Quantitative predictions of a noncarrier model for glucose transport across the human red cell membrane

There is an increasing amount of experimental data on transport across biological membranes which cannot be readily accommodated by classical mobile carrier models. We propose models for membrane transport based upon current concepts in molecular enzymology, in which the membrane component involved in transport is an oligomeric protein which undergoes substrate-induced conformational changes. A number of paradoxical observations on glucose transport in the human erythrocyte are explained if the protein involved is a tetramer possessing two classes of binding sites with different affinities for glucose. We develop in detail a particular model of this type, the internal transfer model, in which transport occurs by transfer of substrate from one subunit to another of the protein. The fit of the predictions of the internal transfer model with most of the experimental data is very good. Those data which cannot be fitted by the model cannot be accounted for by any presently available model. We extend our model qualitatively to include the sodium-activated cotransport systems for sugars and amino acids.     

Logical Probability and Risk Assessment

A risk assessment is intended to provide a statement of current knowledge which is intended to inform a decision-maker of the current state of knowledge in response to a particular concern. Because answering the concerns of decision-makers often requires inferences to be drawn, doubt often arises over how the inference is to be drawn. In quantitative risk assessment, where a mathematical equation or model is used to draw the inference, this uncertainty is referred to as model uncertainty. A two-step process, which is referred to as logical probability, is proposed as a technique for representing model uncertainty in a risk assessment. The first step involves assigning model weights in which the degree of evidential support for each of the alternative models is considered. The second step involves assigning a unique interval in the range of 0 to 1 for each model which reflects the models' weight, to form a probability distribution. While the second step is straightforward, the first step is not. Assigning model weights requires consideration of any line of evidence that may reasonably impact the validity of the assertion of a model. While the development of a procedure for doing so may be expected to be a process which reflects the subjective preferences of whomever is involved in creating it, there are some historical precedents on which to build. Foremost among these are (1) the use of a correlation coefficient or other goodness-of-fit criteria to measure the degree of correspondence between a given model and a set of observations which are used as evidence to support it, and (2) preference given to models which are simpler, which may be ascertained as the number of adjustable parameters the model contains. Additional principles, which have little or no tradition to stand on, must be used to reflect the impact of other empirically supported beliefs on model preference. The procedure proposed is comparable to the procedure known as decision analysis, in which probabilities are assigned to alternative models based on expert or subjective input. The principal difference in the present case is that it is suggested that principles which transcend the decision at hand should be sought and articulated in order to generate a consistent measure of uncertainty arising from interpretation.     

An assessment of a coupled three-state kinetic model for sodium conductance changes.

The behavior of a coupled three-state kinetic scheme is examined to see if it might be a viable model for the conductance changes of sodium channels. It is found that for simulations of experiments which determine the properties of the Hodgkin-Huxley m and h gates, the three-state scheme performs approximately equivalently to the Hodgkin-Huxley model. In particular, the three-state scheme successfully simulates those experiments which the Hodgkin-Huxley model successfully simulates, but fails to simulate those newer voltage clamp experiments which give results anomalous to the H-H model. It is concluded that the three-state scheme is probably as good as the H-H model, but is not a viable successor to it.     

Bayesian predictive inference for time series count data

Correlated count data arise often in practice, especially in repeated measures situations or instances in which observations are collected over time. In this paper, we consider a parametric model for a time series of counts by constructing a likelihood-based version of a model similar to that of Zeger (1988, Biometrika 75, 621-629). The model has the advantage of incorporating both overdispersion and autocorrelation. We consider a Bayesian approach and propose a class of informative prior distributions for the model parameters that are useful for prediction. The prior specification is motivated from the notion of the existence of data from similar previous studies, called historical data, which is then quantified into a prior distribution for the current study. We derive the Bayesian predictive distribution and use a Bayesian criterion, called the predictive L measure, for assessing the predictions for a given time series model. The distribution of the predictive L measure is also derived, which will enable us to compare the predictive ability for each model under consideration. Our methodology is motivated by a real data set involving yearly pollen counts, which is examined in some detail.     

Is there an alternative to the cytoplasmic receptor model for the mechanism of action of steroids?

The essential features of the cytoplasmic receptor model for the mechanism of action of steroids are summarized and the restrictions imposed by such a model are noted. The literature which is not compatible with these restrictions is reviewed, and the necessity for a more general model discussed. A model for the mechanism of steroid action is proposed in which both specificity and saturability of binding in target tissue is both a function of cytoplasmic receptors and nuclear acceptors.     

Some interrelationships among the regression coefficient estimates arising in a class of models appropriate to response-time data.

Interrelationships among three response-time models which incorporate covariate information are explored. The most general of these models is the logistic-exponential in which the log odds of the probability of responding in a fixed interval is assumed to be a linear function of the covariates; this model includes a parameter W for the width of discrete time intervals in which responses occur. As W leads to O this model is equivalent to a continuous time exponential model in which the log hazard is linear in the covariates. As W leads to infininity it is equivalent to a continuous time exponential model in which the hazard itself is a linear function of the covariates. This second model was fitted to the data used in an earlier publication describing the logistic exponential model, and very close agreement of the estimates of the regression coefficients is demonstrated.     

