Character displacement and niche shift analyzed using consumer-resource models of competition

This paper analyzes the adaptive responses to competition (both character displacement and niche shift) in a two consumer-two resource model. The model includes density dependence that is unrelated to the resources that are explicit in the model. This could be due to another resource dimension, parasites, or interference competition. Competitors adapt by changing their relative consumption rate constants on the two resource types. This model can result in mutually divergent, parallel, or mutually convergent displacement of competitors. Parallel displacement may entail net divergence, net convergence, or no net change. Parallel change with net convergence is most likely when the competitors have similar constraints on the possible values of consumption rate constants, unequal allopatric abundances, and significant intraspecific density dependence. Numerical calculations of displacements are presented for several models and the effect of a number of different possible alterations of the model are discussed. The evolution of resource handling and processing efficiency, and displacement in the presence of additional selective pressures on the character are considered in detail. The results have implications for questions about maximization of population size, the relationship of character displacement and the competition coefficient, and "null" models in the study of competition. Differences between this and previous theoretical works are discussed. It is argued that conditions allowing parallel or convergent displacement are not biologically unlikely, and possible examples are discussed. Data on resource partitioning seem to be more consistent with the results reached here than with previous theory.     

Interactions of lipids and proteins: Some general principles

Methods to describe the binding of phospholipids to membrane proteins are described. It is shown that it is difficult to obtain estimates of the number of phospholipids bound to the surface of a membrane protein from ESR experiments in which plots of free to bound spin label (y) vs. molar ratio of lipid to protein are extrapolated toy=0. The relative advantages and disadvantages of ESR and fluorescence methods for measuring relative binding constants of phospholipids to membrane proteins are discussed. The particular problems associated with comparing binding constants of molecules of very different sizes (e.g., fatty acids and cardiolipin) are described and equations are presented to account for these problems. The possible effects of membrane viscosity and thickness on activity of membrane proteins are discussed, but it is concluded that effects of phospholipid structure on activity can only be understood in terms of a reasonably complete kinetic model for the protein.     

The gaussian network model: precise prediction of residue fluctuations and application to binding problems

The single-parameter Gamma matrix of force constants proposed by the Gaussian Network Model (GNM) is iteratively modified to yield native state fluctuations that agree exactly with experimentally observed values. The resulting optimized Gamma matrix contains residue-specific force constants that may be used for an accurate analysis of ligand binding to single or multiple sites on proteins. Bovine Pancreatic Trypsin Inhibitor (BPTI) is used as an example. The calculated off-diagonal elements of the Gamma matrix, i.e., the optimized spring constants, obey a Lorentzian distribution. The mean value of the spring constants is approximately -0.1, a value much weaker than -1 of the GNM. Few of the spring constants are positive, indicating repulsion between residues. Residue pairs with large number of neighbors have spring constants around the mean, -0.1. Large negative spring constants are between highly correlated pairs of residues. The fluctuations of the distance between anticorrelated pairs of residues are subject to smaller spring constants. The importance of the number of neighbors of residue pairs in determining the elements of the Gamma matrix is pointed out. Allosteric effects of binding on a single or multiple residues of BPTI are illustrated and discussed. Comparison of the predictions of the present model with those of the standard GNM shows that the two models agree at lower modes, i.e., those relating to global motions, but they disagree at higher modes. In the higher modes, the present model points to the important contributions from specific residues whereas the standard GNM fails to do so.     

An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

Mathematical Modelling of Lipase Catalysed Resolution of O-Acetyl and O-Formyl Derivatives of Secondary Alcohols

Lipase catalysed transesterification of O-acetyl and O-formyl derivatives of several sterically hindered secondary alcohols with n-butanol is studied. The reaction is mathematically modelled and the rate constants in the model are evaluated by transient parameter estimation procedure. Several models are considered for predicting the enantiomeric excess (ee) of the product and compared with experimental observations. It is found that the maximum ee depends only on the rate constants of the two competing reactions and not on the order of reaction. A second order model is found to predict the observed behaviour well. When the data is fitted to an equilibrium type of model developed by Chen et al. (1987), it is found that the rate of forward reaction is several orders of magnitude larger than the rate of reverse reaction. The effect of acetyl and formyl groups on the reactivity and selectivity are also discussed.     

