Flow analysis in a biological system adopting the buffer capacitor concept.

The flow measurement of each component in each compartment is important in works on transport phenomena in a biological system. The method of flow measurement was studied adopting the capacitor concept derived from network thermodynamics.A biologically active component i in a compartment is defined as follows,

$\text{n}{}_{\mathrm{i}}\text{=n}{}_1\text{+n}{}_2\text{=c}{}_1\text{V+c}{}_2\text{V}$

where the total quantity ni consists of a measurable form n_i (free form, conc. c₁) and concealed form n₂ (conc, c₂). Capacitor for the species i of a compartment is defined as follows,

$\text{C=}\text{d}\text{n}{}_{\mathrm{i}}\text{d}\text{μ}{}_{\mathrm{i}}\text{=}\text{1+}\text{c}{}_2\text{c}{}_1\text{c}{}_1\text{dV}\text{dμ}{}_{\mathrm{i}}\text{+}\text{1+}\text{dc}{}_2\text{dc}{}_1\text{v}\text{dc}{}_1\text{dμ}{}_{\mathrm{i}}$

,

$\text{=Ac}{}_1\text{dV}\text{dμ}{}_{\mathrm{i}}\text{+BV}\text{d}\text{c}{}_1\text{d}\text{t}$

Thus flow of each component is expressed as,

$\text{J}{}_{\mathrm{i}}\text{=}\text{d}\text{n}{}_{\mathrm{i}}\text{d}{}_{\mathrm{i}}\text{=}\text{d}\text{n}{}_{\mathrm{i}}\text{dμ}{}_{\mathrm{i}}\text{dμ}\text{n}{}_{\mathrm{i}}\text{dt}\text{=C}\text{d}\text{μ}{}_{\mathrm{i}}\text{dt}$

,

$\text{=Ac}{}_1\text{d}\text{V}\text{dt}\text{+BV}\text{d}\text{c}{}_1\text{dt}$

Method of determination of capacitor coefficients A and B by titration experiment was also considered. For an experimental case, the capacitance for H⁺ of blood compartment was determined. The relationship between the capacitor concept and the buffer value of Van Slyke was discussed.     

On equivalent forms of Whittaker's similarity index

For two N-species assemblages A, B with specific proportionate abundances of the ith species a_i, b_l respectively, we consider the equality

$\text{∑}\text{t=1}\text{N}\text{c}{}_{\mathrm{i}}\text{= 1?}\text{1}\text{2}\text{∑}\text{t=1}\text{N}\text{|a}{}_{\mathrm{i}}\text{?b}{}_{\mathrm{i}}\text{|, c}{}_{\mathrm{i}}\text{=}\text{a}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i}}\text{? b}{}_{\mathrm{i}}\text{b}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i}}\text{> b}{}_{\mathrm{i}}\text{, 0?a,b,c?1}$

. The left-hand term is known as Sanders' minimum faunal abundance value, while the right side is referred to as Whittaker's similarity index. Both measures are commonly used in community studies. Equality between these two measures obtains only when proportionate abundances are utilized. We develop equivalent formulation which is valid for absolute abundance data, reduces to the Sanders-Whittaker equality when proportionate abundance data are employed, and is more sensitive to differences in species abundance distributions. Namely, we show that

$\text{2}\text{α+β}\text{∑}\text{t=1}\text{N}\text{c}{}_{\mathrm{i}}\text{= 1 ?}\text{1}\text{α+β}\text{∑}\text{t=1}\text{N}\text{|a}{}_{\mathrm{i}}\text{?b}{}_{\mathrm{i}}\text{|}$

, where

$\text{α =}\text{∑}\text{t=1}\text{N}\text{a}{}_{\mathrm{i}}\text{, β =}\text{∑}\text{t=1}\text{N}\text{b}{}_{\mathrm{i}}$

, and the a's, b's c's are as defined above.     

A stable traveling-wave solution of a model of cellular control processes with positive feedback

Edelstein's model

$\text{?E}\text{?τ}\text{=F(M, E)}$

,

$\text{?M}\text{?τ}\text{=G(M, E)+D}\text{?}{}^2\text{M}\text{?s}{}^2$

,

$\text{M(s,0)=?(s)}$

,

$\text{E(s,0)=ψ(s)}$

, where τ ? 0 and ?∞<s<∞, $\text{F(M, E>) = (K}{}_1\text{+M}{}^{\mathrm{m}}\text{)}\text{(K}{}_2\text{+M}{}^{\mathrm{m}}\text{)?k}{}_1\text{E, G(M, E)}\text{= k}{}_1\text{E ? k}{}_2\text{M,}$ m ? 2, describes the behavior of two basic chemical species during the cellular differentiation in a linear ensemble of the same cell type. We prove the existence and uniqueness of a travelling-wavefront solution. We also demonstrate one kind of stability for this solution.     

