Complex dynamics of microbial competition in the gradostat

We examine the conditions necessary for the emergence of complex dynamic behavior in systems of microbial competition. In particular, we study the effect of spatial heterogeneity and substrate-inhibition on the dynamics of such a system. This is accomplished through the study of a mathematical model of two microbial populations competing for a single nutrient in a configuration of two interconnected chemostats. Microbial growth is assumed to follow substrate-inhibited kinetics for both species. Such a system with sterile feed has been shown in a previous work to exhibit stable periodic states. In the present work we study the system for the case of non-sterile feed, i.e., when the two species are present in the feed of the chemostats. The analysis is done by numerical bifurcation theory methods. We demonstrate that, in addition to periodic states, the system possesses stable quasi-periodic states resulting from Neimark-Sacker bifurcations of limit cycles. Also, periodic states may undergo successive period doublings leading to periodic states of increasing period and indicating that chaotic states might be possible. Multistability is also observed, consisting in the coexistence of several stable steady states and possibly stable periodic or quasi-periodic states for given operating conditions. It appears that substrate-inhibition, spatial heterogeneity and presence of microorganisms in the inflow are all necessary conditions for complex dynamics to arise in a microbial system of pure and simple competition.     

Influence of microbial growth kinetics on steady state multiplicity and stability of a two-step nitrification (SHARON) model

In this paper, the influence of microbial growth kinetics on the number and the stability of steady states for a nitrogen removal process is addressed. A two-step nitrification model is studied, in which the maximum growth rate of ammonium oxidizers is larger than the one of nitrite oxidizers. This model describes the behavior of a SHARON reactor for the treatment of wastewater streams with high ammonium concentrations. Steady states are identified through direct calculation using a canonical state space model representation, for several types of microbial kinetics. The stability of the steady states is assessed and the corresponding phase portraits are analyzed. Practical operation of a SHARON reactor aims at reaching ammonium conversion to nitrite while suppressing further conversion to nitrate. Regions in the input space are identified that result in this desired behavior, with only nitrite formation. It is demonstrated that not only the dilution rate plays a role, as is commonly known, but also the influent ammonium concentration. Besides, the type of microbial (inhibition) kinetics has a nonnegligible influence. While the results indicate that product inhibition does not affect the number of steady states of a (bio)reactor model, it is shown that substrate inhibition clearly yields additional steady states. Particular attention is devoted to the physical interpretation of these phenomena.     

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.     

Finite fluctuations and multiple steady states far from equilibrium

The role of finite fluctuations in transitions between nonequilibrium steady states in nonlinear systems is investigated. Attention is focused on a model biochemical system for which the usual deterministic chemical kinetics predicts a far-from-equilibrium region of multiple steady states. A stochastic approach to chemical kinetics is adopted to study explicitly the effect of fluctuations around the coexisting stable states on a predicted hysteresis in the transition between those states. A numerical solution of the stochastic master equation for the system yields results which differ qualitatively from predictions of the purely macroscopic theory. Possible implications of these results are considered, and several important aspects of the computational scheme are discussed in some detail.     

Smooth bistable S-systems

S-systems have been used as models of biochemical systems for over 30 years. One of their hallmarks is that, although they are highly non-linear, their steady states are characterised by linear equations. This allows streamlined analyses of stability, sensitivities and gains as well as objective, mathematically controlled comparisons of similar model designs. Regular S-systems have a unique steady state at which none of the system variables is zero. This makes it difficult to represent switching phenomena, as they occur, for instance, in the expression of genes, cell cycle phenomena and signal transduction. Previously, two strategies were proposed to account for switches. One was based on a technique called recasting, which permits the modelling of any differentiable non-linearities, including bistability, but typically does not allow steady-state analyses based on linear equations. The second strategy formulated the switching system in a piece-wise fashion, where each piece consisted of a regular S-system. A representation gleaned from a simplified form of recasting is proposed and it is possible to divide the characterisation of the steady states into two phases, the first of which is linear, whereas the other is non-linear, but easy to execute. The article discusses a representative pathway with two stable states and one unstable state. The pathway model exhibits strong separation between the stable states as well as hysteresis.     

