Prediction of the Reaction Equilibrium of Biocatalytic Reactions in Aqueous-Organic Two-Phase Systems

Biocatalytic reactions can be carried out in aqueous-organic two-phase systems. Several models to describe the thermodynamically-determined equilibrium position in such systems have appeared in the literature. Some of these models are only valid for dilute systems, whereas others can also be used for nondilute systems. In this paper, these models are described and compared. It is explained in what way the equilibrium constants of each model can be used to predict the product concentration in different organic solvents.     

The effect of multiple parasitoid introductions upon the equilibrium value of host density

Summary Four models of host-parasitoid population variation are examined to study the effect of additional parasitoid introductions. Only one of the models predicts that the equilibrium value of host population increases with such introductions. The other three, that of Holling, of Nicholson, and its recent modification by Hassell and Varley, predict a fall in the equilibrium of host numbers. Thus it would appear that the practice of multiple introductions in biological control is correct when the host-parasitoid relationship can be modelled by these latter systems.     

Population Genetic Models of Genomic Imprinting

The phenomenon of genomic imprinting has recently excited much interest among experimental biologists. The population genetic consequences of imprinting, however, have remained largely unexplored. Several population genetic models are presented and the following conclusions drawn: (i) systems with genomic imprinting need not behave similarly to otherwise identical systems without imprinting; (ii) nevertheless, many of the models investigated can be shown to be formally equivalent to models without imprinting; (iii) consequently, imprinting often cannot be discovered by following allele frequency changes or examining equilibrium values; (iv) the formal equivalences fail to preserve some well known properties. For example, for populations incorporating genomic imprinting, parameter values exist that cause these populations to behave like populations without imprinting, but with heterozygote advantage, even though no such advantage is present in these imprinting populations. We call this last phenomenon "pseudoheterosis." The imprinting systems that fail to be formally equivalent to nonimprinting systems are those in which males and females are not equivalent, i.e., two-sex viability systems and sex-chromosome inactivation.     

The structure and behavior of defoliating insect/forest systems

Using four detailed and complex simulation models we derive a framework for predicting behavior of any defoliating insect/forest system. The framework uses simple and easily gathered biological information on four sets of state variables, each with a characteristic temporal scale, to predict presence, absence or form of key ecological processes acting on or between the variables. The combination of these key processes enables prediction of system equilibrium structure and this structure can be used to derive the temporal behavior of the system. Four qualitatively different classes of system behavior arise from the equilibrium structures. The framework is tested against twelve other systems and field invalidation experiments are outlined. Forest defoliator research and management implications are discussed.     

Mechanisms for the facilitated diffusion of substrates across cell membranes

Two classes of theoretical mechanisms for protein-mediated, passive, transmembrane substrate transport (facilitated diffusion) are compared. The simple carrier describes a carrier protein that exposes substrate influx and efflux sites alternately but never both sites simultaneously. Two-site models for substrate transport describe carrier proteins containing influx and efflux sites simultaneously. Velocity equations describing transport by these mechanisms are derived. These equations take the same general form, being characterized by five experimental constants. Simple carrier-mediated transport is restricted to hyperbolic kinetics under all conditions. Two-site carrier-mediated transport may deviate from hyperbolic kinetics only under equilibrium exchange conditions. When both simple- and two-site carriers display hyperbolic kinetics under equilibrium exchange conditions, these models are indistinguishable by using steady-state transport data alone. Seven sugar transport systems are analyzed. Five of these systems are consistent with both models for sugar transport. Uridine, leucine, and cAMP transport by human red cells are consistent with both simple- and two-site models for transport. Human erythrocyte sugar transport can be modeled by simple- and two-site carrier mechanisms, allowing for compartmentalization of intracellular sugars. In this instance, resolution of the intrinsic properties of the human red cell sugar carrier at 20 degrees C requires the use of submillisecond transport measurements.     

