An Integro-Differential Equation for 1D Cell Migration (original) (raw)
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Damped and persistent oscillations in a simple model of cell crawling
Journal of The Royal Society Interface, 2011
A very simple, one-dimensional, discrete, autonomous model of cell crawling is proposed; the model involves only three or four coupled first-order differential equations. This form is sufficient to describe many general features of cell migration, including both steady forward motion and oscillatory progress. Closed-form expressions for crawling speeds and internal forces are obtained in terms of dimensionless parameters that characterize active intracellular processes and the passive mechanical properties of the cell. Two versions of the model are described: a basic cell model with simple elastic coupling between front and rear, which exhibits stable, steady forward crawling after initial transient oscillations have decayed, and a poroelastic model, which can exhibit oscillatory crawling in the steady state.
Cell crawling on a compliant substrate: A biphasic relation with linear friction
International Journal of Non-Linear Mechanics, 2021
A living cell actively generates traction forces on its environment with its actin cytoskeleton. These forces deform the cell elastic substrate which, in turn, affects the traction forces exerted by the cell and can consequently modify the cell dynamics. By considering a cell constrained to move along a one-dimensional thin track, we take advantage of the problem geometry to explicitly derive the effective law that describes the non-local frictional contact between the cell and the deformable substrate. We then couple such a law with one of the simplest model of the active flow within the cell cytoskeleton. This offers a paradigm that does not invoke any local non-linear friction law to explain that the relation between the cell steady state velocity and the substrate elasticity is non linear as experimentally observed. Additionally, we present an experimental platform to test our theoretical predictions. While our efforts are still not conclusive in this respect as more cell types need to be investigated, our analysis of the coupling between the substrate displacement and the actin flow leads to friction coefficient estimates that are in-line with some previously reported results.
Phase-field modeling and isogeometric analysis of cell crawling
A fascinating feature of eukaryotic cells is their ability to move. Cellular motility controls crucial biological processes such as, e.g., cellular nourishment, wound healing, tissue growth, pathogen removal, or metastatic disease. Cell migration through biological tissues is an exceedingly complex process, which is usually understood as a continuous cycle of five interdependent steps, namely: protrusion and elongation of the leading edge driven by actin polymerization; cell-matrix interaction and formation of focal contacts via transmembrane adhesion proteins; extracellular matrix degradation by cell surface proteases; actomyosin contraction generated by active myosin II bound to actin filaments; and detachment of the trailing edge and slow glide forward. Cell migration may be directed by different external stimuli perceived through the cell’s membrane via membrane proteins. Those stimuli, which may take the form of chemical cues or changes in the physical properties of the environment, produce a cellular response that modifies the motile behavior of the cell. Moreover, motile cells may exhibit a number of morphological variants, called modes of migration, as a function of endogenous and exogenous factors such as, e.g., cell-cell and cell-extracellular matrix adhesion, extracellular matrix degradation, orientation of the extracellular matrix fibers, or the predominant cytoskeleton structure. The prominent modes of individual cell migration are mesenchymal, amoeboid, and blebbing motion. Cells can compensate the loss of a particular motile ability by developing migratory strategies, which include the transition between different modes of cell migration. In this thesis we develop three mathematical models of individual cell migration. The models account for the interactions between the cytosolic, membrane, and extracellular compounds involved in cell motility. The motion of the cell is driven by the actin filament network, which is assumed to be a Newtonian fluid subject to forces caused by the cell motion machinery. Those forces are the surface tension of the membrane, cell-substrate adhesion, actin-driven protrusion, and myosin contraction. Also, repulsive force acting on the cell’s membrane accounts for the interaction with obstacles, which may represent fibers or walls. The models are grounded on the phase-field method, which permits to solve the partial-differential equations posed on the different domains (i.e., the cytosol, the membrane, and the extracellular medium) by using a fixed mesh only. The solution of the higher-order equations derived from the phase-field theory entails a number of challenges. To overcome those challenges, we develop a numerical methodology based on isogeometric analysis, a generalization of the finite element method. For the spatial discretization we employ B-splines as basis functions, which possess higher-order continuity. We propose a time integration algorithm based on the generalized-alpha method. The first model focuses on mesenchymal motion. The model proposes a novel description of the actin phase transformations based on a free-energy functional. The results show that the model effectively reproduces the behavior of actin in keratocytes. The simpler case of cell migration in flat surfaces produces stationary states of motion that are in good agreement with experiments. Also, by considering obstacles, we are able to reproduce complex modes of motion observed in microchannels, such as, e.g., oscillatory and bipedal motion. The second model is used to analyze the spontaneous migration of amoeboid cells. The model accounts for a membrane-bound species that interacts with the cytosolic compounds. The model results show quantitative agreement with experiments of free and confined migration. These results suggest that coupling membrane and intracellular dynamics is crucial to understand amoeboid motion. We also show simulations of a cell moving in a three-dimensional fibrous environment, which we interpret as an initial step toward the computational study of cell migration in the extracellular matrix. The third model focuses on chemotaxis of amoeboid cells. The model captures the interactions between the extracellular chemoattractant, the membrane-bound proteins, and the cytosolic components involved in the signaling pathway that originates cell motility. The two-dimensional results reproduce the main features of chemotactic motion. The simulations unveil a complicated interplay between the geometry of the cell’s environment and the chemoattractant dynamics that tightly regulates cell motility. We also show three-dimensional simulations of chemotactic cells moving on planar substrates and fibrous networks. These examples may constitute a first approach to simulate cell migration through biological tissues.