Model of creation and evolution of stable electropores for DNA delivery

Electroporation, in which electric pulses create transient pores in the cell membrane, is becoming an important technique for gene therapy. To enable entry of supercoiled DNA into cells, the pores should have sufficiently large radii (>10 nm), remain open long enough for the DNA chain to enter the cell (milliseconds), and should not cause membrane rupture. This study presents a model that can predict such macropores. The distinctive features of this model are the coupling of individual pores through membrane tension and the electrical force on the pores, which is applicable to pores of any size. The model is used to explore the process of pore creation and evolution and to determine the number and size of pores as a function of the pulse magnitude and duration. Next, our electroporation model is combined with a heuristic model of DNA uptake and used to predict the dependence of DNA uptake on pulsing parameters. Finally, the model is used to examine the mechanism of a two-pulse protocol, which was proposed specifically for gene delivery. The comparison between experimental results and the model suggests that this model is well-suited for the investigation of electroporation-mediated DNA delivery.     

Stabilizing Control for a Pulsatile Cardiovascular Mathematical Model

In this paper, we develop a pulsatile model for the cardiovascular system which describes the reaction of this system to a submaximal constant workload imposed on a person at a bicycle ergometer test after a period of rest. Furthermore, the model should allow to use measurements for the pulsatile pressure in fingertips which provide information on the diastolic and the systolic pressure for parameter estimation. Based on the assumption that the baroreceptor loop is the essential control loop in this case, we design a stabilizing feedback control for the pulsatile model which is obtained by solving a linear-quadratic regulator problem for the linearization of a non-pulsatile counterpart of the pulsatile model. We also investigate the behavior of the model with respect to changes in the weight of the term in the cost functional for the linear-quadratic regulator problem which penalizes the deviation of the momentary pressure in the aorta from the pressure at the stationary situation which should be obtained.     

Consistent partnership formation: application to a sexually transmitted disease model

We apply a consistent sexual partnership formation model which hinges on the assumption that one gender's choices drives the process (male or female dominant model). The other gender's behavior is imputed. The model is fitted to UK sexual behavior data and applied to a simple incidence model of HSV-2. With a male dominant model (which assumes accurate male reports on numbers of partners) the modeled incidences of HSV-2 are 77% higher for men and 50% higher for women than with a female dominant model (which assumes accurate female reports). Although highly stylized, our simple incidence model sheds light on the inconsistent results one can obtain with misreported data on sexual activity and age preferences.     

Long time evolution of atherosclerotic plaques

The evolution of atherosclerosis in general, and the influence of wall shear stress on the growth of atherosclerotic plaques in particular, is an intricate phenomenon which is still only partly understood. We therefore propose a qualitative mathematical model which consists of a number of ordinary differential equations for the concentrations of the most relevant constituents of the atherosclerotic plaque. These equations were studied both for the case that the wall shear stress is a parameter (model A), and for the case in which the plaque evolution is coupled to the blood flow (model B) which results in a time dependent wall shear stress. We find that both models exhibit a class of marginally stable equilibria, all reflecting states in which the plaque only grows for a short period of time after a perturbation. The uncoupled model A, however, shows bi-stability between this class of equilibria and another equilibrium state in which the plaque experiences unlimited growth in time, if the LDL cholesterol intake exceeds a threshold value. In model B the bi-stability vanishes, but we find that there is still a critical value of the LDL cholesterol intake beyond which the lumen radius drastically decreases. We show that this decrease is quite sensitive to the value of the wall shear stress.     

A model for hyphal growth and branching.

A mathematical model for hyphal growth and branching is described which relates cytological events within hyphae to mycelial growth kinetics. Essentially the model quantifies qualitative theories of hyphal growth in which it is proposed that vesicles containing wall precursors and/or enzymes required for wall synthesis are generated at a constant rate throughout a mycelium and travel to the tips of hyphae where they fuse with the plasma membrane, liberating their contents into the wall and increasing the surface area of the hypha to give elongation. The hypothesis that there is a duplication cycle in hyphae which is equivalent to the cell cycle observed in unicellular micro-organisms is also included in the model. Predictions from the model are compared with experimentally observed growth kinetics of mycelia of Geotrichum candidum and Aspergillus nidulans. The finite difference model which was constructed is capable of predicting changes in hyphal length and in the number and positions of branches and septa on the basis of changes in vesicle and nuclear concentration. Predictions were obtained using the model which were in good agreement with experimentally observed data.     

