A stochastic model for adhesion-mediated cell random motility and haptotaxis

The active migration of blood and tissue cells is important in a number of physiological processes including inflammation, wound healing, embryogenesis, and tumor cell metastasis. These cells move by transmitting cytoplasmic force through membrane receptors which are bound specifically to adhesion ligands in the surrounding substratum. Recently, much research has focused on the influence of the composition of extracellular matrix and the distribution of its components on the speed and direction of cell migration. It is commonly believed that the magnitude of the adhesion influences cell speed and/or random turning behavior, whereas a gradient of adhesion may bias the net direction of the cell movement, a phenomenon known as haptotaxis. The mechanisms underlying these responses are presently not understood.A stochastic model is presented to provide a mechanistic understanding of how the magnitude and distribution of adhesion ligands in the substratum influence cell movement. The receptor-mediated cell migration is modeled as an interrelation of random processes on distinct time scales. Adhesion receptors undergo rapid binding and transport, resulting in a stochastic spatial distribution of bound receptors fluctuating about some mean distribution. This results in a fluctuating spatio-temporal pattern of forces on the cell, which in turn affects the speed and turning behavior on a longer time scale. The model equations are a system of nonlinear stochastic differential equations (SDE's) which govern the time evolution of the spatial distribution of bound and free receptors, and the orientation and position of the cell. These SDE's are integrated numerically to simulate the behavior of the model cell on both a uniform substratum, and on a gradient of adhesion ligand concentration.Furthermore, analysis of the governing SDE system and corresponding Fokker-Planck equation (FPE) yields analytical expressions for indices which characterize cell movement on multiple time scales in terms of cell cytomechanical, morphological, and receptor binding and transport parameters. For a uniform adhesion ligand concentration, this analysis provides expressions for traditional cell movement indices such as mean speed, directional persistence time, and random motility coefficient. In a small gradient of adhesion, a perturbation analysis of the FPE yields a constitutive cell flux expression which includes a drift term for haptotactic directional cell migration. The haptotactic drift contains terms identified as contributions from directional orientation bias (taxis).     

Effects of leukocyte random motility and chemotaxis in tissue inflammatory response.

A rapidly growing body of experimental evidence indicates that defects in leukocyte motility and chemotactic response correlate with increased susceptibility to and severity of bacterial infection in tissue. While this is understandable in qualitative terms, the sensitivity of the correlation is remarkable.In the present study, a theoretical analysis has been developed to relate the dynamics of bacterial growth to the growth and transport parameters of bacteria and leukocytes in tissue. The model considers a local tissue region in the vicinity of a venule and applies continuum unsteady state species conservation equations to the bacterial population, the phagocytic leukocytes, and a chemotactically active chemical mediator assumed to be produced by the bacteria. The analysis quantifies the effects of key parameters, such as leukocyte random motility and chemotactic coefficients, phagocytic and growth rate constants, and leukocyte vessel wall permeability, upon host ability to eliminate the bacteria.As an example, the model's predictions are compared to experimental results correlating inhibition of leukocyte chemotaxis by hemoglobin with its adjuvant action in experimental peritoneal infection by E. coli.     

A stochastic model for chemotaxis based on the ordered extension of pseudopods

Many amoeboid cells move by extending pseudopods. Here I present a new stochastic model for chemotaxis that is based on pseudopod extensions by Dictyostelium cells. In the absence of external cues, pseudopod extension is highly ordered with two types of pseudopods: de novo formation of a pseudopod at the cell body in random directions, and alternating right/left splitting of an existing pseudopod that leads to a persistent zig-zag trajectory. We measured the directional probabilities of the extension of splitting and de novo pseudopods in chemoattractant gradients with different steepness. Very shallow cAMP gradients can bias the direction of splitting pseudopods, but the bias is not perfect. Orientation of de novo pseudopods require much steeper cAMP gradients and can be more precise. These measured probabilities of pseudopod directions were used to obtain an analytical model for chemotaxis of cell populations. Measured chemotaxis of wild-type cells and mutants with specific defects in these stochastic pseudopod properties are similar to predictions of the model. These results show that combining splitting and de novo pseudopods is a very effective way for cells to obtain very high sensitivity to stable gradient and still be responsive to changes in the direction of the gradient.     

A stochastic model of leukocyte rolling.

