A G protein-based model of adaptation in Dictyostelium discoideum

Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer

This special issue on "Multiscale phenomena and patterns in biological systems" is an homage to the seminal contributions of Hans Othmer. He has remained at the forefront of multiscale modelling and pattern formation in biology for over half a century, developing models for molecular signalling networks, the mechanics of cellular movements, the interactions between multiple cells and their contributions to tissue patterning and dynamics. The contributions in this special issue follow Hans' legacy in using advanced mathematics to understand complex biological processes.

Modeling the dynamic behavior of biochemical regulatory networks

Strategies for modeling the complex dynamical behavior of gene/protein regulatory networks have evolved over the last 50 years as both the knowledge of these molecular control systems and the power of computing resources have increased. Here, we review a number of common modeling approaches, including Boolean (logical) models, systems of piecewise-linear or fully non-linear ordinary differential equations, and stochastic models (including hybrid deterministic/stochastic approaches). We discuss the pro's and con's of each approach, to help novice modelers choose a modeling strategy suitable to their problem, based on the type and bounty of available experimental information. We illustrate different modeling strategies in terms of some abstract network motifs, and in the specific context of cell cycle regulation.

Fold-change detection and scale invariance of cell-cell signaling in social amoeba

Cell-cell signaling is subject to variability in the extracellular volume, cell number, and dilution that potentially increase uncertainty in the absolute concentrations of the extracellular signaling molecules. To direct cell aggregation, the social amoebae Dictyostelium discoideum collectively give rise to oscillations and waves of cyclic adenosine 3',5'-monophosphate (cAMP) under a wide range of cell density. To date, the systems-level mechanism underlying the robustness is unclear. By using quantitative live-cell imaging, here we show that the magnitude of the cAMP relay response of individual cells is determined by fold change in the extracellular cAMP concentrations. The range of cell density and exogenous cAMP concentrations that support oscillations at the population level agrees well with conditions that support a large fold-change-dependent response at the single-cell level. Mathematical analysis suggests that invariance of the oscillations to density transformation is a natural outcome of combining secrete-and-sense systems with a fold-change detection mechanism.

A Model for Direction Sensing in Dictyostelium discoideum: Ras Activity and Symmetry Breaking Driven by a Gβγ-Mediated, Gα2-Ric8 -- Dependent Signal Transduction Network

Chemotaxis is a dynamic cellular process, comprised of direction sensing, polarization and locomotion, that leads to the directed movement of eukaryotic cells along extracellular gradients. As a primary step in the response of an individual cell to a spatial stimulus, direction sensing has attracted numerous theoretical treatments aimed at explaining experimental observations in a variety of cell types. Here we propose a new model of direction sensing based on experiments using Dictyostelium discoideum (Dicty). The model is built around a reaction-diffusion-translocation system that involves three main component processes: a signal detection step based on G-protein-coupled receptors (GPCR) for cyclic AMP (cAMP), a transduction step based on a heterotrimetic G protein Gα2βγ, and an activation step of a monomeric G-protein Ras. The model can predict the experimentally-observed response of cells treated with latrunculin A, which removes feedback from downstream processes, under a variety of stimulus protocols. We show that [Formula: see text] cycling modulated by Ric8, a nonreceptor guanine exchange factor for [Formula: see text] in Dicty, drives multiple phases of Ras activation and leads to direction sensing and signal amplification in cAMP gradients. The model predicts that both [Formula: see text] and Gβγ are essential for direction sensing, in that membrane-localized [Formula: see text], the activated GTP-bearing form of [Formula: see text], leads to asymmetrical recruitment of RasGEF and Ric8, while globally-diffusing Gβγ mediates their activation. We show that the predicted response at the level of Ras activation encodes sufficient 'memory' to eliminate the 'back-of-the wave' problem, and the effects of diffusion and cell shape on direction sensing are also investigated. In contrast with existing LEGI models of chemotaxis, the results do not require a disparity between the diffusion coefficients of the Ras activator GEF and the Ras inhibitor GAP. Since the signal pathways we study are highly conserved between Dicty and mammalian leukocytes, the model can serve as a generic one for direction sensing.