Heterozygote frequencies in small subpopulations.

The mean fixation index within subpopulations (F_IS) has been defined as $\text{F}\text{̄}{}_{\mathrm{IS}}\text{= ∑w}{}_{\mathrm{i}}\text{F}{}_{\mathrm{ISi}}\text{or as}\text{F}\text{̂}{}_{\mathrm{IS}}\text{=}\text{∑w}{}_{\mathrm{i}}\text{p}{}_{\mathrm{i}}\text{q}{}_{\mathrm{i}}\text{F}{}_{\mathrm{ISi}}\text{∑w}{}_{\mathrm{i}}\text{p}{}_{\mathrm{i}}\text{q}{}_{\mathrm{i}}$. The latter definition is preferred because it can be obtained from the two other fixation indices, F_ST and F_IT and because it is unaffected by the mean gene frequency. The expected frequency of heterozygotes in small subpopulations of dioecious organisms will exceed Hardy-Weinberg expectations and this can be measured by $\text{F}\text{̂}{}_{\mathrm{IS}}$. In an isolated subpopulation of constant variance effective size $\text{N,}\text{F}\text{̂}{}_{\mathrm{IS}}$ rapidly tends to $\text{1 − 4N}{}^2\text{(N − 1 + [N}{}^2\text{+ 1]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^2$. In the Island model of population structure, $\text{F}\text{̂}{}_{\mathrm{IS}}$ is approximately $\text{−(1 − m)}\text{N}\text{where}\text{m}$ is the immigration rate.When a sample is drawn from a natural population, the observed F_IS will depend upon the genetic structure of the population. The values of F_IS expected in three different types of population structure are discussed.     

Time lag in a model of a biochemical reaction sequence with end product inhibition

The model studied is that of Goodwin, in which all but one of the reactions obey linear kinetics, while the end-product inhibits the first reaction in a term of Michaelis-Menten form, with Hill coefficient ?:

$\text{z=}\text{∫}\text{?∞}\text{t}\text{x}{}_{\mathrm{n}}\text{(T)G(t?T)dt}$

The results obtained relate to time lag in the off diagonal terms in these equations. The time lag is taken in distributed form, for example replacing x_n in the first equation by

$\text{dx}{}_{\mathrm{t}}\text{dt}\text{=k}{}_1\text{x}{}_{\mathrm{t??}}\text{1?b}{}_1\text{x}{}_{\mathrm{t}}\text{, i=2, …n.}$

For any non-negative G, time lag in these terms can not destabilize the equilibrium point in the case ? = 1. For a particular class of functions G one can obtain some insight into the consequences of time lag by relating the model to that with a longer loop of reactions. Then known results can be used for general ? and n.     

Inhibition of chymotrypsin by alkyl phosphonates: a quantitative structure-activity analysis.

A quantitative structure-activity relationship has been formulated for 53 alkyl phosphonates [R₂OPO(CH₃)SR₃] inhibiting chymotrypsin: . In this so-called bilinear model, k_i is the bimolecular rate constant (m^?1 s^?1), β is a disposable parameter evaluated by a computerized iterative procedure, MR is the molar refractivity of a substituent, $\text{σ}{}_3{}^1$ is Taft's polar parameter, and I is an indicator variable for substituents containing a sulfonium group. The correlation coefficient for this equation is 0.985. This quantitative structure-activity relationship is compared with those previously formulated for the action of chymotrypsin on acylamino acid ester substrates.     

On advection in phytoplankton models

Scale analysis is used to deduce a new non-dimensional number, $\text{S =}\text{C}\text{[(A}{}_{\mathrm{H}}\text{τ)}\text{1}\text{2}\text{V}{}_{\mathrm{<mtext>m</mtext>}}\text{]}$ which defines the importance of advection of phytoplankton by organized fluid motion in spatial models of marine food chains. When S ? 1, advection greatly affects biological productivity; when S ? I, advection may be neglected. When S is order one, advection and biological productivity play competing roles in determining the spatial configuration of the plankton biomass.     

