A hybrid stochastic-deterministic mechanochemical model of cell polarization

Implementation of actin polymerization and depolymerization in a two-dimensional cell migration model and its implications on mammalian cell morphology and velocity

Cell migration, a pivotal process in wound healing, immune response, and even cancer metastasis, manifests through intricate interplay between morphology, speed, and cytoskeletal dynamics. Mathematical modeling emerges as a powerful tool to dissect these complex interactions. This work presents a two-dimensional immersed boundary model for mammalian cell migration, incorporating both filamentous actin (F-actin) and monomeric actin (G-actin) to explicitly capture polymerization and depolymerization. This model builds upon our previous one-dimensional efforts, now enabling us to explore the impact of G-actin on not just cell velocity but also morphology. We compare predictions from both models, revealing that while the one-dimensional model captures core dynamics along the cell's axis, the two-dimensional model excels in portraying cell shape evolution and transverse variations in actin concentration and velocity. Our findings highlight the crucial role of including G-actin in shaping cell morphology. Actin velocity aligned with migration direction elongates the cell, while velocity normal to the membrane promotes spreading. Importantly, the model establishes a link between these microscopic aspects and macroscopic observables like cell shape, offering a deeper understanding of cell migration dynamics. This work not only provides a more comprehensive picture of cell migration but also paves the way for future studies exploring the interplay of actin dynamics, cell morphology, and biophysical parameters in diverse biological contexts.

Model supports asymmetric regulation across the intercellular junction for collective cell polarization

Symmetry breaking, which is ubiquitous in biological cells, functionally enables directed cell movement and organized embryogenesis. Prior to movement, cells break symmetry to form a well-defined cell front and rear in a process called polarization. In developing and regenerating tissues, collective cell movement requires the coordination of the polarity of the migration machineries of neighboring cells. Though several works shed light on the molecular basis of polarity, fewer studies have focused on the regulation across the cell-cell junction required for collective polarization, thus limiting our ability to connect tissue-level dynamics to subcellular interactions. Here, we investigated how polarity signals are communicated from one cell to its neighbor to ensure coordinated front-to-rear symmetry breaking with the same orientation across the group. In a theoretical setting, we systematically searched a variety of intercellular interactions and identified that co-alignment arrangement of the polarity axes in groups of two and four cells can only be achieved with strong asymmetric regulation of Rho GTPases or enhanced assembly of complementary F-actin structures across the junction. Our results held if we further assumed the presence of an external stimulus, intrinsic cell-to-cell variability, or larger groups. The results underline the potential of using quantitative models to probe the molecular interactions required for macroscopic biological phenomena. Lastly, we posit that asymmetric regulation is achieved through junction proteins and predict that in the absence of cytoplasmic tails of such linker proteins, the likeliness of doublet co-polarity is greatly diminished.

Limits on the accuracy of contact inhibition of locomotion

Cells that collide with each other repolarize away from contact, in a process called contact inhibition of locomotion (CIL), which is necessary for correct development of the embryo. CIL can occur even when cells make a micron-scale contact with a neighbor-much smaller than their size. How precisely can a cell sense cell-cell contact and repolarize in the correct direction? What factors control whether a cell recognizes it has contacted a neighbor? We propose a theoretical model for the limits of CIL where cells recognize the presence of another cell by binding the protein ephrin with the Eph receptor. This recognition is made difficult by the presence of interfering ligands that bind nonspecifically. Both theoretical predictions and simulation results show that it becomes more difficult to sense cell-cell contact when it is difficult to distinguish ephrin from the interfering ligands, or when there are more interfering ligands, or when the contact width decreases. However, the error of estimating contact position remains almost constant when the contact width changes. This happens because the cell gains spatial information largely from the boundaries of cell-cell contact. We study using statistical decision theory the likelihood of a false-positive CIL event in the absence of cell-cell contact, and the likelihood of a false negative where CIL does not occur when another cell is present. Our results suggest that the cell is more likely to make incorrect decisions when the contact width is very small or so large that it nears the cell's perimeter. However, in general, we find that cells have the ability to make reasonably reliable CIL decisions even for very narrow (micron-scale) contacts, even if the concentration of interfering ligands is ten times that of the correct ligands.

