Non-autonomous logistic equations as models of populations in a deteriorating environment

The non-autonomous logistic equation

$\text{d}\text{x(t)}\text{d}\text{t}\text{= r(t)x(t)[1 ?}\text{x(t)}\text{K(t)}\text{]}$

is studied under conditions that include an environment which is completely deteriorating. In this setting, when the population's growth rate, r, is large on the average, solutions track the environment with a consequent extinction of the population. However, when both r and rK^?1 are small in the sense that they are in L¹[0,∞) then an asymptotic equivalence, where all solutions tend to positive limits as t approaches infinity, results and the population is persistent, independent of initial density. The asymptotic equivalence produces an unreasonable overshoot of carrying capacity which leads to concern about employing the logistic equation in the above form as a population model when growth rates are close to zero.A re-interpretation of the parameters of the logistic equation leads to the alternative logistic formulation

$\text{d}\text{x(t)}\text{d}\text{t}\text{= x(t)[r(t) ?}\text{c}\text{B(t)}\text{x(t)], (c > 0)}$

. A biological interpretation of the parameters is presented and this equation is compared with the classical logistic model in the case where the parameters are constant. If the alternative logistic model is applied in a situation with time-varying parameters, then a deteriorating environment always leads to extinction of the population regardless of the behavior of r. Similarly, a growth rate which is small on the average results in extinction regardless of the behavior of B. Furthermore, r and B have limiting values as t approaches infinity then so does x and the terminal value of x is equal to the terminal value of the carrying capacity of the population. In general, the alternative formulation seems to be the more reasonable model in situations where perturbations lead to severe decreases in environmental quality and growth rates.     

The decay of genetic variability in geographically structured populations. II

The ultimate rate of approach to equilibrium in the infinite stepping-stone model is calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Let f(t, x) be the probability that two homologous genes separated by the vector x in generation t are the same allele. It is supposed that $\text{f(}\text{0, x}\text{) = O(x}{}^{\mathrm{?2?\eta }}\text{), η > 0,}\text{as}\text{x ≡ ¦}\text{x}\text{¦ → ∞}$. In the absence of mutation, f(t, x) tends to unity at the rate $\text{t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ in one dimension and (ln t)^?1 in two dimensions. Thus, the loss of genetic variability in two dimensions is so slow that evolutionary forces not considered in this model would supervene long before a two-dimensional natural population became completely homogeneous. If the mutation rate, u, is not zero f(t, x) asymptotically approaches equilibrium at the rate $\text{(1 ? u)}{}^{\mathrm{2t}}\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ in one dimension and (1 ? u)^2tt^?1(lnt)^?2 in two dimensions. Integral formulas are presented for the spatial dependence of the deviation of f(t, x) from its stationary value as t → ∞, and for large separations this dependence is shown to be (const + x) in one dimension and (const + ln x) in two dimensions. All the results are the same for the Malécot model of a continuously distributed population provided the number of individuals per colony is replaced by the population density. The relatively slow algebraic and logarithmic rates of convergence for the infinite habitat contrast sharply with the exponential one for a finite habitat.     

N-NBD-l-α-Dilauroylphosphatidylethanolamine. A new fluorescent probe to study spontaneous lipid transfer

Migration of the fluorescent phospholipid $\text{N-(7-}\text{nitro-2,1,3-benzoxadiazol-4-yl)-}\text{l}\text{-α-dilauroylphosphatidyl-ethanolamine}$ between small sonicated egg phosphatidylcholine vesicles was studied by use of the fluorescence resonance energy transfer method. Contrary to the results of lipid transfer experiments reported for acyl chain NBD-labeled phospholipids (Nichols, J.W. and Pagano, R.E. (1982) Biochemistry 21, 1720–1728), the migration kinetics of N-NBD-DLPE had to be described by a sum of two exponential functions. The fast component $\text{(t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≈ 38}\text{min}\text{)}$ was assigned to lipid transfer via soluble monomers and the slow component $\text{(t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≈ 400}\text{min}\text{)}$ to transbilayer motion. A reversible four-stage process is suggested as a kinetic model. Mathematical treatment of this scheme is given yielding an analytical expression for the time dependence of NBD emission intensity. The use of N-NBD-DLPE in the resonance energy transfer measurements offers the advantage of simple chemical synthesis of the fluorescent probe and leads to additional information on transbilayer motion which was not available with the NBD-labeled lipids used so far.     

