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23
Mathematics of cell motility: have we got its number?
- J. MATHEMATICAL BIOLOGY
, 2009
"... Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biologic ..."
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Cited by 24 (1 self)
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Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area— cell motility—characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, ‘modular ’ character of motility, and pioneering vision of the area’s experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.
Wave-Pinning and Cell Polarity from a Bistable Reaction-Diffusion System
, 2008
"... ABSTRACT Motile eukaryotic cells polarize in response to external signals. Numerous mechanisms have been suggested to account for this symmetry breaking and for the ensuing robust polarization. Implicated in this process are various proteins that are recruited to the plasma membrane and segregate at ..."
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Cited by 13 (5 self)
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ABSTRACT Motile eukaryotic cells polarize in response to external signals. Numerous mechanisms have been suggested to account for this symmetry breaking and for the ensuing robust polarization. Implicated in this process are various proteins that are recruited to the plasma membrane and segregate at an emergent front or back of the polarizing cell. Among these are PI3K, PTEN, and members of the Rho family GTPases such as Cdc42, Rac, and Rho. Many such proteins, including the Rho GTPases, cycle between active membrane-bound forms and inactive cytosolic forms. In previous work, we have shown that this property, together with appropriate crosstalk, endows a biochemical circuit (Cdc42, Rac, and Rho) with the property of inherent polarizability. Here we show that this property is present in an even simpler system comprised of a single active/inactive protein pair with positive feedback to its own activation. The simplicity of this minimal system also allows us to explain the mechanism using insights from mathematical analysis. The basic idea resides in a well-known property of reaction-diffusion systems with bistable kinetics, namely, propagation of fronts. However, it crucially depends on exchange between active and inactive forms of the chemicals with unequal rates of diffusion, and overall conservation to pin the waves into a stable polar distribution. We refer to these dynamics as wave-pinning and we show that this phenomenon is distinct from Turing-instability-generated pattern formation that occurs in reaction-diffusion systems that appear to be very similar. We explain the mathematical basis of the phenomenon, relate it to spatial segregation of Rho GTPases, and show how it can account for spatial amplification and maintenance of polarity, as well as sensitivity to new stimuli typical in polarization of eukaryotic cells.
Modeling morphogenesis in silico and in vitro: Towards quantitative, predictive, cell-based modeling
- Math. Mod. Nat. Phenom
"... Abstract. Cell-based, mathematical models help make sense of morphogenesis—i.e. cells or-ganizing into shape and pattern—by capturing cell behavior in simple, purely descriptive models. Cell-based models then predict the tissue-level patterns the cells produce collectively. The first step in a cell- ..."
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Abstract. Cell-based, mathematical models help make sense of morphogenesis—i.e. cells or-ganizing into shape and pattern—by capturing cell behavior in simple, purely descriptive models. Cell-based models then predict the tissue-level patterns the cells produce collectively. The first step in a cell-based modeling approach is to isolate sub-processes, e.g. the patterning capabilities of one or a few cell types in cell cultures. Cell-based models can then identify the mechanisms responsible for patterning in vitro. This review discusses two cell culture models of morphogenesis that have been studied using this combined experimental-mathematical approach: chondrogenesis (cartilage patterning) and vasculogenesis (de novo blood vessel growth). In both these systems, radically dif-ferent models can equally plausibly explain the in vitro patterns. Quantitative descriptions of cell behavior would help choose between alternative models. We will briefly review the experimental methodology (microfluidics technology and traction force microscopy) used to measure responses of individual cells to their micro-environment, including chemical gradients, physical forces and neighboring cells. We conclude by discussing how to include quantitative cell descriptions into a cell-based model: the Cellular Potts model.
Redundant Mechanisms for Stable Cell Locomotion Revealed by Minimal Models
, 2011
"... ABSTRACT Crawling of eukaryotic cells on flat surfaces is underlain by the protrusion of the actin network, the contractile activity of myosin II motors, and graded adhesion to the substrate regulated by complex biochemical networks. Some crawling cells, such as fish keratocytes, maintain a roughly ..."
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Cited by 7 (0 self)
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ABSTRACT Crawling of eukaryotic cells on flat surfaces is underlain by the protrusion of the actin network, the contractile activity of myosin II motors, and graded adhesion to the substrate regulated by complex biochemical networks. Some crawling cells, such as fish keratocytes, maintain a roughly constant shape and velocity. Here we use moving-boundary simulations to explore four different minimal mechanisms for cell locomotion: 1), a biophysical model for myosin contraction-driven motility; 2), a G-actin transport-limited motility model; 3), a simple model for Rac/Rho-regulated motility; and 4), a model that assumes that microtubule-based transport of vesicles to the leading edge limits the rate of protrusion. We show that all of these models, alone or in combination, are sufficient to produce half-moon steady shapes and movements that are characteristic of keratocytes, suggesting that these mechanisms may serve redundant and complementary roles in driving cell motility. Moving-boundary simulations demonstrate local and global stability of the motile cell shapes and make testable predictions regarding the dependence of shape and speed on mechanical and biochemical parameters. The models shed light on the roles of membrane-mediated area conservation and the coupling of mechanical and biochemical mechanisms in stabilizing motile cells.