Epistasis of quantitative trait loci under different gene action models

Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham's model, F2-metric, and F(infinity)-metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham's model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F2-metric and F(infinity)-metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F2 population with two unlinked loci, Cockerham's model is reduced to the F2-metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F(infinity)-metric model are not orthogonal but they can be easily translated into those under the F2-metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies.     

Modeling the inhibitor activity and relative binding affinities in enzyme-inhibitor-protein systems: application to developmental regulation in a PG-PGIP system

Within a number of classes of hydrolytic enzymes are certain enzymes whose activity is modulated by a specific inhibitor-protein that binds to the enzyme and forms an inactive complex. One unit of a specific inhibitor-protein activity is often defined as the amount necessary to inhibit one unit of its target enzyme by 50 %. No objective quantitative means is available to determine this point of 50 % inhibition in crude systems such as those encountered during purification. Two models were derived: the first model is based on an irreversible binding approximation, and the second, or equilibrium, model is based on reversible binding. The two models were validated using the inhibition data for the polygalacturonase-polygalacturonase-inhibiting protein (PG-PGIP) system. Theory and experimental results indicate that the first model can be used for inhibitor protein activity determination and the second model can be used for inhibitor protein activity determination as well as for comparison of association constants among enzymes and their inhibitor-proteins from multiple sources. The models were used to identify and further clarify the nature of a differential regulation of expression of polygalacturonase-inhibiting protein in developing cantaloupe fruit. These are the first relations that provide for an objective and quantitative determination of inhibitor-protein activity in both pure and crude systems. Application of these models should prove valuable in gaining insights into regulatory mechanisms and enzyme-inhibitor-protein interactions.     

Hidden assumptions in a maximum-entropy method stacking analyses

Two models have been used to describe indefinite self-association (stacking). In the more popular isodesmic model addition of molecules to the growing stack occurs with the same equilibrium constant, while in the attenuated model successive equilibrium constants decrease in value. In an attempt to choose between the two models application was made of the maximum-entropy method. This paper points out that the conclusions drawn from this method are not proven as the application assumed specific limiting values for experimental values of a molecule in the interior of a stack, and these values are not identical in the two models for the two systems considered.     

Interpretation of Scatchard plots for aggregating receptor systems.

Aggregation of cell surface receptors, with each other or with other membrane proteins, occurs in a variety of experimental systems. The list of systems where receptor aggregation appears to be important in understanding ligand binding and cellular responses is growing rapidly. In this paper we explore the interpretation of equilibrium binding data for aggregating receptor systems. The Scatchard plot is a widely used tool for analyzing equilibrium binding data. The shape of the Scatchard plot is often interpreted in terms of multiple noninteracting receptor populations. Such an analysis does not provide a framework for investigating the role of receptor aggregation and will be misleading if there is a relation between receptor aggregation and ligand binding. We present a general model for the equilibrium binding of a ligand with any number of aggregating receptor populations and derive theoretical expressions for observable Scatchard plot features. These can be used to test particular models and estimate model parameters. We develop particular models and apply the general results in the cases of six aggregating receptor systems where ligand binding and receptor aggregation are related: cross-linking of monovalent cell surface proteins by monoclonal antibodies, cross-linking of cell surface antibodies by bivalent ligand, antibody-induced co-cross-linking of cell surface antibodies and Fc gamma receptors, ligand-enhanced aggregation of identical epidermal growth factor receptors, aggregation of heterologous receptors for interleukin 2 to form a high-affinity receptor, and association of receptors, including those for interleukins 5 and 6, with nonbinding accessory proteins that influence receptor affinity or effector function.     

Displacement analysis of binding inhomogeneities in crude extracts of receptors

Guidelines are given to distinguish different kinds of binding inhomogeneities of non-radiolabeled ligands in crude extracts of receptors if an appropriate 'binding analogue' of the displacers is available in radiolabeled form. Three minimal models for the simplest types of binding inhomogeneities are analysed theoretically. These models include a cooperative system (with two interacting sites on the same receptor molecule) and two non-cooperative systems (one of them with a single-site receptor having two conformational states in equilibrium and the other with two single-site receptors independent of each other). In certain cases one can distinguish these systems experimentally. Furthermore, if a group of displacers is already classified according to the above models, then dissociation constants can be determined. The quantitative comparison of these displacers on the basis of their dissociation constants is more appropriate (e.g. in Quantitative Structure Activity Relationship studies) than on the basis of their ID50 and Ki values or Hill coefficients, which is often done.     