Hydrogen bonding requirements for the insulin-sensitive sugar transport system of rat adipocytes

(1) The $\text{t}\text{1}\text{2}$ for 1.3 mM D-allose uptake and efflux in insulin-stimulated adipocytes is 1.7 ± 0.1 min. In the absence of insulin mediated uptake of D-allose is virtually eliminated and the uptake rate ($\text{t}\text{1}\text{2}\text{= 75.8 ± 4.99}\text{min}$) is near that calculated for nonmediated transport. The kinetic parameters for D-allose zero-trans uptake in insulin-treated cells are K_zt^oi = 271.3 ± 34.2 mM, V_zt^oi = 1.15 ± 0.12 mM · s^?1. (2) A kinetic analysis of the single-gate transporter (carrier) model interacting with two substrates (or substrate plus inhibitor) is presented. The analysis shows that the heteroexchange rates for two substrates interacting with the transporter are not unique and can be calculated from the kinetic parameters for each sugar acting alone with the transporter. This means that the equations for substrate analogue inhibition of the transport of a low affinity substrate such as D-allose can be simplified. It is shown that for the single gate transporter the K_i for a substrate analogue inhibitor should equal the equilibrium exchange K_m for this analogue. (3) Analogues substituted at C-1 show a fused pyranose ring is accepted by the transporter. 1-Deoxy-D-glucose is transported but has low affinity for the transporter. High affinity can be restored by replacing a fluorine in the β-position at C-1. The K_i for $\text{d}\text{-glucose}\text{= 8.62}\text{mM}$; the K_i for $\text{β-}\text{fluoro-}\text{d}\text{-glucose}\text{= 6.87}\text{mM}$. Replacing the ring oxygen also results in a marked reduction in affinity. The K_i for $\text{5-}\text{thio-}\text{d}\text{-glucose}\text{= 42.1}\text{mM}$. (4) A hydroxyl in the gluco configuration at C-2 is not required as 2-deoxy-d-galactose (K_i = 20.75 mM) has a slightly higher affinity than d-galactose (K_i = 24.49 mM). A hydroxyl in the manno configuration at C-2 interferes with transport as d-talose (K_i = 35.4 mM) has a lower affinity than d-galactose. (5) d-Allose (K_m = 271.3 mM) and 3-deoxy-d-glucose (K_i = 40.31 mM) have low affinity but high affinity is restored by substituting a fluorine in the gluco configuration at C-3. The K_i for $\text{3-}\text{fluoro-}\text{d}\text{-glucose}\text{= 7.97}\text{mM}$. (6) Analogues modified at C-4 and C-6 do not show large losses in affinity. However, 6-deoxy-d-glucose (K_i = 11.08 mM) has lower affinity than d-glucose and 6-deoxy-d-galactose K_i = 33.97 mM) has lower affinity than d-galactose. Fluorine substitution at C-6 of d-galactose restores high affinity. The K_i for $\text{6-}\text{fluoro-}\text{d}\text{-galactose}\text{= 6.67}\text{mM}$. Removal of the C-5 hydroxymethyl group results in a large affinity loss. The K_i$\text{d}\text{-xylose}\text{= 45.5}\text{mM}$. The K_i for $\text{l}\text{-arabinose}\text{= 49.69}\text{mM}$. (7) These results indicate that the important hydrogen bonding positions involved in sugar interaction with the insulin-stimulated adipocytes transporter are the ring oxygen, C-1 and C-3. There may be a weaker hydrogen bond to C-6. Sugar hydroxyls in non-gluco configurations may sterically hinder transport.     

