Transient responses and adaptation to steady state in a eukaryotic gene regulation system

Understanding the structure and functionality of eukaryotic gene regulation systems is of fundamental importance in many areas of biology. While most recent studies focus on static or short-term properties, measuring the long-term dynamics of these networks under controlled conditions is necessary for their complete characterization. We demonstrate adaptive dynamics in a well-known system of metabolic regulation, the GAL system in the yeast S. cerevisiae. This is a classic model for a eukaryotic genetic switch, induced by galactose and repressed by glucose. We followed the expression of a reporter gfp under a GAL promoter at single-cell resolution in large population of yeast cells. Experiments were conducted for long time scales, several generations, while controlling the environment in continuous culture. This combination enabled us, for the first time, to distinguish between transient responses and steady state. We find that both galactose induction and glucose repression are only transient responses. Over several generations, the system converges to a single robust steady state, independent of external conditions. Thus, at steady state the GAL network loses its hallmark functionality as a sensitive carbon source rheostat. This result suggests that, while short-term dynamics are determined by specific modular responses, over long time scales inter-modular interactions take over and shape a robust steady state response of the regulatory system.     

On the validity of the steady state assumption of enzyme kinetics

By estimating relevant time scales, a simple new condition can be found that ensures the validity of the steady state assumption for a standard enzyme-substrate reaction. The generality of the approach is demonstrated by applying it to the determination of validity criteria for the steady state assumption applied to an enzyme-substrate-inhibitor system.     

Application of the transition time of metabolic system as a criterion for optimization of metabolic processes

Transition time of metabolic systems in introduced as a suitable optimization criterion for biotechnological processes in which it is desirable to reduce the lag time and minimize the mass contained within the system. Lag time is the time needed for the system to attain the steady state. Results obtained from the sensitivity analysis of this steady state response are presented within the metabolic control analysis and applied to 3 case studies. In all of them the information provided by the transition time control profile allows the implementation of a strategy for biotechnological manipulations aimed at the improvement of the process. (c) 1994 John Wiley & Sons, Inc.     

Nitrogen use efficiency revisited

Nitrogen use efficiency (NUE) was originally defined as the dry mass productivity per unit N taken up from soil. The term was subsequently redefined as the product of nitrogen productivity (NP) and mean residence time of nitrogen (MRT). However, this redefinition was found to contradict the original definition under certain conditions, and confusion arose when the MRT defined for a steady-state system was applied to a system that was actually not at steady state. As MRT is the expected length of time that a unit of N newly taken up from soil is retained before being lost, it can be translated into the plant nitrogen duration (PND) divided by the total N uptake. This MRT is determined equally well for a steady state- and a non-steady state system and is in accordance with the original definition of NUE. It can be applied to a herbaceous perennial stand (that was at a steady state) and to an annual stand (that was not at a steady state) to determine NUE. NUE is also applicable when plant growth and reproduction are analyzed in relation to N use.     

A comparison of tomato (Lycopersicon esculentum) lectin with its deglycosylated derivative

A regime is proposed for the design of coupled enzyme assays in which auxiliary enzymes are added at concentrations proportional to their Km values. Under these conditions it is possible to calculate the complete time course of the assay including the time required for the system to approach its steady state. The consequence of increasing the number of coupling enzymes is shown to be a considerable decrease in time required to reach the steady state provided that the overall transient time remains the same. The method is extended to the general consideration of pathways and shows that pathways of the same length exhibit identical temporal responses provided that the units of concentration and time used are based on the steady-state concentration of intermediates and the transient time respectively. An unexpected finding is that increasing the number of intermediates in a pathway can decrease the time required to enter a steady state.     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

An optimizing start-up strategy for a bio-methanator

This paper presents an optimizing start-up strategy for a bio-methanator. The goal of the control strategy is to maximize the outflow rate of methane in anaerobic digestion processes, which can be described by a two-population model. The methodology relies on a thorough analysis of the system dynamics and involves the solution of two optimization problems: steady-state optimization for determining the optimal operating point and transient optimization. The latter is a classical optimal control problem, which can be solved using the maximum principle of Pontryagin. The proposed control law is of the bang–bang type. The process is driven from an initial state to a small neighborhood of the optimal steady state by switching the manipulated variable (dilution rate) from the minimum to the maximum value at a certain time instant. Then the dilution rate is set to the optimal value and the system settles down in the optimal steady state. This control law ensures the convergence of the system to the optimal steady state and substantially increases its stability region. The region of attraction of the steady state corresponding to maximum production of methane is considerably enlarged. In some cases, which are related to the possibility of selecting the minimum dilution rate below a certain level, the stability region of the optimal steady state equals the interior of the state space. Aside its efficiency, which is evaluated not only in terms of biogas production but also from the perspective of treatment of the organic load, the strategy is also characterized by simplicity, being thus appropriate for implementation in real-life systems. Another important advantage is its generality: this technique may be applied to any anaerobic digestion process, for which the acidogenesis and methanogenesis are, respectively, characterized by Monod and Haldane kinetics.     

