The time course of hydrolysis of a beta-lactam antibiotic by intact gram-negative bacteria possessing a periplasmic beta-lactamase.

An equation is derived from first principles for describing the change in concentration with time of a beta-lactam antibiotic in the presence of intact Gram-negative bacteria possessing a beta-lactamase located in the periplasmic space. The equation predicts a first-order decline in beta-lactam concentration of the form [S] = [Si]e lambda t, where [S] is the exogenous concentration of beta-lactam, [Si] is the value of [S] at time zero, t is the time from mixing of cells and antibiotic and lambda (less than 0) is the decay constant. The value of lambda is exactly described by the theory in terms of experimentally measurable quantities. Quantitative data concerning cephaloridine hydrolysis by intact cells of Haemophilus influenzae agreed well with the theory, as did data concerning the uptake of 2-nitrophenyl galactoside by intact cells of Escherichia coli. Cephalosporin C hydrolysis by intact cells of Pseudomonas aeruginosa did not progress as predicted by the theory. The theory is applicable to any substrate which is acted on by an enzyme that is located solely in the periplasmic space and that obeys the Michaelis-Menten equation of enzyme kinetics.     

Muscular contraction

A molecular theory of muscular contraction, based on the trigger action of the cross bridge between actin and myosin, is postulated. The formation of the cross bridge is followed by a transconformation in contractile protein producing work and liberating heat. The process possesses a mechanochemical character and utilizes the energy liberated by dephosphorylation of ATP. The equation of for tension dependence of muscle power is derived from the theory of reaction rates. The equation of is meaningful after elementary treatment; the physical meaning of the constants in these equations is explained. Quantitative analyses are corroborated by the experimental data.     

Nonlinear poisson equation for heterogeneous media

The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects.     

A Catastrophe Model for the Switch from Vegetative to Reproductive Growth in the Shoot Apex

A mathematical model is constructed to describe the morphopneticswitch that occurs when a vegetative plant apex becomes reproductive.The cusp equation from catastrophe theory is modified, and isused to relate primordial size at initiation to apex size. Theresulting equation may be viewed as an equation of state definingthe allowed organizational modes of the shoot apex. The modelsimulates the growth of the apex from the vetative stage toearly reproductive growth, and makes reasonable predictionsabout apex size and growth rate, primordial sizes, and the lengthsof the plastochron. flowering, mathematical model, catastrophe theory, shoot apex     

Passive Membrane Potentials: A Generalization of the Theory of Electrotonus

The theory of electrotonus, which has been well developed for small cylinders, is extended: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.     

Theory of cell age distribution in suspension culture with feed and/or drain

A mixed problem for the M'Kendrick-von Foerster equation satisfied by the number density function in terms of the age of the viable cells in a suspension culture with feed and/or drain is solved, and a method of calculating the number density function and time-dependent generation time from observed data of cell number and cell mass is presented. This theory is adequate to analyze the growth of cells that undergo binary fission. The equation of mass balance follows as a natural consequence of this treatment. The equation of substrate balance in consideration of the effect of cell volume is derived rigorously.     

pH influence on enzymic activity: the involvement of two active ionized forms of either substrate or enzyme in the reaction.

The interpretation of graphs of pKm as a function of pH by the theory of Dixon (Dixon, 1953) is based on the assumption that each component of the reaction system is active in a particular ionic state and completely inactive in other ionic states. In this paper we analyse the case in which two ionic forms of either the substrate or the enzyme are active to different degrees. After definition of reaction system of this type, a generalized equation is derived connecting the reaction rate to the total substrate concentration. The pH dependence of Km deduced from this equation is discussed and compared with that deriving from the theory of Dixon. The present model predicts that the pKm dependence on pH has the form of a “wave” with two inflexion points, thus providing an alternative interpretation of such behaviour to that afforded by Dixon's formulation.     

Conditional diffusion processes in population genetics

Modern population genetics is involved largely with molecular processes and in particular with the observed outcome of an evolutionary molecular process. The investigation of data of this form requires a mathematical apparatus of conditional diffusion processes. The relevant theory is given in this paper. Many results can be obtained more or less directly: however, the complete theory requires formation of the conditional diffusion equation. The form of this equation, together with several consequences flowing from it, are given.     

