Quantitative relationships between single-cell and cell-population model parameters for chemosensory migration responses of alveolar macrophages to C5a

Bridging hemodynamics, tissue mechanics, and pathophysiology in coronary artery disease: A new agent-based model with tetrahedral mesh integration

We introduce a new multi-physics, multi-scale modeling approach to understand plaque progression during coronary artery disease. Prior works have coupled agent-based models (ABMs) with finite element analysis (FEA) or computational fluid dynamics (CFD) to study the individual contributions of tissue mechanics or hemodynamics to plaque growth. However, these approaches could not simultaneously capture the dynamic interplay between all three domains that drive plaque development. This study aims to present a novel method that merges hemodynamics via CFD, biological processes via ABM, and biomechanics via FEA into a single multi-scale, multi-physics simulation (CAFe). A description of the mechanisms and modeling approaches utilized in the CAFe model is provided, as well as preliminary exploration of the model's capabilities in idealized healthy and stenosed coronary artery models. A volumetric 3D tetrahedral mesh of the artery is shared between CFD, ABM, and FEA to simulate geometrical and biological changes with continuity and consistency. The CFD and FEA modules, implemented with FEBio, calculate the wall shear stress and structural stress and strain, respectively. These biomechanical values are passed to the ABM, implemented in MATLAB, which simulates vascular remodeling using molecular diffusion, cell migration, equations for cellular processes, and volumetric growth to update the geometry. Initial results using CAFe suggest atherosclerotic arteries seek to maintain a hemodynamic threshold through preferential growth and remodeling downstream of a stenosis. The innovative approach described herein marks a significant step forward in predictive modeling of CAD progression and paves the way for powerful coupling of the spatiotemporal-dependent remodeling paradigms exhibited by the disease.

Analysis of the Equilibrium Phase in Immune-Controlled Tumors Provides Hints for Designing Better Strategies for Cancer Treatment

When it comes to improving cancer therapies, one challenge is to identify key biological parameters that prevent immune escape and maintain an equilibrium state characterized by a stable subclinical tumor mass, controlled by the immune cells. Based on a space and size structured partial differential equation model, we developed numerical methods that allow us to predict the shape of the equilibrium at low cost, without running simulations of the initial-boundary value problem. In turn, the computation of the equilibrium state allowed us to apply global sensitivity analysis methods that assess which and how parameters influence the residual tumor mass. This analysis reveals that the elimination rate of tumor cells by immune cells far exceeds the influence of the other parameters on the equilibrium size of the tumor. Moreover, combining parameters that sustain and strengthen the antitumor immune response also proves more efficient at maintaining the tumor in a long-lasting equilibrium state. Applied to the biological parameters that define each type of cancer, such numerical investigations can provide hints for the design and optimization of cancer treatments.

Hybrid models of chemotaxis with application to leukocyte migration

Many chemical and biological systems involve reacting species with vastly different numbers of molecules/agents. Hybrid simulations model such phenomena by combining discrete (e.g., agent-based) and continuous (e.g., partial differential equation- or PDE-based) descriptors of the dynamics of reactants with small and large numbers of molecules/agents, respectively. We present a stochastic hybrid algorithm to model a stage of the immune response to inflammation, during which leukocytes reach a pathogen via chemotaxis. While large numbers of chemoattractant molecules justify the use of a PDE-based model to describe the spatiotemporal evolution of its concentration, relatively small numbers of leukocytes and bacteria involved in the process undermine the veracity of their continuum treatment by masking the effects of stochasticity and have to be treated discretely. Motility and interactions between leukocytes and bacteria are modeled via random walk and a stochastic simulation algorithm, respectively. Since the latter assumes the reacting species to be well mixed, the discrete component of our hybrid algorithm deploys stochastic operator splitting, in which the sequence of the diffusion and reaction operations is determined autonomously during each simulation step. We conduct a series of numerical experiments to ascertain the accuracy and computational efficiency of our hybrid simulations and, then, to demonstrate the importance of randomness for predicting leukocyte migration and fate during the immune response to inflammation.

A size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesis

The recent success of immunotherapies for the treatment of cancer has highlighted the importance of the interactions between tumor and immune cells. Mathematical models of tumor growth are needed to faithfully reproduce and predict the spatiotemporal dynamics of tumor growth. We introduce a mathematical model intended to describe by means of a system of partial differential equations the early stages of the interactions between effector immune cells and tumor cells. The model is structured in size and space, and it takes into account the migration of the tumor antigen-specific cytotoxic effector cells towards the tumor micro-environment by a chemotactic mechanism. We investigate on numerical grounds the role of the key parameters of the model such as the division and growth rates of the tumor cells, and the conversion and death rates of the immune cells. Our main findings are two-fold. Firstly, the model exhibits a possible control of the tumor growth by the immune response; nevertheless, the control is not complete in the sense that the asymptotic equilibrium states keep residual tumors and activated immune cells. Secondly, space heterogeneities of the source of immune cells can significantly reduce the efficiency of the control dynamics, making patterns of remission-recurrence appear.

