Eukaryotic Cell Dynamics from Crawlers to Swimmers.

Movement requires force transmission to the environment, and motile cells are robustly, though not elegantly, designed nanomachines that often can cope with a variety of environmental conditions by altering the mode of force transmission used. As with humans, the available modes range from momentary attachment to a substrate when crawling, to shape deformations when swimming, and at the cellular level this involves sensing the mechanical properties of the environment and altering the mode appropriately. While many types of cells can adapt their mode of movement to their microenvironment (ME), our understanding of how they detect, transduce and process information from the ME to determine the optimal mode is still rudimentary. The shape and integrity of a cell is determined by its cytoskeleton (CSK), and thus the shape changes that may be required to move involve controlled remodeling of the CSK. Motion in vivo is often in response to extracellular signals, which requires the ability to detect such signals and transduce them into the shape changes and force generation needed for movement. Thus the nanomachine is complex, and while much is known about individual components involved in movement, an integrated understanding of motility in even simple cells such as bacteria is not at hand. In this review we discuss recent advances in our understanding of cell motility and some of the problems remaining to be solved.

Development and applications of a model for cellular response to multiple chemotactic cues.

The chemotactic response of a cell population to a single chemical species has been characterized experimentally for many cell types and has been extensively studied from a theoretical standpoint. However, cells frequently have multiple receptor types and can detect and respond chemotactically to more than one chemical. How these signals are integrated within the cell is not known. and we therefore adopt a macroscopic phenomenological approach to this problem. In this paper we derive and analyze chemotactic models based on partial differential (chemotaxis) equations for cell movement in response to multiple chemotactic cues. Our derivation generalizes the approach of Othmer and Stevens [29], who have recently developed a modeling framework for studying different chemotactic responses to a single chemical species. The importance of such a generalization is illustrated by the effect of multiple chemical cues on the chemotactic sensitivity and the spatial pattern of cell densities in several examples. We demonstrate that the model can generate the complex patterns observed on the skin of certain animal species and we indicate how the chemotactic response can be viewed as a form of positional indicator.

A model for individual and collective cell movement in Dictyostelium discoideum.

The cellular slime mold Dictyostelium discoideum is a widely used model system for studying a variety of basic processes in development, including cell-cell signaling, signal transduction, pattern formation, cell motility, and the movement of tissue-like aggregates of cells. Many aspects of cell motion are poorly understood, including how individual cell behavior produces the collective motion of cells observed within the mound and slug. Herein, we describe a biologically realistic model for motile D. discoideum cells that can generate active forces, that interact via surface molecules, and that can detect and respond to chemotactic signals. We model the cells as deformable viscoelastic ellipsoids and incorporate signal transduction and cell-cell signaling by using a previously developed model. The shape constraint restricts the admissible deformations but makes the simulation of a large number of interacting cells feasible. Because the model is based on known processes, the parameters can be estimated or measured experimentally. We show that this model can reproduce the observations on the chemotactic behavior of single cells, streaming during aggregation, and the collective motion of an aggregate of cells driven by a small group of pacemakers. The model predicts that the motion of two-dimensional slugs [Bonner, J. T. (1998) Proc. Natl. Acad. Sci. USA 95, 9355-9359] results from the same behaviors that are exhibited by individual cells; it is not necessary to invoke different mechanisms or behaviors. Our computational experiments also suggest previously uncharacterized phenomena that may be experimentally observable.

A chemotactic model for the advance and retreat of the primitive streak in avian development.

The formation of the primitive streak in early avian development marks the onset of gastrulation, during which large scale cell movement leads to a trilaminar blastoderm comprising prospective endodermal, mesodermal and ectodermal tissue. During streak formation a specialized group of cells first moves anteriorly as a coherent column, beginning from the posterior end of the prospective anterior-posterior axis (a process called progression), and then reverses course and returns to the most posterior point on the axis (a process called regression). To date little is known concerning the mechanisms controlling either progression or regression. Here we develop a model in which chemotaxis directs the cell movement and which is capable of reproducing the principal features connected with progression and regression of the primitive streak. We show that this model exhibits a number of experimentally-observed features of normal and abnormal streak development, and we propose a number of experimental tests which may serve to illuminate the mechanisms. This paper represents the first attempt to model the global features of primitive streak formation, and provides an initial stage in the development of a more biologically-realistic discrete cell model that will allow for variation of properties between cells and control over movement of individual cells.

Subharmonic resonance and chaos in forced excitable systems.

