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

Mathematical modeling of chemotaxis guided amoeboid cell swimming

Cells and microorganisms adopt various strategies to migrate in response to different environmental stimuli. To date, many modeling research has focused on the crawling-basedDictyostelium discoideum(Dd) cells migration induced by chemotaxis, yet recent experimental results reveal that even without adhesion or contact to a substrate, Dd cells can still swim to follow chemoattractant signals. In this paper, we develop a modeling framework to investigate the chemotaxis induced amoeboid cell swimming dynamics. A minimal swimming system consists of one deformable Dd amoeboid cell and a dilute suspension of bacteria, and the bacteria produce chemoattractant signals that attract the Dd cell. We use themathematical amoeba modelto generate Dd cell deformation and solve the resulting low Reynolds number flows, and use a moving mesh based finite volume method to solve the reaction-diffusion-convection equation. Using the computational model, we show that chemotaxis guides a swimming Dd cell to follow and catch bacteria, while on the other hand, bacterial rheotaxis may help the bacteria to escape from the predator Dd cell.

Insight from the maximal activation of the signal transduction excitable network in Dictyostelium discoideum

Cell migration requires the coordination of an excitable signal transduction network involving Ras and PI3K pathways with cytoskeletal activity. We show that expressing activated Ras GTPase-family proteins in cells lacking PTEN or other mutations which increase cellular protrusiveness transforms cells into a persistently activated state. Leading- and trailing-edge markers were found exclusively at the cell perimeter and the cytosol, respectively, of the dramatically flattened cells. In addition, the lifetimes of dynamic actin puncta were increased where they overlapped with actin waves, suggesting a mechanism for the coupling between these two networks. All of these phenotypes could be reversed by inhibiting signal transduction. Strikingly, maintaining cells in this state of constant activation led to a form of cell death by catastrophic fragmentation. These findings provide insight into the feedback loops that control excitability of the signal transduction network, which drives migration.

Mathematical modelling in developmental biology

In recent decades, molecular and cellular biology has benefited from numerous fascinating developments in experimental technique, generating an overwhelming amount of data on various biological objects and processes. This, in turn, has led biologists to look for appropriate tools to facilitate systematic analysis of data. Thus, the need for mathematical techniques, which can be used to aid the classification and understanding of this ever-growing body of experimental data, is more profound now than ever before. Mathematical modelling is becoming increasingly integrated into biological studies in general and into developmental biology particularly. This review outlines some achievements of mathematics as applied to developmental biology and demonstrates the mathematical formulation of basic principles driving morphogenesis. We begin by describing a mathematical formalism used to analyse the formation and scaling of morphogen gradients. Then we address a problem of interplay between the dynamics of morphogen gradients and movement of cells, referring to mathematical models of gastrulation in the chick embryo. In the last section, we give an overview of various mathematical models used in the study of the developmental cycle of Dictyostelium discoideum, which is probably the best example of successful mathematical modelling in developmental biology.

Understanding streaming in Dictyostelium discoideum: theory versus experiments

Recent experimental work involving Dictyostelium discoideum seems to contradict several theoretical models. Experiments suggest that localization of the release of the chemoattractant cyclic adenosine monophosphate to the uropod of the cell is important for stream formation during aggregation. Yet several mathematical models are able to reproduce streaming as the cells aggregate without taking into account localization of the chemoattractant. A careful analysis of the experiments and the theory suggests the two major features of the system which are important to stream formation are random cell motion and chemotaxis to regions of higher cell density. Random cell motion acts to reduce streaming, whereas chemotaxis to regions of higher cell density reinforces streaming. With this understanding, the experimental results can be explained in a manner consistent with the theoretical results. In all the experiments, alterations in the two main factors of random motion and chemotaxis to regions of higher cell density, not the localization of the release of the chemoattractant, can explain the results as they relate to streaming. Additionally, a comparison of results from a mathematical model that simulates cells which localize the chemoattractant and cells which do not shows little difference in the streaming patterns.