Unique pattern of neutrophil migration and function during tumor progression

Although neutrophils have been linked to the formation of the pre-metastatic niche, the mechanism of their migration to distant uninvolved tissues has remained elusive. We report that bone marrow neutrophils from mice with early-stage cancers exhibited much more spontaneous migration to tissues. These cells lacked immunosuppressive activity but had elevated rates of oxidative phosphorylation and glycolysis, and much more production of ATP. Their enhanced spontaneous migration was mediated by the binding of ATP to purinergic receptors. In ectopic tumor models and the late stages of cancers, bone marrow neutrophils demonstrated potent immunosuppressive activity. However, these cells had metabolic and migratory activity indistinguishable from that of control neutrophils. A similar pattern of migration was observed in neutrophils and polymorphonuclear myeloid-derived suppressor cells from patients with cancer. These results elucidate the dynamic changes that neutrophils undergo in cancer and demonstrate the mechanism of neutrophils’ contribution to early tumor dissemination.

Schwann cell durotaxis can be guided by physiologically relevant stiffness gradients

Background

Successful nerve regeneration depends upon directed migration of morphologically specialized repair state Schwann cells across a nerve defect. Although several groups have studied directed migration of Schwann cells in response to chemical or topographic cues, the current understanding of how the mechanical environment influences migration remains largely understudied and incomplete. Therefore, the focus of this study was to evaluate Schwann cell migration and morphodynamics in the presence of stiffness gradients, which revealed that Schwann cells can follow extracellular gradients of increasing stiffness, in a form of directed migration termed durotaxis.

Methods

Polyacrylamide substrates were fabricated to mimic the range of stiffness found in peripheral nerve tissue. We assessed Schwann cell response to substrates that were either mechanically uniform or embedded with a shallow or steep stiffness gradient, respectively corresponding to the mechanical niche present during either the fluid phase or subsequent matrix phase of the peripheral nerve regeneration process. We examined cell migration (velocity and directionality) and morphology (elongation, spread area, nuclear aspect ratio, and cell process dynamics). We also characterized the surface morphology of Schwann cells by scanning electron microscopy.

Results

On laminin-coated polyacrylamide substrates embedded with either a shallow (∼0.04 kPa/mm) or steep (∼0.95 kPa/mm) stiffness gradient, Schwann cells displayed durotaxis, increasing both their speed and directionality along the gradient materials, fabricated with elastic moduli in the range found in peripheral nerve tissue. Uniquely and unlike cell behavior reported in other cell types, the durotactic response of Schwann cells was not dependent upon the slope of the gradient. When we examined whether durotaxis behavior was accompanied by a pro-regenerative Schwann cell phenotype, we observed altered cell morphology, including increases in spread area and the number, elongation, and branching of the cellular processes, on the steep but not the shallow gradient materials. This phenotype emerged within hours of the cells adhering to the materials and was sustained throughout the 24 hour duration of the experiment. Control experiments also showed that unlike most adherent cells, Schwann cells did not alter their morphology in response to uniform substrates of different stiffnesses.

Conclusion

This study is notable in its report of durotaxis of cells in response to a stiffness gradient slope, which is greater than an order of magnitude less than reported elsewhere in the literature, suggesting Schwann cells are highly sensitive detectors of mechanical heterogeneity. Altogether, this work identifies durotaxis as a new migratory modality in Schwann cells, and further shows that the presence of a steep stiffness gradient can support a pro-regenerative cell morphology.

Electronic supplementary material

The online version of this article (10.1186/s40824-018-0124-z) contains supplementary material, which is available to authorized users.

Spatio-temporal Models of Lymphangiogenesis in Wound Healing

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. Lymphangiogenesis is a markedly different process from angiogenesis, occurring at different times and in response to different chemical stimuli. Two main hypotheses have been proposed: (1) lymphatic capillaries sprout from existing interrupted ones at the edge of the wound in analogy to the blood angiogenesis case and (2) lymphatic endothelial cells first pool in the wound region following the lymph flow and then, once sufficiently populated, start to form a network. Here, we present two PDE models describing lymphangiogenesis according to these two different hypotheses. Further, we include the effect of advection due to interstitial flow and lymph flow coming from open capillaries. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from biological data. The models are then solved numerically and the results are compared with the available biological literature.

Deciphering chemokine properties by a hybrid agent-based model of Aspergillus fumigatus infection in human alveoli

The ubiquitous airborne fungal pathogen Aspergillus fumigatus is inhaled by humans every day. In the lung, it is able to quickly adapt to the humid environment and, if not removed within a time frame of 4–8 h, the pathogen may cause damage by germination and invasive growth. Applying a to-scale agent-based model of human alveoli to simulate early A. fumigatus infection under physiological conditions, we recently demonstrated that alveolar macrophages require chemotactic cues to accomplish the task of pathogen detection within the aforementioned time frame. The objective of this study is to specify our general prediction on the as yet unidentified chemokine by a quantitative analysis of its expected properties, such as the diffusion coefficient and the rates of secretion and degradation. To this end, the rule-based implementation of chemokine diffusion in the initial agent-based model is revised by numerically solving the spatio-temporal reaction-diffusion equation in the complex structure of the alveolus. In this hybrid agent-based model, alveolar macrophages are represented as migrating agents that are coupled to the interactive layer of diffusing molecule concentrations by the kinetics of chemokine receptor binding, internalization and re-expression. Performing simulations for more than a million virtual infection scenarios, we find that the ratio of secretion rate to the diffusion coefficient is the main indicator for the success of pathogen detection. Moreover, a subdivision of the parameter space into regimes of successful and unsuccessful parameter combination by this ratio is specific for values of the migration speed and the directional persistence time of alveolar macrophages, but depends only weakly on chemokine degradation rates.

