Effects of heterogeneity in synaptic conductance between weakly coupled identical neurons

A significant degree of heterogeneity in synaptic conductance is present in neuron to neuron connections. We study the dynamics of weakly coupled pairs of neurons with heterogeneities in synaptic conductance using Wang–Buzsaki and Hodgkin–Huxley model neurons which have Types I and II excitability, respectively. This type of heterogeneity breaks a symmetry in the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant when compared to the identical case. For weakly coupled neurons coupled with identical values of synaptic conductance a phase locked solution exists for all values of the synaptic rate constant, α. In particular, in-phase and anti-phase solutions are guaranteed to exist for all α. Heterogeneity in synaptic conductance results in regions where no phase locked solution exists and the general loss of the ubiquitous in-phase and anti-phase solutions of the identically coupled case. We explain these results through examination of interaction functions using the weak coupling approximation and an in-depth analysis of the underlying multiple cusp bifurcation structure of the systems of coupled neurons.     

Breakthrough performance of linear-DNA on ion-exchange membrane columns

Breakthrough performance of linear-DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves were not significantly affected by flow-rate or inlet solution concentration. In the theoretical analysis a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine coupled to the solution of the partial differential model equations was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant and the forward interaction rate-constant, which are the parameters of the membrane transport model. Through this approach a good prediction of the adsorption phenomena is obtained for inlet concentrations and flow rates greater than 0.2 mg/ml and 0.16 ml/min.     

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.     

Theory for modeling the copolymerization of tubulin and tubulin-colchicine complex

Substoichiometric concentrations of tubulin-colchicine complex (TC) inhibits microtubule assembly through a copolymerization reaction between tubulin and TC. We have determined the rates and extent of TC incorporation into bovine brain microtubules and developed a theory that models copolymerization. Our analysis suggests that while the apparent association rate constants for tubulin and TC are similar, the apparent dissociation rate constants for TC are a factor of five or more larger than those of tubulin. Copolymer composition showed only slight changes during assembly despite changes in the solution phase and showed little dependence at high TC upon the initial tubulin concentration. The theory was based on coupled Oosawa-Kasai equations that allow for the co-assembly of two components, tubulin and TC. An expression was derived that relates copolymer composition to reaction mixture composition and to the affinity of microtubule ends for tubulin and TC. This expression predicts copolymer composition at TC concentrations less than 10 microM and correlates composition with assembly inhibition. We perceive copolymerization as a facilitated incorporation of TC requiring the presence of tubulin. TC incorporation was dependent on the ratio of total tubulin to the dissociation constant for TC bound to microtubule ends. The copolymerization reaction is thus characterized by an interplay of two effects (a) where tubulin facilitates the incorporation of TC into the microtubule, and (b) where TC inhibits the assembly of tubulin into microtubules.     

Breakthrough performance of plasmid DNA on ion-exchange membrane columns

Breakthrough performance of plasmid DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves (BTC) were significantly affected by inlet flow rate and solute concentration. In the theoretical analysis, a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine, coupled to the solution of the partial differential model equations, was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant, and the forward interaction rate constant, which are the parameters of the membrane transport model. The analysis shows that as inlet concentration or flow rate increases, the deviation of the model from the experimental behavior decreases. The BTCs displacement as inlet concentration increases was explained in terms of a greater degree of column saturation reached and more efficient operation accomplished. The degree of column saturation was not influenced by inlet flow rate. It was necessary to consider in the column model the slight variation in the BTC produced by the axial dispersion, in order to accomplish the experimental curve dispersion. Consequently, the design criteria that for Pe > 40 the column axial dispersion can be neglected should be taken with precaution.     

An Exact Constant-Field Solution for a Simple Membrane

We show that the exact steady-state solution to the electrodiffusion equations for a simple membrane is the constant electric field solution when the ion environment is electroneutral on both sides of the membrane and the total numbers of ions of the same valence on both sides are equal.     

