Physio-chemical mathematical theory for transitions in cellular metabolism: An aspect of carcinogenesis

It is pointed out that the non-monotonic character of the chemical reaction rate expressions, together with the relative magnitude of the diffusivity constants, is likely to engender a multiplicity of locally stable steady-state solutions to the system of reaction-diffusion equations for the concentration distributions of molecular species through the volume of a living cell. A transition in cellular metabolism, i.e., the dynamical evolution from an initial locally stable steady-state solution for the concentration distributions to another distinct locally stable steady-state solution, can be induced by an etiologic agent which modifies the rate expressions significantly during an interval of time. Global inequality analysis is employed to derive a condition on the modified rate expressions that is sufficient to guarantee the occurrence of such a transition in cellular metabolism. The possibility of a transition induced by a chemical carcinogen is investigated by applying the latter sufficient condition, and it is found that the statistical frequency of carcinogenesis should depend essentially on the magnitude of the grouping (T ^{2 − α} D ^α) for a total doseD of carcinogen administered to an animal at a uniform rate (D/T) over a time interval of durationT, where α is a certain positive number less than 1. This theoretical result is shown to be supported by the available experimental evidence.     

Enzyme kinetics at high enzyme concentration

We re-visit previous analyses of the classical Michaelis-Menten substrate-enzyme reaction and, with the aid of the reverse quasi-steady-state assumption, we challenge the approximation d[C]/dt ≈ 0 for the basic enzyme reaction at high enzyme concentration. For the first time, an approximate solution for the concentrations of the reactants uniformly valid in time is reported. Numerical simulations are presented to verify this solution. We show that an analytical approximation can be found for the reactants for each initial condition using the appropriate quasi-steady-state assumption. An advantage of the present formalism is that it provides a new procedure for fitting experimental data to determine reaction constants. Finally, a new necessary criterion is found that ensures the validity of the reverse quasi-steady-state assumption. This is verified numerically.     

Membrane transport models with fast and slow reactions: General analytical solution for a single relaxation

Summary Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.     

A mathematical model for prokaryotic protein synthesis

A kinetic model for the synthesis of proteins in prokaryotes is presented and analysed. This model is based on a Markov model for the state of the DNA strand encoding the protein. The states that the DNA strand can occupy are: ready, repressed, or having a mRNA chain of length i in the process of being completed. The case i = 0 corresponds to the RNA polymerase attached, but no nucleotides attached to the chain. The Markov model consists of differential equations for the rates of change of the probabilities. The rate of production of the mRNA molecules is equal to the probability that the chain is assembled to the penultimate nucleotide, times the rate at which that nucleotide is attached. Similarly, the mRNA molecules can also be in different states, including: ready and having an amino acid chain of length j attached. The rate of protein synthesis is the rate at which the chain is completed. A Michaelis-Menten type of analysis is done, assuming that the rate of protein degradation determines the ’slow’ time, and that all the other kinetic rates are ‘fast’. In the self-regulated case, this results in a single ordinary differential equation for the protein concentration.     

A note on a paper by Erik Volz: SIR dynamics in random networks

Recent work by Volz (J Math Biol 56:293–310, 2008) has shown how to calculate the growth and eventual decay of an SIR epidemic on a static random network, assuming infection and recovery each happen at constant rates. This calculation allows us to account for effects due to heterogeneity and finiteness of degree that are neglected in the standard mass-action SIR equations. In this note we offer an alternate derivation which arrives at a simpler—though equivalent—system of governing equations to that of Volz. This new derivation is more closely connected to the underlying physical processes, and the resulting equations are of comparable complexity to the mass-action SIR equations. We further show that earlier derivations of the final size of epidemics on networks can be reproduced using the same approach, thereby providing a common framework for calculating both the dynamics and the final size of an epidemic spreading on a random network. Under appropriate assumptions these equations reduce to the standard SIR equations, and we are able to estimate the magnitude of the error introduced by assuming the SIR equations.     

Bioethanol production from ammonia percolated wheat straw

This study examined the effectiveness of ammonia percolation pretreatment of wheat straw for ethanol production. Ground wheat straw at a 10% (w/v) loading was pretreated with a 15% (v/v) ammonia solution. The experiments were performed at treatment temperature of 50∼170°C and residence time of 10∼150 min. The solids treated with the ammonia solution showed high lignin degradation and sugar availability. The pretreated wheat straw was hydrolyzed by a cellulase complex (NS50013) and β-glucosidase (NS50010) at 45°C. After saccharification, Saccharomyces cerevisiae was added for fermentation. The incubator was rotated at 120 rpm at 35°C. As a result of the pretreatment, the delignification efficiency was > 70% (170°C, 30 min) and temperature was found to be a significant factor in the removal of lignin than the reaction time. In addition, the saccharification results showed an enzymatic digestibility of > 90% when 40 FPU/g cellulose was used. The ethanol concentration reached 24.15 g/L in 24 h. This paper reports a total process for bioethanol production from agricultural biomass and an efficient pretreatment of lignocellulosic material.     

