A mathematical model coupling polarity signaling to cell adhesion explains diverse cell migration patterns

LKB1 biology: assessing the therapeutic relevancy of LKB1 inhibitors

Liver Kinase B1 (LKB1), encoded by Serine-Threonine Kinase 11 (STK11), is a master kinase that regulates cell migration, polarity, proliferation, and metabolism through downstream adenosine monophosphate-activated protein kinase (AMPK) and AMPK-related kinase signalling. Since genetic screens identified STK11 mutations in Peutz-Jeghers Syndrome, STK11 mutants have been implicated in tumourigenesis labelling it as a tumour suppressor. In support of this, several compounds reduce tumour burden through upregulating LKB1 signalling, and LKB1-AMPK agonists are cytotoxic to tumour cells. However, in certain contexts, its role in cancer is paradoxical as LKB1 promotes tumour cell survival by mediating resistance against metabolic and oxidative stressors. LKB1 deficiency has also enhanced the selectivity and cytotoxicity of several cancer therapies. Taken together, there is a need to develop LKB1-specific pharmacological compounds, but prior to developing LKB1 inhibitors, further work is needed to understand LKB1 activity and regulation. However, investigating LKB1 activity is strenuous as cell/tissue type, mutations to the LKB1 signalling pathway, STE-20-related kinase adaptor protein (STRAD) binding, Mouse protein 25-STRAD binding, splicing variants, nucleocytoplasmic shuttling, post-translational modifications, and kinase conformation impact the functional status of LKB1. For these reasons, guidelines to standardize experimental strategies to study LKB1 activity, associate proteins, spliced isoforms, post-translational modifications, and regulation are of upmost importance to the development of LKB1-specific therapies. Therefore, to assess the therapeutic relevancy of LKB1 inhibitors, this review summarizes the importance of LKB1 in cell physiology, highlights contributors to LKB1 activation, and outlines the benefits and risks associated with targeting LKB1.

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.

PIEZO1 regulates leader cell formation and cellular coordination during collective keratinocyte migration

The collective migration of keratinocytes during wound healing requires both the generation and transmission of mechanical forces for individual cellular locomotion and the coordination of movement across cells. Leader cells along the wound edge transmit mechanical and biochemical cues to ensuing follower cells, ensuring their coordinated direction of migration across multiple cells. Despite the observed importance of mechanical cues in leader cell formation and in controlling coordinated directionality of cell migration, the underlying biophysical mechanisms remain elusive. The mechanically-activated ion channel PIEZO1 was recently identified to play an inhibitory role during the reepithelialization of wounds. Here, through an integrative experimental and mathematical modeling approach, we elucidate PIEZO1's contributions to collective migration. Time-lapse microscopy reveals that PIEZO1 activity inhibits leader cell formation at the wound edge. To probe the relationship between PIEZO1 activity, leader cell formation and inhibition of reepithelialization, we developed an integrative 2D continuum model of wound closure that links observations at the single cell and collective cell migration scales. Through numerical simulations and subsequent experimental validation, we found that coordinated directionality plays a key role during wound closure and is inhibited by upregulated PIEZO1 activity. We propose that PIEZO1-mediated retraction suppresses leader cell formation which inhibits coordinated directionality between cells during collective migration.

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.

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.

On multistability and constitutive relations of cell motion on fibronectin lanes

Cell motility on flat substrates exhibits coexisting steady and oscillatory morphodynamics, the biphasic adhesion-velocity relation, and the universal correlation between speed and persistence (UCSP) as simultaneous observations common to many cell types. Their universality and concurrency suggest a unifying mechanism causing all three of them. Stick-slip models for cells on one-dimensional lanes suggest multistability to arise from the nonlinear friction of retrograde flow. This study suggests a mechanical mechanism controlled by integrin signaling on the basis of a biophysical model and analysis of trajectories of MDA-MB-231 cells on fibronectin lanes, which additionally explains the constitutive relations. The experiments exhibit cells with steady or oscillatory morphodynamics and either spread or moving with spontaneous transitions between the dynamic regimes, spread and moving, and spontaneous direction reversals. Our biophysical model is based on the force balance at the protrusion edge, the noisy clutch of retrograde flow, and a response function of friction and membrane drag to integrin signaling. The theory reproduces the experimentally observed cell states, characteristics of oscillations, and state probabilities. Analysis of experiments with the biophysical model establishes a stick-slip oscillation mechanism, and explains multistability of cell states and the statistics of state transitions. It suggests protrusion competition to cause direction reversal events, the statistics of which explain the UCSP. The effect of integrin signaling on drag and friction explains the adhesion-velocity relation and cell behavior at fibronectin density steps. The dynamics of our mechanism are nonlinear flow mechanics driven by F-actin polymerization and shaped by the noisy clutch of retrograde flow friction, protrusion competition via membrane tension, and drag forces. Integrin signaling controls the parameters of the mechanical system.

