Time scale and dimension analysis of a budding yeast cell cycle model

Background  

The progress through the eukaryotic cell division cycle is driven by an underlying molecular regulatory network. Cell cycle progression can be considered as a series of irreversible transitions from one steady state to another in the correct order. Although this view has been put forward some time ago, it has not been quantitatively proven yet. Bifurcation analysis of a model for the budding yeast cell cycle has identified only two different steady states (one for G1 and one for mitosis) using cell mass as a bifurcation parameter. By analyzing the same model, using different methods of dynamical systems theory, we provide evidence for transitions among several different steady states during the budding yeast cell cycle.     

Chemostat-cultivated Escherichia coli at high dilution rate: multiple steady states and drift

The representation of metabolic network reaction kinetics in a scaled, polynomial form can allow for the prediction of multiple steady states. The polynomial formalism is used to study chemostat-cultured Escherichia coli which has been observed to exhibit two multiple steady states under ammonium ion-limited growth conditions: a high cell density-low ammonium ion concentration steady state and a low cell density-high ammonium ion concentration steady state. Additionally, the low-cell-density steady state has been observed to drift to the high-cell-density steady state. Inspection of the steady-state rate expressions for the ammonium ion transport/assimilation network (in polynomial form) suggests that at low ammonium ion concentrations, two steady states are possible. One corresponds to heavy use of the glutamine synthetase-glutamate synthase (GLNS-GS) branch and the second to heavy use of the glutamate dehydrogenase (GDH) branch. Realization of the predicted intracellular steady states is also found to be dependent on the parameters of the transport process. Moreover, the two steady states differ in where their energy intensity lies. To explain the drift, GLNS, which is inducible under low ammonium ion concentrations, is suggested to be a "memory element." A chemostat-based model is developed to illustrate that perturbations in dilution rate can lead to drift between the two steady states provided that the disturbance in dilution rate is sufficiently large and/or long in duration.     

A weakly nonlinear analysis of a model of avascular solid tumour growth

 In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations. Received: 25 October 1998 / Revised version: 20 June 1998     

Dynamics and comparative statics of mutualistic communities

Within the network hypothesis proposed by Jerne, the immune response is interpreted as a collective behaviour of different antibody species, interacting through idiotypic recognition. In order to insure the stability of the network, only a few species would be implied in the response to an antigenic challenge.We study a network made up of small cycles of idiotypic units, each element activating the subsequent one and repressing the preceding one. In the recent theoretical models, the kinetics is described by steep sigmoidal functions with a repression threshold lower than the stimulation one. To enable a systematic qualitative analysis of the dynamics, we replace the continuous kinetics by stepfunctions. The antibodies are thus considered as control elements like genes, enzymes or neurones. In order to account for the different thresholds, we use discrete three-level variables.We develop two methods to study the dynamics: the first one, due to Glass, describes the time-evolution of a cycle by a system of piecewise linear (PL) differential equations and the second method is the boolean formalization, applied extensively by Thomas in the field of genetic regulation.These techniques provide complementary informations about the dynamics of the cycle: the PL method establishes a state transition diagram providing all the potential behaviours independently of the parameter values in the model, whereas the purely logical analysis permits a simulation of the trajectories for precise values of the parameters.The state transition diagram presents several steady states. It suggests to interpret the response to an antigenic challenge as a transition from one steady state to another. The multiplicity of the steady states might be associated with the various modes of immune response depending on the doses of antigen injected and on the previous antigenic history of the system.     

On the number of steady states in a multiple futile cycle

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation–dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n − 1 steady states (in particular, this implies that for n = 2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis–Menten quasi-steady state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard Michaelis–Menten quasi-steady state conditions, there is at most one steady state.      

A qualitative study of an idiotypic cyclic network

Within the network hypothesis proposed by Jerne, the immune response is interpreted as a collective behaviour of different antibody species, interacting through idiotypic recognition. In order to insure the stability of the network, only a few species would be implied in the response to an antigenic challenge.We study a network made up of small cycles of idiotypic units, each element activating the subsequent one and repressing the preceding one. In the recent theoretical models, the kinetics is described by steep sigmoidal functions with a repression threshold lower than the stimulation one. To enable a systematic qualitative analysis of the dynamics, we replace the continuous kinetics by stepfunctions. The antibodies are thus considered as control elements like genes, enzymes or neurones. In order to account for the different thresholds, we use discrete three-level variables.We develop two methods to study the dynamics: the first one, due to Glass, describes the time-evolution of a cycle by a system of piecewise linear (PL) differential equations and the second method is the boolean formalization, applied extensively by Thomas in the field of genetic regulation.These techniques provide complementary informations about the dynamics of the cycle: the PL method establishes a state transition diagram providing all the potential behaviours independently of the parameter values in the model, whereas the purely logical analysis permits a simulation of the trajectories for precise values of the parameters.The state transition diagram presents several steady states. It suggests to interpret the response to an antigenic challenge as a transition from one steady state to another. The multiplicity of the steady states might be associated with the various modes of immune response depending on the doses of antigen injected and on the previous antigenic history of the system.     