Total internal reflection fluorescence microscopy: application to substrate-supported planar membranes

The use of total internal reflection illumination in fluorescence microscopy (TIRFM) is reviewed with emphasis on application to fluorescent macromolecules that specifically and reversibly bind to planar model membranes supported on glass or quartz substrates. Several methods for characterizing macromolecular motion and organization are discussed: the measurement of equilibrium binding curves to obtain values for equilibrium binding constants; the measurement of fluorescence photobleaching recovery curves to obtain values of kinetic rate constants and surface diffusion coefficients; and the measurement of fluorescence intensities as a function of the evanescent field polarization to characterize orientational order. Applications to cell-substrate contact regions are summarized and future directions of TIRFM are outlined. Correspondence to: N. L. Thompson     

Chelal growth in pseudoscorpions

There is a linear relationship between the successive positions of named trichobothria on the pedipalpal chelae during the course of post-embryonic development of five British pseudoscorpions belonging to the family Neobisiidae. Growth constants calculated from the trichobothrial data are higher than those calculated for the chelal axis. This discrepancy is resolved in terms of a model, based on growing and non-growing regions, which is a good approximation to the data. The extent of these regions is calculated for all stages and differences between the species are noted. The assumptions implicit in the model, its realism and its predictive value are discussed.     

The steady-state kinetic mechanism of ATP hydrolysis catalyzed by membrane-bound (Na+ + K+)-ATPase from ox brain

The expressions for the kinetic constants corresponding to the steady state model for hydrolysis of ATP catalyzed by (Na+ + K+)-ATPase proposed recently are analyzed with the object of determining the rate constants. The theoretical background for the necessary procedures is described. The results of this analysis are: (1) A small class (four) of rate constants are determined directly by the previously published values of the kinetic constants. (2) For a somewhat larger class of rate constants upper and lower bounds may be established. For several rate constants the upper and lower bounds differ by less than a factor 1.6 (for the "(Na+ + K+)-enzyme", i.e. the enzyme activity with K+ and millimolar substrate concentration) and 1.2 (for the "Na+-enzyme",i.e. the activity at micromolar substrate concentrations). (3) Experiments on inhibition by K+ of the Na+-enzyme at various Mg2+ concentrations are reported and analyzed. With the additional assumption that the rate constants governing the addition to ATP of Mg2+ is independent of whether or not ATP is bound to an enzyme molecule, a set of consistent values for all the 23 rate constants in the mechanism may be obtained. (4) The values of some rate constants lend further support to the contention discussed in a previous paper that the enzyme hydrolyzes ATP along two kinetically distinct pathways, depending on the presence of K+ and on the concentration of substrate, without the necessity of having more than one active substrate site per enzyme unit at any time. (5) The results show that while the two enzyme forms, the "Na+-enzyme" E1 and the "K+-enzyme" E2K, add substrate with (second order) rate constants of the same order of magnitude (differing only by a factor of four in favor of the former), the rate constants for the reverse processes differ by a factor of 100, being largest for the K+-enzyme. This is the main reason for the large difference in the Michaelis constants for the two forms reported previously. (6) Compatibility of the model with the well-known rapid dephosphorylation of the phosphorylated enzyme in the presence of K+ requires the presence, at non-zero steady state concentration, of an enzyme-potassium-phosphate intermediate, which is acid labile and is therefore not detected as a phosphorylated enzyme using conventional methods.     

A linearized and incompressible constitutive model for arteries

In many biomechanical studies, blood vessels can be modeled as pseudoelastic orthotropic materials that are incompressible (volume-preserving) under physiological loading. To use a minimum number of elastic constants to describe the constitutive behavior of arteries, we adopt a generalized Hooke's law for the co-rotational Cauchy stress and a recently proposed logarithmic-exponential strain. This strain tensor absorbs the material nonlinearity and its trace is zero for volume-preserving deformations. Thus, the relationships between model parameters due to the incompressibility constraint are easy to analyze and interpret. In particular, the number of independent elastic constants reduces from ten to seven in the orthotropic model. As an illustratory study, we fit this model to measured data of porcine coronary arteries in inflation-stretch tests. Four parameters, n (material nonlinearity), Young's moduli E? (circumferential), E? (axial), and E? (radial) are necessary to fit the data. The advantages and limitations of this model are discussed.     