Epigenetics as a mediator of plasticity in cancer

The concept of an epigenetic landscape describing potential cellular fates arising from pluripotent cells, first advanced by Conrad Waddington, has evolved in light of experiments showing nondeterministic outcomes of regulatory processes and mathematical methods for quantifying stochasticity. In this Review, we discuss modern approaches to epigenetic and gene regulation landscapes and the associated ideas of entropy and attractor states, illustrating how their definitions are both more precise and relevant to understanding cancer etiology and the plasticity of cancerous states. We address the interplay between different types of regulatory landscapes and how their changes underlie cancer progression. We also consider the roles of cellular aging and intrinsic and extrinsic stimuli in modulating cellular states and how landscape alterations can be quantitatively mapped onto phenotypic outcomes and thereby used in therapy development.

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.

Picking winners in cell-cell collisions: Wetting, speed, and contact

Groups of eukaryotic cells can coordinate their crawling motion to follow cues more effectively, stay together, or invade new areas. This collective cell migration depends on cell-cell interactions, which are often studied by colliding pairs of cells together. Can the outcome of these collisions be predicted? Recent experiments on trains of colliding epithelial cells suggest that cells with a smaller contact angle to the surface or larger speeds are more likely to maintain their direction ("win") upon collision. When should we expect shape or speed to correlate with the outcome of a collision? To investigate this question, we build a model for two-cell collisions within the phase field framework, which allows for cell shape changes. We can reproduce the observation that cells with high speed and small contact angles are more likely to win with two different assumptions for how cells interact: (1) velocity aligning, in which we hypothesize that cells sense their own velocity and align to it over a finite timescale, and (2) front-front contact repolarization, where cells polarize away from cell-cell contact, akin to contact inhibition of locomotion. Surprisingly, though we simulate collisions between cells with widely varying properties, in each case, the probability of a cell winning is completely captured by a single summary variable: its relative speed (in the velocity-aligning model) or its relative contact angle (in the contact repolarization model). Both models are currently consistent with reported experimental results, but they can be distinguished by varying cell contact angle and speed through orthogonal perturbations.

Learning anisotropic interaction rules from individual trajectories in a heterogeneous cellular population

Interacting particle system (IPS) models have proven to be highly successful for describing the spatial movement of organisms. However, it is challenging to infer the interaction rules directly from data. In the field of equation discovery, the weak-form sparse identification of nonlinear dynamics (WSINDy) methodology has been shown to be computationally efficient for identifying the governing equations of complex systems from noisy data. Motivated by the success of IPS models to describe the spatial movement of organisms, we develop WSINDy for the second-order IPS to learn equations for communities of cells. Our approach learns the directional interaction rules for each individual cell that in aggregate govern the dynamics of a heterogeneous population of migrating cells. To sort a cell according to the active classes present in its model, we also develop a novel ad hoc classification scheme (which accounts for the fact that some cells do not have enough evidence to accurately infer a model). Aggregated models are then constructed hierarchically to simultaneously identify different species of cells present in the population and determine best-fit models for each species. We demonstrate the efficiency and proficiency of the method on several test scenarios, motivated by common cell migration experiments.

Sensing the shape of a cell with reaction diffusion and energy minimization

Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell's long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape-and the limits that membrane roughness can place on this process.

Implementing computational modeling in tissue engineering: where disciplines meet

In recent years, the mathematical and computational sciences have developed novel methodologies and insights that can aid in designing advanced bioreactors, microfluidic set-ups or organ-on-chip devices, in optimizing culture conditions, or predicting long-term behavior of engineered tissues in vivo. In this review, we introduce the concept of computational models and how they can be integrated in an interdisciplinary workflow for Tissue Engineering and Regenerative Medicine (TERM). We specifically aim this review of general concepts and examples at experimental scientists with little or no computational modeling experience. We also describe the contribution of computational models in understanding TERM processes and in advancing the TERM field by providing novel insights.