Control of developmental transitions in the cyclic AMP signalling system of Dictyostelium discoideum

In the first few hours after starvation, the developing cAMP secretory system in Dictyostelium discoideum has been observed to be successively in one of four states: (a) quiescent, (b) excitable (capable of relay), (c) autonomously oscillating, and (d) secreting at a high steady level. A theoretical model is presented which demonstrates that the proximal cause of the transitions between different types of behavior may be slow changes in the activities of the enzymes adenylate cyclase and phosphodiesterase. These changes affect the stability properties of the steady state admitted by the cAMP signalling system. Sustained oscillations develop when the steady state is unstable, whereas relay of cAMP signals occurs upon perturbation of a stable steady state for parameter values close to those which produce oscillations. The developmental path suggested in the adenylate cyclase-phosphodiesterase space for the sequential transitions compares with the time course observed for the synthesis of these enzymes after starvation. It is suggested that there is general significance for the understanding of differentiation in the example given of a state-point following a developmental path in parameter space, moving from one behavioral domain to another, and thereby bringing about shifts in qualitative behavior.     

Control of Developmental Transitions in the Cyclic AMP Signalling System of Dictyostelium discoideum

In the first few hours after starvation, the developing cAMP secretory system in Dictyostelium discoideum has been observed to be successively in one of four states: (a) quiescent, (b) excitable (capable of relay), (c) autonomously oscillating, and (d) secreting at a high steady level. A theoretical model is presented which demonstrates that the proximal cause of the transitions between different types of behavior may be slow changes in the activities of the enzymes adenylate cyclase and phosphodiesterase. These changes affect the stability properties of the steady state admitted by the cAMP signalling system. Sustained oscillations develop when the steady state is unstable, whereas relay of cAMP signals occurs upon perturbation of a stable steady state for parameter values close to those which produce oscillations. The developmental path suggested in the adenylate cyclase-phosphodiesterase space for the sequential transitions compares with the time course observed for the synthesis of these enzymes after starvation. It is suggested that there is general significance for the understanding of differentiation in the example given of a state-point following a developmental path in parameter space, moving from one behavioral domain to another, and thereby bringing about shifts in qualitative behavior.     

Bistability in the isocitrate dehydrogenase reaction: an experimentally based theoretical study.

The enzyme isocitrate dehydrogenase (IDH, EC 1.1.1.42) can exhibit activation by one of its products, NADPH. This activation is competitively inhibited by the substrate NADP+, whereas NADPH competes with NADP+ for the catalytic site. Experimental observations briefly presented here have shown that if IDH is coupled to another enzyme, diaphorase (EC 1.8.1.4), which transforms NADPH into NADP+, the system can attain either one of two stable states, corresponding to a low and a high NADPH concentration. The evolution toward either one of these stable states depends on the time of addition of diaphorase to the medium containing IDH and its substrate NADP+. We present a theoretical and numerical analysis of a model for the IDH-diaphorase bienzymatic system, based on the regulatory properties of IDH. The results confirm the occurrence of bistability for parameter values derived from the experiments. Depending on the total concentration of NADP+ plus NADPH and the concentration of IDH, the system can either admit a single steady state or display bistability. We obtain an expression for the critical time t*, before which diaphorase addition leads to the lower steady state and after which addition of the enzyme leads to the upper steady state of NADPH. The analysis is extended to the case where the second substrate of IDH, isocitrate, is consumed in the course of the reaction without being regenerated. Bistability occurs only as a transient phenomenon in these conditions.     

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.     

