Numerical studies of unreactive contractile networks

A conservative algorithm for parabolic problems in domains with moving boundaries

We describe a novel conservative algorithm for parabolic problems in domains with moving boundaries developed for modeling in cell biology. The spatial discretization is accomplished by applying Voronoi decomposition to a fixed rectangular grid. In the vicinity of the boundary, the procedure generates irregular Voronoi cells that conform to the domain shape and merge seamlessly with regular control volumes in the domain interior. Consequently, our algorithm is free of the CFL stability issue due to moving interfaces and does not involve cell-merging or mass redistribution. Local mass conservation is ensured by finite-volume discretization and natural-neighbor interpolation. Numerical experiments with two-dimensional geometries demonstrate exact mass conservation and indicate an order of convergence in space between one and two. The use of standard meshing techniques makes extension of the method to three dimensions conceptually straightforward.

Multiphase flow models of biogels from crawling cells to bacterial biofilms

This article reviews multiphase descriptions of the fluid mechanics of cytoplasm in crawling cells and growing bacterial biofilms. These two systems involve gels, which are mixtures composed of a polymer network permeated by water. The fluid mechanics of these systems is essential to their biological function and structure. Their mathematical descriptions must account for the mechanics of the polymer, the water, and the interaction between these two phases. This review focuses on multiphase flow models because this framework is natural for including the relative motion between the phases, the exchange of material between phases, and the additional stresses within the network that arise from nonspecific chemical interactions and the action of molecular motors. These models have been successful in accounting for how different forces are generated and transmitted to achieve cell motion and biofilm growth and they have demonstrated how emergent structures develop though the interactions of the two phases. A short description of multiphase flow models of tumor growth is included to highlight the flexibility of the model in describing diverse biological applications.

Intelligent behaviors of amoeboid movement based on complex dynamics of soft matter

We review how soft matter is self-organized to perform information processing at the cell level by examining the model organism Physarum plasmodium. The amoeboid organism, Physarum polycephalum, in the class of true slime molds, exhibits the intelligent behavior of foraging in complex situations. When placed in a maze with food sources at two exits, the organism develops tubular structures with its body which connect the food sources along the shortest path so that the rates of nutrient absorption and intracellular communication are maximized. This intelligent behavior results from the organism's control of a dynamic network through which mechanical and chemical information is transmitted. We review experimental studies that explore the development and adaptation of structures that make up the network. Recently a model of the dynamic network has been developed, and we review the formulation of this model and present some key results. The model captures the dynamics of existing networks, but it does not answer the question of how such networks form initially. To address the development of cell shape, we review existing mechanochemical models of the protoplasm of Physarum, present more general models of motile cells, and discuss how to adapt existing models to explore the development of intelligent networks in Physarum.

Characterization of the Nuclear Deformation Caused by Changes in Endothelial Cell Shape

We investigated the mechanotransduction pathway in endothelial cells between their nucleus and adhesions to the extracellular matrix. First, we measured nuclear deformations in response to alterations of cell shape as cells detach from a flat surface. We found that the nuclear deformation appeared to be in direct and immediate response to alterations of the cell adhesion area. The nucleus was then treated as a neo-Hookean compressible material, and we estimated the stress associated with the cytoskeleton and acting on the nucleus during cell rounding. With the obtained stress field, we estimated the magnitude of the forces deforming the nucleus. Considering the initial and final components of this adhesion-cytoskeleton-nucleus force transmission pathway, we found our estimate for the internal forces acting on the nucleus to be on the same order of magnitude as previously measured traction forces, suggesting a direct mechanical link between adhesions and the nucleus.