Biophysical Models of PAR Cluster Transport by Cortical Flow in C. elegans Early Embryogenesis

The clustering of membrane-bound proteins facilitates their transport by cortical actin flow in early Caenorhabditis elegans embryo cell polarity. PAR-3 clustering is critical for this process, yet the biophysical processes that couple protein clusters to cortical flow remain unknown. We develop a discrete, stochastic agent-based model of protein clustering and test four hypothetical models for how clusters may interact with the flow. Results show that the canonical way to assess transport characteristics from single-particle tracking data used thus far in this area, the Péclet number, is insufficient to distinguish these hypotheses and that all models can account for transport characteristics quantified by this measure. However, using this model, we demonstrate that these different cluster-cortex interactions may be distinguished using a different metric, namely the scalar projection of cluster displacement on to the flow displacement vector. Our results thus provide a testable way to use existing single-particle tracking data to test how endogenous protein clusters may interact with the cortical flow to localize during polarity establishment. To facilitate this investigation, we also develop both improved simulation and semi-analytic methodologies to quantify motion summary statistics (e.g., Péclet number and scalar projection) for these stochastic models as a function of biophysical parameters.

Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

Bursting is ubiquitous in cellular excitable rhythms and comes in a plethora of patterns, both experimentally recorded and reproduced through models. As these different patterns may reflect different coding or information properties, it is therefore crucial to develop modeling frameworks that can both capture them and understand their characteristics. In this review, we propose a comprehensive account of the main bursting classification systems that have been developed over the past 40 years, together with recent developments allowing us to extend these classifications. Based upon bifurcation theory and heavily reliant on timescale separation, these schemes take full advantage of the fast subsystem analysis, obtained when slow variables are frozen and considered as bifurcation parameters. We complement this classical view by showing that nontrivial slow subsystem may also encode key informations important to classify bursting rhythms, due to the presence of so-called folded-node singularities. We provide minimal idealized models as well as one generic conductance-based example displaying bursting oscillations that require our extended classification in order to be fully characterized. We also highlight examples of biological data that could be suitably revisited with the lenses of this extended classifications and could lead to new models of complex cellular activity.

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.

Decoding leader cells in collective cancer invasion

Collective cancer invasion with leader-follower organization is increasingly recognized as a predominant mechanism in the metastatic cascade. Leader cells support cancer invasion by creating invasion tracks, sensing environmental cues and coordinating with follower cells biochemically and biomechanically. With the latest developments in experimental and computational models and analysis techniques, the range of specific traits and features of leader cells reported in the literature is rapidly expanding. Yet, despite their importance, there is no consensus on how leader cells arise or their essential characteristics. In this Perspective, we propose a framework for defining the essential aspects of leader cells and provide a unifying perspective on the varying cellular and molecular programmes that are adopted by each leader cell subtype to accomplish their functions. This Perspective can lead to more effective strategies to interdict a major contributor to metastatic capability.

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.

A free boundary mechanobiological model of epithelial tissues

In this study, we couple intracellular signalling and cell-based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction-diffusion equations are derived to describe tissue-level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac-Rho pathway where cell- and tissue-level mechanics are directly related to intracellular signalling. Second, we study an activator-inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell-level processes included in the discrete model.

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.

Periodic propagating waves coordinate RhoGTPase network dynamics at the leading and trailing edges during cell migration

Migrating cells need to coordinate distinct leading and trailing edge dynamics but the underlying mechanisms are unclear. Here, we combine experiments and mathematical modeling to elaborate the minimal autonomous biochemical machinery necessary and sufficient for this dynamic coordination and cell movement. RhoA activates Rac1 via DIA and inhibits Rac1 via ROCK, while Rac1 inhibits RhoA through PAK. Our data suggest that in motile, polarized cells, RhoA–ROCK interactions prevail at the rear, whereas RhoA-DIA interactions dominate at the front where Rac1/Rho oscillations drive protrusions and retractions. At the rear, high RhoA and low Rac1 activities are maintained until a wave of oscillatory GTPase activities from the cell front reaches the rear, inducing transient GTPase oscillations and RhoA activity spikes. After the rear retracts, the initial GTPase pattern resumes. Our findings show how periodic, propagating GTPase waves coordinate distinct GTPase patterns at the leading and trailing edge dynamics in moving cells.

