A numerical model for durotaxis

In silico framework to inform the design of repair constructs for peripheral nerve injury repair

Peripheral nerve injuries affect millions of people per year and cause loss of sensation and muscle control alongside chronic pain. The most severe injuries are treated through a nerve autograft; however, donor site morbidity and poor outcomes mean alternatives are required. One option is to engineer nerve replacement tissues to provide a supportive microenvironment to encourage nerve regeneration as an alternative to nerve grafts. Currently, progress is hampered due to a lack of consensus on how to arrange materials and cells in space to maximize rate of regeneration. This is compounded by a reliance on experimental testing, which precludes extensive investigations of multiple parameters due to time and cost limitations. Here, a computational framework is proposed to simulate the growth of repairing neurites, captured using a random walk approach and parameterized against literature data. The framework is applied to a specific scenario where the engineered tissue comprises a collagen hydrogel with embedded biomaterial fibres. The size and number of fibres are optimized to maximize neurite regrowth, and the robustness of model predictions is tested through sensitivity analyses. The approach provides an in silico tool to inform the design of engineered replacement tissues, with the opportunity for further development to multi-cue environments.

Adhesion strength and contractility enable metastatic cells to become adurotactic

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

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.

The Impact of Elastic Deformations of the Extracellular Matrix on Cell Migration

The mechanical properties of the extracellular matrix, in particular its stiffness, are known to impact cell migration. In this paper, we develop a mathematical model of a single cell migrating on an elastic matrix, which accounts for the deformation of the matrix induced by forces exerted by the cell, and investigate how the stiffness impacts the direction and speed of migration. We model a cell in 1D as a nucleus connected to a number of adhesion sites through elastic springs. The cell migrates by randomly updating the position of its adhesion sites. We start by investigating the case where the cell springs are constant, and then go on to assuming that they depend on the matrix stiffness, on matrices of both uniform stiffness as well as those with a stiffness gradient. We find that the assumption that cell springs depend on the substrate stiffness is necessary and sufficient for an efficient durotactic response. We compare simulations to recent experimental observations of human cancer cells exhibiting durotaxis, which show good qualitative agreement.

Cell migration driven by substrate deformation gradients

Identifying the cues followed by cells is key to understand processes as embryonic development, tissue homeostasis, or several pathological conditions. Based on a durotaxis model, it is shown that cells moving on predeformed thin elastic membrane follow the direction of increasing strain of the substrate. This mechanism, straintaxis, does not distinguish the origin of the strain, but the active stresses produce large strains on cells or tissues being used as substrates. Hence, straintaxis is the natural realization of duratoaxis in vivo. Considering a circular geometry for the substrate cells, it is shown that if the annular component of the active stress component increases with the radial distance, cells migrate toward the substrate cell borders. With appropriate estimation for the different parameters, the migration speeds are similar to those obtained in recent experiments (Reig et al 2017 Nat. Commun. 8 15431). In these, during the annual killifish epiboly, deep cells that move in contact with the epithelial enveloping cell layer (EVL), migrate toward the EVL cell borders with speeds of microns per minute.

Combined Single-Cell Manipulation and Chemomechanical Modeling to Probe Cell Migration Mechanism During Cell-to-Cell Interaction

Spatial presentations of chemical and mechanical information are key parameters for cell migration. However, previous theoretical and experimental studies focus on probing the mechanisms caused by a single type of stimulus, while ignoring the synergetic effects, especially for single cell migration during cell-to-cell interaction. Here we develop a chemomechanical model to assess the biochemical and biophysical modulators of single cell migration during cell-to-cell interaction. This model considers the stimulation of chemoattractant concentration gradient, influence of dynamic adhesion strength and relative motion between cells. The model is validated with single cell manipulation of leukemia cancer cell on stromal cell layer using optical tweezers. Both the modeling and experimental results demonstrate that cell migration velocity caused by chemotaxis can be biased by dynamic adhesion force, which is related to the retrograde flow of stromal cell layer. Besides, the biophysical modulators can influence the effect of drug treatment for specific signaling pathway. Our work provides a quantitative description of single cell migration in a complex environment that is close to realistic in vivo situation and is useful for further exploration of cell signaling pathway during cell-to-cell interactions for investigation of potential therapeutic strategy.

