Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible,partly injectable multicompartment systems

An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.     

A note on the integral-equation description of metabolizing systems

The assumptions latent in the derivation of the integral equation of Branson are rendered explicit and discussed. It is shown that the equation is valid only for systems in which the substance disappears according to a linear rate law.     

Nonlinear generalizations of the Kedem-Katachalsky equations for ionic fluxes

The transport equation of Kedem and Katchalsky for the flux of ions through a membrane is generalized to demonstrate explicitly the role of impermeant ions in determining its mathematical form. Whereas the Kedem-Katchalsky equation is linear in the salt concentrations in the bathing solutions, the more general equation is bilinear (and symmetric) in the ionic concentrations of the permeant species. The Kedem-Katchalsky flux equation is further generalized to include explicitly a term for ion-exchange in systems having more than a single permeant salt. This additional term is also bilinear (and antisymmetric) in the concentrations of the exchanging ionic species. Flux equations are derived for systems having (1) a single mono-monovalent salt, (2) two mono-monovalent salts and (3) an arbitrary number of salts with no restriction upon the valencies of the ionic components. Since it has no effect upon the form of concentration-dependent terms in the flux equations, coupling to volume flow is neglected.     

A model for the study of diffusion and perfusion limitation

On the basis of a very simple model for the association of diffusion and perfusion, an association common to many respiratory gas transfers, a simple equation is described that defines gas partial pressure equilibration in diffusion-perfusion-limited systems as a function of the ratio of D to beta bQ(D = diffusing capacity, beta b = blood capacitance coefficient, Q = perfusion). The equation applies to steady-state conditions and assumes D, beta b, and Q to be independent of gas partial pressures. In spite of the fact that this assumption may represent a gross simplification, the equation can be regarded as a powerful conceptual tool in the analysis of most gas exchange systems.     

对Logistic增长的SIS模型实现反馈线性化和极点配置的一步设计

非线性系统极点配置状态反馈调节器的常用方法为两步设计,而且要求必须满足对合条件,对于一个高于二阶的实际系统很难满足这些严格的条件,精确线性化和极点配置一步设计的方法避免了严格的对合条件,本文将这种方法运用在非线性微分代数系统中,并将此法应用到生物系统上．     

Three-dimensional waves in excitable reaction-diffusion systems

The eikonal equation [5] for excitable media is generalised to three dimensional systems. The main result of the investigation is the demonstration of the existence of toroidal and twisted toroidal scroll waves in the limit of large values of the major radius of the torus. The existence of a helical wave near the z-axis follows from the eidonal equation but its connection with the twisted toroidal scroll remains to be demonstrated. The eikonal equation also predicts a non-uniform rate of rotation of the cross-sectional spiral wave near the toroidal axis. The notion of geometrical stability is introduced for the case of an expanding sphere; in particular it is shown that a discussion of stability of solutions of the eikonal equation must take into account the possible shift in the origin of the coordinate systems with respect to which patterns are defined. On leave of absence from: The Department of Mathematics, Glasgow College of Technology, Cowcaddens Road, Glasgow G4 0BA, UK     

A stability analysis of the power-law steady state of marine size spectra

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a “jump-growth” equation, a first order approximation which is the widely used McKendrick–von Foerster equation, and a second order approximation which is the McKendrick–von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick–von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick–von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.     

一类平面可积二次非Hamilton系统的Poincare’分支

文章研究了一类可逆情况下，可积非Hamilton系统在二次多项式扰动下的Poincar6分支问题，此类问题归于弱化的Hilbert第16问题，等价于研究Abel积分的零点个数．首先采用适当的坐标变换，借助于数学软件Matlab的符号计算功能，将Abel积分写成三个线性无关的函数形式；再利用Picard—Fuchs方程和Riccati方程法，得出此系统的Poincar6分支可以扰动出极限环的个数为2≤n≤3．     