Forced excitable systems arise in a number of biological and physiological applications and have been studied analytically and computationally by numerous authors. Existence and stability of harmonic and subharmonic solutions of a forced piecewise-linear Fitzhugh-Nagumo-like system were studied in Othmer ad Watanabe (1994) and in Xie et al. (1996). The results of those papers were for small and moderate amplitude forcing. In this paper we study the existence of subharmonic solutions of this system under large-amplitude forcing. As in the case of intermediate-amplitude forcing, bistability between 1 : 1 and 2 : 1 solutions is possible for some parameters. In the case of large-amplitude forcing, bistability between 2 : 2 and 2 : 1 solutions, which does not occur in the case of intermediate-amplitude forcing, is also possible for some parameters. We identify several new canonical return maps for a singular system, and we show that chaotic dynamics can occur in some regions of parameter space. We also prove that there is a direct transition from 2 : 2 phase-locking to chaos after the first period-doubling bifurcation, rather than via the infinite sequence of period doublings seen in a smooth quadratic interval map. Coexistence of chaotic dynamics and stable phase-locking can also occur.

Stripe formation in juvenile Pomacanthus explained by a generalized turing mechanism with chemotaxis.

Current interest in pattern formation can be traced to a seminal paper by Turing, who demonstrated that a system of reacting and diffusing chemicals, called morphogens, can interact so as to produce stable nonuniform concentration patterns in space. Recently, a Turing model has been suggested to explain the development of pigmentation patterns on species of growing angelfish such as Pomacanthus semicirculatus, which exhibit readily observed changes in the number, size, and orientation of colored stripes during development of juvenile and adult stages, but the model fails to predict key features of the observations on stripe formation. Here we develop a generalized Turing model incorporating cell growth and movement, we analyze the effects of these processes on patterning, and we demonstrate that the model can explain important features of pattern formation in a growing system such as Pomacanthus. The applicability of classical Turing models to biological pattern formation is limited by virtue of the sensitivity of patterns to model parameters, but here we show that the incorporation of growth results in robustly generated patterns without strict parameter control. In the model, chemotaxis in response to gradients in a morphogen distribution leads to aggregation of one type of pigment cell into a striped spatial pattern.

A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud.

A new model for limb development which incorporates both outgrowth due to cell growth and division, and interactions between morphogens produced in the zone of polarizing activity (ZPA) and the apical epidermal ridge (AER) is developed and analysed. The numerically-computed spatio-temporal distributions of these morphogens demonstrate the importance of interaction between the organizing regions in establishing the morphogenetic terrain on which cells reside, and because growth is explicitly incorporated, it is found that the history of a cell's exposure to the morphogens depends heavily on where the cell originates in the early limb bud. Because the biochemical steps between morphogen(s) and gene activation have not been elucidated, there is no biologically-based mechanism for translating the spatio-temporal distributions of morphogens into patterns of gene expression, but several theoretically plausible functions that bridge the gap are suggested. For example it is shown that interpretation functions based on the history of a cell's exposure to the morphogens can qualitatively account for observed patterns of gene expression. The mathematical model and the associated computational algorithms are sufficiently flexible that other schemes for the interactions between morphogens, and their effect on the spatio-temporal pattern of growth and gene expression, can easily be tested. Thus an additional result of this work is a computational tool that can be used to explore the effects of various mutations and experimental interventions on the growth of the limb and the pattern of gene expression. In future work we will extend the model to a three-dimensional representation of the limb and will incorporate a more realistic description of the rheological properties of the tissue mass, which here is treated as a Newtonian fluid.

The effect of heterogeneously-distributed RyR channels on calcium dynamics in cardiac myocytes.

Calcium plays an essential role in excitation-contraction coupling in muscle, and derangements in calcium handling can produce a variety of potentially harmful conditions, especially in cardiac muscle. In cardiac tissue specialized invaginations of the sarcolemma, called T-tubules, penetrate deep into each sarcomere, and depolarization of the SL leads to an influx of calcium through voltage-sensitive channels in the T-tubules that in turn triggers further calcium release from the sarcoplasmic reticulum via ryanodine-sensitive calcium channels. Under certain conditions, such as elevated external Ca2+, cardiac cells can release calcium from the sarcoplasmic reticulum spontaneously, producing a calcium 'spark' and propagating traveling waves of elevated Ca2+ concentration, without depolarization of the SL (Wier and Blatter, 1991a, Cell Calcium 12, 241-254; Williams, 1993, Cell Calcium 14, 724-735; Cheng et al., 1993a, Science 262, 740-744). However, under normal resting conditions these potentially harmful waves seldom occur. In this paper we investigate the role of the periodic distribution of ryanodine-sensitive channels in determining whether a spark can trigger a wave, using a modification of the kinetic model proposed by Tang and Othmer, 1994b, Biophys. J. 67, 2223-2235, for calcium-induced calcium release. We show that the spatial localization of these channels near the T-tubules has a significant effect on both wave propagation and the onset of oscillations in this system. Spatial localization provides a possible explanation for the differing effects of various experimental protocols on the system's ability to propagate a traveling wave.

