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PubMed1995

Comparative mathematical analyses of freezing in lung and solid tissue.

In contrast to most organs of the body, the lung is composed 80% of air and 20% of tissue. Because freezing of the lung is fundamental to cryomicroscopy, cryopreservation, and cryosurgery, mathematical analyses of freezing in lung and solid tissue were performed to determine differences in freezing behavior resulting from differences in tissue composition. A comparison of the cooling rates of these tissues is presented. At the microscopic level ultrarapid solidification is more rapid in the subpleural region of the lung than in the same region of solid tissue. In this region, the air insulates the lung tissue and prevents transfer of heat from surrounding regions. Cooling rates on the order of 10(6) K/s can be achieved in the pleura and outermost alveolar wall because their aggregate thickness is less than 5 microns in the rat. At the macroscopic level and after steady-state freezing has occurred, one-dimensional analyses show that freezing front propagation in the lung and solid tissue differs by less than 10%. This occurs even though the lung is less conductive than solid tissue; however, by having less heat storage capacity, the thermal diffusive property of lung is similar to that of solid tissue.

Air↗

PubMed2001

Mathematical problem solving and working memory in children with learning disabilities: both executive and phonological processes are important.

The purpose of this investigation was to explore the relationship between working memory (WM) and mathematical problem solving in children with learning disabilities (LD). Children with LD (age 11.5 years) were compared to chronologically age-matched (CA-M) and younger comprehension/computation achievement-matched children (age 8.9 years) on measures of verbal and visual-spatial WM, phonological processing, components of problem solving, and word-problem solving accuracy. The results showed that (1) children with LD were inferior on measures of word solution accuracy, components of problem solving, phonological processing, domain-general WM, and verbal WM when compared to children who were CA-M, (2) children with LD were comparable to younger children on all processing measures, except measures of domain-general WM, visual-spatial WM, phonemic deletion, and identifying problem goals, (3) measures of verbal and visual-spatial WM contributed significant variance to solution accuracy independent of phonological processing, and (4) the influence of WM on solution accuracy was mediated by long-term memory (LTM) processes related to the knowledge of algorithms. The results support the notion that information activated from LTM, rather than phonological processing, mediates the relationship between executive processing and solution accuracy in children with LD.

Child↗

PubMed2001

Ratcheting in post-translational protein translocation: a mathematical model.

We have developed a non-steady-state mathematical model describing post-translational protein translocation across the endoplasmic reticulum membrane. Movement of the polypeptide chain through the channel in the endoplasmic reticulum membrane is considered to be a stochastic process which is biased at the lumenal side of the channel by the binding of BiP (Kar2p), a member of the Hsp70 family of ATPases (ratcheting model). Assuming that movement of the chain through the channel is caused by passive diffusion (Brownian ratchet), the model describes all available experimental data. The optimum set of model parameters indicates that the ratcheting mechanism functions at near-maximum rate, being relatively insensitive to variations of the association or dissociation rate constants of BiP or its concentration. The estimated rate constant for diffusion of a polypeptide inside the channel indicates that the chain makes contact with the walls of the channel. Since fitting of the model to the data required that the backward rate constant be larger than the forward constant during early diffusion steps, translocation must occur against a force. The latter may arise, for example, from the unfolding of the polypeptide chain in the cytosol. Our results indicate that the ratchet can transport polypeptides against a free energy of about 25 kJ/mol without significant retardation of translocation. The modeling also suggests that the BiP ratchet is optimized, allowing fast translocation to be coupled with minimum consumption of ATP and rapid dissociation of BiP in the lumen of the ER. Finally, we have estimated the maximum hydrophobicity of a polypeptide segment up to which lateral partitioning from the channel into the lipid phase does not result in significant retardation of translocation.

Adenosine Triphosphate↗

PubMed1993

A mathematical model for the elimination of carbon monoxide in humans.

Carboxyhaemoglobin (COHb) build-up in the blood as a result of exposures to carbon monoxide (CO) affects human beings. It is important to eliminate CO from the blood for treatment and health safety. A mathematical model is proposed to compute COHb level in the blood as a function of post-exposure time as CO is eliminated. The model takes into account molecular diffusion, facilitated diffusion, convection, non-equilibrium kinetics of CO with haemoglobin and the important physiological parameters, such as ventilation rate, blood flow rate and the total volume of blood in the body. Endogenous production of CO in the body is included in the formulation. The resulting coupled system of non-linear partial differential equations with physiologically relevant initial, entrance and boundary conditions is solved numerically. The COHb levels computed from our model agree with those measured experimentally (Pace et al., 1950; Peterson & Stewart, 1970). The half-life of COHb, i.e. the time required for the blood COHb to decrease from the initial level to its half-value is computed. The half-life values of COHb computed from our model are in good agreement with those based on experimental data collected under different physiological conditions (Pace et al., 1950). Also, the results predicted from our model give better approximation to the experimental values than the CFK equation (Coburn et al., 1965). It is found that the rate of elimination of CO increases with the increase of inspired PO2 and ventilation rate.