Influence of the Selected Antioxidants on the Stability of the Celsior Solution Used for Perfusion and Organ Preservation Purposes

The purpose of the following research was to improve the original Celsior solution in order to obtain a higher degree of stability and effectiveness. The solution was modified by the addition of selected antioxidants such as vitamin C, cysteine, and fumaric acid in the following concentrations: 0.1, 0.3, and 0.5 mmol/l. The solution’s stability was estimated using an accelerated stability test based on changes in histidine concentrations in the solution using Pauly’s method for determining concentrations. Elevated temperatures, the factor accelerating substances’ decomposition reaction rate, were used in the tests. The research was conducted at four temperatures at intervals of 10°C: 60 ± 0.2°C, 70 ± 0.2°C, 80 ± 0.2°C, and 90 ± 0.2°C. It was stated that the studied substances’ decomposition occurred in accordance with the equation for first-order reactions. The function of the logarithmic concentration (log%C) over time was revealed to be rectilinear. This dependence was used to determine the kinetics of decomposition reaction rate parameters (the rate constant of decomposition k, activation energy E _a, and frequency factor A). On the basis of these parameters, the stability of the modified solution was estimated at +5°C. The results obtained show that the proposed antioxidants have a significant effect on lengthening the Celsior solution’s stability. The best results were reached when combining two antioxidants: vitamin C and cysteine in 0.5 mmol/l concentrations. As a result, the Celsior solution’s stability was lengthened from 22 to 299 days, which is 13.5 times. Vitamin C at a concentration of 0.5 mmol/l increased the solution’s stability by 5.2 times (t ₉₀ = 115 days), cysteine at a concentration of 0.5 mmol/l caused a 4.4 times stability increase (t ₉₀ = 96 days), and fumaric acid at a concentration of 0.5 mmol/l extended the stability by 2.1 times (t ₉₀ = 48 days) in relation to the original solution.     

Ion transport and displacement currents with membrane-bound carriers

Summary The standard carrier model for ion transport by a one-to-one mechanism is developed to predict the time-dependent currents for systems that are symmetrical at zero applied potential. The complete solution for ions and carriers bearing any charge is derived by assuming that the concentration of ions in the membrane is low and either that the applied potential is small or that the applied potential affects equally all of the association and dissociation reactions between the ions and the carriers. The response to an abruptly applied potential is then given by the sum of a constant and two declining exponential terms. The time constants of these relaxations are described by the equations derived for neutral carriers by Stark, Ketterer, Benz and Läuger in 1971 (Biophys. J. 11:981). The sum of the amplitudes of the exponentials for small applied potentials obeys a relation like that first derived by Markin and Liberman in 1973 (Biofizika 18:453). For small applied potentials expressions are also provided for the voltage transients in charge-pulse experiments and for the membrane admittance.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

Selective quenching of the fluorescence of core chlorophyll–protein complexes by photochemistry indicates that Photosystem II is partly diffusion limited

The spectral characteristics of fluorescence quenching by open reaction centres in isolated Photosystem II membranes were determined with very high resolution and analysed. Quenching due to photochemistry is maximal near 687 nm, minimal in the chlorophyll b emission interval and displays a distinctive structure around 670 nm. The amplitude of this `quenching hole' is about 0.03 for normalised spectra. On the basis of the absorption spectra of isolated chlorophyll–protein complexes, it is shown that these quenching structures can be exactly described by assuming that photochemistry lowers the fluorescence yield of the reaction centre complex (D1/D2/cytb ₅₅₉) plus CP47, with quenching of the former complex being approximately double that of the latter complex. These data, which qualitatively indicate that there are kinetically limiting processes for primary photochemistry in the antenna, have been analysed by means of several different kinetic models. These models are derived from present structural knowledge of the arrangement of the chlorophyll–protein complexes in Photosystem II and incorporate the reversible charge separation characteristic of the exciton/radical pair equilibration model. In this way it is shown that Photosystem II cannot be considered to be purely trap limited and that exciton migration in the antenna imposes a diffusion limitation of about 30%, irrespective of the structural model assumed. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Generalised Age- and Phase-Structured Model of Human Tumour Cell Populations Both Unperturbed and Exposed to a Range of Cancer Therapies

