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.     

Mathematical models of microbial growth and competition in the chemostat regulated by cell-bound extracellular enzymes

A mathematical model of growth and competitive interaction of microorganisms in the chemostat is analyzed. The growth-limiting nutrient is not in a form that can be directly assimilated by the microorganisms, and must first be transformed into an intermediate product by cell-bound extracellular enzymes. General monotone functions, including Michaelis-Menten and sigmoidal response functions, are used to describe nutrient conversion and growth due to consumption of the intermediate product. It is shown that the initial concentration of the species is an important determining factor for survival or washout. When there are two species whose growth is limited by the same nutrient, three different modes of competition are described. Competitive coexistence steady states are shown to be possible in two of them, but they are always unstable. In all of our numerical simulations, the system approaches a steady state corresponding to the washout of one or both of the species from the chemostat.Research supported by NSF grant DMS-90-96279Research supported by NSERC grant A-9358     

Chemical oscillations and multiple steady states due to variable boundary permeability

An analysis is made of isothermal chemical reactions occurring in a volume bounded by a membrane and immersed in a reservoir of reactants and products at fixed concentrations. The permeability of the membrane to a given species is taken to be a function of the concentration of that species or of another species. This coupling between reaction and permeation provides feedback and, even for extremely simple (one or two step) reaction mechanisms, shows the following phenomena: (i) multiple steady states, some of which are stable, some unstable; (ii) hysteresis effects; (iii) reversible and irreversible transitions between stable branches of steady states induced by variation of one parameter and dependent on the values of other parameters; (iv) oscillatory temporal approach to a stable steady state; and (v) limit cycles. A method is given for predicting in certain limiting conditions the presence of unstable permeation-reaction feedback loops; the procedure is essentially independent of the details of the reaction mechanism. The theory is proposed as an appropriate construct for certain cellular phenomena, and as one example, is compared with some experiments on glycolysis in yeast cells.     

Phytoplankton equilibrium phases during thermal stratification in a deep subtropical reservoir

1. Equilibrium and non-equilibrium hypotheses have often been used to explain observations in community ecology. Published case studies have demonstrated that steady state phytoplankton assemblages are more likely to occur in deep lakes than in shallow mixed ones.
2. Phytoplankton seasonal succession was studied by weekly sampling in Faxinal Reservoir (S Brazil), a subtropical deep, clear, warm monomictic and slightly eutrophic reservoir. This study demonstrated an alternation of steady and non-steady state phases of phytoplankton assemblages with different dominant species during the steady states.
3. During the studied period, three steady states were identified with different dominant algal species: Anabaena crassa (Cyanobacteria), Nephrocytium sp. (green algae) and Asterionella (diatoms).
4. Each steady state in Faxinal Reservoir developed under stratified conditions of the water column according to the predictions of the disturbance concepts. Apparently, the major forces driving the development and persistence of these steady-state phases were closely related to thermal stratification and its consequences.
5. This study is the first report on development of more than one steady state within a year in a stratified water body. The development of three steady states might be the result of the relatively long stratification period in the Faxinal Reservoir and to its unique geochemical features.     

Impossibility of multiple steady states in a family of active membrane transport models

To study the dynamical behavior of active membrane transport models, Vieira and Bisch proposed a complex chemical network (model 3) with two cycles. One cycle involves monomers as pump units while the other cycle uses dimers. In their work, the stoichiometric network analysis was used to study the stability of steady states and the bifurcation analysis was done through numerical methods. They concluded that the possibility of multiple steady states in the model 3 could not be discarded. Here, a zero eigenvalue analysis is applied to prove the impossibility of multiple positive steady states in the model 3. (A positive steady state is one for which all species have positive concentrations.) Moreover, the result is generalized to its family networks. Received: 6 April 1998 / Revised version: 16 October 1998 / Accepted: 28 October 1998     

Unveiling steady-state multiplicity in hybridoma cultures: the cybernetic approach

Mammalian cells grown in suspension produce waste metabolites such as lactate, alanine, and ammonia, which reduce the yield of cell mass and the desired product on the nutrients supplied. Previous studies (Cruz et al., 1999; Europa et al., 2000; Follstad et al., 1999) have shown that the cells can be made to alter their metabolism by starving them on their nutrients in continuous cultures at low dilution rates or starting the culture as a fed-batch. This leads to multiple steady states in continuous reactors, with some states being more favorable than others. Mathematical models that take into account the metabolic regulation that leads to these multiple steady states are invaluable tools for bioreactor control. In this article we present a cybernetic modeling strategy in which Metabolic Flux Analysis (MFA) is used to guide the cybernetic formulation. The hybridoma model presented as a result of this strategy considers the partially substitutable, partially complementary nature of glucose and glutamine. The choice of competitions within the network is guided by MFA and the model is successful in explaining the three multiple steady states observed. The cybernetic model though identified for the hybridoma experiments of Hu and others (Europa et al., 2000) seem generally applicable to mammalian systems as it captures the pathways that are common to mammalian cells grown in suspension. The model presented here could be used for start-up strategies for continuous reactors and model-based feedback control for maintaining high productivity of the reactor.     