Excitable waves and direction-sensing in Dictyostelium discoideum: steps towards a chemotaxis model

In recent years, there have been significant advances in our understanding of the mechanisms underlying chemically directed motility by eukaryotic cells such as Dictyostelium. In particular, the local excitation and global inhibition (LEGI) model has proven capable of providing a framework for quantitatively explaining many experiments that present Dictyostelium cells with tailored chemical stimuli and monitor their subsequent polarization. In their natural setting, cells generate their own directional signals via the detection and secretion of cyclic adenosine monophosphate (cAMP). Here, we couple the LEGI approach to an excitable medium model of the cAMP wave-field that is propagated by the cells and investigate the possibility for this class of models to enable accurate chemotaxis to the cAMP waveforms expected in vivo. Our results indicate that the ultra-sensitive version of the model does an excellent job in providing natural wave rectification, thereby providing a compelling solution to the 'back-of-the-wave paradox' during cellular aggregation.

Evolutionarily conserved coupling of adaptive and excitable networks mediates eukaryotic chemotaxis

Numerous models explain how cells sense and migrate towards shallow chemoattractant gradients. Studies show that an excitable signal transduction network acts as a pacemaker that controls the cytoskeleton to drive motility. Here we show that this network is required to link stimuli to actin polymerization and chemotactic motility and we distinguish the various models of chemotaxis. First, signalling activity is suppressed towards the low side in a gradient or following removal of uniform chemoattractant. Second, signalling activities display a rapid shut off and a slower adaptation during which responsiveness to subsequent test stimuli decline. Simulations of various models indicate that these properties require coupled adaptive and excitable networks. Adaptation involves a G-protein-independent inhibitor, as stimulation of cells lacking G-protein function suppresses basal activities. The salient features of the coupled networks were observed for different chemoattractants in Dictyostelium and in human neutrophils, suggesting an evolutionarily conserved mechanism for eukaryotic chemotaxis.

Cell-to-cell coordination for the spontaneous cAMP oscillation in Dictyostelium

We propose a new cellular dynamics scheme for the spontaneous cAMP oscillations in Dictyostelium discoideum. Our scheme seamlessly integrates both receptor dynamics and G-protein dynamics into our previously developed cellular dynamics scheme. Extensive computer simulation studies based on our new cellular dynamics scheme were conducted in mutant cells to evaluate the molecular network. The validity of our proposed molecular network as well as the controversial PKA-dependent negative feedback mechanism was supported by our simulation studies. Spontaneous cAMP oscillations were not observed in a single mutant cell. However, multicellular states of various mutant cells consistently initiated spontaneous cAMP oscillations. Therefore, cell-to-cell coordination via the cAMP receptor is essential for the robust initiation of spontaneous cAMP oscillations.

The Neurospora photoreceptor VIVID exerts negative and positive control on light sensing to achieve adaptation

Light adaptation in Neurospora is mediated by the photoreceptor VIVID, which exerts both a negative and positive effect on light sensing. These apparently paradoxical roles of VIVID are explained by the dynamics of a network motif that utilizes futile cycling.

The fungus Neurospora detects relative changes in light intensity by adapting to the ambient light level and remaining responsive to increases in light intensity.

Both the downregulation of the acute light response and maintained responsiveness are mediated by the photoreceptor VIVID (VVD).

Data-based mathematical modeling shows that this paradoxical function of VVD can be realized by a futile-cycle network motif that turns feedback inhibition into sensory adaptation.

The light response in Neurospora is mediated by the photoreceptor and circadian transcription factor White Collar Complex (WCC). The expression rate of the WCC target genes adapts in daylight and remains refractory to moonlight, despite the extraordinary light sensitivity of the WCC. To explain this photoadaptation, feedback inhibition by the WCC interaction partner VIVID (VVD) has been invoked. Here we show through data-driven mathematical modeling that VVD allows Neurospora to detect relative changes in light intensity. To achieve this behavior, VVD acts as an inhibitor of WCC-driven gene expression and, at the same time, as a positive regulator that maintains the responsiveness of the photosystem. Our data indicate that this paradoxical function is realized by a futile cycle that involves the light-induced sequestration of active WCC by VVD and the replenishment of the activatable WCC pool through the decay of the photoactivated state. Our quantitative study uncovers a novel network motif for achieving sensory adaptation and defines a core input module of the circadian clock in Neurospora.

