Mathematical theory of biological periodicities: Formulation of then-body case

In a series of papers, L. Danziger and G. Elmergreen (Bull. Math. Biophysics,16, 15–21, 1954;18, 1–13, 1956;19, 9–18, 1957) showed that a non-linear biochemical interaction between the anterior pituary gland and the thyroid gland may result under certain conditions in sustained periodical oscillations of the rates of production and of the blood level of the thyrotropic and of the thyroid hormone. They treated the systems, however, as a homogeneous one. N. Rashevsky (Some Medical Aspects of Mathematical Biology, Springfield, Illinois: Charles C. Thomas, Publisher, 1965;Bull. Math. Biophysics,29, 395–401, 1967) generalized the above results by taking into account the histological structures of the two glands as well as the diffusion coefficients and permeabilities of cells involved. The present paper is the first step toward the theory of interaction of any numbern of glands or, more generally,n components. The differential equations which govern the behavior of such a system represent a system of2n ²+n non-linear first order ordinary equations and involve a total of 7 n ²+3n parameters of partly histological, partly biochemical nature. The requirements of the existence of sustained oscillations demand 4n ²+2n+2 inequalities between those 7n ²+3n parameters.     

Note on an inequality for the coefficients in legendre polynomial expansions and its application to the theory of liquid phase transitions

It is shown that if themth derivative of a function is positive, and it has a Legendre polynomial expansion with coefficients,A _n, then (A _m)/(2m+1)≧(A _n)/(2n+1) forn>m. This result is applied to the theory of liquid phase transitions.     

Refractive index of uranyl-treated bacterial cytoplasm as related to ribonucleic-acid content and growth rate

After fixation and treatment with uranyl acetate solutions, the refractive index (n _D) of the cytoplasm ofCorynebacterium bovis varies with substrate-dependent growth rate and RNA content. The effect, which presumably is due to quantitative binding of uranyl ions by RNA, permits a measurement of the growth rate of single cells in a single-species system by interference microscopy. Temperature-induced changes in growth rate are not reflected in changes ofn _D or RNA content.     

Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.     

The effects of experimental conditions of fluorescence quenching on the binding parameters of apigenin to bovine serum albumin by response surface methods

A fluorescence quenching technique is often used to study interactions between small molecules and serum albumin. However, the results are quite different by using spectroscopic techniques on the same drug‐protein interaction research and they may be affected by different conditions (e.g. working solution of pH and ionic strength). In this research, using apigenin as an example, the effect of experimental conditions of fluorescence quenching on the binding parameters of drug to bovine serum albumin was investigated using a response surface method (RSM). The effect of pH, the concentration of NaCl and the concentration Mg²⁺ on the quenching constant (K_SV), the apparent association constant (K_a) and the number of binding sites (n) was studied by single‐factor experiments with pH, [NaCl] and [Mg²⁺] as independent variables and K_SV, K_a and n as response values. Prediction models were fit to a quadratic polynomial regression equation and the results showed that both K_SV and n displayed a second‐order model, whereas Ka displayed linear relation dependence on pH, [NaCl] and [Mg²⁺]. Under these experimental conditions, [NaCl] was the most significant (p < 0.05) impact factor on K_SV and K_a, whereas n was most affected by pH (p < 0.05). Copyright © 2013 John Wiley & Sons, Ltd.     

Complex ligand-protein systems: a globally convergent iterative method for the n x m case

When n types of univalent ligands are competing for the binding to m types of protein sites, the determination of the system composition at equilibrium reduces to the solving of a non-linear system of n equations in C = [0;1] ⁿ . We present an iterative method to solve such a system. We show that the sequence presented here is always convergent, regardless of the initial value in C. We also prove that the limit of this sequence is the unique solution in C of the non-linear system of equations. Received: 1 November 2000 / Published online: 21 August 2001     

Dynamical machinery of a biochemical clock

Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT _n of then-species limit cycle is given byT _n=nτ_n, where τ_n is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τ_n is, to good approximation, a linear function ofn. For a givenn, τ_n is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.     

Development of a species-specific fractionation equation for Arctic charr (<Emphasis Type="Italic">Salvelinus alpinus</Emphasis> (L.)): an experimental approach

A species-specific fractionation equation for Arctic charr (Salvelinus alpinus (L.)) was developed experimentally for use in ecological studies of temperature-driven phenologies for the species. Juvenile Arctic charr were reared in controlled conditions at different temperatures (2–14°C), with three replicates of each temperature. Otoliths from the fish and water samples from the chambers were analysed for oxygen isotope composition and used to estimate temperature-dependent fractionation equations relating the isotopic ratio to rearing temperature. A linear and a second order polynomial relationship were estimated and validated using comparable Arctic charr data from another study. Temperatures predicted using the polynomial equation were not significantly different from recorded experimental temperatures, whereas with the linear equation there were significant differences between the predicted and measured temperatures. The polynomial equation also showed the least bias as measured by mean predictive error. Statistical comparisons of the polynomial fractionation equation to a similarly estimated equation for brook charr (Salvelinus fontinalis (Mitchill)) indicated significant differences. Results imply the need for species-specific fractionation equations, even for closely related fish. Results further suggest the polynomial form of the fractionation equation will facilitate more accurate characterisation of water temperatures suitable for use in ecological studies of temperature-driven phenologies of Arctic charr.     