Differences in cell shape, motility, and growth reflect chromosomal number variations that can be visualized with live-cell ChReporters

Chromosome numbers often change dynamically in tumors and cultured cells, which complicates therapy as well as understanding genotype-mechanotype relationships. Here we use a live-cell "ChReporter" method to identify cells with a single chromosomal loss in efforts to better understand differences in cell shape, motility, and growth. We focus on a standard cancer line and first show clonal populations that retain the ChReporter exhibit large differences in cell and nuclear morphology as well as motility. Phenotype metrics follow simple rules, including migratory persistence scaling with speed, and cytoskeletal differences are evident from drug responses, imaging, and single-cell RNA sequencing. However, mechanotype-genotype relationships between fluorescent ChReporter-positive clones proved complex and motivated comparisons of clones that differ only in loss or retention of a Chromosome-5 ChReporter. When lost, fluorescence-null cells show low expression of Chromosome-5 genes, including a key tumor suppressor APC that regulates microtubules and proliferation. Colonies are compact, nuclei are rounded, and cells proliferate more, with drug results implicating APC, and patient survival data indicating an association in multiple tumor-types. Visual identification of genotype with ChReporters can thus help clarify mechanotype and mechano-evolution.

The influence of polarized membrane ion carriers and extracellular electrical/pH gradients on cell ionic homeostasis and locomotion

Anisotropic environmental signals or polarized membrane ion/solute carriers can generate spatially-varying intracellular gradients, leading to polarized cell dynamics. For example, directional migration of neutrophils, galvanotaxis of glioblastoma, and water flux in kidney cells, all result from the polarized distribution of membrane ion carriers and other intracellular components. The underlying physical mechanisms behind how polarized ion carriers interact with environmental signals are not well studied. Here, we use a physiologically-relevant, physics-based mathematical model to reveal how ion carriers generate intracellular ionic and voltage gradients. The model is able to discern the contribution of individual ion carriers to the intracellular pH gradient, electric potential, and water flux. We discover that an extracellular pH gradient leads to an intracellular pH gradient via chloride-bicarbonate exchangers, whereas an extracellular electric field leads to an intracellular electric potential gradient via passive potassium channels. In addition, the mechanical-biochemical coupling can modulate actin distribution and flow, and create biphasic dependence of the cell speed on water flux. Moreover, we find that F-actin interaction with NHE alone can generate cell movement, even when other ion carriers are not polarized. Taken together, the model shows the importance of cell ion dynamics in modulating cell migration and cytoskeletal dynamics.

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.

Coupling cell shape and velocity leads to oscillation and circling in keratocyte galvanotaxis

During wound healing, fish keratocyte cells undergo galvanotaxis where they follow a wound-induced electric field. In addition to their stereotypical persistent motion, keratocytes can develop circular motion without a field or oscillate while crawling in the field direction. We developed a coarse-grained phenomenological model that captures these keratocyte behaviors. We fit this model to experimental data on keratocyte response to an electric field being turned on. A critical element of our model is a tendency for cells to turn toward their long axis, arising from a coupling between cell shape and velocity, which gives rise to oscillatory and circular motion. Galvanotaxis is influenced not only by the field-dependent responses, but also cell speed and cell shape relaxation rate. When the cell reacts to an electric field being turned on, our model predicts that stiff, slow cells react slowly but follow the signal reliably. Cells that polarize and align to the field at a faster rate react more quickly and follow the signal more reliably. When cells are exposed to a field that switches direction rapidly, cells follow the average of field directions, while if the field is switched more slowly, cells follow a "staircase" pattern. Our study indicated that a simple phenomenological model coupling cell speed and shape is sufficient to reproduce a broad variety of different keratocyte behaviors, ranging from circling to oscillation to galvanotactic response, by only varying a few parameters.

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.

Actin Dynamics as a Multiscale Integrator of Cellular Guidance Cues

Migrating cells must integrate multiple, competing external guidance cues. However, it is not well understood how cells prioritize among these cues. We investigate external cue integration by monitoring the response of wave-like, actin-polymerization dynamics, the driver of cell motility, to combinations of nanotopographies and electric fields in neutrophil-like cells. The electric fields provide a global guidance cue, and approximate conditions at wound sites in vivo. The nanotopographies have dimensions similar to those of collagen fibers, and act as a local esotactic guidance cue. We find that cells prioritize guidance cues, with electric fields dominating long-term motility by introducing a unidirectional bias in the locations at which actin waves nucleate. That bias competes successfully with the wave guidance provided by the bidirectional nanotopographies.