Carbon Monoxide↗

PubMed1995

Mathematical treatment of the kinetics of binding protein dependent transport systems reveals that both the substrate loaded and unloaded binding proteins interact with the membrane components.

Binding-protein-dependent transport systems in Gram-negative bacteria are multicomponent systems in which a soluble periplasmic binding protein of high substrate binding affinity establishes the major substrate recognition site. Usually, there are two membrane proteins which are thought to interact with the substrate loaded form of the binding protein to allow transport of substrate to occur. Transport is against the concentration gradient and needs energization by an ATP hydrolyzing polypeptide. Overall transport is considered mainly unidirectional owing to the high energy of ATP hydrolysis coupled to transport. We dissected the overall transport process into three individual steps: (i) reversible binding of substrate to the binding protein; (ii) reversible binding of the binding protein to the membrane components forming the translocation complex; (iii) irreversible transport of substrate through the membrane and dissociation of the binding protein from the complex. Two models were considered. In the first, only the substrate-loaded binding protein interacts with the membrane components, while in the second model both the loaded and the unloaded form of the binding protein interact with the membrane components. The mathematical analysis of the second model revealed that the substrate concentration KM at half-maximal rate of transport approaches KD of the binding protein when the last step of transport becomes low and when the concentration of binding protein in the periplasm becomes large. This is usually observed in real systems. Under the same conditions, in model 1 KM approaches zero and is hence considerably smaller than KD. This has never been observed in any real system. In addition, the dependence of the overall rate of transport on the concentration of binding protein in the periplasm follows a sigmoidal curve only when model 2 is considered. The sigmoidal behavior becomes more pronounced when the substrate concentration is low and it is less pronounced when the last step in overall transport is low. This phenomenon has been observed with the Escherichia coli maltose transport system. Thus, at least for the maltose transport system, it seems likely that both the loaded and the unloaded forms of the binding protein interact with the membrane components. We propose that this should generally be considered in binding-protein-dependent transport systems.

Adenosine Triphosphate↗

PubMed1995

Mitochondrial DNA damage and efficiency of ATP biosynthesis: mathematical model.

The role of mitochondrial DNA (mtDNA) damage in ageing processes and in malignant transformation of a cell is discussed. A mathematical model of the mtDNA population in a cell and in tissue is constructed. The model describes the effects of mtDNA damages accumulated during ageing and some features of malignant transformation and regeneration.

Adenosine Triphosphate↗

PubMed1995

A mathematical framework for describing and analysing gene regulatory networks.

This paper presents a mathematical framework for describing and analysing gene regulatory networks by autonomous differential equations. It represents an improvement on existing frameworks in that it may handle a wider range of gene regulatory mechanisms. Gene regulatory networks are frequently threshold-dominated, i.e. genes are activated only when the concentration of certain gene products lie between definite thresholds. Here, the concept of regulatory domain is introduced to describe these regions in the phase space. To each regulatory domain is associated an indicator function whose value is 1 inside and 0 outside the domain. The indicator functions thus reflect the logical structure of the network. The sharp borders between the regulatory domains may be smoothed by replacing the logical step functions by continuous sigmoids or so-called logoid functions. A logoid function coincides with the step function outside a narrow interval around the threshold, and rises continuously from 0 to 1 inside it. Using logoids, the task of finding steady states is considerably simplified. A list of regions in phase space comprising all steady states lying close to a threshold is obtained by examining a certain type of matrix called the Logoid-Jacobian. In addition, this matrix leads to the conditions necessary for stability of the steady states. External signals may be conveniently incorporated in the form of Boolean variables. Thus the framework is well suited for studying gene regulatory networks both in single cells and multicellular systems.

Animals↗

PubMed1995

The inhibition of maltose transport by the unliganded form of the maltose-binding protein of Escherichia coli: experimental findings and mathematical treatment.