Two-dimensional motility of a macrophage cell line on microcontact-printed fibronectin

The ability of macrophages to migrate to sites of infection and inflammation is critical for their role in the innate immune response. Macrophage cell lines have made it possible to study the roles of individual proteins responsible for migration using molecular biology, but it has not been possible to reliably elicit the motility of macrophage cell lines in two dimensions. In the past, measurements of the motility of macrophage cell lines have been largely limited to transwell assays which provide limited quantitative information on motility and limited ability to visualize cell morphology. We used microcontact printing to create polydimethylsiloxane (PDMS) surfaces functionalized with fibronectin that otherwise support little macrophage adhesion. We used these surfaces to measure macrophage migration in two dimensions and found that these cells migrate efficiently in a uniform field of colony-stimulating factor-1, CSF-1. Knockdown of Cdc42 led to a nonstatistically significant reduction in motility, whereas chemical inhibition of PI3K activity led to a complete loss of motility. Inhibition of the RhoA kinase, ROCK, did not abolish the motility of these cells but caused a quantitative change in motility, reducing motility significantly on high concentrations of fibronectin but not on low concentrations. This study illustrates the importance of studying cell motility on well controlled materials to better understand the exact roles of specific proteins on cell migration. © 2014 Wiley Periodicals, Inc.

Analysis of endothelial cell locomotion: Differential effects of motility and contact inhibition

Video microscopy and digital time-lapse recording were used to monitor locomotion and proliferation of bovine pulmonary artery endothelial (BPAE) cells cultured with varying concentrations of basic fibroblast growth factor (bFGF). Cell trajectories were reconstructed using a generalized nearest-neighbor algorithm and analyzed to determine how cell motility is affected by cell-cell collisions, cell divisions, and increasing cell density. The temporal evolution patterns of the average speed of locomotion for all cells in a culture were computed and the effects of varying bFGF concentrations were analyzed. Intermediate concentrations of bFGF (30 and 50 ng/mL) significantly increased the speed of locomotion above the levels we observed with 0 and 100 ng/mL concentrations of bFGF. Increases in cell density due to proliferation were immediately accompanied by a decrease in the average speed of locomotion of the cell population. Finally, the effect of bFGF concentration on the overall cell proliferation rates was assessed. With the addition of 30 or 50 ng/mL of bFGF to the culture media, the observed cell proliferation rates increased significantly. The proliferation rates decreased when the bFGF concentration increased to 100 ng/mL. These results show that bFGF concentrations that increase the motility of BPAE cells also increase the observed cell proliferation rates. (c) 1994 John Wiley & Sons, Inc.

Quantification and regulation of cell migration

Cell migration is essential for many physiological and pathological processes that include embryonic development, the immune response, wound healing, angiogenesis, and cancer metastasis. It is also important for emerging tissue engineering applications such as tissue reconstitution and the colonization of biomedical implants. By summarizing results from recent experimental and theoretical studies, this review outlines the role played by growth factors or substrate-adhesion molecules in modulating cell motility and shows that cell motility can be an important factor in determining the rates of tissue formation. The application of cell motility assays and the use of theoretical models for analyzing cell migration and proliferation are also discussed.

Migration of connective tissue-derived cells is mediated by ultra-low concentration gradient fields of EGF

The directed migration of cells towards chemical stimuli incorporates simultaneous changes in both the concentration of a chemotactic agent and its concentration gradient, each of which may influence cell migratory response. In this study, we utilized a microfluidic system to examine the interactions between epidermal growth factor (EGF) concentration and EGF gradient in stimulating the chemotaxis of connective tissue-derived fibroblast cells. Cells seeded within microfluidic devices were exposed to concentration gradients established by EGF concentrations that matched or exceeded those required for maximum chemotactic responses seen in transfilter migration assays. The migration of individual cells within the device was measured optically after steady-state gradients had been experimentally established. Results illustrate that motility was maximal at EGF concentration gradients between .01- and 0.1-ng/(mL.mm) for all concentrations used. In contrast, the number of motile cells continually increased with increasing gradient steepness for all concentrations examined. Microfluidics-based experiments exposed cells to minute changes in EGF concentration and gradient that were in line with the acute EGFR phosphorylation measured. Correlation of experimental data with established mathematical models illustrated that the fibroblasts studied exhibit an unreported chemosensitivity to minute changes in EGF concentration, similar to that reported for highly motile cells, such as macrophages. Our results demonstrate that shallow chemotactic gradients, while previously unexplored, are necessary to induce the rate of directed cellular migration and the number of motile cells in the connective tissue-derived cells examined.

Chemotactic predator-prey dynamics

A discrete chemotactic predator-prey model is proposed in which the prey secrets a diffusing chemical which is sensed by the predator and vice versa. Two dynamical states corresponding to catching and escaping are identified and it is shown that steady hunting is unstable. For the escape process, the predator-prey distance is diffusive for short times but exhibits a transient subdiffusive behavior which scales as a power law t¹/³ with time t and ultimately crosses over to diffusion again. This allows us to classify the motility and dynamics of various predatory microbes and phagocytes. In particular, there is a distinct region in the parameter space where they prove to be infallible predators.

