Cytoskeleton and Cell Motility

The present article is an invited contribution to the Encyclopedia of Complexity and System Science, Robert A. Meyers Ed., Springer New York (2009). It is a review of the biophysical mechanisms that underly cell motility. It mainly focuses on the eukaryotic cytoskeleton and cell-motility mechanisms. Bacterial motility as well as the composition of the prokaryotic cytoskeleton is only briefly mentioned. The article is organized as follows. In Section III, I first present an overview of the diversity of cellular motility mechanisms, which might at first glance be categorized into two different types of behaviors, namely "swimming" and "crawling". Intracellular transport, mitosis - or cell division - as well as other extensions of cell motility that rely on the same essential machinery are briefly sketched. In Section IV, I introduce the molecular machinery that underlies cell motility - the cytoskeleton - as well as its interactions with the external environment of the cell and its main regulatory pathways. Sections IV D to IV F are more detailed in their biochemical presentations; readers primarily interested in the theoretical modeling of cell motility might want to skip these sections in a first reading. I then describe the motility mechanisms that rely essentially on polymerization-depolymerization dynamics of cytoskeleton filaments in Section V, and the ones that rely essentially on the activity of motor proteins in Section VI. Finally, Section VII is devoted to the description of the integrated approaches that have been developed recently to try to understand the cooperative phenomena that underly self-organization of the cell cytoskeleton as a whole.Comment: 31 pages, 16 figures, 295 reference

Focus on the Physics of Cancer

Despite the spectacular achievements of molecular biology in the second half of the twentieth century and the crucial advances it permitted in cancer research, the fight against cancer has brought some disillusions. It is nowadays more and more apparent that getting a global picture of the very diverse and interlinked aspects of cancer development necessitates, in synergy with these achievements, other perspectives and investigating tools. In this undertaking, multidisciplinary approaches that include quantitative sciences in general and physics in particular play a crucial role. This `focus on' collection contains 19 articles representative of the diversity and state-of-the-art of the contributions that physics can bring to the field of cancer research.Comment: Invited editorial review for the `Focus on the Physics of Cancer' published by the New journal of Physics in 2011--201

Competition between stable equilibria in reaction-diffusion systems: the influence of mobility on dominance

This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front when the symmetry is broken through a small perturbation of the diffusion coefficients is computed. This elementary computation relates to the question, arising from population dynamics, of the influence of mobility on dominance, in reaction-diffusion systems modelling the interaction of two competing species. It is applied to two examples. First a toy example, where it is shown that, depending on the value of a parameter, an increase of the mobility of one of the species may be either advantageous or disadvantageous for this species. Then the Lotka-Volterra competition model, in the bistable regime close to the onset of bistability, where it is shown that an increase of mobility is advantageous. Geometric interpretations of these results are given.Comment: 43 pages, 10 figure

Global behaviour of radially symmetric solutions stable at infinity for gradient systems

This paper is concerned with radially symmetric solutions of systems of the form $$u_t = -\nabla V(u) + \Delta_x u$$ where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of solutions "stable at infinity", that is approaching a spatially homogeneous equilibrium when $|x|$ approaches $+\infty$, is investigated. It is proved that every such solutions approaches a stacked family of radially symmetric bistable fronts travelling to infinity. This behaviour is similar to the one of bistable solutions for gradient systems in one unbounded spatial dimension, described in a companion paper. It is expected (but unfortunately not proved at this stage) that behind these travelling fronts the solution again behaves as in the one-dimensional case (that is, the time derivative approaches zero and the solution approaches a pattern of stationary solutions).Comment: 52 pages, 14 figures. arXiv admin note: substantial text overlap with arXiv:1703.01221. text overlap with arXiv:1604.0200

Global relaxation of bistable solutions for gradient systems in one unbounded spatial dimension

This paper is concerned with spatially extended gradient systems of the form $$u_t=-\nabla V (u) + \mathcal{D} u_{xx}\,,$$ where spatial domain is the whole real line, state-parameter $u$ is multidimensional, $\mathcal{D}$ denotes a fixed diffusion matrix, and the potential $V$ is coercive at infinity. "Bistable" solutions, that is solutions close at both ends of space to stable homogeneous equilibria, are considered. For a solution of this kind, it is proved that, if the homogeneous equilibria approached at both ends belong to the same level set of the potential and if an appropriate (localized in space) energy remains bounded from below when time increases, then the solution approaches, when time approaches infinity, a pattern of stationary solutions homoclinic or heteroclinic to homogeneous equilibria. This result provides a step towards a complete description of the global behaviour of all bistable solutions that is pursued in a companion paper. Some consequences are derived, and applications to some examples are given.Comment: 69 pages, 15 figure

Retracting fronts for the nonlinear complex heat equation

The "nonlinear complex heat equation" $A_t=i|A|^2A+A_{xx}$ was introduced by P. Coullet and L. Kramer as a model equation exhibiting travelling fronts induced by non-variational effects, called "retracting fronts". In this paper we study the existence of such fronts. They go by one-parameter families, bounded at one end by the slowest and "steepest" front among the family, a situation presenting striking analogies with front propagation into unstable states.Comment: 21 pages, 6 figure

A variational proof of global stability for bistable travelling waves

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.Comment: 21 pages, 4 figure

Universal Critical Behavior of Noisy Coupled Oscillators

We study the universal thermodynamic properties of systems consisting of many coupled oscillators operating in the vicinity of a homogeneous oscillating instability. In the thermodynamic limit, the Hopf bifurcation is a dynamic critical point far from equilibrium described by a statistical field theory. We perform a perturbative renormalization group study, and show that at the critical point a generic relation between correlation and response functions appears. At the same time the fluctuation-dissipation relation is strongly violated.Comment: 10 pages, 1 figur