An empirical approach to the analysis of forest stratification

Natural forest communities are made up of different overlapping elementary subpopulations consisting of individuals of different species and ages. To stratify the individuals of a forest stand into elementary subpopulations by using tree height records, a graphical method was empirically proposed employing a derivative of Pearson's type VII distribution as a basis. Arranging all of the individual trees in a stand in the descending order of their height (x), and introducing the rank (N) of any individual in the ordered ranking ofx, finite difference (n) ofN and three values ofx, i.e.,x(N), x(N−n), andx(N+n) labelled byN, N−n, andN+n, respectively, the proposed method used the linear relation betweenx(N) and [x(N−n)+x(N+n)]. On thex(N) vs.[x(N−n)+x(N+n)] diagram, the relation between both the variables could be approximated by a few segmental linear relations and used for stratifying the individuals of a forest stand into subpopulations. The method was applied for analyzing the vertical stratification in forest stands of different forest formations in Japan, Southeast Asia, Africa, and South America. The conclusions on stratification which resulted from the proposed method corresponded well to the conclusions on stratification from profile diagrams.     

Relationship between specific leaf area of aCryptomeria japonica foliage shoot segment and its diameter

Dependency of specific leaf area (SLA) on shoot diameter (x) was studied forCryptomeria japonica foliage shoot segments about 5 cm in length taken from 18 branches at different height levels on 3 trees in August 1987. The leaf area of a shoot segment (S) was defined as half the sum of the needle area (s) estimated by the allometric relationship betweens and needle length. The dry weight of woody tissue (W _w) and needles (W _n) of a segment and itsS were divided by the segment length (L) to give linear densities asW _w/L,W _n/L andS/L, respectively. The densities were related tox by power-form equations.S/L values tended to be constant around 1.2 [cm² cm⁻¹] within the discussed range ofx, whileW _w/L andW _n/L values clearly increased withx. The approximately reciprocal relationship between average needle area (s) and linear density of the number of needles supported the fact thatS/L values were roughly constant regardless ofx.SLA andSNA were defined asS/(W _w+W _n) andS/W _n, respectively. TheSLA-x relationship expected from the average value ofS/L and theW _w/L-x and theW _n/L-x relationships was well fitted to the observed decrease inSLA with increasingx. SNA also decreased asx increased. Variations inSLA andSNA among the shoot segments with similarx were not systematically related to their height levels. An empirical equation with a maximum value ofx (Xmax) was also proposed in order to formulate theSLA-x relationship.     

Some conditions for sustained oscillations in biochemical chains with feedback inhibition

A chain ofn reactions is considered in which the last substance inhibits the production of the first with degreep, p being the order of the inhibition. Maintained oscillations are possible for certain values of the parameters under the following conditions: (1) If there is no time delay, then there must be at least three compartments (n=3) and either the degree of inhibition is sufficiently large (p>8 forn=3) or there must be enzymatic removal from the first compartment, in which casep≥1. (2) If there are time delays, but there is no enzymatic removal, the degree of the inhibition must be greater than or equal to 2 for any value ofn. (3) If there is a time delay in addition to enzymatic removal, one compartment with simple first order inhibition is sufficient. Conditions on the parameters necessary for maintained oscillations are given for many of the cases discussed.     

Bounds on the total particle number for species governed by reaction-diffusion equations in the infinite spatial domain

Differential inequality methods are developed for establishing upper and lower bounds on the total particle numberN(t)=∫θ(x,t) d³ x associated with solutions to nonlinear reaction-diffusion equations of the form ∂θ/∂t=D∇²θ+fθ-gθ ⁿ⁺¹ , whereD(>0),n(>0),f andg are constant parameters. If finite in a neighborhood oft=0,N(t) is bounded below for allt≥0 by a certain derived function oft for equations withg≥0. An upper bound onN(t) is obtained for equations withn=1,f<0 andg<0. These results provide general preservation and extinction criteria for the total particle number.     