A cAMP signaling model explains the benefit of maintaining two forms of phosphodiesterase in Dictyostelium

Starving Dictyostelium cells respond chemotactically to cell-generated waves of cyclic adenosine -3',5'- monophosphate (cAMP) that guide cell aggregation toward a signaling center. In this process, a large number of cells are recruited, resulting in the formation of aggregation territories that are essential for fruiting body formation. The enzyme PdsA phosphodiesterase (PDE), a crucial component of the signaling system, breaks down the external cAMP and can be either membrane-bound or secreted. The existence of two such forms is unusual in cell biology, and it remains to be determined why they have both been maintained through evolution. Here, using a model of the cAMP signaling system, I show that colonies can successfully organize into aggregates over a wider range of initial cell densities when both forms of PDE are present in an appropriately tuned ratio than when only a single form is present. The model indicates that membrane-bound PDE maintains aggregation-territory integrity in colonies with high initial cell density, whereas the secreted form is important for wave propagation at low cell densities. Thus, the ultimate retention of both forms can increase territory size. These findings have implications for other excitable media, including Ca(2+) propagation in cardiac cells and propagation of electrical excitation in nerve axons, since these systems have similar features of spatial nonuniform "release" and "degradation" of the relevant signals.

Modeling species-specific diacylglycerol dynamics in the RAW 264.7 macrophage

A mathematical model of the G protein signaling pathway in RAW 264.7 macrophages downstream of P2Y(6) receptors activated by the ubiquitous signaling nucleotide uridine 5'-diphosphate is developed. The model, which is based on time-course measurements of inositol trisphosphate, cytosolic calcium, and diacylglycerol, focuses particularly on differential dynamics of multiple chemical species of diacylglycerol. When using the canonical pathway representation, the model predicted that key interactions were missing from the current network structure. Indeed, the model suggested that accurate depiction of experimental observations required an additional branch to the signaling pathway. An intracellular pool of diacylglycerol is immediately phosphorylated upon stimulation of an extracellular receptor for uridine 5'-diphosphate and subsequently used to aid replenishment of phosphatidylinositol. As a result of sensitivity analysis of the model parameters, key predictions can be made regarding which of these parameters are the most sensitive to perturbations and are therefore most responsible for output uncertainty.

The group migration of Dictyostelium cells is regulated by extracellular chemoattractant degradation

Starvation of Dictyostelium induces a developmental program in which cells form an aggregate that eventually differentiates into a multicellular structure. The aggregate formation is mediated by directional migration of individual cells that quickly transition to group migration in which cells align in a head-to-tail manner to form streams. Cyclic AMP acts as a chemoattractant and its production, secretion, and degradation are highly regulated. A key protein is the extracellular phosphodiesterase PdsA. In this study we examine the role and localization of PdsA during chemotaxis and streaming. We find that pdsA(-) cells respond chemotactically to a narrower range of chemoattractant concentrations compared with wild-type (WT) cells. Moreover, unlike WT cells, pdsA(-) cells do not form streams at low cell densities and form unusual thick and transient streams at high cell densities. We find that the intracellular pool of PdsA is localized to the endoplasmic reticulum, which may provide a compartment for storage and secretion of PdsA. Because we find that cAMP synthesis is normal in cells lacking PdsA, we conclude that signal degradation regulates the external cAMP gradient field generation and that the group migration behavior of these cells is compromised even though their signaling machinery is intact.

A 3-D model used to explore how cell adhesion and stiffness affect cell sorting and movement in multicellular systems

A three-dimensional mathematical model is used to determine the effects of adhesion and cell signalling on cell movements during the aggregation and slug stages of Dictyostelium discoideum (Dd) and to visualize cell sorting. The building blocks of the model are individual deformable ellipsoidal cells, where movement depends on internal parameter state (cell size and stiffness) and on external cues from the neighboring cells, extracellular matrix, and chemical signals. Cell movement and deformation are calculated from equations of motion using the total force acting on each cell, ensuring that forces are balanced. The simulations show that the sorting patterns of prestalk and prespore cells, emerging during the slug stage, depend critically on the type of cell adhesion and not just on chemotactic differences between cells. This occurs because cell size and stiffness can prevent the otherwise faster cells from passing the slower cells. The patterns are distinctively different when the prestalk cells are more or less adhesive than the prespore cells. These simulations suggest that sorting is not solely due to differential chemotaxis, and that differences in both adhesion strength and type between different cell types play a very significant role, both in Dictyostelium and other systems.

Taxis equations for amoeboid cells

The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues [17,18]. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentally measured statistics can be obtained from the transport equation formalism. We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.