Protein-protein interaction network controlling establishment and maintenance of switchable cell polarity

Cell polarity underlies key processes in all cells, including growth, differentiation and division. In the bacterium Myxococcus xanthus, front-rear polarity is crucial for motility. Notably, this polarity can be inverted, independent of the cell-cycle, by chemotactic signaling. However, a precise understanding of the protein network that establishes polarity and allows for its inversion has remained elusive. Here, we use a combination of quantitative experiments and data-driven theory to unravel the complex interplay between the three key components of the M. xanthus polarity module. By studying each of these components in isolation and their effects as we systematically reconstruct the system, we deduce the network of effective interactions between the polarity proteins. RomR lies at the root of this network, promoting polar localization of the other components, while polarity arises from interconnected negative and positive feedbacks mediated by the small GTPase MglA and its cognate GAP MglB, respectively. We rationalize this network topology as operating as a spatial toggle switch, providing stable polarity for persistent cell movement whilst remaining responsive to chemotactic signaling and thus capable of polarity inversions. Our results have implications not only for the understanding of polarity and motility in M. xanthus but also, more broadly, for dynamic cell polarity.

The asymmetric localization of cellular components (polarity) is at the core of many important cellular functions including growth, division, differentiation and motility. However, important questions still remain regarding the design principles underlying polarity networks and how their activity can be controlled in space and time. We use the rod-shaped bacterium Myxococcus xanthus as a model to study polarity and its regulation. Like many bacteria, in M. xanthus a well-defined front-rear polarity axis enables efficient translocation. This polarity axis is defined by asymmetric polar localization of a switch-like GTPase and its cognate regulators, and can be reversed in response to signaling cues. Here we use a combination of quantitative experiments and data-driven theory to deduce the network of interactions among the M. xanthus polarity proteins and to show how the combination of positive- and negative-feedback interactions give rise to asymmetric polar protein localization. We rationalize this network topology as operating as a spatial toggle switch, providing stable polarity for persistent cell movement whilst remaining responsive to chemotactic signaling and capable of polarity inversions. Our results have broader implications for our understanding of dynamic cell polarity and GTPase regulation in both bacteria and eukaryotic cells.

Tuning Cell Motility via Cell Tension with a Mechanochemical Cell Migration Model

Cell migration is orchestrated by a complicated mechanochemical system. However, few cell migration models take into account the coupling between the biochemical network and mechanical factors. Here, we construct a mechanochemical cell migration model to study the cell tension effect on cell migration. Our model incorporates the interactions between Rac-GTP, Rac-GDP, F-actin, myosin, and cell tension, and it is very convenient in capturing the change of cell shape by taking the phase field approach. This model captures the characteristic features of cell polarization, cell shape change, and cell migration modes. It shows that cell tension inhibits migration ability monotonically when cells are applied with persistent external stimuli. On the other hand, if random internal noise is significant, the regulation of cell tension exerts a nonmonotonic effect on cell migration. Because the increase of cell tension hinders the formation of multiple protrusions, migration ability could be maximized at intermediate cell tension under random internal noise. These model predictions are consistent with our single-cell experiments and other experimental results.

Simple Rho GTPase Dynamics Generate a Complex Regulatory Landscape Associated with Cell Shape

Migratory cells exhibit a variety of morphologically distinct responses to their environments that manifest in their cell shape. Some protrude uniformly to increase substrate contacts, others are broadly contractile, some polarize to facilitate migration, and yet others exhibit mixtures of these responses. Prior studies have identified a discrete collection of shapes that the majority of cells display and demonstrated that activity levels of the cytoskeletal regulators Rac1 and RhoA GTPase regulate those shapes. Here, we use computational modeling to assess whether known GTPase dynamics can give rise to a sufficient diversity of spatial signaling states to explain the observed shapes. Results show that the combination of autoactivation and mutually antagonistic cross talk between GTPases, along with the conservative membrane binding, generates a wide array of distinct homogeneous and polarized regulatory phenotypes that arise for fixed model parameters. From a theoretical perspective, these results demonstrate that simple GTPase dynamics can generate complex multistability in which six distinct stable steady states (three homogeneous and three polarized) coexist for a fixed set of parameters, each of which naturally maps to an observed morphology. From a biological perspective, although we do not explicitly model the cytoskeleton or resulting cell morphologies, these results, along with prior literature linking GTPase activity to cell morphology, support the hypothesis that GTPase signaling dynamics can generate the broad morphological characteristics observed in many migratory cell populations. Further, the observed diversity may be the result of cells populating a complex morphological landscape generated by GTPase regulation rather than being the result of intrinsic cell-cell variation. These results demonstrate that Rho GTPases may have a central role in regulating the broad characteristics of cell shape (e.g., expansive, contractile, polarized, etc.) and that shape heterogeneity may be (at least partly) a reflection of the rich signaling dynamics regulating the cytoskeleton rather than intrinsic cell heterogeneity.

