Signaling networks and cell motility: a computational approach using a phase field description

The processes of protrusion and retraction during cell movement are driven by the turnover and reorganization of the actin cytoskeleton. Within a reaction–diffusion model which combines processes along the cell membrane with processes within the cytoplasm a Turing type instability is used to form the necessary polarity to distinguish between cell front and rear and to initiate the formation of different organizational arrays within the cytoplasm leading to protrusion and retraction. A simplified biochemical network model for the activation of GTPase which accounts for the different dimensionality of the cell membrane and the cytoplasm is used for this purpose and combined with a classical Helfrich type model to account for bending and stiffness effects of the cell membrane. In addition streaming within the cytoplasm and the extracellular matrix is taken into account. Combining these phenomena allows to simulate the dynamics of cells and to reproduce the primary phenomenology of cell motility. The coupled model is formulated within a phase field approach and solved using adaptive finite elements.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

A new necessary condition for Turing instabilities

Reactivity (a.k.a initial growth) is necessary for diffusion driven instability (Turing instability). Using a notion of common Lyapunov function we show that this necessary condition is a special case of a more powerful (i.e. tighter) necessary condition. Specifically, we show that if the linearised reaction matrix and the diffusion matrix share a common Lyapunov function, then Turing instability is not possible. The existence of common Lyapunov functions is readily checked using semi-definite programming. We apply this result to the Gierer-Meinhardt system modelling regenerative properties of Hydra, the Oregonator, to a host-parasite-hyperparasite system with diffusion and to a reaction-diffusion-chemotaxis model for a multi-species host-parasitoid community.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

Linear and Weakly Nonlinear Stability Analyses of Turing Patterns for Diffusive Predator–Prey Systems in Freshwater Marsh Landscapes

We study a diffusive predator–prey model describing the interactions of small fishes and their resource base (small invertebrates) in the fluctuating freshwater marsh landscapes of the Florida Everglades. The spatial model is described by a reaction–diffusion system with Beddington–DeAngelis functional response. Uniform bound, local and global asymptotic stability of the steady state of the PDE model under the no-flux boundary conditions are discussed in details. Sufficient conditions on the Turing (diffusion-driven) instability which induces spatial patterns in the model are derived via linear analysis. Existence of one-dimensional and two-dimensional spatial Turing patterns, including rhombic and hexagonal patterns, are established by weakly nonlinear analyses. These results provide theoretical explanations and numerical simulations of spatial dynamical behaviors of the wetland ecosystems of the Florida Everglades.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Chemical morphogenesis: Turing patterns in an experimental chemical system

Patterns resulting from the sole interplay between reaction and diffusion are probably involved in certain stages of morphogenesis in biological systems, as initially proposed by Alan Turing. Self-organization phenomena of this type can only develop in nonlinear systems (i.e. involving positive and negative feedback loops) maintained far from equilibrium. We present Turing patterns experimentally observed in a chemical system. An oscillating chemical reaction, the CIMA reaction, is operated in an open spatial reactor designed in order to obtain a pure reaction-diffusion system. The two types of Turing patterns observed, hexagonal arrays of spots and parallel stripes, are characterized by an intrinsic wavelength. We identify the origin of the necessary difference of diffusivity between activator and inhibitor. We also describe a pattern growth mechanism by spot splitting that recalls cell division.     

Stochastic amplification of spatial modes in a system with one diffusing species

The problem of pattern formation in a generic two species reaction–diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by working in the generalized setting of a stochastic formulation to the inspected problem, spatially organized patterns can develop, seeded by finite size corrections. General conditions are given for the stochastic patterns to occur. The predictions of the theory are tested for a specific case study.     

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.     