Agonist induction, conformational selection, and mutant receptors

Current models of receptor activation are based on either of two basic mechanisms: agonist induction or conformational selection. The importance of one pathway relative to the other is controversial. In this article, the impossibility of distinguishing between the two mechanisms under a thermodynamic approach is shown. The effect of receptor mutation on the constants governing ligand-receptor equilibria is discussed. The two-state model of agonism both in its original formulation (one cycle) and including multiple active states (multiple cycles) is used. Pharmacological equations for the double (two cycles) two-state model are derived. The simulations performed suggest that the double two-state model of agonism can be a useful model for assessing quantitatively the changes in pharmacological activity following receptor mutation.     

Electron transfer between flavodoxin semiquinone and c-type cytochromes: correlations between electrostatically corrected rate constants, redox potentials, and surface topologies

We have measured the ionic strength dependence of the rate constants for electron transfer from the semiquinone of Clostridium pasteurianum flavodoxin to 12 c-type cytochromes and several inorganic oxidants using stopped-flow methodology. The experimental data were fit quite well by an electrostatic model that represents the interaction domains as parallel disks with a point charge equal to the charge within this region of the protein. The analysis provides an evaluation of the electrostatic interaction energy and the rate constant at infinite ionic strength (k affinity). The electrostatic charge on the oxidant within the interaction site can be obtained from the electrostatic energy, and for most of those reactants for which structures are available, the results are in good agreement with expectation. The k affinity values were found to correlate with redox potential differences, as expected from the theory of adiabatic (or nonadiabatic) outer-sphere electron-transfer reactions. Deviations from the theoretical curves are interpreted in terms of the influence of surface topology on reaction rate constants. In general, we find that electrostatic effects, steric influences, and redox potential all exert a much larger effect on reaction rate constants for the flavodoxin-cytochrome system than has been previously observed for free flavin-cytochrome interactions. The implications of this for determining biological specificity are discussed.     

Effects of diffusion on the stability of the equilibrium in multi-species ecological systems

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.     

Kinetics of macrotetrolide-induced ion transport across lipid bilayer membranes

Ion transport across lipid bilayer membranes in the presence of macrotetrolide antibiotics has been studied by stationary conductance and nonstationary relaxation methods. The results are discussed on the basis of a carrier model which has already been successfully applied to valinomycin induced ion transport. Again a kinetic analysis has been performed from which the single rate constants of the carrier model could be derived. In addition the equilibrium constant of complex formation in the aqueous phase could be determined. Measurements have been made for 4 macrotetrolides, for several ions and for various chain lengths of the lipids molecules composing the membrane.     

Models for hydrogen exchange from folded proteins. II.

The kinetics of hydrogen exchange from folded proteins can be molded as a function of two continuous distributions of rate constants, kcx and kn, representing exchange from the folded and unfolded conformations, respectively. This model can account for the temperature dependence of soybean trypsin inhibitor at pH 3 and pH 6.5. The physical significance of this model, especially the shape and breadth of the kn distribution, are discussed.     

Synthetic polymers as models for enzyme catalysis-a review

Cyclodextrins, crown ethers, cyclophanes, oligopeptides, homopolymers, modified homopolymers, ion exchange resins and various copolymers are considered as models of enzyme catalysis. The importance of hydrophobic, electrostatic and hydrogen-bonding interactions is discussed. Factors, such as complex formation, that contribute to specific binding and catalysis are examined in terms of binding constants and rate enhancement factors. The effectiveness of the various model catalysts is compared with that of natural enzymes. The nature and evolution of enzyme catalytic power is discussed in terms of the relative importance of induced fit, angular relationships, rotational restriction, transition-state binding, strain, coupled group motions, entropy reduction on complex formation and diffusional limitation. It is concluded that much further effort will be required in order to realise model catalyst that are fully characteristic of natural enzymes although impressive progress has been made in recent years. Criteria for future effort are outlined together with some tactical considerations. The potential usefulness and applications of model-enzyme catalysts are discussed briefly.     