Active gel segment behaving as an active particle

We reduce a one-dimensional model of an active segment (AS), which is used, for instance, in the description of contraction-driven cell motility, to a zero-dimensional model of an active particle (AP) characterized by two internal degrees of freedom: position and polarity. Both models give rise to hysteretic force-velocity relations showing that an active agent can support two opposite polarities under the same external force and that it can maintain the same polarity while being dragged by external forces with opposite orientations. This double bistability results in a rich dynamic repertoire which we illustrate by studying static, stalled, motile, and periodically repolarizing regimes displayed by an active agent confined in a viscoelastic environment. We show that the AS and AP models can be calibrated to generate quantitatively similar dynamic responses.

Rac activation is key to cell motility and directionality: An experimental and modelling investigation

Cell migration is a tightly-regulated process that involves protein gradients formed by the Rho family of GTPases, including Rho and Rac. The front (rear) of cells is generally characterized by higher active Rac (Rho) and lower active Rho (Rac) concentrations. Protein clusters, called adhesions, that anchor cells to their external environment have been shown to be dynamic and small (stable and large) at the cell front (rear), forming the force-transmission points necessary for persistent movement. Differences in adhesion sizes and dynamics have been linked to gradients in Rac and Rho activity. Here, we study the effects of Rac activation and gradients in Rac and Rho concentrations and activities on cellular polarity and adhesion size using mathematical and experimental approaches. The former is accomplished by expanding an existing reaction-diffusion model to a 2D domain utilizing stochastic dynamics. The model revealed that a hysteresis between the induced/uninduced states (corresponding to higher/lower Rac concentrations, respectively) along with Rac and Rho activation gradients, generated by chemical cues, were vital for forming polarity. Experimentally, the induced state was generated by increasing the cellular βPIX (a Rac-GEF) level and/or decreasing ROCK (a Rac-GAP effector protein) activity with Y-27632 (a ROCK-inhibitor). In agreement with the simulations, our results showed that cells with elevated RacGTP migrated faster, indicating more robust cellular polarization. However, the directionality of cells was not changed significantly, suggesting that external and/or internal physical or chemical cues were needed. Complementing the faster migration observed, adhesions were smaller, generating the phenotype expected with the induced state.

Mechanochemical Coupling and Junctional Forces during Collective Cell Migration

Cell migration, a fundamental physiological process in which cells sense and move through their surrounding physical environment, plays a critical role in development and tissue formation, as well as pathological processes, such as cancer metastasis and wound healing. During cell migration, dynamics are governed by the bidirectional interplay between cell-generated mechanical forces and the activity of Rho GTPases, a family of small GTP-binding proteins that regulate actin cytoskeleton assembly and cellular contractility. These interactions are inherently more complex during the collective migration of mechanically coupled cells because of the additional regulation of cell-cell junctional forces. In this study, we adapted a recent minimal modeling framework to simulate the interactions between mechanochemical signaling in individual cells and interactions with cell-cell junctional forces during collective cell migration. We find that migration of individual cells depends on the feedback between mechanical tension and Rho GTPase activity in a biphasic manner. During collective cell migration, waves of Rho GTPase activity mediate mechanical contraction/extension and thus synchronization throughout the tissue. Further, cell-cell junctional forces exhibit distinct spatial patterns during collective cell migration, with larger forces near the leading edge. Larger junctional force magnitudes are associated with faster collective cell migration and larger tissue size. Simulations of heterogeneous tissue migration exhibit a complex dependence on the properties of both leading and trailing cells. Computational predictions demonstrate that collective cell migration depends on both the emergent dynamics and interactions between cellular-level Rho GTPase activity and contractility and multicellular-level junctional forces.