Guanine nucleotides modulate cell surface cAMP-binding sites in membranes from Dictyostelium discoideum

D. discoideum contains kinetically distinguishable cell surface cAMP binding sites. One class, S, is slowly dissociating and has high affinity for cAMP (K_d = 15 nM, $\text{t}\text{1}\text{2}\text{= 15 s}$). A second class is fast dissociating ($\text{t}\text{1}\text{2}$ about 1 s) and is composed of high affinity binding sites H (K_d ≈ 60 nM), and low affinity binding sites L (K_d = ≈ 450 nM) which interconvert during the binding reaction. Guanine nucleotides affect these three binding types in membranes prepared by shearing D.discoideum cells through Nucleopore filters. The affinity of S for cAMP is reduced by guanine nucleotides from 13 nM to 25 nM, and the number of S-sites is reduced about 50%. The number of fast dissociating sites is not altered by guanine nucleotides, but these sites are mainly in the low affinity state. Half-maximal effects are obtained at about 1 μM GTP, 2 μM GDP and 10 μM Gpp(NH)p(guanyl-5′-yl-imidodiphosphate); ATP and ADP are without effect up to 1 mM. These results indicate that D.discoideum cells have a functionally active guanine nucleotide binding protein involved in the transduction of extracellular cAMP signals via cell surface cAMP receptors.     

Continous administration of short-lived radioisotope tracers and the analogous Laplace transform.

Short-lived radioactive tracers are used because of the low radiation dose to patients. Another advantage finding increasing use, however, is that the equilibrium activities achieved by continuous administration to a steady state contain kinetic information. This is not the case with long-lived isotopes.The derivation of quantitative kinetic information in the form of rate constants or flows requires the formulation of a model of the system being studied. Several approaches to this have been published based on a model of single compartments with simultaneous arrival of tracer. To deal with more realistic models a method is proposed which uses the analogy between the procedure of continuous administration of short-lived tracer and the Laplace transform. If f(t) is the activity of a long-lived tracer in any part of a system after administration of unit activity to the input, then

$\text{∫}\text{o}\text{∞}\text{e}{}^{\mathrm{?st}}\text{f(t)}\text{d}\text{t=}\text{f}\text{(s)}$

is the “equilibrium” activity of the part after continuous administration of a short-lived tracer of decay constant $\text{f}\text{(f)}$ is also a value of the Laplace transform of (t).This analogy permits all the theorems of Laplace transform theory to be applied to the analysis of measured activities. The basis of the analogy is explained and examples are given of its application to a number of models which represent actual physiology more realistically than single compartment models. In these applications the transformed equations representing the model, with measured values of activity inserted for each transform, are solved to derive the rate constants. This is different from the use of Laplace transforms where the constant coefficients are known and the initial value problem is solved to find the behaviour of the variables.     

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.     

Estimation of mutation rates in cultured mammalian cells

The factors that affect reliable estimations of mutation rates (μ) in cultured mammalian somatic cell populations by fluctuation analysis are studied experimentally and statistically. We analyze the differential effect of the final cell population size in each culture (N_t) and the number of parallel cultures (C) on the variation in the rate estimates ($\text{μ}$) inferred from the P₀ method. The analysis can be made after the derivation of the variance of $\text{μ}$, which is a measure of variation of $\text{μ}$ for a given combination of N_t and C in a number of repeat experiments. The variance of $\text{μ}$ is inversely proportional to C and to the square of N_t. N_t determines the probability of occurrence of mutation in a cell culture. By influencing the size of P₀, N_t also determines whether a rate estimate is obtainable from the experiment. Since P_o is estimated from the fraction of cultures containing no mutation in a set of C cultures, C becomes a determining factor for the accuracy of $\text{μ}$. The rate estimated from $\text{P}\text{?}{}_0$ is biased, but the bias is in general 2 orders of magnitude smaller than $\text{μ}$. By the selection of an appropriate combination of N_t and C for the experiment, this bias can be reduced even further.Based on the notion of comparing two proportions, we propose a test statistic and have applied it to experimental results for a test of equality of mutation rates in different cell lines. This development places the comparison of mutation rates on a statistical basis.     