The tRNA-modifying function of MnmE is controlled by post-hydrolysis steps of its GTPase cycle

MnmE is a homodimeric multi-domain GTPase involved in tRNA modification. This protein differs from Ras-like GTPases in its low affinity for guanine nucleotides and mechanism of activation, which occurs by a cis, nucleotide- and potassium-dependent dimerization of its G-domains. Moreover, MnmE requires GTP hydrolysis to be functionally active. However, how GTP hydrolysis drives tRNA modification and how the MnmE GTPase cycle is regulated remains unresolved. Here, the kinetics of the MnmE GTPase cycle was studied under single-turnover conditions using stopped- and quench-flow techniques. We found that the G-domain dissociation is the rate-limiting step of the overall reaction. Mutational analysis and fast kinetics assays revealed that GTP hydrolysis, G-domain dissociation and P_i release can be uncoupled and that G-domain dissociation is directly responsible for the ‘ON’ state of MnmE. Thus, MnmE provides a new paradigm of how the ON/OFF cycling of GTPases may regulate a cellular process. We also demonstrate that the MnmE GTPase cycle is negatively controlled by the reaction products GDP and P_i. This feedback mechanism may prevent inefficacious GTP hydrolysis in vivo. We propose a biological model whereby a conformational change triggered by tRNA binding is required to remove product inhibition and initiate a new GTPase/tRNA-modification cycle.     

Flash kinetic study of the last steps in the photoinduced reaction cycle of bacteriorhodopsin

The reaction cycle of light adapted bacteriorhodopsin (BR) in aqueous purple membrane suspensions was studied by laser flash photolysis at different temperatures (2–49°C) and pH values (3–10). The activation energy for several reaction steps was determined at pH 7.6. The kinetics of O-bacteriorhodopsin (one of the last intermediates in the cycle) were analyzed in some detail and it was found that the simple consecutive reaction scheme M-BR → O-BR → BR may explain the kinetics of O-bacteriorhodopsin as measured at 680 nm. Since the pH change in neutral aqueous suspensions of purple membrane follows a similar kinetics as O-bacteriorhodopsin it is suggested that protons are released during the reaction M-BR → O-BR and taken up again during the reaction O-BR → BR.Another long-lived intermediate, which absorbs to a greater extent than bacteriorhodopsin at 570 nm and less than bacteriorhodopsin at 420 nm, was identified with the strongly fluorescing species, pseudo- or P-bacteriorhodopsin. The decay of P-bacteriorhodopsin in bacteriorhodopsin had an activation energy of only approx. 1.2 kcal/mol, which suggests that the last step of the photocycle is a relaxation around a single bond.At pH 9–10, the simple first-order kinetics of all the intermediates were changed into a kinetics consisting of two first-order decays. This change of kinetics was accompanied by a drastic decrease in the rotational diffusion relaxation time.To explain the results obtained in this work and those of others, a model involving proton uptake and release by the Schiff base nitrogen combined with an isomerization reaction is finally proposed.     

Dissipative pattern formation in ternary non-linear reaction-electrodiffusion systems with concentration-dependent diffusivities

Conditions for the emergence of ionic dissipative structures after Turing instability of a spatially and temporaly homogeneous stationary concentration distribution are derived for electroneutral ternary ionic reaction systems with concentration-dependent diffusivities. The inhomogeneous solution branch first bifurcating while crossing a critical ratio of diffusivities is given as a series of eigenfunctions of the Laplace operator. The results are specified for a model reaction system and polynomial concentration dependence of diffusion. It is shown that the inclusion of concentration dependence controls the amplitudes of the spatial distribution.     

A new method for choosing the computational cell in stochastic reaction–diffusion systems

How to choose the computational compartment or cell size for the stochastic simulation of a reaction–diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reaction–diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.     

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case,with an Application to Skeletal Pattern Formation

We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.     

Turing patterns inside cells

Concentration gradients inside cells are involved in key processes such as cell division and morphogenesis. Here we show that a model of the enzymatic step catalized by phosphofructokinase (PFK), a step which is responsible for the appearance of homogeneous oscillations in the glycolytic pathway, displays Turing patterns with an intrinsic length-scale that is smaller than a typical cell size. All the parameter values are fully consistent with classic experiments on glycolytic oscillations and equal diffusion coefficients are assumed for ATP and ADP. We identify the enzyme concentration and the glycolytic flux as the possible regulators of the pattern. To the best of our knowledge, this is the first closed example of Turing pattern formation in a model of a vital step of the cell metabolism, with a built-in mechanism for changing the diffusion length of the reactants, and with parameter values that are compatible with experiments. Turing patterns inside cells could provide a check-point that combines mechanical and biochemical information to trigger events during the cell division process.     

Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.     

The regulatory GTPase cycle of turkey erythrocyte adenylate cyclase.

It has recently been suggested that adenylate cyclase activity is controlled by a regulatory cycle consisting of two reactions: a hormone induced formation of the active adenylate cyclase-GTP complex, and a subsequent turn-off reaction in which hydrolysis of the bound nucleotide reverts the system to the inactive state. To test this model each of the two reactions was measured separately and their rate constants were used to estimate the steady state adenylate cyclase and GTPase activities. The first order rate constants were kon = 3 min-1 for the activation reaction and koff = 15 min-1 for the turn-off reaction. Substitution of these rate constants in the steady state equation of the regulatory cycle gave values of hormone stimulated adenylate cyclase and GTPase activities similar to those determined by direct measurements. Treatment of the adenylate cyclase with cholera toxin caused a decrease of 96% in the rate constant of the turn-off reaction. In this case too the activities calculated from the steady state equation were in good agreement with those determined directly.     

The effect of electric field on the spatial-time patterns in the reaction-diffusion system

A model of electrodiffusion processes in the vicinity of cell membrane was developed. The model takes into account chemical reactions, Coulomb interactions between charged particles and the effect of external electric field. It was concluded that the applied electric field can change the characteristics of space-time patterns in the system. Dissipative structures slowly move and this is accompanied by a change in the number of structure elements. The characteristic equation includes odd powers of the wavenumber, which can lead to the appearance of soliton-like structures. The dissipative structures can appear not only due to the Turing diffusion instability but due to the disperse instability under electric field the applied.     

Coordinate regulation of G protein signaling via dynamic interactions of receptor and GAP

Signal output from receptor-G-protein-effector modules is a dynamic function of the nucleotide exchange activity of the receptor, the GTPase-accelerating activity of GTPase-activating proteins (GAPs), and their interactions. GAPs may inhibit steady-state signaling but may also accelerate deactivation upon removal of stimulus without significantly inhibiting output when the receptor is active. Further, some effectors (e.g., phospholipase C-beta) are themselves GAPs, and it is unclear how such effectors can be stimulated by G proteins at the same time as they accelerate G protein deactivation. The multiple combinations of protein-protein associations and interacting regulatory effects that allow such complex behaviors in this system do not permit the usual simplifying assumptions of traditional enzyme kinetics and are uniquely subject to systems-level analysis. We developed a kinetic model for G protein signaling that permits analysis of both interactive and independent G protein binding and regulation by receptor and GAP. We evaluated parameters of the model (all forward and reverse rate constants) by global least-squares fitting to a diverse set of steady-state GTPase measurements in an m1 muscarinic receptor-G(q)-phospholipase C-beta1 module in which GTPase activities were varied by approximately 10(4)-fold. We provide multiple tests to validate the fitted parameter set, which is consistent with results from the few previous pre-steady-state kinetic measurements. Results indicate that (1) GAP potentiates the GDP/GTP exchange activity of the receptor, an activity never before reported; (2) exchange activity of the receptor is biased toward replacement of GDP by GTP; (3) receptor and GAP bind G protein with negative cooperativity when G protein is bound to either GTP or GDP, promoting rapid GAP binding and dissociation; (4) GAP indirectly stabilizes the continuous binding of receptor to G protein during steady-state GTPase hydrolysis, thus further enhancing receptor activity; and (5) receptor accelerates GDP/GTP exchange primarily by opening an otherwise closed nucleotide binding site on the G protein but has minimal effect on affinity (K(assoc) = k(assoc)/k(dissoc)) of G protein for nucleotide. Model-based simulation explains how GAP activity can accelerate deactivation >10-fold upon removal of agonist but still allow high signal output while the receptor is active. Analysis of GTPase flux through distinct reaction pathways and consequent accumulation of specific GTPase cycle intermediates indicate that, in the presence of a GAP, the receptor remains bound to G protein throughout the GTPase cycle and that GAP binds primarily during the GTP-bound phase. The analysis explains these behaviors and relates them to the specific regulatory phenomena described above. The work also demonstrates the applicability of appropriately data-constrained system-level analysis to signaling networks of this scale.