Energetics of coupled Na+ and Cl− entry into epithelial cells of bullfrog small intestine

Na⁺, K⁺ and Cl^? concentrations ($\text{c}{}_{\mathrm{j}}{}^{\mathrm{<mtext>i</mtext>}}$) and activities ($\text{a}{}_{\mathrm{j}}{}^{\mathrm{<mtext>i</mtext>}}$), and mucosal membrane potentials ($\text{E}{}_{\mathrm{<mtext>m</mtext>}}$) were measured in epithelial cells of isolated bullfrog (Rana catesbeiana) small intestine. Segments of intestine were stripped of their external muscle layers, and bathed (at 25°C and pH 7.2) in oxygenated Ringer solutions containing 105 mM Na⁺ and Cl^? and 5.4 mM K⁺. Na⁺ and K⁺ concentrations were determined by atomic absorption spectrometry and Cl^? concentrations by conductometric titration following extraction of the dried tissue with 0.1 M HNO₃. ¹⁴C-labelled inulin was used to determine extracellular volume. $\text{E}{}_{\mathrm{<mtext>m</mtext>}}$ was measured with conventional open tip microelectrodes, $\text{a}{}_{\mathrm{<mtext>Cl</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ with solid-state Cl^?-selective silver microelectrodes and $\text{a}{}_{\mathrm{<mtext>Na</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ and $\text{a}{}_{\mathrm{<mtext>K</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ with Na⁺- and K⁺-selective liquid ion-exchanger microelectrodes. The average $\text{E}{}_{\mathrm{<mtext>m</mtext>}}$ recorded was ?34 mV. $\text{c}{}_{\mathrm{<mtext>Na</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$, $\text{c}{}_{\mathrm{<mtext>K</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ and $\text{c}{}_{\mathrm{<mtext>Cl</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ were 51, 105 and 52 mM. The corresponding values for $\text{a}{}_{\mathrm{<mtext>Na</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$, $\text{a}{}_{\mathrm{<mtext>K</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ and $\text{a}{}_{\mathrm{<mtext>Cl</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ were 18, 80 and 33 mM. These results suggest that a large fraction of the cytoplasmic Na⁺ is ‘bound’ or sequestered in an osmotically inactive form, that all, or virtually all the cytoplasmic K⁺ behaves as if in free solution, and that there is probably some binding of cytoplasmic Cl^?. $\text{a}{}_{\mathrm{<mtext>Cl</mtext>}}{}^{\mathrm{<mtext>i</mtext>}}$ significantly exceeds the level corresponding to electrochemical equilibrium across the mucosal and baso-lateral cell membranes. Earlier studies showed that coupled mucosal entry of Na⁺ and Cl^? is implicated in intracellular Cl^? accumulation in this tissue. This study permitted estimation of the steady-state transapical Na⁺ and Cl^? electrochemical potential differences (Δμ&#x0304;Na and Δμ&#x0304;Cl). Δμ&#x0304;Na (?7000 J · mol^?1; cell minus mucosal medium) was energetically more than sufficient to account for Δμ&#x0304;Cl (1000–2000 J · mol^?1).     

The asymptotic normality of a simple batch epidemic process

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t∈[0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by $\text{[α}{}^2\text{X}{}_{\mathrm{n}}\text{(t){n ?X}{}_{\mathrm{n}}\text{(t) +X}{}_{\mathrm{n}}\text{(0)}]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, t∈[0, ∞), α∈(0, ∞)}$. This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t∈(0, t₀), and t₀ is evaluated.     

Influence of ionic strength upon the proton inventory for the water-catalyzed hydrolysis of N-acetylbenzotriazole

Proton inventory investigations of the hydrolysis N-acetylbenzotriazole at pH 3.0 (or the equivalent point on the pD rate profile) have been conducted at two different temperatures and at ionic strengths ranging from 0 to 3.0 M. The solvent deuterium isotope effects and proton inventories are remarkably similar over this wide range of conditions. The proton inventories suggest a cyclic transition state involving four protons contributing to the solvent deuterium isotope effect for the water-catalyzed hydrolysis. The hydrolysis data are described by the equation $\text{k}{}_{\mathrm{n}}\text{= k}{}_{\mathrm{<mtext>o</mtext>}}\text{(1 ? n + nπ}{}_{\mathrm{<mtext>a</mtext>}}{}^1\text{)}{}^4$ with $\text{π}{}_{\mathrm{<mtext>a</mtext>}}{}^1\text{～ 0.74}$, where k_o is the observed first-order rate constant in protium oxide, n is the atom fraction of deuterium in the solvent, k_n is the rate constant in a protium oxide-deuterium oxide mixture, and $\text{π}{}_{\mathrm{<mtext>a</mtext>}}{}^1$ is the isotopic fractionation factor.     