We develop a general mathematical model for a population of cells differentiated by their position within the cell division cycle. A system of partial differential equations governs the kinetics of cell densities in certain phases of the cell division cycle dependent on time t (hours) and an age-like variable τ (hours) describing the time since arrival in a particular phase of the cell division cycle. Transition rate functions control the transfer of cells between phases. We first obtain a theoretical solution on the infinite domain −∞ < t < ∞. We then assume that age distributions at time t=0 are known and write our solution in terms of these age distributions on t=0. In practice, of course, these age distributions are unknown. All is not lost, however, because a cell line before treatment usually lies in a state of asynchronous balanced growth where the proportion of cells in each phase of the cell cycle remain constant. We assume that an unperturbed cell line has four distinct phases and that the rate of transition between phases is constant within a short period of observation (‘short’ relative to the whole history of the tumour growth) and we show that under certain conditions, this is equivalent to exponential growth or decline. We can then gain expressions for the age distributions. So, in short, our approach is to assume that we have an unperturbed cell line on t ≤ 0, and then, at t=0 the cell line is exposed to cancer therapy. This corresponds to a change in the transition rate functions and perhaps incorporation of additional phases of the cell cycle. We discuss a number of these cancer therapies and applications of the model.     

Further contributions to a probabilistic interpretation of the mathematical biophysics of the central nervous system

Following a previous paper, equations are derived for the most probable time of firing of an efferent neuron in terms of the intensityE of excitation of the afferent pathway, whenE is either constant or any given function of time. The equations are not differential equations, but in integral form. It is suggested that ε, correspondinglyj, represent the number of excitatory, correspondingly inhibitory, terminal bulbs excited within the period of latent addition at a given most probable time. The relation between the suggested theory and the old one, based on differential equations for ε andj is discussed.     

Effect of Human Serum Albumin on the Kinetics of 4-Methylumbelliferyl-β-D-<Emphasis Type="Italic">N-N</Emphasis>′-<Emphasis Type="Italic">N</Emphasis>″ Triacetylchitotrioside Hydrolysis Catalyzed by Hen Egg White Lysozyme

The effect of human serum albumin (HSA) addition on the rate of hydrolysis of the synthetic substrate 4-methylumbelliferyl-β-D-N-N′-N″ triacetylchitotrioside ((NAG)₃-MUF) catalyzed by hen egg white lysozyme has been measured in aqueous solution (citrate buffer 50 mM pH = 5.2 at 37 °C). The presence of HSA leads to a decrease in the rate of the process. The reaction follows a Michaelis–Menten mechanism under all the conditions employed. The catalytic rate constant decreases tenfold when the albumin concentration increases, while the Michaelis constant remains almost constant in the albumin concentration range employed. Ultracentrifugation experiments indicate that the main origin of the observed variation in the kinetic behavior is related to the existence of an HSA–lysozyme interaction. Interestingly, the dependence of the catalytic rate constant with albumin concentration parallels the decrease of the free enzyme concentration. We interpret these results in terms of the presence in the system of two enzyme populations; namely, the HSA associated enzyme which does not react and the free enzyme reacting as in the absence of albumin. Other factors such as association of the substrate to albumin or macromolecular crowding effects due to the presence of albumin are discarded. Theoretical modeling of the structure of the HSA–lysozyme complex shows that the Glu35 and Asp52 residues located in the active site of lysozyme are oriented toward the HSA surface. This conformation will inactivate lysozyme molecules bound to HSA.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Ultracentrifugation studies on the transmembrane domain of the human erythrocyte anion transporter Band 3 in the detergent C12E8

The dilute solution behaviour of the transmembrane domain (TMD) of the human erythrocyte anion exchanger Band 3 was studied by analytical ultracentrifugation. Sedimentation velocity and equilibrium studies of the TMD solubilized with the detergent C₁₂E₈ demonstrate that the protein is a stable dimer in the concentration range 0.1 to 1 mg/ml. There is no evidence of a dissociation at low concentrations or of an association at higher concentrations. Hydrodynamic calculations applying a prolate ellipsoid of revolution and assuming a hydration of w=0.35 result in an asymmetrical particle with an axial ratio (a/b) of ∼3.5. Received: 8 January 1998 / Revised version: 21 April 1998 / Accepted: 22 April 1998     