A fully coupled transient excited state model for the sodium channel

Summary The behavior under voltage clamp conditions of a coupled kinetic scheme for the sodium channel is examined. The scheme is given diagrammatically by: tano Numerical simulations are presented which show that this model fits the voltage clamp data which are well described by the Hodgkin-Huxley equations, but also gives the sorts of behavior anomalous to the Hodgkin-Huxley model which have been seen experimentally. Further, straightforward changes in parameter values are shown to be capable of mimicking the ways in which some axonal preparations differ from others. Detailed, but admittedly heuristic, arguments are presented for the propositions that: 1) the model is minimal; i.e. no simpler kinetic model will fit the array of data simulated, and: 2) the transient excited state is necessary; i.e. no model of comparable simplicity with pure voltage dependent kinetics will fit the array of data simulated.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

Tumor micro-ecology and competitive interactions

Three nested models describing the growth of individual subpopulations in a heterogeneous environment are described. The models represent the dynamics of two populations which compete, to varying degrees, for common resources. The first model describes growth in a totally non-competitive micro-environment, the second model describes an ecology in which competition is proportional to competitor population size, and the third model ecology extends the model described by Jansson & Revesz (1974), which allows one population to emerge from the other. The critical points for each model are defined using the isoclines derived from the Ordinary Differential Equations (ODE's) describing competitive growth. The critical points for each model are characterized by the signs of the eigenvalues of the variational matrix at each point. The theoretical results of the analysis show that a competitive model ecology with Verhulstian logistics allows four critical points: the origin which is a repeller, two competitive exclusion points, and an equilibrium state (Waltman, 1983). The extended model ecology of Jansson & Revesz (1974), allows three critical points: the origin which is a repeller, competitive exclusion of the first population, and an equilibrium point. Data from a human adenocarcinoma of the colon and murine mammary tumors are used as qualitative measures of the dynamics of the three micro-ecologies. Issues such as stochastic extension to model small populations either for clonal extinction or heterogeneous emergence are discussed.     

Sensitivity analysis and parameter estimation in a model of anaerobic waste water treatment processes with substrate inhibition

Anaerobic waste water treatment processes are commonly presented by the fifth order Hill and Barth non-linear model, describing three main stages of anaerobic digestion. The model investigated in the present work is a modified version of the Hill and Barth model, which includes substrate inhibition of growth of methanogenic bacteria. Parameter estimation of this model is a difficult problem because of the high number of parameters to be estimated and the rather restricted information concerning the variables. The aim of the present work is to use sensitivity theory to find a method for selecting the most significant parameters. In particular, we develop a sensitivity model for anaerobic digestion using relative sensitivity functions. Simulation results from the sensitivity model show that the number of parameters to be estimated can be reduced from 12 to 4. We suggest a 2-step procedure for parameter estimation, which is also based on sensitivity analysis. This procedure gives results, which allow for off-line determination of parameter values if the experimental data for biogas production rate are known.     

Comparison of three rat models of cerebral vasospasm

A substantial number of rat models have been used to research subarachnoid hemorrhage-induced cerebral vasospasm; however, controversy exists regarding which method of selection is appropriate for this species. This study was designed to provide extensive information about the three most popular subarachnoid hemorrhage rat models: the endovascular puncture model, the single-hemorrhage model, and the double-hemorrhage model. In this study, the basilar artery and posterior communicating artery were chosen for histopathological examination and morphometric analysis. Both the endovascular puncture model and single-hemorrhage model developed significant degrees of vasospasm, which were less severe when compared with the double-hemorrhage model. The endovascular puncture model and double-hemorrhage model both developed more vasospasms in the posterior communicating artery than in the basilar artery. The endovascular puncture model has a markedly high mortality rate and high variability in bleeding volume. Overall, the present study showed that the double-hemorrhage model in rats is a more suitable tool with which to investigate mechanism and therapeutic approaches because it accurately correlates with the time courses for vasospasm in humans.     

Additive models for dependent cell populations

We propose an additive model for cell population growth which allows for positive correlation between sister cell lifetimes but arbitrary correlations between mother and daughter cell lifetimes. In the model each cell lifetime is the sum of two independent components, one of which is shared with its sister cell and which is also a function of the components of the lifetime of their mother. Assuming that the components follow a gamma distribution, the model is fitted to cell lifetime data of EMT6 cells obtained by the method of time-lapse cinematography.     

The dynamics of two viral infections in a single host population with applications to hantavirus

An SI epidemic model for a host with two viral infections circulating within the population is developed, analyzed, and numerically simulated. The model is a system of four differential equations which includes a state for susceptible individuals, two states for individuals infected with a single virus, one which is vertically transmitted and the other which is horizontally transmitted, and a fourth state for individuals infected with both viruses. A general growth function with density-dependent mortality is assumed. A special case of this model, where there is no coinfection and total cross immunity, is thoroughly analyzed. Several threshold values are defined which determine establishment of the disease and persistence at equilibrium for one or both of the infections within the host population. The model has applications to a hantavirus and an arenavirus that infect cotton rats. The hantavirus is transmitted horizontally whereas the arenavirus is transmitted vertically. It is shown through analysis and numerical simulations that both diseases can be maintained within a single host population, where individuals can be either infected with both viruses or with a single virus.