The effect of redox potential on the kinetics of fluorescence induction in Photosystem II particles from Phormidium laminosum. Sigmoidicity,energy transfer and the slow phase

Fluorescence induction curves in 3-(3,4-dichlorophenyl)-1,1-dimethylurea (DCMU)-inhibited Photosystem (PS) II particles isolated from the blue-green alga Phormidium laminosum have been analysed as a function of redox potential. Redox titration of the initial fluorescence indicated a single component with E_m,7.5 = +30 mV (n = 1) (Bowes, J., Horton, P. and Bendall, D.S. (1981) FEBS Lett. 135, 261–264). Despite this simplified electron acceptor system and the small number of chlorophylls per reaction centre, a sigmoidal induction curve was nevertheless seen. Sigmoidicity decreased as Q was reduced potentiometrically prior to induction such that the induction was exponential when the ratio $\text{F}{}_{\mathrm{<mtext>i</mtext>}}\text{F}{}_{\mathrm{<mtext>m</mtext>}}\text{= 0.64}$. These particles also showed a slow (β) phase of induction which titrated with an E_m value slightly more positive than that of the major quencher. It is concluded that the sigmoidal shape of the fluorescence induction curve observed in Phormidium PS II particles is not a consequence of a requirement for two photons to close the PS II reaction centre, but is generated as a result of energy transfer between photosynthetic units comprising one reaction centre per approx. 50 chlorophylls. Also, the existence of PS II heterogeneity (PS II_α, PS II_β centres) does not require a structurally differentiated chloroplast, but may only indicate the extent of aggregation of PS II centres.     

The determination of positive and negative co-operativity with allosteric enzymes and the interpretation of sigmoid curves and non-linear double reciprocal plots for the MWC and KNF models

(i) It is proved that only four independent constants can ever be obtained by extrapolation procedures applied to non-hyperbolic steady-state or binding data, (ii) Analysis of the algebraic graphs $\text{y}\text{x}$, $\text{(1/y)}\text{(1/x)}$, $\text{y}\text{(}\text{y}\text{x}\text{)}$ and ($\text{x}\text{y}\text{)/x}$ is shown to require a knowledge of the sign of six curve shape determinants. In each case, the sign is a necessary and sufficient condition for a specific curve shape feature, (iii) The precise graphical effect of positive and negative co-operativity then requires the definition of two reference curves, the osculating hyperbola at zero substrate concentration, OH(0), and the osculating hyperbola at infinite substrate concentration OH(∞). These are better first order approximations than the Hill equation, (iv) Rules for determining unambiguously the sign of initial, final and overall co-operativity coefficients by inspection of non-hyperbolic binding curves are then possible, (v) These rules require that saturation data for:

$\text{y=}\text{∑}\text{i=1}\text{n}\text{a}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}\text{∑}\text{i=0}\text{n}\text{β}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}$

be fitted by computer for low concentrations to the hyperbola:

$\text{OH}\text{(o)=}\text{(}\text{-a}{}_1{}^2\text{ψ}{}_{11}{}^{20}\text{)x}\text{[(}\text{-a}{}_1\text{β}{}_0\text{ψ}{}_{11}{}^{20}\text{)+x]}$

while regression of high substrate concentration data is to:

$\text{OH}\text{(∞)=}\text{(}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)x}\text{[(}\text{φ}{}_{\mathrm{n,n-1}}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)+x]}$

. Comparisons of the best fit pseudo-kinetic constants then gives the type of co-operativity present in an unambiguous way with no assumptions as to molecular mechanism, (vi) These rules are then applied to the MWC and KNF allosteric models of ligand binding and the constraints necessary for specific curve shape effects are given, (vii) The graphical expression of positive or negative final co-operativity depends only on events at high substrate concentration but overall and initial co-operativities produce specific geometric effects depending upon the difference between behaviour of saturation data at both extremes of concentration, (viii) This apparent anomaly is explained by a discussion of the relationships between the osculating hyperbolae, the theoretical parent hyperbola and the Hill plot asymptotes.     

Ligand-dependent tryptic inactivation of the ouabain sensitivity of ADP-ATP exchange catalyzed by canine renal Na+, K+-ATPase.