A general pre-steady-state solution to complex kinetic mechanisms

We have developed a general method for solving transient kinetic equations using Laplace transforms. Laplace transforms can be used to transform systems of differential equations that describe pre-steady-state kinetics to systems of linear algebraic equations. The general form of the pre-steady-state solution is (formula; see text) where I(t) is the time dependence of the physically observed property of the system, n is the number of intermediates, lambda i are the observed rate constants (reciprocals of the relaxation times), t is time, and Ii are the amplitude coefficients associated with each observed rate constant. We have written a program in compiled BASIC to run on a personal computer to evaluate Ii and lambda i. The program will evaluate the rate constants and coefficients of a mechanism with eight intermediates and seven relaxation times in 4 s on an 8-MHz PC-XT equipped with a math coprocessor. The most complex mechanism that we have solved, a mechanism containing 20 intermediates and 19 relaxation times, required approximately 5 min. We believe that this method will be useful to evaluate the differences in transient properties of complex biochemical mechanisms.     

Stability analysis for a peri-implant osseointegration model

We investigate stability of the solution of a set of partial differential equations, which is used to model a peri-implant osseointegration process. For certain parameter values, the solution has a ‘wave-like’ profile, which appears in the distribution of osteogenic cells, osteoblasts, growth factor and bone matrix. This ‘wave-like’ profile contradicts experimental observations. In our study we investigate the conditions, under which such profile appears in the solution. Those conditions are determined in terms of model parameters, by means of linear stability analysis, carried out at one of the constant solutions of the simplified system. The stability analysis was carried out for the reduced system of PDE’s, of which we prove, that it is equivalent to the original system of equations, with respect to the stability properties of constant solutions. The conclusions, derived from the linear stability analysis, are extended for the case of large perturbations. If the constant solution is unstable, then the solution of the system never converges to this constant solution. The analytical results are validated with finite element simulations. The simulations show, that stability of the constant solution could determine the behavior of the solution of the whole system, if certain initial conditions are considered.     

A steady state model for analyzing the cellular binding, internalization and degradation of polypeptide ligands

We demonstrate that the interaction of polypeptide ligands with cells under physiological conditions can be described by a set of steady state equations. These equations include four new rate constants: V_r, the rate of insertion of receptors into the cell membrane; K_e, the endocytotic rate constant of occupied receptors; K_t, the turnover rate constant of unoccupied receptors; and K_h, the rate constant of hydrolysis of internalized ligand. Several simple procedures are described for determining these constants. In experiments in which epidermal growth factor and human fibroblasts were used, the cell-ligand interactions followed the predictions of the steady state model. The utility of the steady state equations is demonstrated by establishing the kinetic basis of the commonly observed “down regulation” phenomenon and by quantitating the effect of methylamine on the endocytotic and degradation rates of epidermal growth factor. We also show that the slope of a “Scatchard plot” of steady state binding data is a complex constant including terms for the endocytotic rate of both occupied and unoccupied receptors. The X-intercept of such a plot is a function of the insertion rate of new receptors, the internalization rate of occupied receptors and the degradation rate of the internalized ligand. The steady state equations allow one to predict changes in cellular ligand binding resulting from alterations in the four rate constants. They also provide a foundation for computer simulations of ligand-cell interactions, which closely correspond to experimental data. These approaches should facilitate studies on the control of cellular activities by these polypeptide ligands.     

Two models of interfering predators in impulsive biological control

In this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters.     

Hydrodynamics of macromolecular complexes. II. Rotation

We have used the modified Oseen hydrodynamic interaction tensor along with iterative numerical solution of the coupled hydrodynamic interaction equations to calculate the rotational diffusion coefficients of macromolecular complexes composed of nonidentical spherical subunits. For the one structure, a prolate ellipsoid of revolution, for which exact solutions are available, a subunit model with the same length and volume gives asymptotic agreement with the Perrin equations. Other structures considered include plane polygonal rings, lollipops, and dumbbells.     