Modeling stochastic noise in gene regulatory systems

The Master equation is considered the gold standard for modeling the stochastic mechanisms of gene regulation in molecular detail, but it is too complex to solve exactly in most cases, so approximation and simulation methods are essential. However, there is still a lack of consensus about the best way to carry these out. To help clarify the situation, we review Master equation models of gene regulation, theoretical approximations based on an expansion method due to N.G. van Kampen and R. Kubo, and simulation algorithms due to D.T. Gillespie and P. Langevin. Expansion of the Master equation shows that for systems with a single stable steady-state, the stochastic model reduces to a deterministic model in a first-order approximation. Additional theory, also due to van Kampen, describes the asymptotic behavior of multistable systems. To support and illustrate the theory and provide further insight into the complex behavior of multistable systems, we perform a detailed simulation study comparing the various approximation and simulation methods applied to synthetic gene regulatory systems with various qualitative characteristics. The simulation studies show that for large stochastic systems with a single steady-state, deterministic models are quite accurate, since the probability distribution of the solution has a single peak tracking the deterministic trajectory whose variance is inversely proportional to the system size. In multistable stochastic systems, large fluctuations can cause individual trajectories to escape from the domain of attraction of one steady-state and be attracted to another, so the system eventually reaches a multimodal probability distribution in which all stable steady-states are represented proportional to their relative stability. However, since the escape time scales exponentially with system size, this process can take a very long time in large systems.     

Modeling the kinetics of the pectin methylesterase catalyzed de-esterfication of pectin in frozen systems

The applicability of the William, Landel, and Ferry (WLF) equation with a modification to take into account the effect of melt-dilution and an empirical log-logistic equation were evaluated to model the kinetics of diffusion-controlled reactions in frozen systems. Kinetic data for the pectin methylesterase catalyzed hydrolysis of pectin in four model systems with different glass transition temperatures: sucrose, maltodextrin (DE = 16.5-19.5), carboxymethylcellulose (CMC) and fructose in a temperature range of -24 to 0 degrees C were used. The modified WLF equation was evaluated with a concentration-dependent glass transition temperature (T(g)) as well as the glass transition temperature of the maximally freeze-concentrated matrix (T(g)') as reference temperatures. The equation with temperature-dependent T(g) described the reaction kinetics reasonably well in all the model systems studied. However the kinetics was better described by a linear relationship between log(V(0)/V(0ref)) and (T - T(ref)) in all cases except CMC. The log-logistic equation also described the kinetics reasonably well. The effect of melt-dilution on reactant concentration was found to be minimal in all cases.     

Evaluation of a simplified generic bi-substrate rate equation for computational systems biology

The evaluation of a generic simplified bi-substrate enzyme kinetic equation, whose derivation is based on the assumption of equilibrium binding of substrates and products in random order, is described. This equation is much simpler than the mechanistic (ordered and ping-pong) models, in that it contains fewer parameters (that is, no K(i) values for the substrates and products). The generic equation fits data from both the ordered and the ping-pong models well over a wide range of substrate and product concentrations. In the cases where the fit is not perfect, an improved fit can be obtained by considering the rate equation for only a single set of product concentrations. Due to its relative simplicity in comparison to the mechanistic models, this equation will be useful for modelling bi-substrate reactions in computational systems biology.     

On the Accuracy of Numerical Solutions for Some Stereological Problems such as the Wicksell Corpuscle Problem

Two stereological problems—the estimation of sphere diameter distributions and the estimation of sheet thickness distributions from linear or plane sections—are considered. Their numerical solution consists in the solution of linear equation systems. To compare the quality of various methods theoretically, not only by experiments, the condition numbers of the matrices of the corresponding equation systems are determined.     