Enzyme kinetics: Inhibition equations derived from the inelastic collision hypothesis

The inelastic collision hypothesis of enzyme action has been proposed by G. Medwedew. This theory has been here extended to the cases of competitive and non-competitive inhibitors, which form enzyme-inhibitor complexes, and to the cases of competitive and non-competitive substrates. The resulting equations are discussed and contrasted to those derived from the classical enzyme-substrate hypothesis. The original formulation of Medwedew neglects the presence of the enzyme-substrate complex although the general theory admits the formation of this complex. The formulation has been revised and extended to include complex formation. The resulting equation is discussed in terms of the usual criteria used to evaluate the Michaelis-Menten-Briggs-Haldane equation. Taken in part from a thesis submitted by Robert Katzman to the faculty of the Department of Physiology, University of Chicago, in partial fulfillment of the requirements for the Degree of Master of Science.     

Energy transfer in photosynthesis

A theoretical model is presented to account for the physical mechanism of energy transfer from antenna molecules to the reaction centers in photosynthesis. The energy transfer is described by a generalized transport equation or “master equation”. The solution of this equation for the proposed model gives a relationship between the antennae interaction energy and the transfer rate. The results are shown to be in agreement with inter-antenna transfer rates calculated from experimental fluorescence lifetimes. Previous theories were based either on the Förster mechanism, which is valid for very small interaction energies, or an exciton model valid for very large interactions, but experimental results seemed to indicate that the actual situation was intermediate between these two. The Förster theory and the exciton model are limiting cases of the master equation.     

The δ-function counterion condensate around a line charge derived from the Fuoss,Katchalsky, and Lifson polyelectrolyte theory

It has recently been proven that the counterion condensate around an isolated line charge in an electrolyte, as characterized by nonlinear Poisson-Boltzmann theory, is an encapsulating δ-function. Here the identical result is shown to hold in the framework of the polyelectrolyte theory of Fuoss, Katchalsky, and Lifson. The proof fully exploits analytic solutions to the differential equation which are not available for the nonlinear, cylindrical Poisson-Boltzmann equation.     

On the theory of cell division from the point of view of the principle of maximum energy exchange

An analysis is made of the equation of cellular elongation during division, which was derived by N. Rashevsky from the principle of maximum energy exchange. The method used is the same as that employed by H. D. Landahl in discussing a similar equation deduced from the theory of diffusion drag forces. The differential equation is expanded in a power series of the relative elongation, and in this way is reduced to the form studied by H. D. Landahl, which has been shown to agree very well with experimental data. An estimate of the order of magnitude of the universal constant τ, which appears in the generalized Hamiltonian principle, is made, and τ is found to be of the order of 10⁻⁴ sec.     

Integral equation theories for predicting water structure around molecules

Water plays a crucial role in the structure and function of proteins and other biological macromolecules; thus, theories of aqueous solvation for these molecules are of great importance. However, water is a complex solvent whose properties are still not completely understood. Statistical mechanical integral equation theories predict the density distribution of water molecules around a solute so that all particles are fully represented and thus potentially both molecular and macroscopic properties are included. Here we discuss how several theoretical tools we have developed have been integrated into an integral equation theory designed for globular macromolecular solutes such as proteins. Our approach predicts the three-dimensional spatial and orientational distribution of water molecules around a solute. Beginning with a three-dimensional Ornstein-Zernike equation, a separation is made between a reference part dependent only on the spatial distribution of solvent and a perturbation part dependent also on the orientational distribution of solvent. The spatial part is treated at a molecular level by a modified hypernetted chain closure whereas the orientational part is treated as a Boltzmann prefactor using a quasi-continuum theory we developed for solvation of simple ions. A potential energy function for water molecules is also needed and the sticky dipole models of water, such as our recently developed soft-sticky dipole (SSD) model, are ideal for the proposed separation. Moreover, SSD water is as good as or better than three point models typically used for simulations of biological macromolecules in structural, dielectric and dynamics properties and yet is seven times faster in Monte Carlo and four times faster in molecular dynamics simulations. Since our integral equation theory accurately predicts results from Monte Carlo simulations for solvation of a variety of test cases from a single water or ion to ice-like clusters and ion pairs, the application of this theory to biological macromolecules is promising.     