Mathematical modelling of cell migration: stiffness dependent jump rates result in durotaxis

Durotaxis, the phenomena where cells migrate up a gradient in substrate stiffness, remains poorly understood. It has been proposed that durotaxis results from the reinforcement of focal adhesions on stiff substrates. In this paper we formulate a mathematical model of single cell migration on elastic substrates with spatially varying stiffness. We develop a stochastic model where the cell moves by updating the position of its adhesion sites at random times, and the rate of updates is determined by the local stiffness of the substrate. We investigate two physiologically motivated mechanisms of stiffness sensing. From the stochastic model of single cell migration we derive a population level description in the form of a partial differential equation for the time evolution of the density of cells. The equation is an advection–diffusion equation, where the advective velocity is proportional to the stiffness gradient. The model shows quantitative agreement with experimental results in which cells tend to cluster when seeded on a matrix with periodically varying stiffness.

Label-free cell sorting strategies via biophysical and biochemical gradients

Isolating active mesenchymal stem cells from a heterogeneous population is an essential step that determines the efficacy of stem cell therapy such as for osteoarthritis. Nowadays, the gold standard of cell sorting, fluorescence-activated cell sorting, relies on labelling surface markers via antibody-antigen reaction. However, sorting stem cells with high stemness usually requires the labelling of multiple biomarkers. Moreover, the labelling process is costly, and the high operating pressure is harmful to cell functionality and viability. Although label-free cell sorting, based on physical characteristics, has gained increasing interest in the past decades, it has not shown the ability to eliminate stem cells with low stemness. Cell motility, as a novel sorting marker, is hence proposed for label-free sorting active stem cells. Accumulating evidence has demonstrated the feasibility in manipulating directional cell migration through patterning the biophysical, biochemical or both gradients of the extracellular matrix. However, applying those findings to label-free cell sorting has not been well discussed and studied. This review thus first provides a brief overview about the effect of biophysical and biochemical gradients of the extracellular matrix on cell migration. State-of-the-art fabrication techniques for generating such gradients of hydrogels are then introduced. Among current research, the authors suggest that hydrogels with dual-gradients of biochemistry and biophysics are potential tools for accurate label-free cell sorting with satisfactory selectivity and efficiency.

TRANSLATIONAL POTENTIAL OF THIS ARTICLE

The reviewed label-free cell sorting approaches enable us to isolate active cell for cytotherapy. The proposed system can be further modified for single-cell analysis and drug screening.

An investigation of the distribution and location of mast cells affected by the stiffness of substrates as a mechanical niche

The distribution and location of mast cells are closely related to their physiological and pathological functions, such as allergic responses, immunity, and fibrosis, and are used in acupuncture. In this study, the distribution of mast cells in vivo was observed, and mechanical clues for understanding their distribution based on mechanical niches were explored. By toluidine blue staining and immunohistochemical staining, we examined the distribution and location of mast cells in rat skin and found that mast cells are distributed in a spatially nonuniform manner, preferring to locate at regions in the tissue and extracellular matrix with stiffness changes. In vitro experiments for studying the distribution of rat basophilic leukemia (RBL-2H3) mast cell line on poly-di-methyl-siloxane (PDMS) substrates with stiffness variations were performed. It was found that RBL-2H3 cells migrate and tend to remain in the areas with stiffness variations. The present research suggests that changing the stiffness of local tissues may stimulate mast cell recruitment, which may be the method by which some traditional Chinese medicine treatments, such as acupuncture. On the basis of the origin of mast cells and our experimental results, we predict that mast cells exist in tissues that contain permeable capillaries and prefer regions with stiffness changes. We discussed this prediction using examples of specific tissues from some cases.