Vascular smooth muscle cell durotaxis depends on substrate stiffness gradient strength

Mechanical compliance is emerging as an important environmental cue that can influence certain cell behaviors, such as morphology and motility. Recent in vitro studies have shown that cells preferentially migrate from less stiff to more stiff substrates; however, much of this phenomenon, termed durotaxis, remains ill-defined. To address this problem, we studied the morphology and motility of vascular smooth muscle cells on well-defined stiffness gradients. Baselines for cell spreading, polarization, and random motility on uniform gels with moduli ranging from 5 to 80 kPa were found to increase with increasing stiffness. Subsequent analysis of the behavior of vascular smooth muscle cells on gradient substrata (0-4 kPa/100 mum, with absolute moduli of 1-80 kPa) demonstrated that the morphology on gradient gels correlated with the absolute modulus. In contrast, durotaxis (evaluated quantitatively as the tactic index for a biased persistent random walk) and cell orientation with respect to the gradient both increased with increasing magnitude of gradient, but were independent of the absolute modulus. These observations provide a foundation for establishing quantitative relationships between gradients in substrate stiffness and cell response. Moreover, these results reveal common features of phenomenological cell response to chemotactic and durotactic gradients, motivating further mechanistic studies of how cells integrate and respond to multiple complex signals.

Experimental verification of the behavioral foundation of bacterial transport parameters using microfluidics

We present novel microfluidic experiments to quantify population-scale transport parameters (chemotactic sensitivity chi(0) and random motility mu) of a population of bacteria. Previously, transport parameters have been derived theoretically from single-cell swimming behavior using probabilistic models, yet the mechanistic foundations of this upscaling process have not been verified experimentally. We designed a microfluidic capillary assay to generate and accurately measure gradients of chemoattractant (alpha-methylaspartate) while simultaneously capturing the swimming trajectories of individual Escherichia coli bacteria using videomicroscopy and cell tracking. By measuring swimming speed and bias in the swimming direction of single cells for a range of chemoattractant concentrations and concentration gradients, we directly computed the chemotactic velocity VC and the associated chemotactic sensitivity chi(0). We then show how mu can also be readily determined using microfluidics but that a population-scale microfluidic approach is experimentally more convenient than a single-cell analysis in this case. Measured values of both chi(0) [(12.4 +/- 2.0) x 10(-4) cm(2) s(-1)] and mu [(3.3 +/- 0.8) x 10(-6) cm(2) s(-1)] are comparable to literature results. This microscale approach to bacterial chemotaxis lends experimental support to theoretical derivations of population-scale transport parameters from single-cell behavior. Furthermore, this study shows that microfluidic platforms can go beyond traditional chemotaxis assays and enable the quantification of bacterial transport parameters.

A computational model of chemotaxis-based cell aggregation

We present a computational model that successfully captures the cell behaviors that play important roles in 2-D cell aggregation. A virtual cell in our model is designed as an independent, discrete unit with a set of parameters and actions. Each cell is defined by its location, size, rates of chemoattractant emission and response, age, life cycle stage, proliferation rate and number of attached cells. All cells are capable of emitting and sensing a chemoattractant chemical, moving, attaching to other cells, dividing, aging and dying. We validated and fine-tuned our in silico model by comparing simulated 24-h aggregation experiments with data derived from in vitro experiments using PC12 pheochromocytoma cells. Quantitative comparisons of the aggregate size distributions from the two experiments are produced using the Earth Mover's Distance (EMD) metric. We compared the two size distributions produced after 24 h of in vitro cell aggregation and the corresponding computer simulated process. Iteratively modifying the model's parameter values and measuring the difference between the in silico and in vitro results allow us to determine the optimal values that produce simulated aggregation outcomes closely matched to the PC12 experiments. Simulation results demonstrate the ability of the model to recreate large-scale aggregation behaviors seen in live cell experiments.

Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration

Limited cell ingrowth is a major problem for tissue engineering and the clinical application of porous biomaterials as bone substitutes. As a first step, migration and proliferation of an interacting cell population can be studied in two-dimensional culture. Mathematical modelling is essential to generalize the results of these experiments and to derive the intrinsic parameters that can be used for predictions. However, a more thorough evaluation of theoretical models is hampered by limited experimental observations. In this study, experiments and image analysis methods were developed to provide a detailed spatial and temporal picture of how cell distributions evolve. These methods were used to quantify the migration and proliferation of skeletal cell types including MG63 and human bone marrow stromal cells (HBMSCs). The high level of detail with which the cell distributions were mapped enabled a precise assessment of the correspondence between experimental results and theoretical model predictions. This analysis revealed that the standard Fisher equation is appropriate for describing the migration behaviour of the HBMSC population, while for the MG63 cells a sharp front model is more appropriate. In combination with experiments, this type of mathematical model will prove useful in predicting cell ingrowth and improving strategies and control of skeletal tissue regeneration.

A simple immune system simulation reveals optimal movement and cell density parameters for successful target clearance

We report here a simple simulation of the immune system in which we analysed the behaviour of responder cells in the presence of target cells. Variable parameters determined the behaviour of the cells within the simulation, and many simulations using the same parameters ensured that statistical variability was achieved. The model demonstrated that high mobility of the target or responder cells produced a more robust response, and that clearance by the immune system was favoured when effector cells moved rapidly compared with the target cells. Therefore, the high motility coefficients exhibited by T cells studied in vivo may play a role in optimizing the effector response to pathogens. Surprisingly, when the number density of responding cells was increased, target cell numbers were limited more effectively, but there was an increased likelihood of a prolonged response.