Modeling and measuring signal relay in noisy directed migration of cell groups

We develop a coarse-grained stochastic model for the influence of signal relay on the collective behavior of migrating Dictyostelium discoideum cells. In the experiment, cells display a range of collective migration patterns, including uncorrelated motion, formation of partially localized streams, and clumping, depending on the type of cell and the strength of the external, linear concentration gradient of the signaling molecule cyclic adenosine monophosphate (cAMP). From our model, we find that the pattern of migration can be quantitatively described by the competition of two processes, the secretion rate of cAMP by the cells and the degradation rate of cAMP in the gradient chamber. Model simulations are compared to experiments for a wide range of strengths of an external linear-gradient signal. With degradation, the model secreting cells form streams and efficiently transverse the gradient, but without degradation, we find that model secreting cells form clumps without streaming. This indicates that the observed effective collective migration in streams requires not only signal relay but also degradation of the signal. In addition, our model allows us to detect and quantify precursors of correlated motion, even when cells do not exhibit obvious streaming.

Collective cell migration is observed in various biological processes including angiogenesis, gastrulation, fruiting body formation, and wound healing. Dictyostelium discoideum, for example, exhibits highly dynamic patterns such as streams and clumps during its early phases of collective motion and has served as a model organism for the study of collective migration. In this study, facilitated by experiments, we develop a conceptual, minimalistic, computational model to analyze the dynamical processes leading to the emergence of collective patterns and the associated dependence on the external injection of a cAMP signal, the intercellular cAMP secretion rate, and the cAMP degradation rate. We demonstrate that degradation is necessary to reproduce the experimentally observed collective migration patterns, and show how our model can be utilized to uncover basic dependences of migration modes on cell characteristics. Our numerical observations elucidate the different possible types of motion and quantify the onset of collective motion. Thus, the model allows us to distinguish noisy motion guided by the external signal from weakly correlated motion.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

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.

Biology by numbers: mathematical modelling in developmental biology

In recent years, mathematical modelling of developmental processes has earned new respect. Not only have mathematical models been used to validate hypotheses made from experimental data, but designing and testing these models has led to testable experimental predictions. There are now impressive cases in which mathematical models have provided fresh insight into biological systems, by suggesting, for example, how connections between local interactions among system components relate to their wider biological effects. By examining three developmental processes and corresponding mathematical models, this Review addresses the potential of mathematical modelling to help understand development.

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 the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons

Cultured gonadotropin-releasing hormone (GnRH) neurons have been shown to express GnRH receptors. GnRH binding to its receptors activates three types of G-proteins at increasing doses. These G-proteins selectively activate or inhibit GnRH secretion by regulating the intracellular levels of Ca2+ and cAMP. Based on these recent observations, we build a model in which GnRH plays the roles of a feedback regulator and a diffusible synchronizing agent. We show that this GnRH-regulated GnRH-release mechanism is sufficient for generating pulsatile GnRH release. The model reproduces the observed effects of some key drugs that disturb the GnRH pulse generator in specific ways. Simulations of 100 heterogeneous neurons revealed that the synchronization mediated by a common pool of diffusible GnRH is robust. The population can generate synchronized pulsatile signals even when all the individual GnRH neurons oscillate at different amplitudes and peak at different times. These results suggest that the positive and negative effects of the autocrine regulation by GnRH on GnRH neurons are sufficient and robust in generating GnRH pulses.

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.

cAMP signaling in Dictyostelium. Complexity of cAMP synthesis, degradation and detection

cAMP plays a pivotal role in control of cell movement, differentiation and response to stress in all phases of the Dictyostelium life cycle. The multitudinous functions of cAMP require precise spatial and temporal control of its production, degradation and detection. Many novel proteins have recently been identified that critically modulate the cAMP signal. We focus in this review on the properties and functions of the three adenylyl cyclases and the three cAMP-phosphodiesterases that are present in Dictyostelium, and the network of proteins that regulate the activity of these enzymes. We also briefly discuss the two modes of detection of cAMP.