Immobilized enzymes: Electrokinetic effects on reaction rates under external diffusion

The rates of reactions catalyzed by enzymes immobilized on a nonporous solid surface have been computed employing a Nernst film model. The Nernst-Planck equations for the transport of the charged substrate and product species in the film and the Poisson equation for the distribution of electrical potential are solved numerically with the appropriate boundary conditions. The electrical charge at the surface is assumed to arise from the dissociation equilibria of the acidic and basic surface groups of the enzyme. The pH at the surface affects both the surface charge as well as the intrinsic kinetics of the enzyme-catalyzed reaction. Factors which determine the pH at the surface include the pH in the bulk solution and the release of H(+) ions in the enzyme-catalyzed reaction. The latter causes a lowering of pH at the surface, causing the reaction rate to differ from that computed assuming an equilibrium distribution of electrical potential. Another kind of nonequilibrium contribution is caused by unequal charges or diffusivities of the substrate and products, which results in a diffusion potential being set up. Two moduli are introduced to evaluate the significance of the reaction-generated lowering of pH and the diffusion potential effect. The effect of changing various parameters, e.g., reaction rate constant, substrate concentration, enzyme concentration, pH, etc., on the overall reaction rate are studied.     

On the Fokker-Planck equation in the stochastic theory of mortality: II

This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.     

Reaction center and UQH2:cytc 2 oxidoreductase act as independent enzymes inRps. sphaeroides

Turnover of the ubiquinol oxidizing site of the UQH₂:cyt c₂ oxidoreductase (b/c ₁ complex) ofRps. sphaeroides can be assayed by measuring the rate of reduction of cytb ₅₆₁ in the presence of antimycin (AA). Oxidation of ubiquinol is a second-order process, with a value ofk ₂ of about 3 × 10⁵ M^–1. The reaction shows saturation at high quinol concentrations, with an apparentK _m of about 6–8 mM (with respect to the concentration of quinol in the membrane). When the quinone pool is oxidized before illumination, reduction of the complex shows a substantial lag (about 1 ms) after a flash, indicating that the quinol produced as a result of the photochemical reactions is not immediately available to the complex. We have suggested that the lag may be due to several factors, including the leaving time of the quinol from the reaction center, the diffusion time to the complex, and the time for the head group to cross the membrane. We have suggested aminimal value for the diffusion coefficient of ubiquinone in the membrane (assuming that the lag is due entirely to diffusion) of about 10^–9 cm^–2 sec^–1. The lag is reduced to about 100 µsec when the pool is significantly reduced, showing that quinol from the pool is more rapidly available to the complex than that from the reaction center. With the pool oxidized, similar kinetics are seen when the reduction of cytb ₅₆₁ occurs through the AA-sensitive site (with reactions at the quinol oxidizing site blocked by myxothiazol). These results show that there is no preferential reaction pathway for transfer of reducing equivalents from reaction center tob/c ₁ complex. Oxidation of cytb ₅₆₁ through the AA-sensitive site can be assayed from the slow phase of the carotenoid electrochromic change, and by comparison with the kinetics of cytb ₅₆₁. As long as the quinone pool is significantly oxidized, the reaction is not rate-determining for the electrogenic process. On reduction of the pool below 1 quinone per complex, a slowing of the electrogenic process occurs, which could reflect a dependence on the concentration of quinone. If the process is second-order, the rate constant must be about 2–5 times greater than that for quinol oxidation, since the effect on rate is relatively small compared with the effect seen at the quinol oxidizing site when the quinol concentration is changed over theE _h range where the first few quinols are produced on reductive titration. When the quinone pool is extracted (experiments in collaboration with G. Venturoli and B. A. Melandri), the slowing of the electrochromic change on reduction of the pool is not enhanced; we assume that this is due to the fact that a minimum of one quinone per active complex is produced by turnover of the quinol oxidizing site. Two lines of research lead us to revise our previous estimate for the minimal value of the quinone diffusion coefficient. These relate to the relation between the diffusion coefficient and the rate constants for processes involving the quinones: (a) The estimated rate constant for reaction of quinone at the AA-site approaches the calculated diffusion limited rate constant, implying an improbably efficient reaction. (b) From a preliminary set of experiments, the activation energy determined by measuring the variation of the rate constant for quinol oxidation with temperature, is about 8 kcal mol^–1. Although we do not know the contribution of entropic terms to the pre-exponential factor, the result is consistent with a considerably larger value for the diffusion coefficient than that previously suggested.