New method for modeling connective-tissue cell migration: improved accuracy on motility parameters

Mathematical models of cell migration based on persistent random walks have been successfully applied to describe the motility of several cell types. However, the migration of slowly moving connective-tissue cells, such as fibroblasts, is difficult to observe experimentally and difficult to describe theoretically. We identify two primary sources of this difficulty. First, cells such as fibroblasts tend to migrate slowly and change shape during migration. This makes accurate determination of cell position difficult. Second, the cell population is considerably heterogeneous with respect to cell speed. Here we develop a method for fitting connective-tissue cell migration data to persistent random walk models, which accounts for these two significant sources of error and enables accurate determination of the cell motility parameters. We demonstrate the usefulness of this method for modeling both isotropic cell motility and biased cell motility, where the migration of a population of cells is influenced by a gradient in a surface-bound adhesive peptide. This method can discern differences in the motility of populations of cells at different points along the peptide gradient and can therefore be used as a tool to quantify the effects of peptide concentration and gradient magnitude on cell migration.

A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment

Glioblastoma, the most malignant form of brain cancer, is responsible for 23% of primary brain tumors and has extremely poor outcome. Confounding the clinical management of glioblastomas is the extreme local invasiveness of these cancer cells. The mechanisms that govern invasion are poorly understood. To gain insight into glioblastoma invasion, we conducted experiments on the patterns of growth and dispersion of U87 glioblastoma tumor spheroids in a three-dimensional collagen gel. We studied two different cell lines, one with a mutation to the EGFR (U87DeltaEGFR) that is associated with increased malignancy, and one with an endogenous (wild-type) receptor (U87WT). We developed a continuum mathematical model of the dispersion behaviors with the aim of identifying and characterizing discrete cellular mechanisms underlying invasive cell motility. The mathematical model quantitatively reproduces the experimental data, and indicates that the U87WT invasive cells have a stronger directional motility bias away from the spheroid center as well as a faster rate of cell shedding compared to the U87DeltaEGFR cells. The model suggests that differences in tumor cell dispersion may be due to differences in the chemical factors produced by cells, differences in how the two cell lines remodel the gel, or different cell-cell adhesion characteristics.

Morphology matters in immune cell chemotaxis: membrane asymmetry affects amplification

A key mediator of eukaryotic chemotaxis is the asymmetric accumulation of phosphatidylinositol-3,4,5-triphosphate (PIP3) on the cell membrane. Recent work has focused on understanding how a shallow external gradient of chemoattractant leads to an amplified internal gradient of PIP3. In this paper we dissect what fraction of this amplification is derived biochemically by the signal transduction network and how much arises entirely from the effects of cell morphology. Here we identify and formalize the role of morphology in signal detection and demonstrate its effects through simulation and experiments. Our key result is that an asymmetric distribution of membrane accounts for approximately one-half of the measured amplification from ligand concentration to PIP3 production. We also show that the underlying biochemical network behaves as a linear amplifier in the micropipette assay.

Symmetry-breaking in mammalian cell cohort migration during tissue pattern formation: role of random-walk persistence

Coordinated, cohort cell migration plays an important role in the morphogenesis of tissue patterns in metazoa. However, individual cells intrinsically move in a random walk-like fashion when studied in vitro. Hence, in the absence of an external orchestrating influence or template, the emergence of cohort cell migration must involve a symmetry-breaking event. To study this process, we used a novel experimental system in which multiple capillary endothelial cells exhibit spontaneous and robust cohort migration in the absence of chemical gradients when cultured on micrometer-scale extracellular matrix islands fabricated using microcontact printing. A computational model suggested that directional persistence of random-walk and dynamic mechanical coupling of adjacent cells are the critical control parameters for this symmetry-breaking behavior that is induced in spatially-constrained cell ensembles. The model predicted our finding that fibroblasts, which exhibit a much shorter motility persistence time than endothelial cells, failed to undergo symmetry breaking or produce cohort migration on the matrix islands. These findings suggest that cells have intrinsic motility characteristics that are tuned to match their role in tissue patterning. Our results underscore the importance of studying cell motility in the context of cell populations, and the need to address emergent features in multicellular organisms that arise not only from cell-cell and cell-matrix interactions, but also from properties that are intrinsic to individual cells.

Stochastic models for cell motion and taxis

Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic differential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. Directional cell motion in response to a stimulus, termed taxis, has previously been modeled at a phenomenological level using the Keller-Segel diffusion equation. The Keller-Segel model cannot distinguish certain modes of taxis, and this motivates the introduction of a richer class of models which is nevertheless still amenable to statistical analysis. A state space model formulation is used to link models proposed for cell velocity to observed data. Sequential Monte Carlo methods enable parameter estimation via maximum likelihood for a range of applicable models. One particular experimental situation, involving the effect of an electric field on cell behavior, is considered in detail. In this case, an Ornstein- Uhlenbeck model for cell velocity is found to compare favorably with a nonlinear diffusion model.