Viscoelasticity, Like Forces, Plays a Role in Mechanotransduction

Viscoelasticity and its alteration in time and space has turned out to act as a key element in fundamental biological processes in living systems, such as morphogenesis and motility. Based on experimental and theoretical findings it can be proposed that viscoelasticity of cells, spheroids and tissues seems to be a collective characteristic that demands macromolecular, intracellular component and intercellular interactions. A major challenge is to couple the alterations in the macroscopic structural or material characteristics of cells, spheroids and tissues, such as cell and tissue phase transitions, to the microscopic interferences of their elements. Therefore, the biophysical technologies need to be improved, advanced and connected to classical biological assays. In this review, the viscoelastic nature of cytoskeletal, extracellular and cellular networks is presented and discussed. Viscoelasticity is conceptualized as a major contributor to cell migration and invasion and it is discussed whether it can serve as a biomarker for the cells' migratory capacity in several biological contexts. It can be hypothesized that the statistical mechanics of intra- and extracellular networks may be applied in the future as a powerful tool to explore quantitatively the biomechanical foundation of viscoelasticity over a broad range of time and length scales. Finally, the importance of the cellular viscoelasticity is illustrated in identifying and characterizing multiple disorders, such as cancer, tissue injuries, acute or chronic inflammations or fibrotic diseases.

Forced into shape: Mechanical forces in Drosophila development and homeostasis

Mechanical forces play a central role in shaping tissues during development and maintaining epithelial integrity in homeostasis. In this review, we discuss the roles of mechanical forces in Drosophila development and homeostasis, starting from the interplay of mechanics with cell growth and division. We then discuss several examples of morphogenetic processes where complex 3D structures are shaped by mechanical forces, followed by a closer look at patterning processes. We also review the role of forces in homeostatic processes, including cell elimination and wound healing. Finally, we look at the interplay of mechanics and developmental robustness and discuss open questions in the field, as well as novel approaches that will help tackle them in the future.

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.

Dynamic intracellular mechanical cues facilitate collective signaling responses

Collective behavior emerges in diverse life machineries, e.g., the immune responses to dynamic stimulations. The essential questions that arise here are that whether and how cells in vivo collectively respond to stimulation frequencies higher than their intrinsic natural values, e.g., the acute inflammation conditions. In this work, we systematically studied morphological and signaling responses of population fibroblasts in an interconnected cell monolayer and uncovered that, besides the natural NF-κB oscillation frequency of 1/90 min⁻¹, collective signaling response emerges in the cell monolayer at 1/20 min⁻¹ TNF-α input periodicity as well. Using a customized microfluidic device, we independently induced dynamic chemical stimulation and cytoskeleton reorganization on the stand-alone cells to exclude the effect of cell-cell communication. Our results reveal that, at this particular frequency, chemical stimulation is translated into dynamic intracellular mechanical cues through RAC1-medicated induction of dynamic cell-cell connections and cytoskeleton reorganizations, which synergize with chemical input to facilitate collective signaling responses.

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Dynamic intracellular mechanical cues facilitate collective cellular responses

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The dynamic chemical stimulations are translated into intracellular mechanical cues

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The synergy between dynamic mechanical and chemical signal plays crucial roles

Primary cilium: a paradigm for integrating mathematical modeling with experiments and numerical simulations in mechanobiology

Primary cilia are non-motile, solitary (one per cell) microtubule-based organelles that emerge from the mother centriole after cells have exited the mitotic cycle. Identified as a mechanosensing organelle that responds to both mechanical and chemical stimuli, the primary cilium provides a fertile ground for integrative investigations of mathematical modeling, numerical simulations, and experiments. Recent experimental findings revealed considerable complexity to the underlying mechanosensory mechanisms that transmit extracellular stimuli to intracellular signaling many of which include primary cilia. In this invited review, we provide a brief survey of experimental findings on primary cilia and how these results lead to various mathematical models of the mechanics of the primary cilium bent under an external forcing such as a fluid flow or a trap. Mathematical modeling of the primary cilium as a fluid-structure interaction problem highlights the importance of basal anchorage and the anisotropic moduli of the microtubules. As theoretical modeling and numerical simulations progress, along with improved state-of-the-art experiments on primary cilia, we hope that details of ciliary regulated mechano-chemical signaling dynamics in cellular physiology will be understood in the near future.

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.