Binding protein-dependent transport systems in Gram-negative enteric bacteria are multicomponent systems in which a soluble periplasmic binding protein of high substrate binding affinity establishes the major substrate recognition site. Usually, there are two integral membrane proteins which are thought to interact with the substrate loaded form of the binding protein to allow transport of substrate to occur. Transport is against the concentration gradient and needs energization by an ATP hydrolizing polypeptide. Overall transport is considered mainly unidirectional due to the high energy of ATP hydrolysis coupled to transport. In the study reported here, maltose transport in membrane vesicles in the presence of varying concentrations of unliganded maltose-binding protein but with constant amounts of maltose was measured. The conditions were chosen such that the concentration of maltose was always smaller than that of the binding protein and the initial concentration of the liganded binding protein was essentially constant. It was found that the initial rate of transport went through a maximum with increasing amounts of binding protein and declined thereafter. This finding strongly supports the conclusion that both the liganded and the unliganded forms of the binding protein interact with the membrane components of the transport system. The mathematical treatment of the experimental data allowed the ratio of the affinities for the membrane components of the substrate loaded and unloaded binding protein to be estimated. Published data on the binding protein-dependent transport of histidine in membrane vesicles of Salmonella typhimurium were also used. The data allowed the ratio of the binding affinity of the membrane components to the substrate-loaded and free binding protein to be determined. In addition, the KM of transport to the KD of binding protein was approximated.

ATP-Binding Cassette Transporters↗

PubMed1996

On the dangers of adjusting the parameters values of mechanism-based mathematical models.

Mechanism-based mathematical models describe systems in terms of identifiable physical processes, and the parameters are assumed to have fundamental physical significance. Ideally, the parameter values are measured independent of the system being modeled, but these values are often adjusted to give the best fit of model predictions to experimental data. A systematic investigation of the effects of such parameter adjustment was conducted by developing a model system comprising a known reaction mechanism and known rate constants. Simulations of experiments were run, and then attempts were made to model the system under a variety of problematic, but realistic, conditions. (1) When one rate constant was seriously in error, adjustment of a different rate constant gave the greatest improvement in the model fit. (2) When a contaminant was present in the experiment, the effects could be hidden by the adjustment of the rate constants. (3) When an incorrect reaction mechanism was assumed, the error could be hidden by parameter adjustment if the concentrations of only one of the reacting species were considered or if an unweighted fit was used for the optimization. (4) Parameter values adjusted for one set of experimental conditions gave a poorer fit than did the unadjusted parameter values when attempting to model a new set of experimental condition (addition of an inhibitor). These results show the potential dangers of adjusting parameter values and the importance of measuring as many variables as possible in a complex system.

Animals↗

PubMed1998

Mathematical analysis of antiretroviral therapy aimed at HIV-1 eradication or maintenance of low viral loads.

Motivated by the ability of combinations of antiretroviral agents to sustain viral suppression in HIV-1-infected individuals, we analyse the transient and steady-state behavior of a mathematical model of HIV-1 dynamics in vivo in order to predict whether these drug regimens can eradicate HIV-1 or maintain viral loads at low levels. The model incorporates two cell types (CD4+ T cells and a long-lived pool of cells), two strains of virus (drug-sensitive wild type and drug-resistant mutant) and two types of antiretroviral agents (reverse transcriptase and protease inhibitors). The transient behavior of the cells and virus and the eventual eradication of the virus are determined primarily by the strength of the combination therapy against the mutant strain and the maximum achievable increase in the uninfected CD4+ T cell concentration. We also predict, if the parameters of the model remain constant during therapy, that less intensive maintenance regimens will be unable to maintain low viral loads for extensive periods of time. However, if the reduction in viral load produced by therapy reduces the state of activation of the immune system, the number of cells susceptible for HIV-1 infection may decrease even though total CD4+ T cells increase. Our model predicts that if this occurs strong inductive therapy that reduces viral load followed by weaker maintenance regimes may succeed.

Antiviral Agents↗

PubMed2000

A cell cycle model for somitogenesis: mathematical formulation and numerical simulation.

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119-130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation.

Animals↗

PubMed2002

The mathematics of metabolic control analysis revisited.

In this minireview, several different approaches to derivation of the theorems and relationships of Metabolic Control Analysis (MCA) are discussed and an alternative approach is presented. This new approach consists of solving the steady-state mass balances for the intracellular metabolites using linearized kinetics. The application of linearized kinetics reflects the fact that MCA is based on linearization of the system equations in a reference steady state. Our derivation is valid for metabolic networks of arbitrary complexity, including those containing conserved moieties and branches. The value of our approach is its simplicity: the derivation is straightforward and therefore easy to follow. It can serve as a compact introduction to the mathematical basis of MCA.

Biomedical Engineering↗

PubMed2001

Mathematical analysis of hemoglobin spectrophotometry in microvessels.