Cytoplasm dynamics and cell motion: two-phase flow models

The motion of amoeboid cells is characterized by cytoplasmic streaming and by membrane protrusions and retractions which occur even in the absence of interactions with a substratum. Cell translocation requires, in addition, a transmission mechanism wherein the power produced by the cytoplasmic engine is applied to the substratum in a highly controlled fashion through specific adhesion proteins. Here we present a simple mechano-chemical model that tries to capture the physical essence of these complex biomolecular processes. Our model is based on the continuum equations for a viscous and reactive two-phase fluid model with moving boundaries, and on force balance equations that average the stochastic interactions between actin polymers and membrane proteins. In this paper we present a new derivation and analysis of these equations based on minimization of a power functional. This derivation also leads to a clear formulation and classification of the kinds of boundary conditions that should be specified at free surfaces and at the sites of interaction of the cell and the substratum. Numerical simulations of a one-dimensional lamella reveal that even this extremely simplified model is capable of producing several typical features of cell motility. These include periodic 'ruffle' formation, protrusion-retraction cycles, centripetal flow and cell-substratum traction forces.

On the mechanics of the first cleavage division of the sea urchin egg

We describe a continuum model of the sea urchin egg during the first cleavage division. Using estimated values of the relevant mechanical parameters we then carry out numerical simulations of cytokinesis and conduct a systematic comparison of these computations with a variety of published experimental data.

Stochastic model of receptor-mediated cytomechanics and dynamic morphology of leukocytes

The proposed mathematical model investigates the simplified cytomechanics of cell shape change driven by stochastic stimulation from chemosensory receptors. The cytomechanical component of our model describes the dynamical distribution of F-actin and associated forces in an idealized cortical actin network around the cell periphery. The chemosensory component describes the distribution of chemotactic receptors in the cell membrane surrounding the cortex, where bound receptors give rise to an intracellular signal which modulates some property of the cortical network. As in our earlier models, an account is made for (1) the reactive, contractive properties of cortical actin, but here also for a stress induced by curvature of the cortex-membrane complex which carries an effective surface tension, and (2) statistical fluctuations in receptor binding, but generalized here to include statistical fluctuations in the spatial distribution of receptors, entirely determined by the additional prescription of membrane diffusion coefficients along with total receptor number, receptor binding rate constants and the local concentration field of chemotactic factor. We simplify the analysis by restricting the model to a prototype in which viscous stresses in the cortical network are negligible and the radial extension of the cell cortex is a prescribed function of the cortical actin concentration. We assume in particular that the assembly rate of cortical actin depends on the local density of bound receptors. These assumptions lead to a 4th-order parabolic differential equation on the unit circle coupled to a system of stochastic differential equations. We characterize via bifurcation analysis, stochastic simulations, and analytical correlation functions the spatial-temporal pattern of cell morphology under the influence of fluctuations in the bound receptor distribution for the case of a uniform concentration field of chemotactic factor. In addition to addressing the biological significance of our model, we remark on its relevance to the generic problem of the influence of correlated stochastic perturbations on spatial patterns in morphogenetic media.

The fundamental motor of the human neutrophil is not random: evidence for local non-Markov movement in neutrophils

The search for a fundamental mechano-chemical process that results in net cell motion has led investigators to fit neutrophil tracking data to well described physical models in hopes of understanding the functional form of the driving force. The Ornstein-Uhlenbeck (OU) equation for mean square displacement describes a locally persistent and globally random process and is often used as a starting point for analysis of neutrophil displacements. Based upon the apparently close fit of neutrophil tracking data to this equation and the nature of its derivation, biologists have inferred that the motor of the neutrophil is best represented as a random process. However, 24 of 37 neutrophil paths that we investigated preferentially display programmatic rather than Markov short term correlations between displacements or turn angles. These correlations reflect a bimodal rather than a uniform distribution of subpath correlations in the two variables, and are strongly sampling rate-dependent. Significant periodic components of neutrophil shape change are also detected at the same time scale using either Fourier or elliptical Fourier transform-based descriptors of the neutrophil perimeter. Oscillations in neutrophil velocity have the same period. Taken together, these data suggest a nonstochastic, and perhaps periodic, component to the process driving neutrophil movement.