Selectin-mediated leukocyte rolling under flow is an important process in leukocyte recruitment during inflammation. The rolling motion of individual cells has been observed to fluctuate randomly both in vivo and in vitro. This paper presents a stochastic model of the micromechanics of cell rolling and provides an analytical method of treating experimental data. For a homogeneous cell population, the velocity distribution is obtained in an analytical form, which is in good agreement with experimentally determined velocity histograms obtained previously. For a heterogeneous cell population, the model provides a simple, analytical method of separating the contributions of temporal fluctuations and population heterogeneity to the variance of measured rolling velocities. The model also links the mean and variance of rolling velocities to the molecular events underlying the observed cellular motion, allowing characterization of the distribution and release rate of the clusters of molecular bonds that tether the cell to substratum. Applying the model to the analysis of data obtained for neutrophils rolling on an E-selectin-coated surface at a wall shear stress of 1.2 dyn/cm2 yields estimations of the average distance between bond clusters (approximately micron) and the average time duration of a bond cluster resisting the applied fluid force (approximately 0.5 s).     

Measurement of leukocyte motility and chemotaxis parameters with a linear under-agarose migration assay

The interpretation of quantitative assays for leukocyte chemotactic migration is usually made in terms of measurements such as leading front distance, total migrating cells, and leukotactic index. These quantities allow comparison of cellular migration behavior under specified conditions. They are not useful; however, for comparisons between systems or for correlation with in vivo performance, because they depend upon specific physical aspects of the assay system, such as the geometry, chemoattractant concentration and diffusivity, and observation time. It would be more helpful to measure intrinsic properties of cell movement that could be used for comparison between systems, for correlation with in vivo studies, and to increase our understanding of the cell physiology. In this paper we demonstrate a means of quantitating leukocyte random motility, chemokinesis, and chemotaxis in terms of parameters that do characterize intrinsic cell properties. These parameters are the random motility coefficient and the chemotaxis coefficient, which appear in theoretical models of cell migration. We examine how well such a model describes the leukocyte density profile data observed in a modified under-agarose assay having a linear geometry. Furthermore, we obtain values for the random motility coefficient (and its dependence upon the concentration of the attractant peptide FNLLP) and for the chemotaxis coefficient for leukocytes responding to FNLLP.     

A spatially extended stochastic model of the bacterial chemotaxis signalling pathway

We have combined two distinct but related stochastic approaches to model the Escherichia coli chemotaxis pathway. Reactions involving cytosolic components of the pathway were assumed to obey the laws of conventional stochastic chemical kinetics, while the clustered membrane receptors were represented in two-dimensional arrays similar to the Ising model. Receptors were assumed to flip between an active and an inactive state with probabilities dependent upon three energy inputs: ligand binding, methylation level due to adaptation, and the activity of neighbouring receptors. Examination of models with different lattice size and geometry showed that the sensitivity to stimuli increases with lattice size and the nearest-neighbour coupling strength up to a critical point, but this amplification was also accompanied by a proportional increase in steady-state noise. Multiple methylation of receptors resulted in diminished signal-to-noise ratio, but showed improved stability to variation in the coupling strength and increased gain. Under the best conditions the simulated output of a coupled lattice of receptors closely matched the time-course and amplitude found experimentally in living bacteria. The model also has some of the properties of a cellular automaton and shows an unexpected emergence of spatial patterns of methylation within the receptor lattice.     

Simple molecular model for sensing and adaptation based on receptor modification with application to bacterial chemotaxis

A series of simple models to explain adaptation in a sensory system based on reversible covalent modification is developed. The models are applied to the reversible methylation of chemoreceptors in bacteria and by analogy to other sensory transduction systems. The receptor modification system exhibits sensing and adaptation, i.e. raising the stimulus to a new level generates a transient response followed by a return to prestimulus behavior. By means of an analytical solution of the kinetic equation that governs the evolution of the receptor system. an exact expression is obtained for the time required for adaptation. The results account for the most conspicuous properties of the bacterial sensory system; namely, the response times in relation to stimulus changes, the proportionality of receptor modification to receptor occupancy, and the additivity of response times. The analysis indicates how these properties depend upon the parameters of the system, e.g. the rates of covalent modification and demodification, the accuracy of the detector, and the molecular nature of the response regulator. The theory developed for analysis of the bacterial system revealed properties that will be applicable to any system processing sensory information.     