Fabrication of Chitosan/Polypyrrole‐coated poly(L‐lactic acid)/Polycaprolactone aligned fibre films for enhancement of neural cell compatibility and neurite growth

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Axon growth regulation by a bistable molecular switch

For the brain to function properly, its neurons must make the right connections during neural development. A key aspect of this process is the tight regulation of axon growth as axons navigate towards their targets. Neuronal growth cones at the tips of developing axons switch between growth and paused states during axonal pathfinding, and this switching behaviour determines the heterogeneous axon growth rates observed during brain development. The mechanisms controlling this switching behaviour, however, remain largely unknown. Here, using mathematical modelling, we predict that the molecular interaction network involved in axon growth can exhibit bistability, with one state representing a fast-growing growth cone state and the other a paused growth cone state. Owing to stochastic effects, even in an unchanging environment, model growth cones reversibly switch between growth and paused states. Our model further predicts that environmental signals could regulate axon growth rate by controlling the rates of switching between the two states. Our study presents a new conceptual understanding of growth cone switching behaviour, and suggests that axon guidance may be controlled by both cell-extrinsic factors and cell-intrinsic growth regulatory mechanisms.

Rhythmicity and waves in the cortex of single cells

Emergence of dynamic patterns in the form of oscillations and waves on the cortex of single cells is a fascinating and enigmatic phenomenon. Here we outline various theoretical frameworks used to model pattern formation with the goal of reducing complex, heterogeneous patterns into key parameters that are biologically tractable. We also review progress made in recent years on the quantitative and molecular definitions of these terms, which we believe have begun to transform single-cell dynamic patterns from a purely observational and descriptive subject to more mechanistic studies. Specifically, we focus on the nature of local excitable and oscillation events, their spatial couplings leading to propagating waves and the role of active membrane. Instead of arguing for their functional importance, we prefer to consider such patterns as basic properties of dynamic systems. We discuss how knowledge of these patterns could be used to dissect the structure of cellular organization and how the network-centric view could help define cellular functions as transitions between different dynamical states. Last, we speculate on how these patterns could encode temporal and spatial information.

This article is part of the theme issue ‘Self-organization in cell biology’.

Paxillin phosphorylation at serine 273 and its effects on Rac, Rho and adhesion dynamics

Focal adhesions are protein complexes that anchor cells to the extracellular matrix. During migration, the growth and disassembly of these structures are spatiotemporally regulated, with new adhesions forming at the leading edge of the cell and mature adhesions disassembling at the rear. Signalling proteins and structural cytoskeletal components tightly regulate adhesion dynamics. Paxillin, an adaptor protein within adhesions, is one of these proteins. Its phosphorylation at serine 273 (S273) is crucial for maintaining fast adhesion assembly and disassembly. Paxillin is known to bind to a GIT1-βPIX-PAK1 complex, which increases the local activation of the small GTPase Rac. To understand quantitatively the behaviour of this system and how it relates to adhesion assembly/disassembly, we developed a mathematical model describing the dynamics of the small GTPases Rac and Rho as determined by paxillin S273 phosphorylation. Our model revealed that the system possesses bistability, where switching between uninduced (active Rho) and induced (active Rac) states can occur through a change in rate of paxillin phosphorylation or PAK1 activation. The bistable switch is characterized by the presence of memory, minimal change in the levels of active Rac and Rho within the induced and uninduced states, respectively, and the limited regime of monostability associated with the uninduced state. These results were validated experimentally by showing the presence of bimodality in adhesion assembly and disassembly rates, and demonstrating that Rac activity increases after treating Chinese Hamster Ovary cells with okadaic acid (a paxillin phosphatase inhibitor), followed by a modest recovery after 20 min washout. Spatial gradients of phosphorylated paxillin in a reaction-diffusion model gave rise to distinct regions of Rac and Rho activities, resembling polarization of a cell into front and rear. Perturbing several parameters of the model also revealed important insights into how signalling components upstream and downstream of paxillin phosphorylation affect dynamics.

Cellular migration is crucial in both physiological and pathological functions. Maintenance of proper migration and development of aberrant migration are effectuated by cellular machinery involving protein complexes, called adhesions, that anchor the cell to its environment. Over time, these adhesions assemble at the leading edge, as the cell extends forward, anchoring the front of the cells to its substrate, while those at the cell rear disassemble, allowing detachment and forward movement. Their dynamics are controlled by a number of regulatory factors, occurring on both cell-wide and adhesion-level scales. The coordination of these regulatory factors is complex, but insights about their dynamics can be gained from the use of mathematical modeling techniques which integrate many of these components together. Here, we developed several molecularly explicit models to explore how local regulation of paxillin, an adhesion protein, interacts with the activities of Rac and Rho to produce cell-wide polarization associated with motility and directionality. By altering paxillin phosphorylation/dephosphorylation within such models, we have advanced our understanding of how a shift from a non-motile state to a highly motile state occurs. Deciphering these key processes quantitatively thus helped us gain insight into the subcellular factors underlying polarity and movement.