Aggregation kinetics of well and poorly differentiated human prostate cancer cells

Aggregation of attachment-dependent animal cells represents a series of motility, collision, and adhesion events applicable to such diverse fields as tissue engineering, bioseparations, and drug testing. Aggregation of human prostate cancer cells in liquid-overlay culture was modeled using Smoluchowski's collision theory. Using well (LNCaP) and poorly differentiated (DU 145 and PC 3) cell lines, the biological relevance of the model was assessed by comparing aggregation rates with diffusive and adhesive properties. Diffusion coefficients ranged from 5 to 90 microm(2)/min for single LNCaP and PC 3 cells, respectively. Similar diffusivities were predicted by the persistent random walk model and Einstein relation, indicating random motion. LNCaP cells were the most adhesive in our study with reduced cell shedding, 100% adhesion probability, and enhanced expression of E-cadherin. There was an increase in DU 145 cells staining positive for E-cadherin from nearly 20% of single cells to uniform staining across the surface of all aggregates; under 30% of PC 3 aggregates stained positive. Aggregation rates were more consistent with adhesive properties than with motilities, suggesting that aggregation in our study was reaction-controlled. Relative to other assays employed here, aggregation rates were more sensitive to phenotypic differences in cell lines and described size-dependent changes in aggregation at a finer resolution. In particular, model results suggest similar aggregation rates for two-dimensional DU 145 and PC 3 aggregates and upwards of 4-fold higher rates for larger three-dimensional DU 145 spheroids, consistent with expression of E-cadherin. The kinetic model has application to spheroid production, to cell flocculation and as an adhesion assay.

Substrate microtopography can enhance cell adhesive and migratory responsiveness to matrix ligand density

The regulation of cell motility by ligand density on substrates with variable microtopography is not well understood. In this report, we studied the adhesion and motility behavior of HepG2 cells on microtextured poly(glycolic-co-lactic)acid (PGLA) copolymer substrates, whose surface bioactivity was differentially modified through the adsorption of 0-5.5 ng/cm(2) collagen. Microtextured PGLA substrates were fabricated as thin films with a uniform surface distribution of micropores of median size of 3.1 +/- 1.5 microm and three-dimensional root mean squared roughness of 0.253 microm. Even in the absence of collagen, cells on microtextured substrates responded to substrate topography by exhibiting a 200% increase in adhesion strength compared with untextured controls and ventral localization of the intracellular adhesion protein vinculin. Further enhancement in adhesion strength (420% over untextured, untreated substrates) was demonstrated with bioactivated, microtextured surfaces, indicating that cell adhesion responses to topography and surface ligand density were cooperative. Our motility studies of cells on untextured substrates adsorbed with different levels of collagen demonstrated that a classical biphasic relationship between the cell population averaged migration rate, mu, and the collagen ligand density was preserved. However, comparison of cell motility responses between untextured and microtextured substrates indicates that the motility versus ligand density curve shifted, such that equivalent levels of cell motility were achieved at lower ligand density on microtextured surfaces. Furthermore, the maximum mu values achieved on the microtextured substrates exceeded those on untextured substrates by twofold. Taken together, we show that the magnitude of subcellular scale microtexture of a polymer substrate can sensitize the cell motility responsiveness to substrate ligand concentration; we suggest that the underlying mechanisms involve alteration in the degree of cell-substrate adhesivity as well as changes in the nature of ligand-induced cell activation processes.

Migration of lymphocytes on fibronectin-coated surfaces: temporal evolution of migratory parameters

Lymphocytes typically interact with implanted biomaterials through adsorbed exogenous proteins. To provide a more complete characterization of these interactions, analysis of lymphocyte migration on adsorbed extracellular matrix proteins must accompany the commonly performed adhesion studies. We report here a comparison of the migratory and adhesion behavior of Jurkat cells (a T lymphoblastoid cell line) on tissue culture treated and untreated polystyrene surfaces coated with various concentrations of fibronectin. The average speed of cell locomotion showed a biphasic response to substrate adhesiveness for cells migrating on untreated polystyrene and a monotonic decrease for cells migrating on tissue culture-treated polystyrene. A modified approach to the persistent random walk model was implemented to determine the time dependence of cell migration parameters. The random motility coefficient showed significant increases with time when cells migrated on tissue culture-treated polystyrene surfaces, while it remained relatively constant for experiments with untreated polystyrene plates. Finally, a cell migration computer model was developed to verify our modified persistent random walk analysis. Simulation results suggest that our experimental data were consistent with temporally increasing random motility coefficients.

Monomeric monocyte chemoattractant protein-1 (MCP-1) binds and activates the MCP-1 receptor CCR2B

To address the role of dimerization in the function of the monocyte chemoattractant protein-1, MCP-1, we mutated residues that comprise the core of the dimerization interface and characterized the ability of these mutants to dimerize and to bind and activate the MCP-1 receptor, CCR2b. One mutant, P8A*, does not dimerize. However, it has wild type binding affinity, stimulates chemotaxis, inhibits adenylate cyclase, and stimulates calcium influx with wild type potency and efficacy. These data suggest that MCP-1 binds and activates its receptor as a monomer. In contrast, Y13A*, another monomeric mutant, has a 100-fold weaker binding affinity, is a much less potent inhibitor of adenylate cyclase and stimulator of calcium influx, and is unable to stimulate chemotaxis. Thus Tyr13 may make important contacts with the receptor that are required for high affinity binding and signal transduction. We also explored whether a mutant, [1+9-76]MCP-1 (MCP-1 lacking residues 2-8), antagonizes wild type MCP-1 by competitive inhibition, or by a dominant negative mechanism wherein heterodimers of MCP-1 and [1+9-76]MCP-1 bind to the receptor but are signaling incompetent. Consistent with the finding that MCP-1 can bind and activate the receptor as a monomer, we demonstrate that binding of MCP-1 in the presence of [1+9-76]MCP-1 over a range of concentrations of both ligands fits well to a simple model in which monomeric [1+9-76]MCP-1 functions as a competitive inhibitor of monomeric MCP-1. These results are crucial for elucidating the molecular details of receptor binding and activation, for interpreting mutagenesis data, for understanding how antagonistic chemokine variants function, and for the design of receptor antagonists.