Quantification of random motility and chemotaxis bacterial transport coefficients using individual-cell and population-scale assays

A number of individual-cell and population-scale assays have been introduced to quantify bacterial motility and chemotaxis. The transport coefficients reported in the literature, however, span several orders of magnitude, making it difficult to ascertain to what degree variations in bacterial species/strain, growth medium, growth and experimental conditions, and experiment type contribute to the reported differences in coefficient values. We quantified the random motility of Escherichia coli AW405 using the capillary assay, stopped-flow diffusion chamber (SFDC), and tracking microscope. We obtained good agreement for the random motility coefficient between these assays when using the same bacterial strain and consistent growth and experimental conditions. Chemotaxis of E. coli toward the attractant alpha-methylaspartate was quantified using the SFDC and capillary assay. Good agreement for the chemotactic sensitivity coefficient between the SFDC and the capillary assay was obtained across a limited attractant concentration range. Three different mathematical models were considered for analyzing capillary assay data to obtain a chemotactic sensitivity coefficient. These models differed by their treatment of the bacterial concentration in the chamber and the attractant concentration at the mouth. Results from our study indicate that the capillary assay, the most commonly used bacterial random motility and chemotaxis assay, can be used to accurately quantify bacterial transport coefficients over a limited range of attractant concentrations, provided experiments are performed carefully and appropriate mathematical models are used to interpret the experimental data.     

A mechanical role for the chemotaxis system in swarming motility

The chemotaxis system, but not chemotaxis, is essential for swarming motility in Salmonella enterica serovar Typhimurium. Mutants in the chemotaxis pathway exhibit fewer and shorter flagella, downregulate class 3 or 'late' motility genes, and appear to be less hydrated when propagated on a surface. We show here that the output of the chemotaxis system, CheY approximately P, modulates motor bias during swarming as it does during chemotaxis, but for a distinctly different end. A constitutively active form of CheY was found to promote swarming in the absence of several upstream chemotaxis components. Two point mutations that suppressed the swarming defect of a cheY null mutation mapped to FliM, a protein in the motor switch complex with which CheY approximately P interacts. A common property of these suppressors was their increased frequency of motor reversal. These and other data suggest that the ability to switch motor direction is important for promoting optimal surface wetness. If the surface is sufficiently wet, exclusively clockwise or counterclockwise directions of motor rotation will support swarming, suggesting also that the bacteria can move on a surface with flagellar bundles of either handedness.     

Measurement of bacterial random motility and chemotaxis coefficients: I. Stopped-flow diffusion chamber assay

Bacterial chemotaxis, the directed movement of a cell population in response to a chemical gradient, plays a critical role in the distribution and dynamic interaction of bacterial populations in nonmixed systems. Therefore, in order to make reliable predictions about the migratory behavior of bacteria within the environment, a quantitative characterization of the chemotactic response in terms of intrinsic cell properties is needed.The design of the stopped-flow diffusion chamber (SFDC) provides a well-characterized chemical gradient and reliable method for measuring bacterial migration behavior. During flow through the chamber, a step change in chemical concentration is imposed on a uniform suspension of bacteria. Once flow is stopped, diffusion causes a transient chemical gradient to develop, and bacteria respond by forming a band of high cell density which travels toward higher concentrations of the attractant. Changes in bacterial spatial distributions observed through light scattering are recorded on photomicrographs during a 10-min period. Computer-aided image analysis converts absorbance of the photographic negatives to a digital representation of bacterial density profiles. A mathematical model (part II) is used to quantitatively characterize these observations in terms of intrinsic cell parameters: a chemotactic sensitivity coefficient, mu(0), from the aggregate cell density accumulated in the band and a random motility coefficient, mu, from population dispersion in the absence of a chemical gradient.Using the SFDC assay and an individual-cell-based mathematical model, we successfully determined values for both of these population parameters for Escherichia coli K12 responding to fucose. The values obtained were mu = 1.1 +/- 0. 4 x 10(-5) cm(2)/s and chi(o) = 8 +/- 3 +/- 10(-5) cm(2)/s. We have demonstrated a method capable of determining these parameter values from the now validated mathematical model which will be useful for predicting bacterial migration in application systems.     

Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis.