Model of Polarization and Bistability of Cell Fragments
, 2007
"... ABSTRACT Directed cell motility is preceded by cell polarization—development of a front-rear asymmetry of the cytoskeleton and the cell shape. Extensive studies implicated complex spatial-temporal feedbacks between multiple signaling pathways in establishing cell polarity, yet physical mechanisms of ..."
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Cited by 5 (1 self)
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ABSTRACT Directed cell motility is preceded by cell polarization—development of a front-rear asymmetry of the cytoskeleton and the cell shape. Extensive studies implicated complex spatial-temporal feedbacks between multiple signaling pathways in establishing cell polarity, yet physical mechanisms of this phenomenon remain elusive. Based on observations of lamellipodial fragments of fish keratocyte cells, we suggest a purely thermodynamic (not involving signaling) quantitative model of the cell polarization and bistability. The model is based on the interplay between pushing force exerted by F-actin polymerization on the cell edges, contractile force powered by myosin II across the cell, and elastic tension in the cell membrane. We calculate the thermodynamic work produced by these intracellular forces, and show that on the short timescale, the cell mechanics can be characterized by an effective energy profile with two minima that describe two stable states separated by an energy barrier and corresponding to the nonpolarized and polarized cells. Cell dynamics implied by this energy profile is bistable—the cell is either disk-shaped and stationary, or crescent-shaped and motile—with a possible transition between them upon a finite external stimulus able to drive the system over the macroscopic energy barrier. The model accounts for the observations of the keratocyte fragments ’ behavior and generates quantitative predictions about relations between the intracellular forces ’ magnitudes and the cell geometry and motility.
ASYMPTOTIC AND BIFURCATION ANALYSIS OF WAVE-PINNING IN A REACTION-DIFFUSION MODEL FOR CELL POLARIZATION
"... Abstract. We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops i ..."
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Abstract. We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second (“inactive”) species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.
Quantitative Analysis of G-Actin Transport in Motile Cells
, 2008
"... ABSTRACT Cell migration is based on an actin treadmill, which in turn depends on recycling of G-actin across the cell, from the rear where F-actin disassembles, to the front, where F-actin polymerizes. To analyze the rates of the actin transport, we used the Virtual Cell software to solve the diffus ..."
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ABSTRACT Cell migration is based on an actin treadmill, which in turn depends on recycling of G-actin across the cell, from the rear where F-actin disassembles, to the front, where F-actin polymerizes. To analyze the rates of the actin transport, we used the Virtual Cell software to solve the diffusion-drift-reaction equations for the G-actin concentration in a realistic three-dimensional geometry of the motile cell. Numerical solutions demonstrate that F-actin disassembly at the cell rear and assembly at the front, along with diffusion, establish a G-actin gradient that transports G-actin forward ‘‘globally’ ’ across the lamellipod. Alternatively, if the F-actin assembly and disassembly are distributed throughout the lamellipod, F-/G-actin turnover is local, and diffusion plays little role. Chemical reactions and/or convective flow of cytoplasm of plausible magnitude affect the transport very little. Spatial distribution of G-actin is smooth and not sensitive to F-actin density fluctuations. Finally, we conclude that the cell body volume slows characteristic diffusion-related relaxation time in motile cell from;10 to;100 s. We discuss biological implications of the local and global regimes of the G-actin transport.
A computational model of cell polarization and motility coupling mechanics and biochemistry. Multiscale Modeling
- Simulation
"... Abstract. The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from mole-cular to tissue) as well as disparate time scales, with reaction kinetics on the order of secon ..."
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Abstract. The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from mole-cular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deforma-tion and motion of the cell occurring on the order of minutes. The computational difficulty, even in two dimensions, resides in the fact that the problem is inherently one of deforming, nonstationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed-boundary method to address this problem. We adopt a simple reaction-diffusion (RD) system that represents an internal regulatory mechanism controlling the polarization of a cell and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the RD system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells. Key words. 2D eukaryotic cell motility, simulation, immersed-boundary method, tension, protrusion, reaction-diffusion system, deforming domain, polarization, wave-pinning, elastic perimeter
Review Cell-Biomaterial Mechanical Interaction in the Framework of Tissue Engineering: Insights, Computational Modeling and Perspectives
, 2011
"... Abstract: Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to ..."
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Cited by 2 (0 self)
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Abstract: Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.