Mathematical model of the primary CD8 T cell immune response: stability analysis of a nonlinear age-structured system

The primary CD8 T cell immune response, due to a first encounter with a pathogen, happens in two phases: an expansion phase, with a fast increase of T cell count, followed by a contraction phase. This contraction phase is followed by the generation of memory cells. These latter are specific of the antigen and will allow a faster and stronger response when encountering the antigen for the second time. We propose a nonlinear mathematical model describing the T CD8 immune response to a primary infection, based on three nonlinear ordinary differential equations and one nonlinear age-structured partial differential equation, describing the evolution of CD8 T cell count and pathogen amount. We discuss in particular the roles and relevance of feedback controls that regulate the response. First we reduce our system to a system with a nonlinear differential equation with a distributed delay. We study the existence of two steady states, and we analyze the asymptotic stability of these steady states. Second we study the system with a discrete delay, and analyze global asymptotic stability of steady states. Finally, we show some simulations that we can obtain from the model and confront them to experimental data.     

Two-Species Migration and Clustering in Two-Dimensional Domains

We extend two-species models of individual aggregation or clustering to two-dimensional spatial domains, allowing for more realistic movement of the populations compared with one spatial dimension. We assume that the domain is bounded and that there is no flux into or out of the domain. The motion of the species is along fitness gradients which allow the species to seek out a resource. In the case of competition, species which exploit the resource alone will disperse while avoiding one another. In the case where one of the species is a predator or generalist predator which exploits the other species, that species will tend to move toward the prey species, while the prey will tend to avoid the predator. We focus on three primary types of interspecies interactions: competition, generalist predator–prey, and predator–prey. We discuss the existence and stability of uniform steady states. While transient behaviors including clustering and colony formation occur, our stability results and numerical evidence lead us to believe that the long-time behavior of these models is dominated by spatially homogeneous steady states when the spatial domain is convex. Motivated by this, we investigate heterogeneous resources and hazards and demonstrate how the advective dispersal of species in these environments leads to asymptotic steady states that retain spatial aggregation or clustering in regions of resource abundance and away from hazards or regions or resource scarcity.     

A Mathematical Biologist’s Guide to Absolute and Convective Instability

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator–prey systems.     

Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.     

Effect of wall growth on scale-up problems and dynamic operating characteristics of the biological reactor

If inhibitory substrates are being utilized in a well-stirred biological reactor, microbiological growth on the walls of the reactor can create a scale-up problem. A simple model is proposed which shows that without such growth, of the three existing steady states only one is stable and nontrivial, but with wall growth the trivial, stable, steady state (washout) is impossible. In addition, wall growth reduces the region over which three steady states are feasible and reduces the minimum residence time for which there is only one steady state that corresponds to a high conversion. Thus, a laboratory process with a high surface area to volume ratio can give an over optimistic prediction of both necessary residence; time and stability of the full scale process unless wall growth is accounted for.     

Transient behavior of a continuous stirred tank biological reactor utilizing phenol as an inhibitory substrate

Transient experiments were conducted on a Pseudomomas utilizing phenol in a continuous culture by disturbing the influent substrate concentration and dilution rate. Two stable steady states existed for some ranges of the parameters. Highly damped oscillations were observed in approaching a new high conversion steady state or in returning to a new high conversion steady state following a small disturbance. When a large disturbance was applied there was a smooth (overdamped) approach to a new low conversion steady state. The observed oscillatory behavior for small disturbances was predicted by a modified Powell-Ierusalemskii bottleneck model, but could not be predicted by a Monod-Haldane model; neither model was accurate for predicting the effect of large disturbances. A constant wall growth factor was used to account for microbial film activity, and the existence of two stable states was directly due to the presence of the film.     

Decision-making neural circuits mediating social behaviors

We propose a mathematical model of a continuous attractor network that controls social behaviors. The model is examined with bifurcation analysis and computer simulations. The results show that the model exhibits stable steady states and thresholds for steady state transitions corresponding to some experimentally observed behaviors, such as aggression control. The performance of the model and the relation with experimental evidence are discussed.     