A kinetic description of sequential,reversible, Michaelis-Menten reactions: practical application of theory to metabolic pathways

Equations are presented which describe a linear coupled system of reactions that utilize a single substrate and convert it to product by way of several intermediate enzyme catalysed steps. The present analysis extends previous results by assuming that the enzymes obey reversible Michaelis-Menten kinetics. In order for the system to reach steady state one must assume that the initial substrate concentration and the final product concentration are buffered to a constant value. Using the present analysis it can be shown that the system will not enter a steady state if the maximal velocity of any forward reaction is less than the steady state flux through the system. This condition represents a practical test for determining if a system will enter steady state but is valid only when the rate of the primary enzyme is not affected allosterically be intermediates in the pathway. The equations are used to analyse a portion of the rat liver glycogenic pathway that catalyses the conversion of glucose to fructose 1,6-bisphosphate.     

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.     

Density-dependent migration and stability in a system of linked populations

The effect of adding density-dependent migration between nearest neighbour populations of a single discrete-generation species in a chain of habitat fragments is investigated. The larger the population on a particular habitat fragment, the greater the fraction of inhabitants who migrate before reproducing. It has previously been shown for similar models with density-independent migration that coupling populations in this way has no effect on the stability of these populations. Here, it is demonstrated that this effect is also generally true if migration is density-dependent. However, if the migration rate is large enough and has density dependence of the correct form, then the steady state (with all the populations remaining at the same constant value through time) can be destabilised. The conditions for this to occur are obtained analytically. When this “destabilisation” occurs, the system settles down to an alternative steady state where half of the populations take one constant value which is below that of an equivalent isolated system, and the other populations all share a population value which is greater than the steady state of the isolated populations. Once this configuration is reached, the population size on each patch remains constant over time. hence the change might more properly be described as a decrease in homogeneity rather than in stability.     

Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.     

Stimulus-response method for flows and volumes in slightly perturbed constant parameter systems

The area and the first time moment of tracer kinetic data on steady state systems contains information on flows and volumes of distribution in the system. A simple and general method of obtaining such information is given. The method consists of equating the ratio of the time integrals of appropriate stimulus function and response function in an actually performed finite amount tracer experiment to the ratio of stimulus and response that would exist in the tracer steady state. The generality of the method is illustrated by deriving the volume of osmotically exchangeable water in an organ from measurement of its transient bulk water exchange during a small osmotic perturbation.     

Unsteady-state operation of continuous fermentor for enhancement of cell mass production

The transient behavior of continuous fermentation is studied concentrating on the time scale intrinsic to the system. The time scale is the time required for the fermentorto reach a stable steady state after the disturbance of cell mass is introduced. When the cell concentration is disturbed from the steady-state value, in particular, at the dilution rate near washout, the transient period becomes extended significantly, and the steady state is resumed sluggishly. This sluggish transient behavior could be turned to an advantage for enhancing the cell mass output rate. The proposed transient operation is a continuous fermentation whereby a positive disturbance in the cell mass is introduced, so that the cell concentration is higher than the steady-state value for an extended transient period. It is shown that a significantly higher cell mass production than that from the steady-state continuous fermentation can be achieved. Simple experiments were performed to demonstrate the improvement of cell (Candida utilis) productivity.     

Oscillatory viral dynamics in a delayed HIV pathogenesis model

We consider an HIV pathogenesis model incorporating antiretroviral therapy and HIV replication time. We investigate the existence and stability of equilibria, as well as Hopf bifurcations to sustained oscillations when drug efficacy is less than 100%. We derive sufficient conditions for the global asymptotic stability of the uninfected steady state. We show that time delay has no effect on the local asymptotic stability of the uninfected steady state, but can destabilize the infected steady state, leading to a Hopf bifurcation to periodic solutions in the realistic parameter ranges.     

A Simple Self-Maintaining Metabolic System: Robustness,Autocatalysis, Bistability

A living organism must not only organize itself from within; it must also maintain its organization in the face of changes in its environment and degradation of its components. We show here that a simple (M,R)-system consisting of three interlocking catalytic cycles, with every catalyst produced by the system itself, can both establish a non-trivial steady state and maintain this despite continuous loss of the catalysts by irreversible degradation. As long as at least one catalyst is present at a sufficient concentration in the initial state, the others can be produced and maintained. The system shows bistability, because if the amount of catalyst in the initial state is insufficient to reach the non-trivial steady state the system collapses to a trivial steady state in which all fluxes are zero. It is also robust, because if one catalyst is catastrophically lost when the system is in steady state it can recreate the same state. There are three elementary flux modes, but none of them is an enzyme-maintaining mode, the entire network being necessary to maintain the two catalysts.