On the mathematical theory of rumor spread

The applicability of the theory of random nets to the theory of rumor spread is shown. In particular the “weak connectivity” of the net appears as the saturation fraction of “knowers” in a thoroughly mixed population through which a message diffuses where each knower tells the message to a finite average number of individuals. Further it is shown how the time course equation of rumor spread, where time is measured by the number of “removes” from the starters, can be translated into an ordinary continuous time course equation if the distribution of the telling intervals is known.     

Mathematical biophysics of the galvanic skin response

Beginning with Rashevsky's equation for the development of the excitatory state in a nerve fiber, an equation for the change in skin resistance upon the presentation of an instantaneous stimulus is derived. The mechanism assumed is in conformity with the existing evidence of neuro-physiology. Certain deductions from the equations are made and experimental problems suggested for testing the theory.     

Electroosmotic Flow in Nanoscale Parallel-plate Channels: Molecular Simulation Study and Comparison with Classical Poisson–Boltzmann Theory

Molecular dynamics simulations have been carried out for simple electrolyte systems to study the electrokinetically driven osmotic flow in parallel-plate channels of widths ～10–120?nm. The results are compared with the classical theory predictions based on the solution to the Poisson–Boltzmann equation. We find that despite some of the limitations in the Poisson–Boltzmann equation, such as assumption of the Boltzmann distribution for the ions, the classical theory captures the general trend of the variations of the osmotic flow with channel width, as characterized by the mobility of the fluid in channels between ～10 and 120?nm at moderate to low ion concentration. At moderate concentration (corresponding to relatively low surface potential), the classical theory is almost quantitative. The theory and simulation show more disagreement at low concentration, primarily caused by the high surface potential where the assumption of Boltzmann distribution becomes inaccurate. We discuss the limitations of the Poisson–Boltzmann equation as applied to the nanoscale channels.     

Theoretical considerations on the C-D effect in self-thinning plant populations

A model for describing the competition–density (C-D) effect in self-thinning populations was developed on the basis of the following three basic assumptions: (1) the growth of mean phytomass follows a general logistic equation; (2) final yield is independent of initial population density; and (3) there exists a functional relationship between actual and initial population densities at any given time. The resultant equation takes the same reciprocal form as the reciprocal equation of the C-D effect derived from Shinozaki–Kira's theory (i.e., the logistic theory of the C-D effect), which deals with the density effect in nonself-thinning populations. However, one of the two time-dependent coefficients is quite different in mathematical interpretation between the two reciprocal equations. The reciprocal equation for self-thinning populations is essentially the same as the reciprocal equation assumed in the derivation of the functional relationship between actual and initial population densities. The establishment of the reciprocal equation is supported by the empirical facts that the reciprocal relationship between mean phytomass and population density is discernible in not only nonself-thinning populations but also in self-thinning populations. The present model is expected to systematically interpret underlying mechanisms between the C-D effect, which is observed at a time constant among populations with various initial densities, and self-thinning, which is observed along a time continuum in a given population. Received: August 5, 1998 / Accepted: January 7, 1999     

Low Reynolds number steady state flow through a branching network of rigid vessels: I. A mixture theory

A mixture theory has been used to formulate a theory of blood perfusion. By means of a formal averaging procedure the discrete network of microvessels is transformed into a continuum. During this procedure, the distinction between arterioles, capillaries and venules is preserved by means of an arteriovenous parameter. In this paper, two equations are derived for the case of low Reynolds number steady-state flow through a rigid vessel network: the extended Darcy equation and the continuity equation. A verification of the theory is presented, on the basis of a network analysis.     

The matching of thermal fields surrounding countercurrent microvessels and the closure approximation in the Weinbaum-Jiji equation

A second order perturbation theory is developed to show the difference between the average artery-vein temperature Tm and the local average tissue temperature theta. This theory demonstrates that the closure approximation in the Weinbaum-Jiji bioheat equation does not require that theta = Tm and that although the difference between these two temperatures is second order the magnitude of the countercurrent convection terms in the Weinbaum-Jiji equation can be of order unity. The theory also shows that to second order this new bioheat equation is the same as the simplified set of one-dimensional model equations used in Baish et al.