Fractal model of ion-channel kinetics

Markov models with discrete states, such as closed in equilibrium with closed in equilibrium with open have been widely used to model the kinetics of ion channels in the cell membrane. In these models the transition probabilities per unit time (the kinetic rate constants) are independent of the time scale on which they are measured. However, in many physical systems, a property, L, depends on the scale, epsilon, at which it is measured such that L(epsilon) alpha epsilon 1-D where D is the fractal dimension. Such systems are said to be 'fractal'. Based on the assumption that the kinetic rates are given by k(t) alpha t1-D we derive a fractal model of ion-channel kinetics. This fractal model has fewer adjustable parameters, is more consistent with the dynamics of protein conformations, and fits the single-channel recordings from the corneal endothelium better than the discrete-state Markov model.     

The representation of root processes in models addressing the responses of vegetation to global change

The representation of root activity in models is here confined to considerations of applications assessing the impacts of changes in climate or atmospheric [CO₂]. Approaches to modelling roots can be classified into four major types: models in which roots are not considered, models in which there is an interplay between only selected above-ground and below-ground processes, models in which growth allocation to all parts of the plants depends on the availability and matching of the capture of external resources, and models with explicit treatments of root growth, architecture and resource capture. All models seem effective in describing the major root activities of water and nutrient uptake, because these processes are highly correlated, particularly at large scales and with slow or equilibrium dynamics. Allocation models can be effective in providing a deeper, perhaps contrary, understanding of the dynamic underpinning to observations made only above ground. The complex and explicit treatment of roots can be achieved only in small-scale highly studied systems because of the requirements for many initialized variables to run the models.     

Paradoxical persistence through mixed-system dynamics: towards a unified perspective of reversal behaviours in evolutionary ecology

Counterintuitive dynamics of various biological phenomena occur when composite system dynamics differ qualitatively from that of their component systems. Such composite systems typically arise when modelling situations with time-varying biotic or abiotic conditions, and examples range from metapopulation dynamics to population genetic models. These biological, and related physical, phenomena can often be modelled as simple financial games, wherein capital is gained and lost through gambling. Such games have been developed and used as heuristic devices to elucidate the processes at work in generating seemingly paradoxical outcomes across a spectrum of disciplines, albeit in a field-specific, ad hoc fashion. Here, we propose that studying these simple games can provide a much deeper understanding of the fundamental principles governing paradoxical behaviours in models from a diversity of topics in evolution and ecology in which fluctuating environmental effects, whether deterministic or stochastic, are an essential aspect of the phenomenon of interest. Of particular note, we find that, for a broad class of models, the ecological concept of equilibrium reactivity provides an intuitive necessary condition that must be satisfied in order for environmental variability to promote population persistence. We contend that further investigations along these lines promise to unify aspects of the study of a range of topics, bringing questions from genetics, species persistence and coexistence and the evolution of bet-hedging strategies, under a common theoretical purview.     

Global dispersal reduces local diversity

Metapopulation models and stepping-stone models in genetics are based on very different underlying dispersal structures, yet it can be difficult to distinguish the behaviour of the two kinds of models. We demonstrate a striking qualitative difference in the equilibrium behaviour possible with these two kinds of dispersal. If, in a local patch, there are multiple stable equilibria (and consequently an unstable equilibrium), we demonstrate that, for the spatial system with a metapopulation structure, at equilibrium every patch has to be near one of the stable equilibria. This contrasts with the clinal structure possible with a stepping-stone or continuous space model; thus the result can be used to deduce qualitative information about the form of dispersal from observations of allele frequencies.     