Computational modeling of single-cell mechanics and cytoskeletal mechanobiology

Cellular cytoskeletal mechanics plays a major role in many aspects of human health from organ development to wound healing, tissue homeostasis and cancer metastasis. We summarize the state-of-the-art techniques for mathematically modeling cellular stiffness and mechanics and the cytoskeletal components and factors that regulate them. We highlight key experiments that have assisted model parameterization and compare the advantages of different models that have been used to recapitulate these experiments. An overview of feed-forward mechanisms from signaling to cytoskeleton remodeling is provided, followed by a discussion of the rapidly growing niche of encapsulating feedback mechanisms from cytoskeletal and cell mechanics to signaling. We discuss broad areas of advancement that could accelerate research and understanding of cellular mechanobiology. A precise understanding of the molecular mechanisms that affect cell and tissue mechanics and function will underpin innovations in medical device technologies of the future. WIREs Syst Biol Med 2018, 10:e1407. doi: 10.1002/wsbm.1407 This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Computational Modeling of the Dynamics of Spatiotemporal Rho GTPase Signaling: A Systematic Review

The Rho family of GTPases are known to play pivotal roles in the regulation of fundamental cellular processes, ranging from cell migration and polarity to wound healing and regulation of actin cytoskeleton. Over the past decades, accumulating experimental work has increasingly mapped out the mechanistic details and interactions between members of the family and their regulators, establishing detailed interaction circuits within the Rho GTPase signaling network. These circuits have served as a vital foundation based on which a multitude of mathematical models have been developed to explain experimental data, gain deeper insights into the biological phenomenon they describe, as well as make new testable predictions and hypotheses. Due to the diverse nature and purpose of these models, they often vary greatly in size, scope, complexity, and formulation. Here, we provide a systematic, categorical, and comprehensive account of the recent modeling studies of Rho family GTPases, with an aim to offer a broad perspective of the field. The modeling limitations and possible future research directions are also discussed.

Deciphering Epithelial-Mesenchymal Transition Regulatory Networks in Cancer through Computational Approaches

Epithelial–mesenchymal transition (EMT), the process by which epithelial cells can convert into motile mesenchymal cells, plays an important role in development and wound healing but is also involved in cancer progression. It is increasingly recognized that EMT is a dynamic process involving multiple intermediate or “hybrid” phenotypes rather than an “all-or-none” process. However, the role of EMT in various cancer hallmarks, including metastasis, is debated. Given the complexity of EMT regulation, computational modeling has proven to be an invaluable tool for cancer research, i.e., to resolve apparent conflicts in experimental data and to guide experiments by generating testable hypotheses. In this review, we provide an overview of computational modeling efforts that have been applied to regulation of EMT in the context of cancer progression and its associated tumor characteristics. Moreover, we identify possibilities to bridge different modeling approaches and point out outstanding questions in which computational modeling can contribute to advance our understanding of pathological EMT.

Mechanochemical feedback underlies coexistence of qualitatively distinct cell polarity patterns within diverse cell populations

Cell polarization and directional cell migration can display random, persistent, and oscillatory dynamic patterns. However, it is not clear whether these polarity patterns can be explained by the same underlying regulatory mechanism. Here, we show that random, persistent, and oscillatory migration accompanied by polarization can simultaneously occur in populations of melanoma cells derived from tumors with different degrees of aggressiveness. We demonstrate that all of these patterns and the probabilities of their occurrence are quantitatively accounted for by a simple mechanism involving a spatially distributed, mechanochemical feedback coupling the dynamically changing extracellular matrix (ECM)-cell contacts to the activation of signaling downstream of the Rho-family small GTPases. This mechanism is supported by a predictive mathematical model and extensive experimental validation, and can explain previously reported results for diverse cell types. In melanoma, this mechanism also accounts for the effects of genetic and environmental perturbations, including mutations linked to invasive cell spread. The resulting mechanistic understanding of cell polarity quantitatively captures the relationship between population variability and phenotypic plasticity, with the potential to account for a wide variety of cell migration states in diverse pathological and physiological conditions.