A coupled bulk-surface model for cell polarisation.

Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behavior of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions.

Regimes of wave type patterning driven by refractory actin feedback: transition from static polarization to dynamic wave behaviour.

Patterns of waves, patches, and peaks of actin are observed experimentally in many living cells. Models of this phenomenon have been based on the interplay between filamentous actin (F-actin) and its nucleation promoting factors (NPFs) that activate the Arp2/3 complex. Here we present an alternative biologically-motivated model for F-actin-NPF interaction based on properties of GTPases acting as NPFs. GTPases (such as Cdc42, Rac) are known to promote actin nucleation, and to have active membrane-bound and inactive cytosolic forms. The model is a natural extension of a previous mathematical mini-model of small GTPases that generates static cell polarization. Like other modellers, we assume that F-actin negative feedback shapes the observed patterns by suppressing the trailing edge of NPF-generated wave-fronts, hence localizing the activity spatially. We find that our NPF-actin model generates a rich set of behaviours, spanning a transition from static polarization to single pulses, reflecting waves, wave trains, and oscillations localized at the cell edge. The model is developed with simplicity in mind to investigate the interaction between nucleation promoting factor kinetics and negative feedback. It explains distinct types of pattern initiation mechanisms, and identifies parameter regimes corresponding to distinct behaviours. We show that weak actin feedback yields static patterning, moderate feedback yields dynamical behaviour such as travelling waves, and strong feedback can lead to wave trains or total suppression of patterning. We use a recently introduced nonlinear bifurcation analysis to explore the parameter space of this model and predict its behaviour with simulations validating those results.

Reverse engineering an amyloid aggregation pathway with dimensional analysis and scaling.

Human islet amyloid polypeptide (hIAPP) is a cytotoxic protein that aggregates into oligomers and fibrils that kill pancreatic ß-cells. Here we analyze hIAPP aggregation in vitro, measured via thioflavin-T fluorescence. We use mass-action kinetics and scaling analysis to reconstruct the aggregation pathway, and find that the initiation step requires four hIAPP monomers. After this step, monomers join the nucleus in pairs, until the first stable nucleus (of size approximately 20 monomers) is formed. This nucleus then elongates by successive addition of single monomers. We find that the best-fit of our data is achieved when we include a secondary fibril-dependent nucleation pathway in the reaction scheme. We predict how interventions that change rates of fibril elongation or nucleation rates affect the accumulation of potentially cytotoxic oligomer species. Our results demonstrate the power of scaling analysis in reverse engineering biochemical aggregation pathways.

Non-local models for the formation of hepatocyte-stellate cell aggregates.

Liver cell aggregates may be grown in vitro by co-culturing hepatocytes with stellate cells. This method results in more rapid aggregation than hepatocyte-only culture, and appears to enhance cell viability and the expression of markers of liver-specific functions. We consider the early stages of aggregate formation, and develop a new mathematical model to investigate two alternative hypotheses (based on evidence in the experimental literature) for the role of stellate cells in promoting aggregate formation. Under Hypothesis 1, each population produces a chemical signal which affects the other, and enhanced aggregation is due to chemotaxis. Hypothesis 2 asserts that the interaction between the two cell types is by direct physical contact: the stellates extend long cellular processes which pull the hepatocytes into the aggregates. Under both hypotheses, hepatocytes are attracted to a chemical they themselves produce, and the cells can experience repulsive forces due to overcrowding. We formulate non-local (integro-partial differential) equations to describe the densities of cells, which are coupled to reaction-diffusion equations for the chemical concentrations. The behaviour of the model under each hypothesis is studied using a combination of linear stability analysis and numerical simulations. Our results show how the initial rate of aggregation depends upon the cell seeding ratio, and how the distribution of cells within aggregates depends on the relative strengths of attraction and repulsion between the cell types. Guided by our results, we suggest experiments which could be performed to distinguish between the two hypotheses.

Mutual interactions, potentials, and individual distance in a social aggregation.

We formulate a Lagrangian (individual-based) model to investigate the spacing of individuals in a social aggregate (e.g., swarm, flock, school, or herd). Mutual interactions of swarm members have been expressed as the gradient of a potential function in previous theoretical studies. In this specific case, one can construct a Lyapunov function, whose minima correspond to stable stationary states of the system. The range of repulsion (r) and attraction (a) must satisfy r < a for cohesive groups (i.e., short range repulsion and long range attraction). We show quantitatively how repulsion must dominate attraction ( Rr(d+1) > cAa(d+1) where R, A are magnitudes, c is a constant of order 1, and d is the space dimension) to avoid collapse of the group to a tight cluster. We also verify the existence of a well-spaced locally stable state, having a characteristic individual distance. When the number of individuals in a group increases, a dichotomy occurs between swarms in which individual distance is preserved versus those in which the physical size of the group is maintained at the expense of greater crowding.

Modelling perspectives on aging: can mathematics help us stay young?