A continuum analysis of the chemotactic signal seen by Dictyostelium discoideum.

We developed a mathematical model of cell-to-cell-signalling in Dictyostelium discoideum that predicts the cAMP signal seen by individual cells in early aggregation. The model employs two cells on a plane and is designed to predict the space-time characteristics of both the extracellular cAMP signal seen by one cell when a nearby cell relays, and the intracellular cAMP response produced by the stimulus in the receiving cell. The effect of membrane bound phosphodiesterase is studied and it is shown that cells can orient effectively even in its absence. Our results give a detailed picture of how the spatio-temporal characteristics of the extracellular signal can be transduced into a time- and space-dependent intracellular gradient, and they suggest a plausible mechanism for orientation in a natural chemotactic wave.

A model of excitation and adaptation in bacterial chemotaxis.

Bacterial chemotaxis is widely studied because of its accessibility and because it incorporates processes that are important in a number of sensory systems: signal transduction, excitation, adaptation, and a change in behavior, all in response to stimuli. Quantitative data on the change in behavior are available for this system, and the major biochemical steps in the signal transduction/processing pathway have been identified. We have incorporated recent biochemical data into a mathematical model that can reproduce many of the major features of the intracellular response, including the change in the level of chemotactic proteins to step and ramp stimuli such as those used in experimental protocols. The interaction of the chemotactic proteins with the motor is not modeled, but we can estimate the degree of cooperativity needed to produce the observed gain under the assumption that the chemotactic proteins interact directly with the motor proteins.

A discrete cell model with adaptive signalling for aggregation of Dictyostelium discoideum.

Dictyostelium discoideum (Dd) is a widely studied model system from which fundamental insights into cell movement, chemotaxis, aggregation and pattern formation can be gained. In this system aggregation results from the chemotactic response by dispersed amoebae to a travelling wave of the chemoattractant cAMP. We have developed a model in which the cells are treated as discrete points in a continuum field of the chemoattractant, and transduction of the extracellular cAMP signal into the intracellular signal is based on the G protein model developed by Tang & Othmer. The model reproduces a number of experimental observations and gives further insight into the aggregation process. We investigate different rules for cell movement the factors that influence stream formation the effect on aggregation of noise in the choice of the direction of movement and when spiral waves of chemoattractant and cell density are likely to occur. Our results give new insight into the origin of spiral waves and suggest that streaming is due to a finite amplitude instability.

Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics.

We study the models for calcium (Ca) dynamics developed in earlier studies, in each of which the key component is the kinetics of intracellular inositol-1,4,5-trisphosphate-sensitive Ca channels. After rapidly equilibrating steps are eliminated, the channel kinetics in these models are represented by a single differential equation that is linear in the state of the channel. In the reduced kinetic model, the graph of the steady-state fraction of conducting channels as a function of log10(Ca) is a bell-shaped curve. Dynamically, a step increase in inositol-1,4,5-trisphosphate induces an incremental increase in the fraction of conducting channels, whereas a step increase in Ca can either potentiate or inhibit channel activation, depending on the Ca level before and after the increase. The relationships among these models are discussed, and experimental tests to distinguish between them are given. Under certain conditions the models for intracellular calcium dynamics are reduced to the singular perturbed form epsilon dx/d tau = f(x, y, p), dy/d tau = g(x, y, p). Phase-plane analysis is applied to a generic form of these simplified models to show how different types of Ca response, such as excitability, oscillations, and a sustained elevation of Ca, can arise. The generic model can also be used to study frequency encoding of hormonal stimuli, to determine the conditions for stable traveling Ca waves, and to understand the effect of channel properties on the wave speed.

Frequency encoding in excitable systems with applications to calcium oscillations.