Deconstructing (and reconstructing) cell migration

An overriding objective in cell biology is to be able to relate properties of particular molecular components to cell behavioral functions and even physiology. In the "traditional" mode of molecular cell biology, this objective has been tackled on a molecule-by-molecule basis, and in the "future" mode sometimes termed "functional genomics," it might be attacked in a high-throughput, parallel manner. Regardless of the manner of approach, the relationship between molecular-level properties and cell-level function is exceedingly difficult to elucidate because of the large number of relevant components involved, their high degree of interconnectedness, and the inescapable fact that they operate as physico-chemical entities-according to the laws of kinetics and mechanics-in space and time within the cell. Cell migration is a prominent representative example of such a cell behavioral function that requires increased understanding for both scientific and technological advance. This article presents a framework, derived from an engineering perspective regarding complex systems, intended to aid in developing improved understanding of how properties of molecular components influence the function of cell migration. That is, cell population migration behavior can be deconstructed as follows: first in terms of a mathematical model comprising cell population parameters (random motility, chemotaxis/haptotaxis, and chemokinesis/haptokinesis coefficients), which in turn depend on characteristics of individual cell paths that can be analyzed in terms of a mathematical model comprising individual cell parameters (translocation speed, directional persistence time, chemotactic/haptotactic index), which in turn depend on cell-level physical processes underlying motility (membrane extension and retraction, cell/substratum adhesion, cell contractile force, front-vs.-rear asymmetry), which in turn depend on molecular-level properties of the plethora of components involved in governance and regulation of these processes. Hence, the influence of any molecular component on cell population migration can be understood by reconstructing these relationships from the molecular level to the physical process level to the individual cell path level to the cell population distribution level. This approach requires combining experimental, theoretical, and computational methodologies from molecular biology, biochemistry, biophysics, and bioengineering.

Characterization of single-cell migration using a computer-aided fluorescence time-lapse videomicroscopy system

Single-cell assays of cell migration, while yielding dynamic measurements of cell position and morphology, are predominantly limited by the time required for data collection and analysis. Computer-aided fluorescence time-lapse videomicroscopy (CAFTiV) was developed in order to facilitate the tracking and rapid examination of large numbers of motile cells. The system combines time-lapse videomicroscopy with epifluorescence capability, which allows full automation of image capture, sorting, and analysis due to the low background in the fluorescence images. Utilizing the CAFTiV system, data analysis time was reduced from over 125 h to less than 1 labor minute. In addition, fluorescence imaging permits cell tracking in small-volume chambers (<100 microL), which is useful should the addition of expensive reagents be required. It is anticipated that the ability to characterize both biochemical and biophysical properties responsible for cell movement will be enhanced by this methodology.

A model for the kinetics of homotypic cellular aggregation under static conditions

We present the formulation and testing of a mathematical model for the kinetics of homotypic cellular aggregation. The model considers cellular aggregation under no-flow conditions as a two-step process. Individual cells and cell aggregates 1) move on the tissue culture surface and 2) collide with other cells (or aggregates). These collisions lead to the formation of intercellular bonds. The aggregation kinetics are described by a system of coupled, nonlinear ordinary differential equations, and the collision frequency kernel is derived by extending Smoluchowski's colloidal flocculation theory to cell migration and aggregation on a two-dimensional surface. Our results indicate that aggregation rates strongly depend upon the motility of cells and cell aggregates, the frequency of cell-cell collisions, and the strength of intercellular bonds. Model predictions agree well with data from homotypic lymphocyte aggregation experiments using Jurkat cells activated by 33B6, an antibody to the beta 1 integrin. Since cell migration speeds and all the other model parameters can be independently measured, the aggregation model provides a quantitative methodology by which we can accurately evaluate the adhesivity and aggregation behavior of cells.

Cytokine-stimulated chemotaxis of human neutrophils in a 3-D conjoined fibrin gel assay

The ability of neutrophils to migrate through three-dimensional (3-D) tissues in response to chemical stimuli is critical to their host defense function. However, studies characterizing stimulated migration in vitro have been largely limited to two-dimensional (2-D) surfaces. In this study, we have employed direct observation methods to quantify human neutrophil migration in 3-D fibrin gel using time-lapse video microscopy and automated cell tracking methods. A novel 3-D conjoined gel assay was developed to establish experimentally quantifiable and theoretically predictable diffusion gradients of chemotactic factors. This assay was used to measure objective migration parameters, namely the random motility and chemotaxis coefficients, in response to the cytokine, interleukin-8 (IL-8). The random motility coefficient, mu, showed a biphasic dependence on IL-8 concentration with a maximum of 1.1 x 10(-8) cm2/s at 5 x 10(-8) M IL-8; no significant motility was observed in the absence of IL-8. We further established the dependence of cell orientation bias, phi, on the concentration and gradient steepness (i.e., specific gradient, SG) of IL-8. Results indicate that phi increases with increasing SG, provided the concentration is maintained sufficiently low, which we conjecture to result from minimizing IL-8 receptor down-regulation. The chemotaxis coefficient, chi, was maximum at an intermediate SG for both IL-8 concentrations studied. We also examined the applicability of this assay to estimate mu and chi from indirect measurements of chemotaxis, namely the simpler measurement of cell redistribution after a prescribed incubation time, as opposed to direct cell tracking measurements. By virtue of measuring chi, this is the first quantitatively objective study of mammalian cell chemotaxis in a physiologically relevant 3-D gel and, in particular, of neutrophil chemotaxis on any substratum in response to the physiologically relevant chemotactic factor, IL-8.