Leukocyte biophysics. An invited review

The biophysical properties of leukocytes in the passive and active state are discussed. In the passive unstressed state, leukocytes are spherical with numerous membrane folds. Passive leukocytes exhibit viscoelastic properties, and the stress is carried largely by the cell cytoplasm and the nucleus. The membrane is highly deformable in shearing and bending, but resists area expansion. Membrane tension can usually be neglected but plays a role in cases of large deformation when the membrane becomes unfolded. The constant membrane area constraint is a determinant of phagocytic capacity, spreading of cells, and passage through narrow pores. In the active state, leukocytes undergo large internal cytoplasmic deformation, pseudopod projection, and granule redistribution. Several different measurements for assessment of biophysical properties and the internal cytoplasmic deformation in form of strain and strain rate tensors are presented. The current theoretical models for active cytoplasmic motion in leukocytes are discussed in terms of specific macromolecular reactions.

Mechanics and control of the cytoskeleton in Amoeba proteus

Many models of the cytoskeletal motility of Amoeba proteus can be formulated in terms of the theory of reactive interpenetrating flow (Dembo and Harlow, 1986). We have devised numerical methodology for testing such models against the phenomenon of steady axisymmetric fountain flow. The simplest workable scheme revealed by such tests (the minimal model) is the main preoccupation of this study. All parameters of the minimal model are determined from available data. Using these parameters the model quantitatively accounts for the self assembly of the cytoskeleton of A. proteus: for the formation and detailed morphology of the endoplasmic channel, the ectoplasmic tube, the uropod, the plasma gel sheet, and the hyaline cap. The model accounts for the kinematics of the cytoskeleton: the detailed velocity field of the forward flow of the endoplasm, the contraction of the ectoplasmic tube, and the inversion of the flow in the fountain zone. The model also gives a satisfactory account of measurements of pressure gradients, measurements of heat dissipation, and measurements of the output of useful work by amoeba. Finally, the model suggests a very promising (but still hypothetical) continuum formulation of the free boundary problem of amoeboid motion. by balancing normal forces on the plasma membrane as closely as possible, the minimal model is able to predict the turgor pressure and surface tension of A. proteus. Several dynamical factors are crucial to the success of the minimal model and are likely to be general features of cytoskeletal mechanics and control in amoeboid cells. These are: a constitutive law for the viscosity of the contractile network that includes an automatic process of gelation as the network density gets large; a very vigorous cycle of network polymerization and depolymerization (in the case of A. proteus, the time constant for this reaction is approximately 12 s); control of network contractility by a diffusible factor (probably calcium ion); and control of the adhesive interaction between the cytoskeleton and the inner surface of the plasma membrane.

The mechanics of motility in dissociated cytoplasm

We stimulate the dynamical behavior of dissociated cytoplasm using the Reactive Flow Model (Dembo, M., and F. Harlow, 1986, Biophys. J., 50:109-121). We find that for the most part the predicted dynamical behavior of the cytoplasm is governed by three nondimensional numbers. Several other nondimensional parameters, the initial conditions, and boundary conditions are found to have lesser effects. Of the three major nondimensional parameters, one (D#) controls the percentage of ectoplasm, the second (C#) controls the sharpness of the endoplasm-ectoplasm boundary, and the third (R#) controls the topological complexity of the endoplasm-ectoplasm distribution. If R# is very small, then the cytoplasm contracts into a single uniform mass, and there is no bulk streaming. If R# is very large, then the cytoplasmic mass breaks up into a number of clumps scattered throughout the available volume. Between these clumps the solution undergoes turbulent or chaotic patterns of streaming. Intermediate values of R# can be found such that the mass of cytoplasm remains connected and yet undergoes coherent modes of motility similar to flares (Taylor, D.L., J.S. Condeelis, P.L. Moore, and R.D. Allen, 1973, J. Cell Biol., 59:378-394) and rosettes (Kuroda, K., 1979, Cell Motility: Molecules and Organization, 347-362).