A number of excitable cell types respond to a constant hormonal stimulus with a periodic oscillation in intracellular calcium. The frequency of oscillation is often proportional to the hormonal stimulus, and one says that the stimulus is frequency encoded. Here we develop a theory of frequency encoding in excitable systems and apply it to intracellular calcium oscillations that results from increases in the intracellular level of inositol 1,4,5-triphosphate.

Excitation, oscillations and wave propagation in a G-protein-based model of signal transduction in Dictyostelium discoideum.

In an earlier paper (Tang & Othmer 1994 Math. Biosci 120, 25-76), we developed a G-protein-based model for signal transduction in the cellular slime mould Dictyostelium discoideum and showed that it can account for the results from perfusion experiments done by Devreotes and coworkers (Devreotes et al. 1979 J. Cell. 80, 300-309; Devreotes & Steck 1979 J. Cell Biol. 80, 300-309; Dinauer et al. 1980 J. Cell Biol. 86, 537-561). The primary experimental observables are the amounts of cAMP secreted and the time scale of adaptation in response to various stimuli, and we showed that the predictions of the model agree well with the observations. Adaptation in the model arises from dual receptor-mediated pathways, one of which produces a stimulatory G protein Gs and the other of which produces an inhibitory G protein Gi. In this paper we use the model to simulate the suspension experiments of Gerisch & Wick (1975 Biochem. biophys. Res. Commun. 65, 364-370) and the experiments done in cell cultures on Petri dishes (Tomchik & Devreotes 1981 Science, Wash. 212, 443-446). The model predicts excitation to cAMP stimuli, sustained oscillations, or spiral waves and target patterns, depending on the developmental stage of the cells and experimental conditions. The interaction between different pacemakers is also studied.

A model of calcium dynamics in cardiac myocytes based on the kinetics of ryanodine-sensitive calcium channels.

The ryanodine-sensitive calcium channels are pivotal to signal transduction and cell function in many cell types, including cardiac myocytes. In this paper a kinetic model is proposed for these channels. In the model there are two Ca regulatory sites on the channel protein, one positive and the other negative. Cytoplasmic Ca binds to these regulatory sites independently It is assumed that the binding of Ca to the positive site is a much faster process than binding to the negative site. At steady state, the channel opening as a function of the Ca concentration is a bell-shaped curve. The model predicts the adaptation of channels to constant Ca stimulus. When this model is applied to cardiac myocytes, it predicts excitability with respect to Ca perturbations, smoothly graded responses, and Ca oscillations in certain pathological circumstances. In a spatially distributed system, traveling Ca waves in individual myocytes exist under certain conditions. This model can also be applied to other systems where the ryanodine-sensitive channels have been identified.

A G protein-based model of adaptation in Dictyostelium discoideum.

A new model is proposed based on signal transduction via G proteins for adaptation of the signal relay process in the cellular slime mold Dictyostelium discoideum. The kinetic constants involved in the model are estimated from Dictyostelium discoideum and other systems. A qualitative analysis of the model shows how adaptation arises, and numerical computations show that the model agrees with observations in both perfusion and suspension experiments. Several experiments that can serve to test the model are suggested.

Cyclic AMP oscillations in suspensions of Dictyostelium discoideum.

A model developed previously for signal relay and adaptation in the cellular slime mould Dictyostelium discoideum is shown to account for the observed oscillations of calcium and cyclic AMP in cellular suspensions. A qualitative argument is given which explains how the oscillations arise, and numerical computations show how characteristics such as the period and amplitude of the periodic solutions depend on parameters in the model. Several extensions of the basic model are investigated, including the effect of cell aggregation and the effect of time delays in the activation and adaptation processes. The dynamics of mixed cell populations in which only a small fraction of the cells are capable of autonomous oscillation are also studied.

Pacemakers in aggregation fields of Dictyostelium discoideum: does a single cell suffice?

In this paper we address the following question: can a single cell of the cellular slime mold Dictyostelium discoideum serve as a pacemaker for the aggregation phase? Whether or not this is possible is determined by the relative importance of cyclic AMP production due to self-stimulation as compared to diffusion of cyclic AMP away from the cell and extracellular degradation. We determine the conditions under which a single cell on an infinite place can emit periodic signals of cyclic AMP using a model developed previously for signal relay and adaptation in Dictyostelium. Elsewhere it has been shown that this model provides an accurate representation of the stimulus-response behavior of Dictyostelium for a variety of experimental conditions.

Models of dispersal in biological systems.

In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of "runs" separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.