Studies on the interaction of palmatine hydrochloride with bovine hemoglobin

The interaction between bovine hemoglobin (BHb) and palmatine hydrochloride (PMT) was investigated at different temperatures using multispectroscopy, as well as the effect of common metal ions (Ca²⁺, Mg²⁺, Zn²⁺, Cu²⁺, Fe²⁺, Fe³⁺, Co²⁺, Ni²⁺) on the BHb–PMT system. Results showed that the quenching mechanism of PMT on BHb was a static process. The electrostatic force played an important role in the conjugation reaction between BHb and PMT. The order of magnitude of the binding constants (K_a) was 10⁴, and the number of binding sites (n) in the binary system was ~ 1. The binding distance (r) was ~ 2.44 nm and the primary binding for PMT was located at β‐37 tryptophan in the hydrophobic cavity of BHb. In addition, the Hill's coefficients were ~ 1. Synchronous and circular dichroism spectra revealed that the microenvironment and the conformation of BHb were changed during the binding reaction. Copyright © 2013 John Wiley & Sons, Ltd.     

Peroxynitrite-scavenging constituents from the brown alga<Emphasis Type="Italic">Sargassum thunbergii</Emphasis>

Peroxynitrite formationin vivo is implicated in numerous human diseases and there is considerable interest in the use of antioxidants and natural products for their treatment. The three components (1–3) isolated fromSargassum thunbergii as well as the organic solventsoluble fractions and the aqueous layer ofS. thunbergii were evaluated for their potential to scavenge authentic ONOO and ONOO derived from 3-morpholinosydnonimine (SIN-1). The antioxidant activity of the individual fractions was in the order of 85% aquaous (aq.) MeOH>n-BuOH>n-hexane> H₂O. The three known compounds sargahydroquinoic acid (1), sargaquinoic acid (2) and sargachromenol (3) showed peroxynitrite-scavenging activities comparable to those of L-ascorbic acid and penicillamine. These results showed a possible antioxidant activity in major constituents ofS. thunbergii.     

C. botulinum inactivation kinetics implemented in a computational model of a high‐pressure sterilization process

High‐pressure, high‐temperature (HPHT) processing is effective for microbial spore inactivation using mild preheating, followed by rapid volumetric compression heating and cooling on pressure release, enabling much shorter processing times than conventional thermal processing for many food products. A computational thermal fluid dynamic (CTFD) model has been developed to model all processing steps, including the vertical pressure vessel, an internal polymeric carrier, and food packages in an axis‐symmetric geometry. Heat transfer and fluid dynamic equations were coupled to four selected kinetic models for the inactivation of C. botulinum; the traditional first‐order kinetic model, the Weibull model, an nth‐order model, and a combined discrete log‐linear nth‐order model. The models were solved to compare the resulting microbial inactivation distributions. The initial temperature of the system was set to 90°C and pressure was selected at 600 MPa, holding for 220 s, with a target temperature of 121°C. A representation of the extent of microbial inactivation throughout all processing steps was obtained for each microbial model. Comparison of the models showed that the conventional thermal processing kinetics (not accounting for pressure) required shorter holding times to achieve a 12D reduction of C. botulinum spores than the other models. The temperature distribution inside the vessel resulted in a more uniform inactivation distribution when using a Weibull or an nth‐order kinetics model than when using log‐linear kinetics. The CTFD platform could illustrate the inactivation extent and uniformity provided by the microbial models. The platform is expected to be useful to evaluate models fitted into new C. botulinum inactivation data at varying conditions of pressure and temperature, as an aid for regulatory filing of the technology as well as in process and equipment design. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

A fully nonequilibrium concerted model for enzymes

The concerted model of Monod, Wyman, and Changeux is generalized so that all effects of interactions for an enzyme operating at a nonequilibrium stationary state are considered. In contrast to the original model, which is based on an analogy to equilibrium ligand binding, the generalization may show both “positive” and “negative cooperativity” in both catalytic binding and conformational processes. Furthermore, in contrast to any equilibrium binding model, the Hill coefficients may be greater than the number of sites n. For catalysis, the maximum value is 2n, and for conformational changes, n + 1. These points are illustrated by two cases that yield simpler analytic expressions. The first obtains when catalysis occurs on a much faster time scale than the conformational changes, and the second, when this situation is reversed.     

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.     

Some remarks on the Kedem-Katchalsky equations for non-electrolytes

The Kedem-Katchalsky equation for the flow of a non-electrolyte through a homogeneous membrane is shown to be a first order expansion of an exact integral of the Spiegler-Bearman-Kirkwood frictional equations under the assumption that the partial frictional coefficients, ζ _ij , are concentration independent. The equations are solved in terms of volume flow; there are no water-to-volume flow correction terms for the permeability, ω, or the reflection coefficient, σ. The precision of the expansion depends upon the magnitude of the water flow. The frictional coefficientsf _sm andf _sw are given as functions of the experimentally determined parameters ω and σ; the frictions, are shown to be independent ofL _p .