Phosphate transporter of bovine heart mitochondria was purified by solubilization of submitochondrial particles with octylglucoside and fractionation of the extract with ammonium sulfate. After reconstitution into liposomes the purified protein catalyzed phosphate transport which was sensitive to mersalyl and other SH reagents. Transport measured either as $\text{P}{}_{\mathrm{i}}\text{OH}$ or $\text{P}{}_{\mathrm{i}}\text{P}{}_{\mathrm{i}}$ exchange was proportional to protein concentration and time. The $\text{P}{}_{\mathrm{i}}\text{OH}$ but not the $\text{P}{}_{\mathrm{i}}\text{P}{}_{\mathrm{i}}$ exchange was stimulated several fold by valinomycin plus nigericin in the presence of K⁺. The reconstituted system provides a suitable assay during purification of the mitochondrial phosphate transporter.     

Transport kinetics of dipicrylamine through lipid bilayer membranes. Effects of membrane structure

Charge-pulse current-relaxation studies have been performed with lipid bilayer membranes in the presence of the hydrophobic ion dipicrylamine. From the analysis of the relaxation times and amplitudes the translocation rate constant $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ of dipicrylamine as well as the partition coefficient β between membrane surface and water could be evaluated. In a first series of experiments membranes made from monoolein or dioleoylphosphatidylcholine in a number of different $\text{n-}\text{alkane}$ solvents were studied, as well as virtually solvent-free bilayer membranes made from monolayers. The thickness $\text{d}$ of the hydrocarbon layer of these membranes varied between 5.0 and 2.5 nm. While β was almost insensitive to variations in $\text{d}$, a strong decrease of $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ with increasing membrane thickness was found; the observed dependence of $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ on $\text{d}$ approximately agreed with the theoretically expected influence of membrane thickness on the height of the dielectric barrier. No specific differences between Mueller-Rudin films and solvent-free (Montal-Mueller) membranes other than differences in thickness were found. In a further series of experiments the chemical structure of the lipid was systematically varied (number and position of double bonds in the hydrocarbon chain, nature of the polar head group). The translocation rate constant $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ was much larger in phosphatidylethanolamine membranes than in phosphatidylcholine membranes. A strong increase of $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ was found when the number of double bonds in the hydrocarbon chain was increased from one to three. These changes were discussed in terms of membrane fluidity and dielectric barrier height. Much higher values of $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ were observed in lipids with ester linkage between hydrocarbon chain and glycerol backbone, as compared with the corresponding ether analogs. This finding is qualitatively consistent with determinations of dipolar potentials in monolayers of ester and ether lipids. When cholesterol is added to phosphatidylcholine membranes, the translocation rate constant $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ increases up to five-fold, while the partition coefficient β remains virtually constant. The variation of $\text{k}{}_{\mathrm{<mtext>i</mtext>}}$ in this case can be largely accounted for by a decrease in membrane thickness and a concomitant reduction in dielectric barrier height. In membranes made from the negatively charged lipid phosphatidylserine the partition coefficient of dipicrylamine strongly increased with ionic strength, as expected from the Gouy-Chapman theory of the surface potential.     

Theoretical analysis of the effects of two pH regulation patterns on the temperature sensitivities of biological systems in nonhomeothermic animals.

When protonation reactions comprise a part of biochemical reaction schemes in vivo, the temperature sensitivities of chemical equilibria and of enzymatic rates are modulated according to the variation of pH with T_B (body temperature). Two patterns of pH regulation have been established, each pattern controlling the ionization states of different titratable groups as T_B changes: $\text{dpH}\text{dT}{}_{\mathrm{<mtext>B</mtext>}}\text{? 0}$ pH unit/° C for the blood of heterothermic mammals (constant charge for phosphate groups, e.g.), and $\text{dpH}\text{dT}{}_{\mathrm{<mtext>B</mtext>}}$ ? ?0.017 pH unit/° C for intracellular and blood compartments of vertebrate and invertebrate poikilotherms (constant charge for histidine imidazole and -SH groups). Calculations demonstrate the feasibility of the following. (i) A large effect of T_B on a metabolic branch point is possible when the competition of two pathways for common substrate is governed by two titratable groups with large differences in ΔH_i^o values (ionization enthalpy), e.g., phosphate and alpha amino groups. (ii) The effect of temperature on the amount of free energy released from equilibrium chemical reactions (as might occur for reactions not controlled by rate-limiting enzymes) would depend strongly on the value of ΔH_i⁰, and on which of the two pH regulation patterns was present. (iii) If free energy released from ionization reactions were coupled to activation processes of enzyme catalysis, so as to effectively decrease free energy barriers, the temperature sensitivities of reaction rates would be smaller in pH environments in which pH varied inversely with T_B than in environments of constant pH, regardless of ΔH_i⁰. (iv) Different mechanisms for the modulation of metabolism as a function of T_B might have evolved for animals with different values of $\text{dpH}\text{dT}{}_{\mathrm{B}}$.