The growth of new capillary blood vessels, or angiogenesis, is a prominent component of numerous physiological and pathological conditions. An understanding of the co-ordination of underlying cellular behaviors would be helpful for therapeutic manipulation of the process. A probabilistic mathematical model of angiogenesis is developed based upon specific microvessel endothelial cell (MEC) functions involved in vessel growth. The model focuses on the roles of MEC random motility and chemotaxis, to test the hypothesis that these MEC behaviors are of critical importance in determining capillary growth rate and network structure. Model predictions are computer simulations of microvessel networks, from which questions of interest are examined both qualitatively and quantitatively. Results indicate that a moderate MEC chemotactic response toward an angiogenic stimulus, similar to that measured in vitro in response to acidic fibroblast growth factor, is necessary to provide directed vascular network growth. Persistent random motility alone, with initial budding biased toward the stimulus, does not adequately provide directed network growth. A significant degree of randomness in cell migration direction, however, is required for vessel anastomosis and capillary loop formation, as simulations with an overly strong chemotactic response produce network structures largely absent of these features. The predicted vessel extension rate and network structure in the simulations are quantitatively consistent with experimental observations of angiogenesis in vivo. This suggests that the rate of vessel outgrowth is primarily determined by MEC migration rate, and consequently that quantitative in vitro migration assays might be useful tools for the prescreening of possible angiogenesis activators and inhibitors. Finally, reduction of MEC speed results in substantial inhibition of simulated angiogenesis. Together, these results predict that both random motility and chemotaxis are MEC functions critically involved in determining the rate and morphology of new microvessel network growth.     

A model of IP3 receptor with a luminal calcium binding site: stochastic simulations and analysis

We have constructed a stochastic model of the inositol 1,4,5-trisphosphate receptor-Ca2+ channel that is based on quantitative measurements of the channel's properties. It displays the observed dependence of the open probability of the channel with cytosolic [Ca2+] and [IP3] and gives values for the dwell times that agree with the observations. The model includes an explicit dependence of channel gating with luminal calcium. This not only explains several observations reported in the literature, but also provides a possible explanation of why the open probabilities and shapes of the bell-shaped curves reported in [Nature 351 (1991) 751] and in [Proc. Natl. Acad. Sci. U.S.A. 269 (1998) 7238] are so different.     

A calcium model with random absorption: a stochastic approach.

Absorption of calcium, or any mineral, by the body is subject to the random fluctuations typical of diffusion through membranes. In this paper we consider the absorption of calcium from the gut as a white noise process added to the deterministic model of Sen & Mohr (1990, J. theor. Biol. 142, 179-188). The first two moments for the amount of calcium in the extracellular fluid (ECF) have been derived using the Ito Calculus. A confidence interval for the total amount of calcium in the ECF is constructed. The equations for the first two moments of the fraction of dose calcium in the ECF are also given. Suggestions are made for the collection of experimental data in a form which should be helpful in investigating the magnitude of the stochastic effect.     

A model for sperm-egg binding and fusion based on ADAMs and integrins

Once a sperm meets an egg, several events must occur in order for fertilization to proceed. Sperm must bind to the zona pellucida, undergo the acrosome reaction, penetrate the zona pellucida and then bind to and fuse with the egg plasma membrane. Shortly thereafter, the egg must be activated for zygotic development. This review focuses on mammalian sperm-egg plasma membrane binding and fusion, and in particular on the roles of two families of cell-adhesion molecules, ADAMs and integrins, in this important union.     

A new set of chemotaxis homologues is essential for Myxococcus xanthus social motility

Myxococcus xanthus cells aggregate and develop into multicellular fruiting bodies in response to starvation. A new M. xanthus locus, designated dif for defective in fruiting, was identified by the characterization of a mutant defective in fruiting body formation. Molecular cloning, DNA sequencing and sequence analysis indicate that the dif locus encodes a new set of chemotaxis homologues of the bacterial chemotaxis proteins MCPs (methyl-accepting chemotaxis proteins), CheW, CheY and CheA. The dif genes are distinct genetically and functionally from the previously identified M. xanthus frz chemotaxis genes, suggesting that multiple chemotaxis-like systems are required for the developmental process of M. xanthus fruiting body formation. Genetic analysis and phenotypical characterization indicate that the M. xanthus dif locus is required for social (S) motility. This is the first report of a M. xanthus chemotaxis-like signal transduction pathway that could regulate or co-ordinate the movement of M. xanthus cells to bring about S motility.