Integrin-binding peptide in solution inhibits or enhances endothelial cell migration, predictably from cell adhesion

We have examined the effects of an integrin-binding competitor, echistatin, in solution on adhesion and migration of rat microvessel endothelial cells on fibronectin in vitro. A biphasic dependence of cell motility on fibronectin surface density was observed, with a peak random motility coefficient of about 8 x 10(-9) cm2/sec occurring below 0.3 microgram/cm2 fibronectin. In the presence of echistatin at 0.5 microM, the peak random motility coefficient was similar but occurred at the significantly greater fibronectin surface density of 1.2 micrograms/cm2. Hence, the same concentration of this soluble integrin-binding competitor inhibited migration on low fibronectin densities but enhanced migration on high fibronectin densities. At the same time, echistatin decreased adhesiveness on all fibronectin surface densities. When motility was correlated explicitly with adhesiveness, a single biphasic relationship was obtained for both absence and presence of echistatin with peak motility occurring in both cases at identical adhesiveness. Both the inhibiting and enhancing effects of the soluble integrin-binding competitor on motility are predictable from its effect on adhesion, consistent with the theoretical models of Lauffenburger (15) and DiMilla et al. (3).

Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model

More than 20 years after its proposal, Keller and Segel's model (1971, J. theor. Biol., 30, 235-248) remains by far the most popular model for chemical control of cell movement. However, before the Keller-Segel equations can be applied to a particular system, appropriate functional forms must be specified for the dependence on chemical concentration of the cell transport coefficients and the chemical degradation rate. In the vast majority of applications, these functional forms have been chosen using simple intuitive criteria. We focus on the particular case of eukaryotic cell movement, and derive an approximation to the detailed model of Sherratt et al. (1993, J. theor. Biol., 162, 23-40). The approximation consists of the Keller-Segel equations, with specific forms predicted for the cell transport coefficients and chemical degradation rate. Moreover, the parameter values in these functional forms can be directly measured experimentally. In the case of the much studied neutrophil-peptide system, we test our approximation using both the Boyden chamber and under-agarose assays. Finally, we show that for other cell-chemical interactions, a simple comparison of time scales provides a rapid check on the validity of our Keller-Segel approximation.

Three-dimensional cell cultures mimic tissues

Three-dimensional cell culture using gels of type I collagen is a flexible method for studying cell behavior in a tissuelike environment. With only small changes in the basic protocol, we were able to encapsulate neutrophils, hepatocytes, and PC12 cells. As demonstrated by cell-specific assays for migration, protein secretion, and growth factor induction, the encapsulated cells were viable and functional. In future studies, we will focus on using these cell cultures to study cell movement, cell growth, and cell function in carefully controlled tissuelike environments.

Quantification of human neutrophil motility in three-dimensional collagen gels. Effect of collagen concentration

Leukocytes must migrate through tissues to fulfill their role in the immune response, but direct methods for observing and quantifying cell motility have mostly been limited to migration on two-dimensional surfaces. We have now developed methods for examining neutrophil movement in a three-dimensional gel containing 0.1 to 0.7 mg/ml rat tail tendon collagen. Neutrophil-populated collagen gels were formed within flat glass capillary tubes, permitting direct observation with light microscopy. By following the tracks of individual cells over a 13.5-min observation period and comparing them to a stochastic model of cell movement, we quantified cell speed within a given gel by estimating a random motility coefficient (mu) and persistence time (P). The random motility coefficient changed significantly with collagen concentration in the gel, varying from 1.6 to 13.3 x 10(-9) cm2/s, with the maximum occurring at a collagen gel concentration of 0.3 mg/ml. The methods described may be useful for studying tissue dynamics and for evaluating the mechanism of cell movement in three-dimensional gels of extracellular matrix (ECM) molecules.

Computer simulations of cell-target encounter including biased cell motion toward targets: single and multiple cell-target simulations in two dimensions

In order for immune cells to carry out many of their functions, including clearance of infectious agents from tissue, they must first encounter their targets in the tissue. This encounter process is often the rate-limiting step in the overall function. Most immune cells exhibit chemotactic ability, and previous continuum models for encounter rates and dynamics have shown that chemotaxis can be a great advantage to cells by greatly increasing encounter rates relative to those for randomly moving cells. This paper describes computer simulations of discrete cell-target encounter events in two dimensions, for the two cases considered by the continuum models: where only a single cell and a single target are present, and where many cells and targets are present. The results of these simulations verify our previous model predictions that a small amount of chemotactic bias dramatically decreases the encounter time, while further increases in the amount of bias have a much smaller effect. Chemotactic ability is shown to be an important determinant of the kinetics of target clearance, and its effects depend on the initial cell-target ratio and the initial distributions of cells and targets. To the best of our knowledge, this work provides the first computer simulations of particle-target encounter in which there is biased motion of particles toward their targets, and is therefore of general interest beyond specific application to immune cell function.