Selection and self-organization of self-reproducing macromolecules under the constraint of constant flux.

We investigate the dynamic behavior of a set of self-reproducing macromolecules (e.g., polynucleotides) under conditions such that the fluxes of all monomer units into the system are kept constant. Such conditions might prevail in an evolution reactor or in certain naturally occurring situations. A general set of equations is developed to describe the behavior of both the macromolecule and the monomer concentrations. The question of how the rate of macromolecule synthesis varies with the monomer levels is discussed briefly. With the help of several physically reasonable approximations, we obtain an exact solution for a simplified constant flux system. Comparison with the corresponding system under the constraint of constant overall organization reveals important similarities, most notably in the existance and composition of quasispecies. Given the same set of physical and chemical parameters, a system subject to constant flux will always evolve toward selective equilibrium more slowly than under the constraint of constant organization.     

Theoretical Analysis of the Footprinting Experiment

Abstract

In the footprinting experiment, an end-radiolabeled DNA restriction fragment is subjected to digest by an endonuclease in the presence and absence of a ligand which alters the endonuclease cleavage rate at sites of ligand-DNA contact. The location of these sites, and the strength of the ligand binding, are then deduced from the measured concentrations of the different oligonucleotides produced by the digest. We analyze the experiment in terms of coupled kinetic equations which take into account the cutting rates of endonuclease for sites with ligand present and absent, and the rates of binding and dissociation of the ligand to a site. As long as the ligand concentration remains essentially constant (which occurs, for example, if digest is terminated early enough to assure that all fragments result from single cuts by the endonuclease), the oligonucleotide concentrations reflect only the ligand binding equilibrium constant (ratio of rate constants) and the cutting rates in the presence and absence of ligand. We also show how the measured oligonucleotide concentrations (from, e.g. an autoradiogram) can be used to deduce the ligand equilibrium binding constants for the various sites on the polymer.     

Synaptic Vesicle Dynamics: A Simple Model of Phasic Release

We present a simple model of phasic neurotransmitter release whichreproduces the salient features of chemical neurotransmission. The synapticvesicle cycle has been modelled as a set of biochemical reactionsrepresented by a system of coupled differential equations. These equationshave been solved analytically to obtain the time dependent behaviour of thesystem on perturbation from the steady state. The scheme of the synapticvesicle network has been emphasized and its role in determining some of themajor experimentally observed properties of synaptic transmission has beendiscussed, which includes the biphasic decay of the rate neurotransmitterrelease even under sustained stimulation. Another interesting outcome ofthis theoretical exercise is the saturation of total release with thecalcium dependent rate constant. The theoretically calculated values oftotal release fit very well into a sigmoidal saturating function with afourth order cooperativity exponent similar to the empiricalDodge–Rahamimoff equation. It appears that the synaptic vesiclenetwork itself is responsible for some of the major properties associatedwith chemical neurotransmission.     

Steady state kinetics of consecutive enzyme catalysed reactions involving single substrates: procedures for the interpretation of coupled assays

Sets of differential rate equations are written describing a linear sequence of reactions occurring in solution each catalysed by a control enzyme or one of the Michaelis-Menten type. It is shown that the solutions of these equations may be formulated as a set of Maclaurin polynomials, expressing the concentration of each reactant and of final product as a function of time. From arrays of such polynomials, general expressions are induced for the first non-zero term of the series. These are used to formulate a procedure (illustrated with an example simulated by numerical integration) by which results of coupled enzymic assays may be analysed in terms of maximal velocities and apparent Michaelis constants: correlation is made with other established methods for conducting coupled assays. The present procedure assumes a steady state of enzyme-substrate complexes but not of intermediate reactants.     

