Chemical Reaction Systems with Toric Steady States

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.     

Logical identification of all steady states: The concept of feedback loop characteristic states

Biological regulatory systems can be described in terms of non-linear differential equations or in logical terms (using an “infinitely non-linear” approximation). Until recently, only part of the steady states of a system could be identified on logical grounds. The reason was that steady states frequently have one or more variable located on a threshold (see below); those steady states were not detected because so far no logical status was assigned to threshold values. This is why we introduced logical scales with values 0,¹θ, 1²θ, 2, ..., in which¹θ,²θ, ... are the logical values assigned to the successive thresholds of the scale. We thus have, in addition to the regular logical states,singular states in which one or more variables is located on a threshold. This permits identifyingall the steady states on logical grounds. It was noticed that each feedback loop (or reunion of disjointed loops) can be characterized by a logical state located at the thresholds at which the variables of the loop operate. This led to the concept ofloop-characteristic state, which, as we will see, enormously simplifies the analysis.The core of this paper is a formal demonstration that among the singular states of a system, only loop-characteristic states can be steady. Reciprocally, given a loop-characteristic state, there are parameter values for which this state is steady; in this case, the loop is effective (i.e. it generates multistationarity if it is a positive loop, homeostasis if it is a negative loop). This not only results in the above-mentioned radical simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.     

Cooperative hunting in a discrete predator–prey system II

ABSTRACT

We investigate a discrete-time predator–prey system with cooperative hunting in the predators proposed by Chow et al. by determining local stability of the interior steady states analytically in certain parameter regimes. The system can have either zero, one or two interior steady states. We provide criteria for the stability of interior steady states when the system has either one or two interior steady states. Numerical examples are presented to confirm our analytical findings. It is concluded that cooperative hunting of the predators can promote predator persistence but may also drive the predator to a sudden extinction.     

Dynamics of commensalistic systems with self- and cross-inhibition

Up to three stable steady states are possible in a simple commensalistic system, taking place in an open-loop mixed reactor when the growth rates of the two species are inhibited by the substrates they prey on (Self-inhibition). Two stable states are possible in a system with noncompetitive inhibition of the species by the substrate they are not preying on (cross-inhibition). A large number of steady states as well as oscillatory states are possible when both self- and cross-inhibition are strong. Multiplicity of steady states is also possible in a reactor with biomas recirculation for these kinetics. Yet, the latter is more stable than the open-loop reactor in the sense that the domain of steady-state multiplicity is narrower. The stability of steady states and the dynamics of the systems for each of the investigated kinetics are summarized in a qualitative phase plane. The importance of the analysis for improving the selectively and yield of the system and for predicting the response of the system to changes in the operating conditions, is discussed.     

Analyzing steady states of dynamics of bio-molecules from the structure of regulatory networks

Regulatory relations between biological molecules constitute complex network systems and realize diverse biological functions through the dynamics of molecular activities. However, we currently have very little understanding of the relationship between the structure of a regulatory network and its dynamical properties. In this paper we introduce a new method, named “linkage logic” to analyze the dynamics of network systems. By this method, we can restrict possible steady states of a given complex network system from the knowledge of regulatory linkages alone. The regulatory linkage simply specifies the list of variables that affect the dynamics of each variable. We formalize two aspects of the linkage logic: the “Principle of Compatibility” determines the upper limit of the diversity of possible steady states of the dynamics realized by a given network; the “Principle of Dependency” determines the possible combinations of states of the system. By combining these two aspects, (i) for a given network, we can identify a cluster of nodes that gives an alternative representation of the steady states of the whole system, (ii) we can reduce a given complex network into a simpler one without loss of the ability to generate the diversity of steady states, (iii) we can examine the consistency between the structure of network and observed set of steady states, and (iv) sometimes we can predict unknown states or unknown regulations from an observed set of steady states alone. We illustrate the method by several applications to an experimentally determined regulatory network for biological functions.     

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.     