We survey several types of mathematical models that keep track of age distributions in a population, or follow some aspects of aging, such as loss of replicative potential of stem cells. The properties of a class of linear models of this type are discussed and compared. We illustrate the applicability of such models with a simple example based on hypothetical stem cell dynamics developed to address age-related telomere loss in the human granulocyte pool. We then describe the contrasting behaviour of nonlinear systems. Examples are drawn from the class of "dynamical diseases" to illustrate some of the aspects of nonlinear systems. Applications of these, and other models to the problems of aging and replicative aging are discussed.

A model for actin-filament length distribution in a lamellipod.

A mathematical model is derived to describe the distributions of lengths of cytoskeletal actin filaments, along a 1 D transect of the lamellipod (or along the axis of a filopod) in an animal cell. We use the facts that actin filament barbed ends are aligned towards the cell membrane and that these ends grow rapidly in the presence of actin monomer as long as they are uncapped. Once a barbed end is capped, its filament tends to be degraded by fragmentation or depolymerization. Both the growth (by polymerization) and the fragmentation by actin-cutting agents are depicted in the model, which takes into account the dependence of cutting probability on the position along a filament. It is assumed that barbed ends are capped rapidly away from the cell membrane. The model consists of a system of discrete-integro-PDE's that describe the densities of barbed filament ends as a function of spatial position and length of their actin filament "tails". The population of capped barbed ends and their trailing filaments is similarly represented. This formulation allows us to investigate hypotheses about the fragmentation and polymerization of filaments in a caricature of the lamellipod and compare theoretical and observed actin density profiles.

Models for spatial polymerization dynamics of rod-like polymers.

We investigate the polymerization kinetics of rod-like polymer filaments interacting with a distribution of monomer in one spatial dimension (e.g. along a narrow tube). We consider a variety of possible cases, including competition by the filament tips for the available monomer, and behaviour analogous to "treadmilling" in which the polymer adds subunits to one end and loses them at the other end so as to maintain a constant length. Applications to biological polymers such as actin filaments and microtubules are discussed.

Complexity, pattern, and evolutionary trade-offs in animal aggregation.

One of the most striking patterns in biology is the formation of animal aggregations. Classically, aggregation has been viewed as an evolutionarily advantageous state, in which members derive the benefits of protection, mate choice, and centralized information, balanced by the costs of limiting resources. Consisting of individual members, aggregations nevertheless function as an integrated whole, displaying a complex set of behaviors not possible at the level of the individual organism. Complexity theory indicates that large populations of units can self-organize into aggregations that generate pattern, store information, and engage in collective decision-making. This begs the question, are all emergent properties of animal aggregations functional or are some simply pattern? Solutions to this dilemma will necessitate a closer marriage of theoretical and modeling studies linked to empirical work addressing the choices, and trajectories, of individuals constrained by membership in the group.

A mathematical approach to cytoskeletal assembly.

The cytoskeleton is a fundamental and important part of cell's structure, and is known to play a large role in controlling the shape, function, division, and motility of the cell. In recent years, the traditional biological and biophysical experimental work on the cytoskeleton has been enhanced by a variety of theoretical, physical and mathematical approaches. Many of these approaches have been developed in the traditional frameworks of physicochemical and statistical mechanics or equilibrium thermodynamic principles. An alternative is to use kinetic modelling and couch the analysis in terms of differential equations which describe mean field properties of cytoskeletal networks or assemblies. This paper describes two such recent efforts. In the first part of the paper, a summary of work on the kinetics of polymerization, fragmentation, and dynamics of actin and polymers in the presence of gelsolin (which nulceates, fragments, and caps the filaments) is given. In the second part, some of the kinetic models aimed at elucidating the spatio-angular density distribution of actin filaments interacting via crosslinks is described. This model given insight into effects that govern the formation of clusters and bundles of actin filaments, and their spatial distribution.

Models for the length distributions of actin filaments: I. Simple polymerization and fragmentation.

We studied mathematical models for the length distributions of actin filaments under the effects of polymerization/depolymerization, and fragmentation. In this paper, we emphasize the effects of these two processes acting alone. In this case, simple discrete and continuous models can be derived and solved explicitly (in several special cases), making the problem interesting from a modeling and pedagogical point of view. In a companion paper (Ermentrout and Edelstein-Keshet, 1998, Bull. Math. Biol. 60, 477-503) we investigate what happens when the processes act together, with particular attention to fragmentation by gelsolin, and with a greater level of biological detail.

Models for the length distributions of actin filaments: II. Polymerization and fragmentation by gelsolin acting together.

In a previous paper, we studied elementary models for polymerization, depolymerization, and fragmentation of actin filaments (Edelstein-Keshet and Ermentrout, 1988, Bull. Math. Biol. 60, 449-475). When these processes act together, more complicated dynamics occur. We concentrate on a particular case study, using the actin-fragmenting protein gelsolin. A set of biological parameter values (drawn from the experimental literature) is used in computer simulations of the kinetics of gelsolin-mediated actin filament fragmentation.