Persistence-Driven Durotaxis: Generic, Directed Motility in Rigidity Gradients

Cells move differently on substrates with different rigidities: the persistence time of their motion is higher on stiffer substrates. We show that this behavior-in and of itself-results in a net flux of cells directed up a soft-to-stiff gradient. Using simple random walk models with varying persistence and stochastic simulations, we characterize the propensity to move in terms of the durotactic index also measured in experiments. A one-dimensional model captures the essential features and highlights the competition between diffusive spreading and linear, wavelike propagation. Persistence-driven durokinesis is generic and may be of use in the design of instructive environments for cells and other motile, mechanosensitive objects.

A Cellular Potts Model of single cell migration in presence of durotaxis

Cell migration is a fundamental biological phenomenon during which cells sense their surroundings and respond to different types of signals. In presence of durotaxis, cells preferentially crawl from soft to stiff substrates by reorganizing their cytoskeleton from an isotropic to an anisotropic distribution of actin filaments. In the present paper, we propose a Cellular Potts Model to simulate single cell migration over flat substrates with variable stiffness. We have tested five configurations: (i) a substrate including a soft and a stiff region, (ii) a soft substrate including two parallel stiff stripes, (iii) a substrate made of successive stripes with increasing stiffness to create a gradient and (iv) a stiff substrate with four embedded soft squares. For each simulation, we have evaluated the morphology of the cell, the distance covered, the spreading area and the migration speed. We have then compared the numerical results to specific experimental observations showing a consistent agreement.

Mechanical boundary conditions bias fibroblast invasion in a collagen-fibrin wound model

Because fibroblasts deposit the collagen matrix that determines the mechanical integrity of scar tissue, altering fibroblast invasion could alter wound healing outcomes. Anisotropic mechanical boundary conditions (restraint, stretch, or tension) could affect the rate of fibroblast invasion, but their importance relative to the prototypical drivers of fibroblast infiltration during wound healing--cell and chemokine concentration gradients--is unknown. We tested whether anisotropic mechanical boundary conditions affected the directionality and speed of fibroblasts migrating into a three-dimensional model wound, which could simultaneously expose fibroblasts to mechanical, structural, steric, and chemical guidance cues. We created fibrin-filled slits in fibroblast-populated collagen gels and applied uniaxial mechanical restraint along the short or long axis of the fibrin wounds. Anisotropic mechanical conditions increased the efficiency of fibroblast invasion by guiding fibroblasts without increasing their migration speed. The migration behavior could be modeled as a biased random walk, where the bias due to multiple guidance cues was accounted for in the shape of a displacement orientation probability distribution. Taken together, modeling and experiments suggested an effect of strain anisotropy, rather than strain-induced fiber alignment, on fibroblast invasion.

Computational modelling of multi-cell migration in a multi-signalling substrate

Cell migration is a vital process in many biological phenomena ranging from wound healing to tissue regeneration. Over the past few years, it has been proven that in addition to cell-cell and cell-substrate mechanical interactions (mechanotaxis), cells can be driven by thermal, chemical and/or electrical stimuli. A numerical model was recently presented by the authors to analyse single cell migration in a multi-signalling substrate. That work is here extended to include multi-cell migration due to cell-cell interaction in a multi-signalling substrate under different conditions. This model is based on balancing the forces that act on the cell population in the presence of different guiding cues. Several numerical experiments are presented to illustrate the effect of different stimuli on the trajectory and final location of the cell population within a 3D heterogeneous multi-signalling substrate. Our findings indicate that although multi-cell migration is relatively similar to single cell migration in some aspects, the associated behaviour is very different. For instance, cell-cell interaction may delay single cell migration towards effective cues while increasing the magnitude of the average net cell traction force as well as the local velocity. Besides, the random movement of a cell within a cell population is slightly greater than that of single cell migration. Moreover, higher electrical field strength causes the cell slug to flatten near the cathode. On the other hand, as with single cell migration, the existence of electrotaxis dominates mechanotaxis, moving the cells to the cathode or anode pole located at the free surface. The numerical results here obtained are qualitatively consistent with related experimental works.