Maximization of steady-state bacterial production in a chemostat with pH and substrate control.

This analytical study deals with the steady-state behavior and control of microbial growth in continuous cultures. A second order Haldane-Monod model of continuous cultures is used as a basis for study of the effects of the adjustment of pH by the addition of acidic (or basic) materials. The treatment of a hydrogen ion concentration, in addition to substrate and microbial concentrations as state variables, results in a third order system of equations describing the process. The analysis of the system in equilibrium yields several admissible steady states, that is, steady states which satisfy all constraints. An optimal control problem is formulated and subsequently solved to maximize steady-state microbial production.     

From bistability to oscillations in a model for the isocitrate dehydrogenase reaction

Considered is a bienzymatic system consisting of isocitrate dehydrogenase (IDH, EC 1.1.1.42), which transforms NADP(+) into NADPH, and of diaphorase (DIA, EC 1.8.1.4), which catalyzes the reverse reaction. Experimental evidence as well as a theoretical model show the possibility of a coexistence between two stable steady states in this reaction system. The phenomenon originates from the regulatory properties of IDH. We extend the analysis of a theoretical model proposed for the IDH-DIA bienzymatic system and investigate the occurrence of different modes of bistability, with or without hysteresis, i.e. in the presence of two or only one limit point bounding the domain of multiple steady states. The analysis indicates that the two types of bistability may sometimes be observed sequentially as a given control parameter is progressively increased. We further obtain conditions in which sustained oscillations develop in the model. These results establish the isocitrate dehydrogenase reaction coupled to diaphorase as a suitable candidate for further experimental and theoretical studies of bistability and oscillations in biochemical systems.     

A conceptual model of the dynamics of the Syamozero Lake community

We present a conceptual mathematical model of the dynamics of a spatially heterogeneous population system whose prototype is the fish community of Lake Syamozero. Analysis of the solutions of this model is used to demonstrate that interactions between the predator and prey populations in two neighboring biotopes (the pelagic and coastal zones) may result in either undamped oscillations or steady states of the population sizes. The model population densities are of the same order of magnitude as the values obtained in long-term observations of the Syamozero biota. It is also demonstrated that the transition to steady states may be accompanied by long-term (dozens or hundreds of years) damped oscillations of the prey and predator population densities. Under natural conditions, long transitional periods may prevent fish communities from reaching stationary modes.     

Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Erratum to: Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.     

Dynamic effects related to steady-state multiplicity in continuous Saccharomyces cerevisiae cultivations

The behavioral differences between chemostat and productostat cultivation of aerobic glucose-limited Saccharomyces cerevisiae were investigated. Three types of experiments were conducted: a chemostat, where the dilution rate was shifted up or down in stepwise manner; and a productostat, with either stepwise changed or a rampwise increased ethanol setpoint, i.e., an accelero-productostat. The transient responses from chemostat and productostat experiments were interpreted using a simple metabolic flux model. In a productostat it was possible to obtain oxido-reductive steady states at dilution rates far below Dcrit due to a strong repression of the respiratory system. However, these steady states could not be obtained in a chemostat, since a dilution rate shift-down from an oxido-reductive steady state led to a derepression of the respiratory system. It can therefore be concluded that the range of dilution rates where steady-state multiplicity can be obtained differs depending on the operation mode and that this dilution rate multiplicity range may appear larger in a productostat than in a chemostat. A more narrow multiplicity range, however, was obtained when the productostat was operated as an accelero-productostat.     

Modeling the adaptive immune response in HBV infection

The aim of this work is to investigate a new mathematical model that describes the interactions between Hepatitis B virus (HBV), liver cells (hepatocytes), and the adaptive immune response. The qualitative analysis of this as cytotoxic T lymphocytes (CTL) cells and the antibodies. These outcomes are (1) a disease free steady state, which its local stability is characterized as usual by R ₀ < 1, (2) and the existence of four endemic steady states when R ₀ > 1. The local stability of these steady states depends on functions of R ₀. Our study shows that although we give conditions of stability of these steady states, not all conditions are feasible. This rules out the local stability of two steady states. The conditions of stability of the two other steady states (which represent the complete failure of the adaptive immunity and the persistence of the disease) are formulated based on the domination of CTL cells response or the antibody response.     

