) of cells.

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.

Membrane Tension Can Enhance Adaptation to Maintain Polarity of Migrating Cells

Migratory cells are known to adapt to environments that contain wide-ranging levels of chemoattractant. Although biochemical models of adaptation have been previously proposed, here, we discuss a different mechanism based on mechanosensing, in which the interaction between biochemical signaling and cell tension facilitates adaptation. We describe and analyze a model of mechanochemical-based adaptation coupling a mechanics-based physical model of cell tension coupled with the wave-pinning reaction-diffusion model for Rac GTPase activity. The mathematical analysis of this model, simulations of a simplified one-dimensional cell geometry, and two-dimensional finite element simulations of deforming cells reveal that as a cell protrudes under the influence of high stimulation levels, tension-mediated inhibition of Rac signaling causes the cell to polarize even when initially overstimulated. Specifically, tension-mediated inhibition of Rac activation, which has been experimentally observed in recent years, facilitates this adaptation by countering the high levels of environmental stimulation. These results demonstrate how tension-related mechanosensing may provide an alternative (and potentially complementary) mechanism for cell adaptation.

Cell Shape and Durotaxis Explained from Cell-Extracellular Matrix Forces and Focal Adhesion Dynamics

Many cells are small and rounded on soft extracellular matrices (ECM), elongated on stiffer ECMs, and flattened on hard ECMs. Cells also migrate up stiffness gradients (durotaxis). Using a hybrid cellular Potts and finite-element model extended with ODE-based models of focal adhesion (FA) turnover, we show that the full range of cell shape and durotaxis can be explained in unison from dynamics of FAs, in contrast to previous mathematical models. In our 2D cell-shape model, FAs grow due to cell traction forces. Forces develop faster on stiff ECMs, causing FAs to stabilize and, consequently, cells to spread on stiff ECMs. If ECM stress further stabilizes FAs, cells elongate on substrates of intermediate stiffness. We show that durotaxis follows from the same set of assumptions. Our model contributes to the understanding of the basic responses of cells to ECM stiffness, paving the way for future modeling of more complex cell-ECM interactions.