The synthesis of methyl 4-O-(4-O-α-d-galactopyranosyl-β-d-galactopyranosyl)-β-d-glucopyranoside: The methyl β-glycoside of the trisaccharide related to Fabry's disease

As part of a program to synthesize the ceramide trisaccharide (1) related to Fabry's disease, methyl 4-O-(4-O-α-$\text{d}$-galactopyranosyl-β-$\text{d}$-galactopyranosyl)-β-$\text{d}$-glucopyranoside (12) was prepared. Methyl β-lactoside (2) was converted into methyl 4-O-(4,6-O-benzylidene-β-$\text{d}$-galactopyranosyl)-β-$\text{d}$-glucopyranoside (4). Methyl 2,3,6-tri-O-benzoyl-4-O-(2,3,6-tri-O-benzoyl-β-$\text{d}$-galactopyranosyl)-β-$\text{d}$-glucopyranoside (7) was synthesized from 4 through the intermediates methyl 2,3,6-tri-O-benzoyl-4-O-(4,6-O-benzylidene-2,3-di-O-benzoyl-β-$\text{d}$-galactopyranosyl)-β-$\text{d}$-glucopyranoside (5) and methyl 2,3,6-tri-O-benzoyl-4-O-(2,3-di-O-benzoyl-β-$\text{d}$-galactopyranosyl)-β-$\text{d}$-glucopyranoside (6). The halide-catalyzed condensation of 7 with 2,3,4,6-tetra-O-benzyl-$\text{d}$-galactopyranosyl bromide (8) gave methyl 2,3,6-tri-O-benzoyl-4-O-[2,3,6-tri-O-benzoyl-4-O-(2,3,4,6-tetra-O-benzyl-α-$\text{d}$-galactopyranosyl)- β-$\text{d}$-galactopyranosyl]-β-$\text{d}$-glucopyranoside (10). Stepwise deprotection of 10 led to 12, the methyl β-glycoside of the trisaccharide related to Fabry's disease.     

Index measures for assessing the mode of inheritance of continuously distributed traits: I, theory and justifications

A class of indices that may be applied to quantitative data on nuclear families and that can help to assess degrees of mode of inheritance is developed. Given phenotype values of spouses x⁽¹⁾ and x⁽²⁾ and offspring y, the deviation of an offspring value from the midparent is $\text{¦y ?}\text{1}\text{2}\text{(x}{}^{\mathrm{(1)}}\text{+ x}{}^{\mathrm{(2)}}\text{)¦}$, and those from the separate parents are $\text{¦y ? x}{}^{\mathrm{(1)}}\text{¦}$ and $\text{¦y ? x}{}^{\mathrm{(2)}}\text{¦}$. The indices called major-gene indices (MGI) investigated are functions of the deviations from midparental values compared to corresponding symmetric functions of the deviations from separate parents. Major-gene indices exceeding 1 may indicate some extent of major-gene inheritance, whereas an MGI less than 1 is suggestive of relatively more polygenic inheritance. Superposition of assortative mating and environmental effects will tend not to shift the MGI greater than 1 for polygenic inheritance, nor will they shift the MGI less than 1 for major-gene factors. The reliance on the proposed indices is reinforced on the basis of a hierarchy of representative models of monogenic and multifactorial inheritance. Extensions of the method to deal with multigenerational pedigrees are briefly discussed.     