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.     

A theory of oscillatory instability in two-state systems in relation to nerve excitation

The oscillatory aspect in a system having two steady states is studied theoretically using a model of excitable nerve membrane. The condition for the occurrence of oscillatory instability is discussed on the basis of the kinetic picture of nerve excitation in consideration of the non-Markoffian effect caused by ion transport in the system. Small oscillations around a steady state as well as a giant fluctuation between two states are obtained. Results are compared with experiments carried out with squid giant axons perfused intracellularly.     

Variable elimination in post-translational modification reaction networks with mass-action kinetics

We define a subclass of chemical reaction networks called post-translational modification systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop a mathematical framework based on the notion of a cut (a particular subset of species in the system), which provides a linear elimination procedure to reduce the number of variables in the system to a set of core variables. The steady states are parameterized algebraically by the core variables, and graphical conditions for when steady states with positive core variables imply positivity of all variables are given. Further, minimal cuts are the connected components of the species graph and provide conservation laws. A criterion for when a (maximal) set of independent conservation laws can be derived from cuts is given.     

How to make a biological switch

Some biological regulatory systems must "remember" a state for long periods of time. A simple type of system that can accomplish this task is one in which two regulatory elements negatively regulate one another. For example, two repressor proteins might control one another's synthesis. Qualitative reasoning suggests that such a system will have two stable states, one in which the first element is "on" and the second "off", and another in which these states are reversed. Quantitative analysis shows that the existence of two stable steady states depends on the details of the system. Among other things, the shapes of functions describing the effect of one regulatory element on the other must meet certain criteria in order for two steady states to exist. Many biologically reasonable functions do not meet these criteria. In particular, repression that is well described by a Michaelis-Menten-type equation cannot lead to a working switch. However, functions describing positive cooperativity of binding, non-additive effects of multiple operator sites, or depletion of free repressor can lead to working switches.     

Multiple steady states in a continuous stirred tank reactor: an experimental case study for hydrolysis of sucrose by invertase

The purpose of this work was to validate experimentally that multiple steady states may be achieved in a continuous stirred tank reactor (CSTR) during hydrolysis of sucrose by invertase. Experiments were done with four initial sucrose concentrations (0.1, 0.175, 0.584 and 1 M) to study their effect on residual sucrose and reaction rate at steady state. Two different steady states (S=0.7 M, r=9×10⁻⁴ mol/l min and S=0.135 M, r=1.54×10⁻³ mol/l min) were found depending on initial concentration of sucrose in the reactor. Two stable steady states were possible in a CSTR using invertase for the hydrolysis of sucrose. A third possible steady state can be derived theoretically, but it should be a metastable condition because any small disturbance in the system will result in transitory states stabilizing at sugar concentrations of either 0.135 or 0.7 M.     

Multiplicity of steady states in a bioreactor during the production of 1,3-propanediol by <Emphasis Type="Italic">Clostridium butyricum</Emphasis>

In recent decades, the production of compounds from microorganisms has increased significantly. Glycerol as a source of substrate appears to have great potential, due to its large supply because of the increase in biodiesel production. This paper will discuss the multiplicity of steady states for the production of 1,3-propanediol from glycerol by Clostridium butyricum, employing a model that takes into account inhibition by fermentation products. The theoretical study of bifurcation enabled us to make a qualitative adjustment to the various experimental steady states, using the theoretical steady states obtained from the AUTO2007 program. The theoretical model parameters were varied to fit qualitatively the values of the experimental steady states. In addition, this work is a qualitative study, using experimental steady states that can be used as an initial study for more advanced work on optimizing the production of 1,3-propanediol.