Determination of kinetic parameters from chemical relaxation data discrimination between coupled two-step binding reactions differing in stoichiometry

The chemical relaxation times of two different two-step equilibrium reactions, characterized by a 1:1 binding process followed by a subsequent rearrangement step and a stepwise 1:2 binding reaction, are analyzed for the purpose of qualitative model discrimination and quantitative determination of kinetic parameters. The equations describing the dependences of the two reciprocal relaxation times on suitable concentrations are given for both models in the general case as well as for four different limiting situations which are characterized by well separated relaxation times. The conditions corresponding to the limiting cases are expressed in terms of strong, weak and no coupling between the two partial equilibrium steps involved in both models. The coupling strength depends on the rate constants as well as on the total concentrations of the reactants. Criteria to discriminate between these two reaction models under defined limiting conditions are developed. In the general case, the product of both reciprocal relaxation times can be used to distinguish both models. If only one relaxation time can be resolved experimentally, it is possible under conditions described to determine only a reduced set of individual rate constants for most of the limiting cases considered. If both relaxation times are observed, all rate constants are determinable in the general case as well as in most of the limiting cases discussed.     

Assessment of formal and low structured kinetic modeling of polyhydroxyalkanoate synthesis from complex substrates

A formal kinetic mathematical model for poly-(3-hydroxybutyrate-co-3-hydroxyvalerate-co-4-hydroxybutyrate) [P(3HB-co-3HV-co-4HB)] terpolyester synthesis from glucose and galactose derived from whey permeate supplemented with gamma-butyrolactone by the archaeon Haloferax mediterranei was created. Further, a low structured mathematical model for poly-3-hydroxybutyrate synthesis from whey permeate by Pseudomonas hydrogenovora was developed. In both cases, biosyntheses for obtaining the experimental data used for compiling the models were performed via fed-batch cultivations. The model developed for H. mediterranei consists of 10 differential and 11 algebraic equations, including 27 kinetic constants. The model compiled for P. hydrogenovora encompasses 10 differential and 3 algebraic equations, including 36 kinetic constants. Both models were solved by Runge-Kuta variable step numerical integration with Monte Carlo parameter optimization procedure. Difficulties arising from the modeling of redirection of metabolic fluxes from biomass growth toward polyhydroxyalkanoate synthesis and byproducts are discussed.     

Direct evidence for GTP and GDP-Pi intermediates in microtubule assembly

Identification of the kinetic intermediates in GTP hydrolysis on microtubules and characterization of their assembly properties is essential in understanding microtubule dynamics. By using an improved glass filter assay that selectively traps microtubules with a dead time of 2 s and monitoring taxol-induced rapid assembly of microtubules from [gamma-32P,3H]GTP-tubulin 1:1 complex, direct evidence has been obtained for GTP- and GDP-Pi-microtubule transient states in the early stages of the polymerization process. A simple kinetic analysis of GTP hydrolysis on microtubules within two sequential pseudo-first-order processes led to apparent first-order rate constants of 0.065 s-1 for the cleavage of the gamma-phosphate and 0.02 s-1 for the liberation of Pi, assuming a simple random model. Apparent rate constants for GTP hydrolysis and Pi release were independent of the composition of the buffer used to polymerize tubulin. The significance of these values with respect to those derived from previous studies from this and other laboratories and the possibility of a vectorial model for GTP hydrolysis are discussed.     

Hormone receptor-coupling factor-adenylate cyclase interaction: theoretical considerations

The steady-state response properties of two current models for receptor/nucleotide coupling protein/adenylate cyclase systems are examined by computer modeling techniques. In the model of Levitzki (Trends Pharmac. Sci., May, 1982, pp. 203-208), a ligand may give rise to full or partial agonist behavior only. In the model of Stadel, DeLean, and Lefkowitz (1982) configurations of the rate constants can be found which lead not only to full or partial agonist behavior, but also to varying degrees of inhibition at sufficiently high concentrations of ligand, as observed experimentally in a variety of adenylate cyclase systems. In the latter model, it is also possible to find configurations of the rate constants for which addition of a ligand will lead to inhibition of adenylate cyclase activity. The nature of partial agonism and reasons as to why it may be expected to occur for a wide variety of ligands are discussed.