Spectrophotometry of hemoglobin in microvessels is commonly performed by collecting light either from a small region around the vessel centerline or from the entire lumen of the vessel. In the latter instance, parallel rays of light may not encounter the same amount of absorbing species. Hence, a phenomenon similar to the sieve effect reported in the literature on hemoglobin spectrophotometry may be expected to occur. Although it has been observed that under such circumstances nonlinearities in calibration characteristics arise, the implications of this effect on the interpretation of the spectrophotometric mean concentration have never been addressed so far. Mathematical analysis of hemoglobin spectrophotometry in microvessels, performed in this study, reveals that for practical situations the calibration curve is indeed nonlinear. Moreover, the spectrophotometric mean oxygen saturation is an overestimate of the mean oxygen saturation during oxygenation and an underestimate of the mean oxygen saturation during deoxygenation. These deviations depend upon the manner in which the total heme concentration is distributed within the lumen. Application of the analysis to artificial microvessels showed that the observed superior oxygen transport characteristics of flowing erythrocyte suspensions and hemoglobin solution mixtures could in part be due to the assumptions underlying the procedure used to interpret the experimental results. The implications of this result on models for oxygen transport in microvessels are discussed along with possible resolutions.

Calibration↗

PubMed1988

Mathematical analysis of network topology in the cerebrocortical microvasculature.

The three-dimensional branching pattern of deep cerebrocortical capillary networks was reconstructed from histological sections. The distribution of blood flow in a mathematical model of the reconstructed network was calculated. The transit of red blood cells through the network was simulated by computer, and the total path length traveled by the cells was estimated. The results support both anatomical and hemodynamic heterogeneity of the cerebrocortical microvascular system.

Animals↗

PubMed1982

Identification of viable regions in "in vitro" spheroidal tumors: a mathematical investigation.

In the treatment of solid tumors by hyperthermia, a major question is how to obtain an a priori knowledge of which tumors can be effectively treated with this modality. The key question is; given a solid tumor, what parameters in the various regions of the tumor, need to be measured so that a tumor-tissue model can provide a meaningful real time simulation of the hyperthermic treatment. This paper addresses the former question as a mathematical investigation, and the latter as a consequence of the former.

Animals↗

PubMed1986

A mathematical model applying the random-walk method to the environment of a neuron.

A mathematical model describing the supply and demand relationships existing in the environment of a brain cell (neuron of the cerebral cortex) was developed. The stochastic random-walk technique was applied to the representation and solution of the system which consisted of a neuron being supplied with nutrients by an adjoining capillary. The random-walk method incorporated a uniformly generated random number which was weighted by the normal distribution curve to determine the random walk of a molecule. The resultant weighted value was designated as defining the motion of any particular species in space. The distribution curve was a function of diffusivity and time. The method allowed the tracking of individual molecules as they proceeded through the metabolic reactions in the cell. Oxygen, glucose, carbon dioxide and lactate were selected as the primary components of study, since they represent the major input and output parameters of metabolism inside the cell. The consumption and/or production of these components were dependent on probability values assigned to each metabolic reaction into which they entered. The solution of the model was based on the number of molecules existing in the tissue as a function of PO2, (partial pressure of oxygen), glucose level, etc. The model was very sensitive to perturbations of metabolic scheme parameters and to PO2 levels in the capillary. The model predicted an excess of O2, (oxygen) in the tissue. The effects of edema on intercapillary distances as well as changes in the size and number of mitochondria within the neuron were examined using the model.(ABSTRACT TRUNCATED AT 250 WORDS)

Animals↗

PubMed1984

Biphasic positive inotropic actions of ouabain on rat, guinea-pig and cat heart: a mathematical description.

The inotropic action of ouabain on isolated perfused hearts of rat, guinea-pig, and cat was studied over a wide concentration range (10(-12)-5 X 10(-3) M). In all three species used, the positive inotropic effect (PIE) of ouabain appeared to be biphasic in character. However, there was a remarkable difference in the course of the logdose-response curves of ouabain on guinea-pig and cat heart as compared with than on rat heart. The first two species showed, at very low concentrations of ouabain (guinea-pig heart: 10(-9) M and cat heart: 10(-10) M), a typical bell-shaped increase in cardiac contractile activity, while at higher concentrations (10(-8)-10(-6) M and 10(-9)-10(-7) M, respectively) the normally observed S-shaped increase in contractile activity occurred. On the contrary, rat hearts showed a flat S-curve between 10(-8) and 10(-6) M and a steep one between 10(-6) and 10(-4) M of ouabain. In order to explain the biphasic action of ouabain a hypothetical model for the mechanism(s) of action of ouabain is discussed. Mathematical description of this model is based on the existence of two different receptor-types for ouabain. It is suggested that sarcolemma-bound calcium may play an important role in both mechanisms of inotropic action of ouabain.

Animals↗

PubMed1991

Mathematical analysis of a basic model for epidermal wound healing.

The stimuli for the increase in epidermal mitosis during wound healing are not fully known. We construct a mathematical model which suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The numerical results of the model compare well with experimental data. We investigate the model analytically by making biologically relevant approximations. We then obtain travelling wave solutions which provide information about the accuracy of these approximations and clarify the roles of the various model parameters.

Animals↗