The effect of experimentally induced triploidy on the lethal expression of the t12 mutation in mouse embryos

Diploid embryos which are homozygous for the t¹² mutation die at the morula stage. In the current studies, ova from heterozygous ($\text{+}\text{t}{}^{12}$) females were fertilized in vitro with spermatozoa from $\text{+}\text{t}{}^{12}$ males. The fertilized ova were immediately placed into media containing cytochalasin B to prevent second polar body formation, producing +/+/+, +/+/t¹², +/t¹²/t¹², and t¹²/t¹²/t¹² embryos. The subsequent development of these triploid embryos was compared with that of diploid +/+, $\text{+}\text{t}{}^{12}$, and $\text{t}{}^{12}\text{t}{}^{12}$ embryos developing from ova which were also fertilized in vitro with spermatozoa from $\text{+}\text{t}{}^{12}$ males but which were not treated with cytochalasin B immediately following gamete coincubation. The data show that those triploid embryos which possess a wild-type allele and two mutant alleles are phenotypically wild type while those possessing three mutant alleles are not phenotypically distinguishable from their diploid ($\text{t}{}^{12}\text{t}{}^{12}$) counterparts. Like $\text{t}{}^{12}\text{t}{}^{12}$ embryos, t¹²/t¹²/t¹² embryos die at the morula stage, prior to blastocoelic cavity formation.     

Spatial requirements for insulin-sensitive sugar transport in rat adipocytes

(1) Alkyl sugar inhibition of d-allose uptake into adipocytes has been used to explore the spatial requirements of the external sugar transport site in insulin-treated cells. α-methyl and β-methyl glucosides show low affinity indicating very little space around C-1. The high affinity of d-glucosamine (K_i = 9.05 ± 0.66 mM) is lost by N-acetylation. $\text{N-}\text{Acetyl-}\text{d}\text{-glucosamine}$ shows no detectable affinity, indicating that a bulky group at C-2 is not accepted. Similarly $\text{2,3-}\text{di}\text{-O-}\text{methyl-}\text{d}\text{-glucose}$ (K_i = 42.1 ± 7.5 mM) has lower affinity than $\text{3-O-}\text{methyl-}\text{d}\text{-glucose}$ (K_i = 5.14 ± 0.32 mM) indicating very little space around C-2 but much more around C-3. A reduction in affinity does occur if a propyl group is introduced into the C-3 position. The K_i for $\text{3-O-}\text{propyl-}\text{d}\text{-glucose}$ is 11.26 ± 2.12 mM. $\text{6-O-}\text{Methyl-}\text{d}\text{-galactose}$ (K_i = 87.2 ± 17.9 mM) and $\text{6-O-}\text{propyl-}\text{d}\text{-glucose}$ (K_i = 78.07 ± 12.6 mM) show low affinity compared with d-galactose and d-glucose, indicating steric constraints around C-6. High affinity is restored in $\text{6-O-}\text{pentyl-}\text{d}\text{-galactose}$ (K_i = 4.66 ± 0.23 mM) possibly indicating a hydrophobic binding site around C-6). (2) In insulin treated cells $\text{4,6-O-}\text{ethylidene-}\text{d}\text{-glucose}$ (K_i = 6.11 ± 0.5 mM) and maltose (K_i = 23.5 ± 2.1 mM) are well accommodated by the site but trehalose shows no detectable inhibition. These results indicate that the site requires a specific orientation of the sugar as it approaches the transporter from the external solution. C-1 faces the inside while C-4 faces the external solution. (3) To determine the spatial and hydrogen bonding requirements for basal cells 40 μM $\text{3-O-}\text{methyl-}\text{d}\text{-glucose}$ was used as the substrate. Poor hydrogen bonding analogues and analogues with sterically hindering alkyl groups showed similar K_i values to those determined for insulin-treated cells. These results indicate that insulin does not change the specificity of the adipocyte transport system.     