A model for a G-protein-mediated mechanism for synaptic channel modulation

Neurons communicate with other neurons via specialized structures called synapses, at which the digital voltage signal encoded in an action potential is converted into an analog chemical signal. An action potential that arrives at the presynaptic face triggers release of neurotransmitter from vesicles in a calcium-dependent manner, and the neurotransmitter diffuses across the synaptic cleft and binds to receptors on the post-synaptic face, where it may trigger a postsynaptic action potential. Calcium is a critical component of the release process, and its spatio-temporal dynamics can control the release and can lead to facilitation or augmentation. However, how cells regulate cytoplasmic calcium so that exocytosis can be triggered successfully is still not completely understood. We propose a mechanism, based upon the experimental findings of Barrett and Rittenhouse [C.F. Barrett, A.R. Rittenhouse, Modulation of N-type calcium channel activity by G-proteins and protein kinase C, J. Gen. Physiol. 115 (3) (2000) 277], for the regulation of calcium influx through N-type channels in the presynaptic terminal by PKC and downstream effectors of G-protein activation. This proposed modulatory mechanism consists of a feedback loop involving cytoplasmic calcium, neurotransmitters and G-protein-coupled receptors. We study the dynamics of each component separately and then we address how kinetic properties of the components and the frequency of the stimuli affect the regulatory mechanisms presented here.

Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates

In this paper, we use singular perturbation methods to study the structure of travelling waves for some reaction-diffusion models obtained from the Martiel-Goldbeter and Goldbeter-Segel's models of cAMP signalling in Dictyostelium discoideum. As a consequence, we derive analytic formulae for quantities like wave speed, maximum concentration and other magnitudes in terms of the different biochemical constants that appear in the model.

Mathematically modelling the effects of counting factor in Dictyostelium discoideum

Size regulation is a crucial feature in many biological systems, with misregulation leading to dysplasia or hyperplasia. The recent discovery of counting factor (CF) in Dictyostelium discoideum will lead to a greater understanding of how the system regulates the size of a group of cells. In this paper we mathematically model the known effects of CF using two different models: a cellular automata model and a discrete continuum hybrid model. With the use of these models we are able to understand how modulation of adhesion and motile forces by CF can facilitate stream breakup. In addition, the modelling suggests a new possible mechanism for stream breakup involving the frequency of cell reorientation.

Spatial Turing-type Pattern Formation in a Model of Signal Transduction Involving Membrane-based Receptors Coupled by G Proteins

In this paper, a model of signaling pathways involving G proteins is investigated. The model incorporates reaction-diffusion mechanisms in which various reactants participate inside and on the extra-cellular surface membrane. The messenger molecules may diffuse over the surface of the cell membrane and signal transduction across the cell membrane is mediated by membrane receptor bound proteins which connect the genetically controlled biochemical intra-cellular reactions to the production of the second messenger, leading to desired functional responses. Dynamic and steady-state properties of the model are then investigated through weakly nonlinear stability analysis. Turing-type patterns are shown to form robustly under different delineating conditions on the system parameters. The theoretical predictions are then discussed in the context of some recently reported experimental evidence.

Kinetic models for chemotaxis: hydrodynamic limits and spatio-temporal mechanisms

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.

How cellular movement determines the collective force generated by the Dictyostelium discoideum slug

How the collective motion of cells in a biological tissue originates in the behavior of a collection of individuals, each of which responds to the chemical and mechanical signals it receives from neighbors, is still poorly understood. Here we study this question for a particular system, the slug stage of the cellular slime mold Dictyostelium discoideum (Dd). We investigate how cells in the interior of a migrating slug can effectively transmit stress to the substrate and thereby contribute to the overall motive force. Theoretical analysis suggests necessary conditions on the behavior of individual cells, and computational results shed light on experimental results concerning the total force exerted by a migrating slug. The model predicts that only cells in contact with the substrate contribute to the translational motion of the slug. Since the model is not based specifically on the mechanical properties of Dd cells, the results suggest that this behavior will be found in many developing systems.

Motifs, modules and games in bacteria

Global explorations of regulatory network dynamics, organization and evolution have become tractable thanks to high-throughput sequencing and molecular measurement of bacterial physiology. From these, a nascent conceptual framework is developing, that views the principles of regulation in term of motifs, modules and games. Motifs are small, repeated, and conserved biological units ranging from molecular domains to small reaction networks. They are arranged into functional modules, genetically dissectible cellular functions such as the cell cycle, or different stress responses. The dynamical functioning of modules defines the organism's strategy to survive in a game, pitting cell against cell, and cell against environment. Placing pathway structure and dynamics into an evolutionary context begins to allow discrimination between those physical and molecular features that particularize a species to its surroundings, and those that provide core physiological function. This approach promises to generate a higher level understanding of cellular design, pathway evolution and cellular bioengineering.