Characterization of a cAMP-stimulated cAMP phosphodiesterase in Dictyostelium discoideum

A cyclic nucleotide phosphodiesterase, PdeE, that harbors two cyclic nucleotide binding motifs and a binuclear Zn(2+)-binding domain was characterized in Dictyostelium. In other eukaryotes, the Dictyostelium domain shows greatest homology to the 73-kDa subunit of the pre-mRNA cleavage and polyadenylation specificity factor. The Dictyostelium PdeE gene is expressed at its highest levels during aggregation, and its disruption causes the loss of a cAMP-phosphodiesterase activity. The pdeE null mutants show a normal cAMP-induced cGMP response and a 1.5-fold increase of cAMP-induced cAMP relay. Overexpression of a PdeE-yellow fluorescent protein (YFP) fusion construct causes inhibition of aggregation and loss of the cAMP relay response, but the cells can aggregate in synergy with wild-type cells. The PdeE-YFP fusion protein was partially purified by immunoprecipitation and biochemically characterized. PdeE and its Dictyostelium ortholog, PdeD, are both maximally active at pH 7.0. Both enzymes require bivalent cations for activity. The common cofactors Zn(2+) and Mg(2+) activated PdeE and PdeD maximally at 10 mm, whereas Mn(2+) activated the enzymes to 4-fold higher levels, with half-maximal activation between 10 and 100 microm. PdeE is an allosteric enzyme, which is approximately 4-fold activated by cAMP, with half-maximal activation occurring at about 10 microm and an apparent K(m) of approximately 1 mm. cGMP is degraded at a 6-fold lower rate than cAMP. Neither cGMP nor 8-Br-cAMP are efficient activators of PdeE activity.

Modeling the cell's guidance system

Cell locomotion can be directed by external gradients of diffusible substances leading to chemotaxis. Recently, the mechanisms of gradient sensing, the cell guidance system, came under scrutiny both in experimental analysis and computational modeling. Here, we review several recent computational models of gradient sensing in eukaryotic cells, demonstrating why some of them predict little sensitivity to changes in the gradient and response "locking," whereas others predict high gradient sensitivity at the expense of signal gain. We also propose a way to view chemotaxis regulation as a highly coupled combination of semi-independent control modules, leading to simplifying modeling of this complex cellular behavior.

Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils

Eukaryotic cells can detect shallow gradients of chemoattractants with exquisite precision and respond quickly to changes in the gradient steepness and direction. Here, we describe a set of models explaining both adaptation to uniform increases in chemoattractant and persistent signaling in response to gradients. We demonstrate that one of these models can be mapped directly onto the biochemical signal-transduction pathways underlying gradient sensing in amoebae and neutrophils. According to this scheme, a locally acting activator (PI3-kinase) and a globally acting inactivator (PTEN or a similar phosphatase) are coordinately controlled by the G-protein activation. This signaling system adapts perfectly to spatially homogeneous changes in the chemoattractant. In chemoattractant gradients, an imbalance between the action of the activator and the inactivator results in a spatially oriented persistent signaling, amplified by a substrate supply-based positive feedback acting through small G-proteins. The amplification is activated only in a continuous presence of the external signal gradient, thus providing the mechanism for sensitivity to gradient alterations. Finally, based on this mapping, we make predictions concerning the dynamics of signaling. We propose that the underlying principles of perfect adaptation and substrate supply-based positive feedback will be found in the sensory systems of other chemotactic cell types.

Modeling the model organism Dictyostelium discoideum

The cellular slime mold Dictyostelium discoideum is a fascinating organism, not only for biologists, but also for physicists. Since the Belousov-Zhabotinskii reaction pattern, a well-known non-linear phenomenon in chemistry, was observed during aggregation of Dictyostelium amoebae, Dictyostelium has been one of the major subjects of non-linear dynamics studies. Macroscopic theory, such as continuous cell density approximation, has been a common approach to studying pattern formation since the pioneering work of Turing. Recently, promising microscopic approaches, such as the cellular dynamics method, have emerged. They have shown that Dictyostelium is useful as a model system in biology, The synchronization mechanism of oscillatory production of cyclic adenosine 3',5'-monophosphate in Dictyostelium is discussed in detail to show how it is a universal feature that can explain synchronization in other organisms.