A mathematical model for cellular locomotion exhibiting chemotaxis
Computational and Mathematical Methods in Medicine - COMPUT MATH METHOD MED, 2001
A fundamental problem of cellular biology is to understand the mechanisms underlying cellular locomotion. Bacterial organisms may use appendages such as flagellae or cilia to facilitate motion. Amoeboid motion [6], exhibited by eucaryotic cells are seen to flatten onto surfaces and extend thin sheets of cytosol called lamellipodia. These in turn make attachments to the surface and by the initiation of internal contractions within the cell, a forward motion is achieved. The processes which govern this behaviour are extremely complex, however, key ingredients have been identified which may provide a sufficient basis for persistent cellular motion. These factors are osmotichydrostatic expansion and cellular contraction mediated by intracellular calcium ca2+. In this paper, we develop a simple two dimensional model for a non-muscle motile cell based on these two key factors. We show it is capable of producing persistent cellular motion and chemotactic behaviour. *
1987
Cell locomotion begins with a protrusion from the leading periphery of the cell. What drives this extension? Here we present a model for the extension of cell protuberances that unifies certain aspects of this phenomenon, and is based on the hypothesis that osmotic pressure drives cell extensions. This pressure arises from membrane-associated reactions, which liberate osmoticallv active particles, and from the swelling of the actin network that underlies the membrane. in t r o d u c t io n : w hat are the forces driving cell motility? Cells possess several mechanisms for exerting forces on their surroundings. In particular, a number of mechanochemical enzymes have been identified, including myosin, dynein and kinesin; other molecules will probably be identified in the future. All of these molecules share a common characteristic: they enable the cell to exert only contractile forces. This is a puzzling situation, since in order to move about cells must also be capable of generating protrusive forces. Placing cells in hypertonic media seems to suppress all protrusive activity, suggesting that protrusive force generation may be produced by simple osmotic pressure (Harris, 1973; Trinkaus, 1984 Trinkaus, , 1985. But osmotic pressure is an isotropic force: it acts equally in all directions. Therefore, in order to use pressure for protrusion, the cell must devise means to focus the force in particular directions. In the next section we propose a mechanism by which osmotic forces drive cell protrusion, and which is coordinated with the polymerization of the actin network that fills such protrusions. This model is an extension of two previous models we have proposed for lamellipod and acrosomal extension.
Alternating regimes of motion in a model with cell-cell interactions
Physical Review E
Cellular movement is a complex dynamic process, resulting from the interaction of multiple elements at the intra-and extracellular levels. This epiphenomenon presents a variety of behaviors, which can include normal and anomalous diffusion or collective migration. In some cases, cells can get neighborhood information through chemical or mechanical cues. A unified understanding about how such information can influence the dynamics of cell movement is still lacking. In order to improve our comprehension of cell migration we have considered a cellular Potts model where cells move actively in the direction of a driving field. The intensity of this driving field is constant, while its orientation can evolve according to two alternative dynamics based on the Ornstein-Uhlenbeck process. In one case, the next orientation of the driving field depends on the previous direction of the field. In the other case, the direction update considers the mean orientation performed by the cell in previous steps. Thus, the latter update rule mimics the ability of cells to perceive the environment, avoiding obstacles and thus increasing the cellular displacement. Different cell densities are considered to reveal the effect of cell-cell interactions. Our results indicate that both dynamics introduce temporal and spatial correlations in cell velocity in a friction-coefficient and cell-density-dependent manner. Furthermore, we observe alternating regimes in the mean-square displacement, with normal and anomalous diffusion. The crossovers between diffusive and directed motion regimes are strongly affected by both the driving field dynamics and cell-cell interactions. In this sense, when cell polarization update grants information about the previous cellular displacement, the duration of the diffusive regime decreases, particularly in high-density cultures.
A full computational model of cell motility: Early spreading, cell migration and competing taxis
Cell motility represents one of the most fundamental function in mechanobiology. Cell motility is directly implicated in development, cancer or tissue regeneration, but it also plays a key role in the future of tissue and biomedical engineering. Here, we derived a computational model of cell motility that incorporates the most important mechanisms toward cell motility: cell protrusion, polarization and retrograde flow. We first validate our model to explain two important types of cell migration, i.e. confined and ameboid cell migration, as well as all phases of the latter cell migration type, i.e. symmetric cell spreading, cell polarization and latter migration. Then, we use our model to investigate durotaxis and chemotaxis. The model predicts that chemotaxis alone induces larger migration velocities than durotaxis and that durotaxis is activated in soft matrices but not in stiff ones. More importantly, we analyze the competition between chemical and mechanical signals. We show that...