Long-range signal transmission in autocrine relays

Intracellular signaling induced by peptide growth factors can stimulate secretion of these molecules into the extracellular medium. In autocrine and paracrine networks, this can establish a positive feedback loop between ligand binding and ligand release. When coupled to intercellular communication by autocrine ligands, this positive feedback can generate constant-speed traveling waves. To demonstrate that, we propose a mechanistic model of autocrine relay systems. The model is relevant to the physiology of epithelial layers and to a number of in vitro experimental formats. Using asymptotic and numerical tools, we find that traveling waves in autocrine relays exist and have a number of unusual properties, such as an optimal ligand binding strength necessary for the maximal speed of propagation. We compare our results to recent observations of autocrine and paracrine systems and discuss the steps toward experimental tests of our predictions.

Computational approaches to cellular rhythms

Oscillations arise in genetic and metabolic networks as a result of various modes of cellular regulation. In view of the large number of variables involved and of the complexity of feedback processes that generate oscillations, mathematical models and numerical simulations are needed to fully grasp the molecular mechanisms and functions of biological rhythms. Models are also necessary to comprehend the transition from simple to complex oscillatory behaviour and to delineate the conditions under which they arise. Examples ranging from calcium oscillations to pulsatile intercellular communication and circadian rhythms illustrate how computational biology contributes to clarify the molecular and dynamical bases of cellular rhythms.

A model for individual and collective cell movement in Dictyostelium discoideum

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

Complexity in biological signaling systems

Biological signaling pathways interact with one another to form complex networks. Complexity arises from the large number of components, many with isoforms that have partially overlapping functions; from the connections among components; and from the spatial relationship between components. The origins of the complex behavior of signaling networks and analytical approaches to deal with the emergent complexity are discussed here.

Modeling chemotactic cell sorting during Dictyostelium discoideum mound formation

Coordinated cell movement is a major mechanism of the multicellular development of most organisms. The multicellular morphogenesis of the slime mould Dictyostelium discoideum, from single cells into a multicellular fruiting body, results from differential chemotactic cell movement. During aggregation cells differentiate into prestalk and prespore cells that will form the stalk and spores in the fruiting body. These cell types arise in a salt and pepper pattern after what the prestalk cells chemotactically sort out to form a tip. The tip functions as an organizer because it directs the further development. It has been difficult to get a satisfactory formal description of the movement behavior of cells in tissues. Based on our experiments, we consider the aggregate as a drop of a viscous fluid and show that this consideration is very well suited to mathematically describe the motion of cells in the tissue. We show that the transformation of a hemispherical mound into an elongated slug can result from the coordinated chemotactic cell movement in response to scroll waves of the chemoattractant cAMP. The model calculations furthermore show that cell sorting can result from differences in chemotactic cell movement and cAMP relay kinetics between the two cell types. During this process, the faster moving and stronger signaling cells collect on the top of the mound to form a tip. The mound then extends into an elongated slug just as observed in experiments. The model is able to describe cell movement patterns in the complex multicellular morphogenesis of Dictyostelium rather well and we expect that this approach may be useful in the modeling of tissue transformations in other systems.

Cyclic AMP oscillations in Dictyostelium discoideum: models and observations

Oscillations in intra- and extracellular cyclic AMP are believed to underlie aggregation and morphogenesis in Dictyostelium discoideum. Upon comparing mathematical models with observations we find that the models are, qualitatively speaking, quite successful. At the same time many features remain unexplained. A strong case can be made for cyclic AMP-independent oscillations whose basis remains to be explored.

Propagating waves control Dictyostelium discoideum morphogenesis

The morphogenesis of Dictyostelium results from the coordinated movement of starving cells to form a multicellular aggregate (mound) which transforms into a motile slug and finally a fruiting body. Cells differentiate in the mound and sort out to form an organised pattern in the slug and fruiting body. During aggregation, cell movement is controlled by propagating waves of the chemo-attractant cAMP. We show that mounds are also organised by propagating waves. Their geometry changes from target or single armed spirals during aggregation to multi-armed spiral waves in the mound. Some mounds develop transiently into rings in which multiple propagating wave fronts can still be seen. We model cell sorting in the mound stage assuming cell type specific differences in cell movement speed and excitability. This sorting feeds back on the wave geometry to generate twisted scroll waves in the slug. Slime mould morphogenesis can be understood in terms of wave propagation directing chemotactic cell movement.

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

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