Sustained oscillations and threshold phenomena in an operon control circuit

A genetic regulatory model involving a positive feedback (via induction) and a negative feedback (via catabolite repression) is analyzed and applied to the problem of the lac operon regulation in E. coli. Damped and sustained oscillations of the limit cycle type are found along with threshold phenomena corresponding to multiple limit cycles or to multiple steady states, for values of the parameters compatible with experimental data. A comparison With the observations of Knorre and Goodwin is outlined.     

Analysis of Discrete Bioregulatory Networks Using Symbolic Steady States

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.     

System-level feedbacks control cell cycle progression

Repetitive cell cycles, which are essential to the perpetuation of life, are orchestrated by an underlying biochemical reaction network centered around cyclin-dependent protein kinases (Cdks) and their regulatory subunits (cyclins). Oscillations of Cdk1/CycB activity between low and high levels during the cycle trigger DNA replication and mitosis in the correct order. Based on computational modeling, we proposed that the low and the high kinase activity states are alternative stable steady states of a bistable Cdk-control system. Bistability is a consequence of system-level feedback (positive and double-negative feedback signals) in the underlying control system. We have also argued that bistability underlies irreversible transitions between low and high Cdk activity states and thereby ensures directionality of cell cycle progression.     

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.     

Characterizing multistationarity regimes in biochemical reaction networks

Switch like responses appear as common strategies in the regulation of cellular systems. Here we present a method to characterize bistable regimes in biochemical reaction networks that can be of use to both direct and reverse engineering of biological switches. In the design of a synthetic biological switch, it is important to study the capability for bistability of the underlying biochemical network structure. Chemical Reaction Network Theory (CRNT) may help at this level to decide whether a given network has the capacity for multiple positive equilibria, based on their structural properties. However, in order to build a working switch, we also need to ensure that the bistability property is robust, by studying the conditions leading to the existence of two different steady states. In the reverse engineering of biological switches, knowledge collected about the bistable regimes of the underlying potential model structures can contribute at the model identification stage to a drastic reduction of the feasible region in the parameter space of search. In this work, we make use and extend previous results of the CRNT, aiming not only to discriminate whether a biochemical reaction network can exhibit multiple steady states, but also to determine the regions within the whole space of parameters capable of producing multistationarity. To that purpose we present and justify a condition on the parameters of biochemical networks for the appearance of multistationarity, and propose an efficient and reliable computational method to check its satisfaction through the parameter space.     

Experimental investigations of multiple steady states in aerobic continuous cultivations of Saccharomyces cerevisiae

The steady-state behavior of a glucose-limited, aerobic, continuous cultivation of Saccharomyces cerevisiae CEN.PK113-7D was investigated around the critical dilution rate. Oxido-reductive steady states were obtained at dilution rates up to 0.09 h(-1) lower than the critical dilution rate by operating the bioreactor as a productostat, where the dilution rate was controlled on the basis of an ethanol measurement. Thus, the experimental investigations revealed that multiple steady states exist in a region of dilution rates below the critical dilution rate. The existence of multiple steady states was attributed to two distinct physiological effects occurring when growth changed from oxidative to oxido-reductive: (i) a decrease in the efficiency of ATP production and utilization (at ethanol concentrations below 3 g/L) and (ii) repression of the oxidative metabolism (at higher ethanol concentrations). The first effect was best observed at low ethanol concentrations, where multiple steady states were observed even when no repression of the oxidative metabolism was evident, i.e., the oxidative capacity was constant. However, at higher ethanol concentrations repression of the oxidative metabolism was observed (the oxidative capacity decreased), and this resulted in a broader range of dilution rates where multiple steady states could be found.     