Kinetic effects of simultaneous inhibition by substrate and product

The starting point for the present investigations was the finding that increasing influent concentrations from 10 to 380 mmol/L glucose decreased the attainable growth rate of an acidogenic population in continuous culture from 0.52 to 0.05 h(-1) To account for this phenomenon, a new kinetic model is developed that combines substrate and product inhibition. Both effects are connected through the product yield, giving rise to a complex dependency of the growth rate on the substrate concentration. As a main feature, the maximum attainable growth rate decreases almost hyperbolically above some optimal substrate concentration in the influent. Furthermore, under certain conditions the kinetic model predicts the existence of three steady states: a high-conversion and a low-conversion state that are both stable and a metastable intermediate state. The latter states from the multiple-steady-state region are to be avoided, and eventual transitions to these states may have important consequences for the stability and the operation of such reaction systems. Substrate as well as product inhibition is reported for Propionibacterium freundenreichii and recently could be demonstrated for the above-mentioned acidogenic population. The proposed model allows optimization of anaerobic wastewater treatment processes and is applicable also to other fermentations.     

Fluorescence characteristics of 5-carboxytetramethylrhodamine linked covalently to the 5'' end of oligonucleotides: multiple conformers of single-stranded and double-stranded dye-DNA complexes.

Fluorescence steady-state and lifetime experiments have been carried out on duplex and single-stranded DNA molecules labeled at the 5' ends with 5-carboxytetramethylrhodamine (TMRh). The temperature and ionic strength of the solutions were varied over large ranges. The results reveal at least three well-defined states of the TMRh-DNA molecules for the single-stranded as well as for the double-stranded DNA molecules. Two states are fluorescent, with lifetimes in the range of 0.5-1 ns and 2.5-3 ns. A third state of TMRh-DNA does not fluoresce (a dark species of TMRh-DNA). The distribution of the TMRh-DNA molecules among these three states is strongly temperature and ionic strength dependent. Estimates are made of some reaction parameters of the multistate model. The results are discussed in terms of the photophysics of TMRh, and consequences of the multiple conformers of TMRh-DNA for studies involving fluorescence studies with TMRh-labeled DNA are considered.     

Heat-induced unfolding of neocarzinostatin, a small all-beta protein investigated by small-angle X-ray scattering

Neocarzinostatin is an all-beta protein, 113 amino acid residues long, with an immunoglobulin-like fold. Its thermal unfolding has been studied by small-angle X-ray scattering. Preliminary differential scanning calorimetry and fluorescence measurements suggest that the transition is not a simple, two-state transition. The apparent radius of gyration is determined using three different approaches, the validity of which is critically assessed using our experimental data as well as a simple, two-state model. Similarly, each step of data analysis is evaluated and the underlying assumptions plainly stated. The existence of at least one intermediate state is formally demonstrated by a singular value decomposition of the set of scattering patterns. We assume that the pattern of the solution before the onset of the transition is that of the native protein, and that of the solution at the highest temperature is that of the completely unfolded protein. Given these, actually not very restrictive, boundary constraints, a least-squares procedure yields a scattering pattern of the intermediate state. However, this solution is not unique: a whole class of possible solutions is derived by adding to the previous linear combination of the native and completely unfolded states. Varying the initial conditions of the least-squares calculation leads to very similar solutions. Whatever member of the class is considered, the conformation of this intermediate state appears to be weakly structured, probably less than the transition state should be according to some proposals. Finally, we tried and used the classical model of three thermodynamically well-defined states to account for our data. The failure of the simple thermodynamic model suggests that there is more than the single intermediate structure required by singular value decomposition analysis. Formally, there could be several discrete intermediate species at equilibrium, or an ensemble of conformations differently populated according to the temperature. In the latter case, a third state would be a weighted average of all non native and not completely unfolded states of the protein but, since the weights change with temperature, no meaningful curve is likely to be derived by a global analysis using the simple model of three thermodynamically well-defined states.     

Multiple steady-state phenomena within enzyme reactors: The enzyme reaction with two substrates

The theoretical dynamic characteristics of an isothermal continuous flow stirred tank enzyme reactor (CFSTER) operating on two substrates are investigated. Under certain conditions multiple steady states are possible; namely, with an enzyme which binds with the two substrates sequentially. The occurrence of multiple steady states is found to be primarily dictated by three dimensionless parameters which incorporate rate law constants. The global stability of certain steady states is examined by numerically solving the transient material balance on the CFSTER. The effect of recycle on the dynamics of an isothermal plug flow enzyme reactor (PFER) is also studied. A general conclusion indicated by this work is that any open isothermal reaction system wherein the reaction rate law passes through a maximum with increasing substrate concentration and where back mixing occurs with exhibit multiple steady-state behavior in some operating range.     

The three-state model for the elementary process of energy conversion in muscle.

A three-state model for the elementary process of energy conversion in striated muscle is analysed; in two of the three states, the crossbridge is attached to an actin site, while the third is a detached state. This model accounts for the mechanical properties of steady shortening and lengthening processes as well as those of isometric and isotonic transient processes in a quantitative way. Moreover, qualitative agreement is obtained for the total energy liberation from muscle. Biochemical properties are also computed for transient processes. Comparisions are made with other models with "three states".     

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.