Thermodynamic modeling of the partitioning of biomolecules in aqueous two-phase systems using a modified Flory–Huggins equation

A modified Flory–Huggins equation accounting for the solvation of polymer molecules by water molecules was used to model the phase behavior of aqueous two-phase systems (ATPS) formed by poly(ethylene glycol) (PEG) and dextran. The parameters of the equation were obtained by fitting experimental equilibrium data either accounting for or disregarding dextran polidispersity. The modified equation was subsequently applied to calculate partition coefficients of biomolecules in these systems. It was found that accounting for polidispersity did not affect significantly the calculated phase equilibrium, but increased the agreement of calculated partition coefficients with experimental data. Further improvement was obtained by using a size dependent interaction parameter for dextran pseudo-components.     

Solvation energy density occlusion approximation for evaluation of desolvation penalties in biomolecular interactions

In calculations involving many displacements of an interacting pair of biomolecules, such as brownian dynamics, the docking of a substrate/ligand to an enzyme/receptor, or the screening of a large number of ligands as prospective inhibitors for a particular receptor site, there is a need for rapid evaluation of the desolvation penalties of the interacting pair. Although continuum electrostatic treatments with distinct dielectric constants for solute and solvent provide an account of the electrostatics of solvation and desolvation, it is necessary to re-solve the Poisson equation, at considerable computational cost, for each displacement of the interacting pair. We present a new method that uses a formulation of continuum electrostatic solvation in terms of the solvation energy density and approximates desolvation in terms of the occlusion of this density. We call it the SEDO approximation. It avoids the need to re-solve the Poisson equation, as desolvation is now estimated by an integral over the occluded volume. Test calculations are presented for some simple model systems and for some real systems that have previously been studied using the Poisson equation approach: MHC class I protein-peptide complexes and a congeneric series of human immunodeficiency virus type 1 (HIV-1) protease--ligand complexes. For most of the systems considered, the trends and magnitudes of the desolvation component of interaction energies obtained using the SEDO approximation are in reasonable correlation with those obtained by re-solving the Poisson equation. In most cases, the error introduced by the SEDO approximation is much less than that of the often-used test-charge approximation for the charge-charge components of intermolecular interactions. Proteins 2001;43:12-27.     

A mathematical description of metabolizing systems: II

Some of the laboratory procedures available for determining the functions in the integral equation established in part I are discussed. The tracer or tagged molecule technique is shown to be especially promising including the use of “double tracer” molecules. Conversely, the integral equation may be a convenient device for correlating and integrating some of the work now being done with tracer molecules in biological systems.     

The distribution of wearout over evolved reliability structures

A multiple-integral equation, termed the wearout equation, describes the distribution of wearout (or aging) over evolved reliability structures, such as organisms and self-replicating machines, and thus statistically governs virtually all aging properties of the systems. The equation is applied to the computation of ab initio ("from the beginning") life tables for four natural populations of ungulates--wild boar, Dall sheep, African buffalo, and hippopotamus--which represent a broad range of survival characteristics. The good agreement of the ab initio and empirical tables, the best available for testing the theory, demonstrates the basic realism of the wearout equation. If the equation withstands further experimental testing, its analysis may provide insight into fundamental questions in the biology of aging.     

Kinetics of the alkaline phosphatase catalyzed hydrolysis of disodium p-nitrophenyl phosphate in frozen model systems

The alkaline phosphatase catalyzed hydrolysis of disodium-p-nitrophenyl phosphate was studied in four model systems comprising sucrose, maltodextrin, carboxymethylcellulose (CMC), and CMC-lactose in a temperature range of -28 to 20 degrees C. In the maltodextrin and CMC-lactose model systems, the reaction rate decreased to a very low value as the glass transition temperature was approached. In the CMC and CMC-lactose systems with low initial solute concentration, as a consequence of freeze-concentration, a rate maximum around the initial freezing temperature was observed. The Arrhenius equation described the temperature dependence of the reaction rate both in the liquid and the glassy states in all systems studied, while a slightly curved Arrhenius plot was observed in the "rubbery" state of the CMC and CMC-lactose systems. The WLF equation with system-dependent coefficients described the kinetics in the rubbery state of all the model systems except sucrose, excluding the short temperature range where reaction rate enhancement with decreasing temperature was observed.