Testing a model for the dynamics of actin structures with biological parameter values.

A simple mathematical model for the dynamics of network-bundle transitions in actin filaments has been previously proposed and some of its mathematical properties have been described. Other models in this class have since been considered and investigated mathematically. In this paper, we have made the first steps in connecting parameters in the model with biologically measurable quantities such as published values of rate constants for filament-crosslinker association. We describe how this connection was made, and give some preliminary numerical simulation results for the behavior of the model under biologically realistic parameter regimes. A key result is that filament length influences the bundle-network transition.

Selecting a common direction. II. Peak-like solutions representing total alignment of cell clusters.

The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of delta-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

Modelling the dynamics of F-actin in the cell.

The regulation of the interactions between the actin binding proteins and the actin filaments are known to affect the cytoskeletal structure of F-actin. We develop a model depicting the formation of actin cytoskeleton, bundles and orthogonal networks, via activation or inactivation of different types of actin binding proteins. It is found that as the actin filament density increases in the cell, a spontaneous tendency to organize into bundles or networks occurs depending on the active actin binding protein concentration. Also, a minute change in the relative binding affinity of the actin binding proteins in the cell may lead to a major change in the actin cytoskeleton. Both the linear stability analysis and the numerical results indicate that the structures formed are highly sensitive to changes in the parameters, in particular to changes in the parameter phi, denoting the relative binding affinity and concentration of the actin binding proteins.

Cellular automata approaches to biological modeling.

We review a number of biologically motivated cellular automata (CA) that arise in models of excitable and oscillatory media, in developmental biology, in neurobiology, and in population biology. We suggest technical and theoretical arguments that permit greater speed and enhanced realism, and apply these to several classical examples of pattern formation. We also describe CA that arise in models for fibroblast aggregation, branching networks, trail following, and neuronal maps.

Contact response of cells can mediate morphogenetic pattern formation.

Theories of morphogenetic pattern formation have included Turing's chemical prepatterns, mechanochemical interactions, cell sorting, and other mechanisms involving guided motion or signalling of cells. Many of these theories presuppose long-range cellular communication or other controls such as chemical concentration fields. However, the possibility that direct interactions between cells can lead to order and structure has not been seriously investigated in mathematical models. In this paper we consider this possibility, with emphasis on cells that reorient and align with each other when they come into contact. We show that such contact responses can account for the formation of multicellular patterns called parallel arrays. These patterns typically occur in tissue cultures of fibroblasts, and consist of clusters of cells sharing a common axis of orientation. Using predictions of a mathematical model and computer simulations of cell motion and interactions we show that contact responses alone, in the absence of other global controls, can promote the formation of these patterns. We suggest other situations in which patterns may result from direct cellular communication. Previous theories of morphogenesis are briefly reviewed and compared with this proposed mechanism.

Models for contact-mediated pattern formation: cells that form parallel arrays.

Kinetic continuum models are derived for cells that crawl over a 2D substrate, undergo random reorientation, and turn in response to contact with a neighbor. The integro-partial differential equations account for changes in the distribution of orientations in the population. It is found that behavior depends on parameters such as total mass, random motility, adherence, and sloughing rates, as well as on broad aspects of the contact response. Linear stability analysis, and numerical, and cellular automata simulations reveal that as parameters are varied, a bifurcation leads to loss of stability of a uniform (isotropic) steady state, in favor of an (anisotropic) patterned state in which cells are aligned in parallel arrays.

Characterization of tumour temperature distributions in hyperthermia based on assumed mathematical forms.

Assessing the efficacy of hyperthermia treatments involves three distinct problems: (1) adequately sampling the spatial temperature distribution in a region; (2) defining (a set of) 'descriptors', numerical values which could be used in comparing distinct treatments; (3) testing whether the predictions of prognosis are statistically significant. This paper addresses the first two problems. We use simple assumptions about the tumour geometry and heating pattern to obtain convenient mathematical representations of a temperature distribution, which are then used in defining scalar descriptors such as weighted average temperature TV, and the fraction of tumour volume heated above a given temperature VT/V. Two extreme cases are discussed. In the first, tumour geometry plays the dominant role, and in the second the specific absorption rate (SAR) distribution is assumed to have the greatest influence on the temperature distribution.

Clinical application of thermal isoeffect dose.

Clinically, there is strong rationale for developing a method which will provide a scientific basis for comparing the efficacy of one hyperthermia treatment with another. In order to accomplish this goal, methods must first be developed which will allow the clinician to know the three-dimensional temperature distribution in heated tissue. In this paper, examples of how this goal can be achieved are presented. Techniques for compensating for various modifiers of hyperthermia effectiveness are proposed. The limitations and advantages of these approaches are described and directions for future research are discussed.