Thermodynamics and kinetics of the glyoxylate cycle

Because the standard Gibbs energies of formation of all the species of reactants in the glyoxylate cycle are known at 298.15 K, it is possible to calculate the apparent equilibrium constants of the five reactions in the cycle in the pH range 5-9 and ionic strengths from 0 to approximately 0.35 M. In making calculations on such a system, it is convenient to specify concentrations of coenzymes like NADox and NADred because they are involved in many reactions and may be in steady states. Calculations are given for [NADox] = 1000[NADred] and [NADox] = 10[NADred]. Equilibrium compositions are calculated using computer programs when all the reactants are present initially and when only glyoxylate and CoA are present initially. The kinetics of the reactions in the glyoxylate cycle at specified concentrations of NADox and NADred are calculated by numerical solution of the steady-state rate equations for the case where the reactant concentrations are below their Michaelis constants and only glyoxylate and CoA are present initially.     

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.     

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.     

Dissipation and maintenance of stable states in an enzymatic system: analysis and simulation

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. An essential assumption for constraint-based analysis is the formation of a stable steady state. This work investigates dissipation and maintenance of stable states in a simple reversible enzymatic reaction with substrate inhibition. Under mass-action kinetics, the conditions under which the reaction maintains a stable steady state are analytically derived and numerically confirmed. It is shown that, in order to maintain a steady state in the regulated reaction, maximal enzyme activity must be much higher than input rate. Moreover, it is revealed that requirements for large enzyme activity are due to substrate inhibition. It is suggested that high activities of enzymes may play a vital role in protecting a stable state from its catastrophic collapse, giving an additional explanation to an intriguing problem—why the activities of some enzymes greatly exceed the flux capacity of a pathway. In addition, dissipation of the enzymatic reaction is analysed. It is shown that the collapse of stable states is always associated with a point at which dissipation is the highest. Therefore, in order to maintain a stable state, dissipation of the reaction must be less than a critical value. Moreover, although external forcing may not change net mass flow, it may lead to collapse of stable states. Furthermore, when stable states collapse at a critical forcing amplitude and period, dissipation also reaches a highest value. It is concluded that collapse of stable steady state in the enzyme system with substrate inhibition always corresponds to critical points at which dissipation is highest, regardless if the reaction is forced or not. Therefore, for the substrate inhibited reaction, maintenance of stable states is intrinsically related to level of dissipation.     

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.     

Species selection in a reactor-settler system

The competition between flocculating and nonflocculating microorganisms was investigated in a continuous reactor-settler system (e.g. activated sludge). Co existence states were found to be possible, over a certain domain of operating conditions, even with simple monotonic kinetics and simple competition. Multiple solutions exist when coexistence states are unstable. Coexistence solutions are stable when the flocculating bacteria grow faster at feed conditions as in the activated sludge problem. The analysis applies to one or several mixed or plug flow reactors. Other effects, such as enrichment of the recycle stream by the flocculating microorganism or substrate adsorption and storage, may change the structure of solution.     

Boolean analysis of cell regulation networks

A comparison is made between the predictions of the Boolean and continuous analysis of a regulation model when the formation of two mediators interacting by cross-inhibition is stimulated by one or two specific signals. For such a system, the Boolean analysis reproduces the characteristics of behaviour previously predicted by continuous analysis (multiple stable states of opposite type, discontinuous transition, and associated hysteresis phenomenon). The qualitative agreement between the two methods allows a qualitative but rigorous treatment of regulation systems in which the Boolean analysis is applicable. From a general schematic representation of interaction in bidirectional control systems, we analyse by the Boolean method a large range of possible systems of increasing complexities which could theoretically apply. Previously unforeseen consequences of some systems are described. After that, we give a logical analysis of a well-known system (negative loop grafted with additional external controls) and discuss the application of such a system to explain certain oscillatory phenomena in the cell, showing the disrupting role of an additional control on the expected behaviour. Thus, when the analysis of a model including a negative loop does not indicate the possibility of experimentally suggested oscillations, we propose other simple logical structures which can predict this behaviour. Finally, we show a logical analysis of an opposite type of example of cell regulation where the biochemical observations can be accounted for simply by a negative loop grafted with one input variable.     