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

Coupled reaction of immobilized aspartate aminotransferase and malate dehydrogenase. A plausible model for the cellular behaviour of these enzymes

To study the effect of facilitated diffusion of the intermediate metabolite, oxaloacetate, on the coupled reaction of aspartate aminotransferase (L-aspartate: 2-oxoglutarate aminotransferase, EC 2.6.1.1) and malate dehydrogenase (L-malate:NAD+ oxidoreductase, EC 1.1.1.37), these enzymes were co-immobilized on the surface of a collagen film. The kinetic properties of the immobilized enzymes were compared with those observed with the enzymes in solution. Since the reactions correspond to the cytosolic enzymes, they have been studied in the direction aspartate aminotransferase toward malate dehydrogenase. Coupled enzymes in solution showed classical behaviour. A lag-time was observed before they reached a steady state and this lag-time was dependent on the kinetic properties of the second enzyme, malate dehydrogenase. The same lag-time was observed when malate dehydrogenase in solution was coupled with aspartate aminotransferase bound to the film. When aspartate aminotransferase in solution was coupled with malate dehydrogenase bound to the collagen film, a very long lag-time was observed. Theoretical considerations showed that in the latter case, the lag-time was dependent on the kinetic properties of the second enzyme and the transport coefficient of the intermediate substrate through the boundary layer near the surface of the film. Then both enzymes were co-immobilized on the collagen film. The coupled activity of aspartate aminotransferase and malate dehydrogenase was compared for films with an activity ratio of 5 and 0.8. In both cases, a highly efficient coupling was observed. In the former case, where malate dehydrogenase was rate-limiting, 81% of this limiting activity was observed. In the latter case, aspartate aminotransferase was rate-limiting and 82% of its rate was obtained for the final product formation. The linear increase of product formation with time corresponded fairly well to the theoretical equations developed in the paper. To interpret these rate equations, one should assume that the intermediate substrate oxaloacetate formed by aspartate aminotransferase was used by malate dehydrogenase in the diffusion layer near the film, before diffusing in the bulk solution.     

Theoretical analysis of the footprinting experiment

In the footprinting experiment, an end-radiolabeled DNA restriction fragment is subjected to digest by an endonuclease in the presence and absence of a ligand which alters the endonuclease cleavage rate at sites of ligand-DNA contact. The location of these sites, and the strength of the ligand binding, are then deduced from the measured concentrations of the different oligonucleotides produced by the digest. We analyze the experiment in terms of coupled kinetic equations which take into account the cutting rates of endonuclease for sites with ligand present and absent, and the rates of binding and dissociation of the ligand to a site. As long as the ligand concentration remains essentially constant (which occurs, for example, if digest is terminated early enough to assure that all fragments result from single cuts by the endonuclease), the oligonucleotide concentrations reflect only the ligand binding equilibrium constant (ratio of rate constants) and the cutting rates in the presence and absence of ligand. We also show how the measured oligonucleotide concentrations (from, e.g. an autoradiogram) can be used to deduce the ligand equilibrium binding constants for the various sites on the polymer.     

One-dimensional steady inviscid flow through a stenotic collapsible tube

A one-dimensional inviscid solution for flow through a compliant tube with a stenosis is presented. The model is used to represent an artery with an atherosclerotic plaque and to investigate a range of conditions for which arterial collapse may occur. The coupled equations for flow through collapsible tubes are solved using a Runge-Kutta finite difference scheme. Quantitative results are given for specific physiological parameters including inlet and outlet pressure, flow rate, stenosis size, length and stiffness. The results suggest that high-grade stenotic arteries may exhibit collapse with typical physiological pressures. Critical stenoses may cause choking of flow at the throat followed by a transition to supercritical flow with tube collapse downstream. Greater amounts of stenosis produced a linear reduction of flow rate and a shortening of the collapsed region. Changes in stenosis length created proportional changes in the length of collapse. Increasing the stiffness of the stenosis to a value greater than the nominal tube stiffness caused a greater amount of flow limitation and more negative pressures, compared to a stenosis with constant stiffness. These findings assist in understanding the clinical consequences of flow through atherosclerotic arteries.