Mixed micelle formation between gramicidin-S and a nonionic detergent: a nuclear magnetic resonance model study of peptide/detergent aggregation

The interaction of the cyclic decapeptide antibiotic gramicidin-S (GrS) with the nonionic detergent octaethylene glycol mono-n-dodecyl ether was studied by NMR spectroscopy. Detergent binding led to a slightly altered average conformation in the d-Phe side chains of the peptide. The changing diamagnetic shielding of nearby protons resulted in chemical shift variations, the largest effect being observed for the d-Phe C _α proton. The continuous upfield shift of this proton resonance, indicating rapid exchange of the peptide between detergent-associated and unassociated states, was employed for an evaluation of the detergent/peptide aggregation equilibria. The nonlinear binding plot thus obtained was attributed to essentially different aggregational states, depending on the detergent/peptide ratio. The almost linear dependence of the spin-lattice relaxation rate and of the hydrogen-deuterium exchange rate on the fraction of detergent-associated GrS could be reconciled with a simple model, comprising binding of detergent monomers and cooperative binding of micelles at low and high detergent/peptide molar ratios, respectively. Thus, GrS provides a useful model for a study of backbone dynamics and water penetration in detergent- and membrane-bound peptides and proteins. The results will also be discussed with reference to the interaction of GrS with biological membranes. Received: 22 June 1998 / Revised version: 5 October 1998 / Accepted: 9 October 1998     

Identification of all steady states in large networks by logical analysis

The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.     

Application of the metabolic control theory to the study of the dynamics of substrate cycles

Substrate cycles are ubiquitous structures of the cellular metabolism (e.g. Krebs cycle, fatty acids -oxydation cycles, etc... ). Moiety-conserved cycles (e.g. adenine nucleotides and NADH/NAD, etc...) are also important.The role played by such cycles in the metabolism and its regulation is not clearly understood so far. However, it was shown that these cycles can generate multistationarity (bistability), irreversible transitions, enhancement of sensitivity, temporal oscillations and chaotic motions (Hervagault & Canu, 1987; Hervagault & Cimino, 1989; Reich & Sel'kov, 1981; Ricard & Soulié, 1982). Fig. 1: Scheme of the open binary substrate cycle under study. The substrate S is converted into P with a net rate v₂. Substrate P is converted in turn into S with a net rate v₃. Step v₂ is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input of S at rates \ga₀(v₁) and first order outputs of S and P at rates \gaS and \gaP(v₄), respectively.The metabolic control theory (see also Fell, 1990), which shows how a metabolic network reacts to small perturbations in the vicinity of a steady state, and is formulated with the so-called control coefficients, was applied to such a cycle in order to get a better knowledge on the importance of each step at the regulatory point of view.The behaviour of a binary substrate cycle (fig. 1) in which one of the enzymes may be subjected to inhibition by excess of its substrate (v₂) was studied theoretically. The flux and concentration control coefficients were calculated for various steady states of the system. The evolution of the different control coefficients is compared to the evolution of the steady states. We mainly focused our study on situations for which the steady states are stable.     

Oscillations in Biochemical Reaction Networks Arising from Pairs of Subnetworks

Biochemical reaction models show a variety of dynamical behaviors, such as stable steady states, multistability, and oscillations. Biochemical reaction networks with generalized mass action kinetics are represented as directed bipartite graphs with nodes for species and reactions. The bipartite graph of a biochemical reaction network usually contains at least one cycle, i.e., a sequence of nodes and directed edges which starts and ends at the same species node. Cycles can be positive or negative, and it has been shown that oscillations can arise as a result of either a positive cycle or a negative cycle. In earlier work it was shown that oscillations associated with a positive cycle can arise from subnetworks with an odd number of positive cycles. In this article we formulate a similar graph-theoretic condition, which generalizes the negative cycle condition for oscillations. This new graph-theoretic condition for oscillations involves pairs of subnetworks with an even number of positive cycles. An example of a calcium reaction network with generalized mass action kinetics is discussed in detail.     

Two’s company,three (or more) is a simplex

As a candidate mechanism of neural representation, large numbers of synfire chains can efficiently be embedded in a balanced recurrent cortical network model. Here we study a model in which multiple synfire chains of variable strength are randomly coupled together to form a recurrent system. The system can be implemented both as a large-scale network of integrate-and-fire neurons and as a reduced model. The latter has binary-state pools as basic units but is otherwise isomorphic to the large-scale model, and provides an efficient tool for studying its behavior. Both the large-scale system and its reduced counterpart are able to sustain ongoing endogenous activity in the form of synfire waves, the proliferation of which is regulated by negative feedback caused by collateral noise. Within this equilibrium, diverse repertoires of ongoing activity are observed, including meta-stability and multiple steady states. These states arise in concert with an effective connectivity structure (ECS). The ECS admits a family of effective connectivity graphs (ECGs), parametrized by the mean global activity level. Of these graphs, the strongly connected components and their associated out-components account to a large extent for the observed steady states of the system. These results imply a notion of dynamic effective connectivity as governing neural computation with synfire chains, and related forms of cortical circuitry with complex topologies.