The role of behavioral dynamics in determining the patch distributions of interacting species

The effect of the behavioral dynamics of movement on the population dynamics of interacting species in multipatch systems is studied. The behavioral dynamics of habitat choice used in a range of previous models are reviewed. There is very limited empirical evidence for distinguishing between these different models, but they differ in important ways, and many lack properties that would guarantee stability of an ideal free distribution in a single-species system. The importance of finding out more about movement dynamics in multispecies systems is shown by an analysis of the effect of movement rules on the dynamics of a particular two-species-two-patch model of competition, where the population dynamical equilibrium in the absence of movement is often not a behavioral equilibrium in the presence of adaptive movement. The population dynamics of this system are explored for several different movement rules and different parameter values, producing a variety of outcomes. Other systems of interacting species that may lack a dynamically stable distribution among patches are discussed, and it is argued that such systems are not rare. The sensitivity of community properties to individual movement behavior in this and earlier studies argues that there is a great need for empirical investigation to determine the applicability of different models of the behavioral dynamics of habitat selection.     

Floquet theory: a useful tool for understanding nonequilibrium dynamics

Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Creative dynamics approach to neural intelligence

The thrust of this paper is to introduce and discuss a substantially new type of dynamical system for modelling biological behavior. The approach was motivated by an attempt to remove one of the most fundamental limitations of artificial neural networks — their rigid behavior compared with even simplest biological systems. This approach exploits a novel paradigm in nonlinear dynamics based upon the concept of terminal attractors and repellers. It was demonstrated that non-Lipschitzian dynamics based upon the failure of Lipschitz condition exhibits a new qualitative effect — a multi-choice response to periodic external excitations. Based upon this property, a substantially new class of dynamical systems — the unpredictable systems — was introduced and analyzed. These systems are represented in the form of coupled activation and learning dynamical equations whose ability to be spontaneously activated is based upon two pathological characteristics. Firstly, such systems have zero Jacobian. As a result of that, they have an infinite number of equilibrium points which occupy curves, surfaces or hypersurfaces. Secondly, at all these equilibrium points, the Lipschitz conditions fails, so the equilibrium points become terminal attractors or repellers depending upon the sign of the periodic excitation. Both of these pathological characteristics result in multi-choice response of unpredictable dynamical systems. It has been shown that the unpredictable systems can be controlled by sign strings which uniquely define the system behaviors by specifying the direction of the motions in the critical points. By changing the combinations of signs in the code strings the system can reproduce any prescribed behavior to a prescribed accuracy. That is why the unpredictable systems driven by sign strings are extremely flexible and are highly adaptable to environmental changes. It was also shown that such systems can serve as a powerful tool for temporal pattern memories and complex pattern recognition. It has been demonstrated that new architecture of neural networks based upon non-Lipschitzian dynamics can be utilized for modelling more complex patterns of behavior which can be associated with phenomenological models of creativity and neural intelligence.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Equilibrium and Nonequilibrium Models in Ecological Anthropology: An Evaluation of "Stability" in Maring Ecosystems in New Guinea

Three models pertaining to the stability of Maring ecosystems have been proposed. The first is the local stability model, in which a population seeks its own equilibrium state; the second is the regional stability model, in which each population is ultimately unstable, but populations persist somewhere in space; and the third is the disequilibrium model, in which neither stability nor population regulation is attained. In the disequilibrium model, exogenous factors prevent a population, which is moving toward some equilibrium state, from reaching it. The large number of quantitative anthropological and ecological studies in Highlands New Guinea has not shown clearly which of these three models best describes reality. Simulation of shifting agriculture in New Guinea shows that the Highlands systems are equilibrium-seeking, but have such limited recovery rates from disturbance that even small perturbations are sufficient to keep them from reaching equilibrium. When the influences of technological innovation, environmental change, and social-cultural evolution are taken into account, the disequilibrium model is the model of choice. These systems remain away from their stable equilibrium points most of the time, if those exist at all. Thus, New Guinea agroecosystems can be stable or unstable depending upon how stability is defined.