Biological Control Through Intraguild Predation: Case Studies in Pest Control,Invasive Species and Range Expansion

Intraguild predation (IGP), the interaction between species that eat each other and compete for shared resources, is ubiquitous in nature. We document its occurrence across a wide range of taxonomic groups and ecosystems with particular reference to non-indigenous species and agricultural pests. The consequences of IGP are complex and difficult to interpret. The purpose of this paper is to provide a modelling framework for the analysis of IGP in a spatial context. We start by considering a spatially homogeneous system and find the conditions for predator and prey to exclude each other, to coexist and for alternative stable states. Management alternatives for the control of invasive or pest species through IGP are presented for the spatially homogeneous system. We extend the model to include movement of predator and prey. In this spatial context, it is possible to switch between alternative stable steady states through local perturbations that give rise to travelling waves of extinction or control. The direction of the travelling wave depends on the details of the nonlinear intraguild interactions, but can be calculated explicitly. This spatial phenomenon suggests means by which invasions succeed or fail, and yields new methods for spatial biological control. Freshwater case studies are used to illustrate the outcomes.     

Exergy as an indicator: Observations of an aquatic ecosystem model

Exergy is considered as a goal function or ecological orientor. Normally at the edge of oscillation exergy reaches to its maximum value when the ecosystem had no adaptation on it. To study the variation of exergy in different states of ecosystem, a simple three species (phytoplankton, zooplankton and fish) food chain model has been considered. From the model it is shown that the system moves from steady state to chaotic state by decreasing zooplankton body volume in turn increasing its grazing rate. By the property of self-adaptability the system tries to overcome this situation. Two such possible processes are described here: (i) by the toxic effect of phytoplankton and (ii) by reducing half saturation constant of fishes. In both this cases exergy value reduces and the system reaches to stable state. Through the analysis of exergy variation in all these situations this paper shows that the system chose the process in which the reduction of exergy will be the minimum.     

Static characteristics of a continuous flow bioreactor containing antibiotic-resistant recombinant cells

The steady-state behavior of a continuous bioreactor containing antibiotic-resistant recombinant cells has been investigated. Only the plasmid-free cell is susceptible to and killed by antibiotics. A Monod form of specific death rate was found to simulate quite well the experimental death rates of various cells due to antibiotics. The stability characteristics, including bifurcation of the possible steady states, are examined. Appropriate numerical illustrations for the steady-state characteristics have been provided. Theoretically, two coexistence steady states (CO), three partial washout steady states (PW), and one total washout steady state (TW) are feasible, but only one CO, one PW, and one TW were realized. When antibiotic consumption is not extremely significant the CO can exist over one or two ranges of dilution rates depending upon the antibiotic concentration in the feed. The CO is globally stable. Whenever the PW and/or the TW exist(s) together with the CO they are unstable. Sensitivity analyses for several key kinetic parameters have been made. The rate at which the plasmid-bearing cells revert to the plasmid-free cells has the most significant effect on the antibiotic susceptibility of the system. Some simplified optimization calculations for maximum profit have been carried out.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.     

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.     

Biodegradation of 3,4-dichloroaniline in a fluidized bed bioreactor and a steady-state biofilm Kinetic model

Mixed culture of microorganisms immobilized onto Celite diatomaceous earth particles were used to degrade 3,4-dichloroaniline (34DCA) in a three-phase draft tube fluidized bed bioreactor. Biodegradation was confirmed as the dominant removal mechanism by measurements of the concomitant chloride ion evolution. Degradation efficiencies of 95% were obtained at a reactor retention time of 1.25 h. A mathematical model was used to describe the simultaneous diffusion and reaction of 34DCA and oxygen in the biofilms on the particles in the reactor. The parameters describing freely suspended cell growth on 34DCA were obtained in batch experiments. The model was found to describe the system well for three out of four steady states and to predict qualitatively the experimentally observed transition in the biofilm kinetics from 34DCA to oxygen limitation.     

Evolution in bioids: hypercompetitivity as a source of bistability and a possible role of metal complexes as prenucleoprotic mediators of molecular asymmetry.

Spontaneous production of optically active compounds can occur through kinetic instability of an asymmetric steady state in open systems, in which two enantiomeric autocatalysts compete for a common prochiral substrate in a stereoselective reaction of order n greater than 2. For the case of n=3, a proof of instability of a symmetric reacting state in the general case, and functions of reaction parameters ('Chemical Reynolds Numbers') governing the existence and stability of 7 different steady states are derived. The 'extinct state' (without autocatalyst) is stable; a finite amount of products is required to shift it into one of the reacting steady states. A mutation from one state into another in such system ('bioids') involves an amplification of different 'kinds of information', as 'stochastic' (noise into dissipative structures), 'molecular' (autocatalysts), and 'stoichimetric' information. Stereospecific third order kinetics are believed to be realizable on octahedral metal complexes with two-dentated ligands and to have played a role in the prebiological evolution of optically active compounds.