Migration and thermotaxis of dictyostelium discoideum slugs, a model study

Dictyostelium discoideum slugs show a pronounced thermotaxis. We have modelled the motion of the D. discoideum slug in the absence and in the presence of a thermal gradient. Our model is an extension of the hybrid cellular automata/partial differential equation model, as formulated by Savill and Hogeweg [J. theor. Biol., (1997) 184, 229-235]. The modelled slugs maintain their shape and crawl, with a velocity depending on slug size, as found in experiments. Moreover, they show thermotactic behaviour: independent of the initial orientation, after some transient process, the slugs start moving along the temperature gradient. The slug behaviour in our model is due to the collective behaviour of the amoebae. Individual amoebae can neither respond to a shallow temperature gradient, nor show differentiation in motion velocity. The behaviour is achieved by a modification of the cyclic AMP waves: differences in temperature alter the excitability of the cell, and thereby the shape of the cyclic AMP wave. Chemotaxis towards cyclic AMP causes the slug to turn. We show that the mechanism still functions at very low signal-to-noise ratios. Copyright 1999 Academic Press.

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.

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.

Modeling oscillations and waves of cAMP in Dictyostelium discoideum cells

We examine the theoretical aspects of temporal and spatiotemporal organization in the cAMP signaling system of Dictyostelium discoideum amoebae which aggregate in a wavelike manner after starvation, in response to pulses of cAMP emitted with a periodicity of several minutes by cells behaving as aggregation centers. We first extend the model based on receptor desensitization, previously proposed by Martiel and Goldbeter, by incorporating the role of G proteins in signal transduction. The extended model accounts for observations on the response of the signaling system to successive step increases in extracellular cAMP. In the presence of the positive feedback loop in cAMP synthesis, this model generates sustained oscillations in cAMP and in the fraction of active cAMP receptor, similar to those obtained in the simpler model where the role of the G proteins is not taken into account explicitly. We use the latter model to address the formation of concentric and spiral waves of cAMP in the course of D. discoideum aggregation. Previous analyses of the model showed that a progressive increase in the activity of adenylate cyclase and phosphodiesterase can account for the transitions no relay-relay-oscillations-relay observed in the experiments. We show that the degree of cellular synchronization on such a developmental path in parameter space markedly affects the nature of the spatial patterns generated by the model. These patterns range from concentric waves to a small number of large spirals, and finally to a large number of smaller spirals, as the degree of developmental desynchronization between cells increases.

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.

Positive genetic feedback governs cAMP spiral wave formation in Dictyostelium

The aggregation stage of the life cycle of Dictyostelium discoideum is governed by the chemotactic response of individual amoebae to excitable waves of cAMP. We modeled this process through a recently introduced hybrid automata-continuum scheme and used computer simulation to unravel the role of specific components of this complex developmental process. Our results indicated an essential role for positive feedback between the cAMP signaling and the expression of the genes encoding the signal transduction and response machinery.

Dual role of cAMP during Dictyostelium development

cAMP plays an essential role during Dictyostelium development both outside and inside the cell. Membrane-bound receptors and adenylyl cyclase are responsible for sensing and producing extracellular cAMP, whereas a phosphodiesterase is responsible for maintaining a low basal level. The molecular events underlying this type of hormone like signalling, which are now beginning to be deciphered, will be presented, in the light of cAMP analogue studies. The importance of intracellular cAMP for cell differentiation has been demonstrated by the central role of the cAMP dependent protein kinase. Mutants as well as strains obtained by reverse genetics will be reviewed which lead to our current understanding of the role of intracelluar cAMP in the differentiation of both stalk and spore cells.

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

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

Dictyostelium discoideum: cellular self-organization in an excitable biological medium

The dynamics which govern the establishment of pattern and form in multicellular organisms remain a key problem of developmental biology. We study this question in the case of morphogenesis during aggregation of the slime mould Dictyostelium discoideum. Here detailed experimental information allows the formulation of a mechanistic model in which the central element is the coupling of the previously much-studied intracellular cyclic AMP signalling with the chemotactic cell response in cyclic AMP gradients. Numerical simulations of the model show quantitatively how signal relay, chemotactic movement and adaptation orchestrate the collective modes of cell signalling and migration in the aggregating cell layer. The interaction of chemotaxis with the cyclic AMP excitation waves causes the initially homogeneous cell layer to become unstable towards the formation of a branching cell stream pattern with close cell-cell contacts as observed in situ. The evolving cell morphology in turn leads to a pattern of non-homogeneous excitability of the medium and thus feeds back into the cAMP dynamics. This feedback can explain the decrease in signalling period and propagation speed with time, as well as observations on the structure of the spiral wave core in this self-organized excitable medium.

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

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