The role of cytochrome b-566 in the electron-transfer chain of Rhodopseudomonas sphaeroides

Spectrophotometric, kinetic, thermodynamic and stoichiometric properties of the low-potential b-type cytochrome of chromatophores from Rhodopseudomonas sphaeroides are reported. Cytochrome b-566 has a double α-band with maxima at 559 and 566 nm. Resolution of the spectrum by full-spectral redox potentiometry showed no indication that the two peaks represent more than one component. The component titrated with E_m,7 ≈ ?80 ± 10 mV. By appropriate choice of wavelength pairs and by subtraction of the contribution due to other components, the kinetics of cytochrome b-566 absorbance changes following flash excitation have been resolved from those of other components. Time-resolved flash spectra corrected for the contributions of other components are consistent with the behavior of both peaks of the α-band as a single kinetic species. The kinetics of cytochrome b-566 in the presence of antimycin show that the reduction of this cytochrome occurred only if cytochrome b-561 was reduced before the flash, either chemically, by poising the ambient redox potential (E_h) below the E_m of cytochrome b-561 (E_m,7 ≈ 50 mV), or photochemically at higher redox potentials by a previous flash. The rate of reduction of cytochrome b-566 varied with E_h. At low E_h (approx. 0 mV) reduction on the first flash showed $\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≈ 1.25}\text{ms}$; at high E_h (approx. 180 mV) reduction on the second flash showed $\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≈ 10}\text{ms}$. In the absence of antimycin at E_h ≈ 0 mV, cytochrome b-566 was observed to become rapidly reduced ($\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≈ 500 μ}\text{s}$) and then reoxidized ($\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≈ 2}\text{ms}$) after a single flash. At higher redox potentials (E_h > 80 mV) no kinetic changes which could be unambiguously attributed to cytochrome b-566 were observed following a single flash. The results are interpreted in terms of a Q-cycle mechanism in which the reductant for cytochrome b-566 is the semiquinone formed on oxidation of ubiquinol from the quinone pool. The oxidation of the ubiquinol occurs by a concerted reaction in which one electron is accepted by the Rieske-type FeS center and the other by cytochrome b-566. We suggest that the kinetic characteristics may indicate a pathway for reduction of the b-type cytochromes in which cytochrome b-566 is the immediate electron acceptor and donates to cytochrome b-561 in a serial pathway. The experimental results in the presence of antimycin are compared with data from a computer simulation of the thermodynamic behavior of the chain, and the computer model is shown to provide an excellent fit.     

Deamination of 2-amino-2-deoxyhexitols and of their per-O-methylated derivatives with nitrous acid

The products of nitrous acid deamination of per-O-methylated 2-amino-2-deoxy-d-glucitol and $\text{2-}\text{amino}\text{-2-}\text{deoxy}\text{-3-O-β-}\text{d}\text{-galactopyranosyl-}\text{d}\text{-glucitol}$ and its per-O-methylated derivative have been characterized by g.l.c.—mass spectrometry after treatment with sodium borodeuteride and further substitution by acetylation, methylation, or (trideuteriomethyl)ation. The results confirm that the most important reaction pathway (1) involves a 1 → 2-hydride shift to give $\text{2-}\text{deoxy-}\text{d}\text{-arabino-}\text{hexoses}$, but that significant side-reactions include (2) solvolytic displacement at C-2, (3) a 3 → 2-hydride shift, to give $\text{2-}\text{deoxy-}\text{d}\text{-erythro-3-}\text{hexuloses}$, and (4) a C-4→C-2 migration to give $\text{2-}\text{deoxy}\text{-2-C-}\text{(hydroxymethyl)-}\text{d}\text{-ribose}$ and -d-arabinose. Reactions (3) and (4) result in elimination of the original 3-O-substituents, with the exposure of new reducing groups, from oligosaccharides terminated by 3-O-substituted 2-amino-2-deoxyhexitols.     

Estimation of the digestibility of low-quality hays by cattle from measurements made with sheep

Seventeen sets of apparent digestibility data derived from 10 experiments, in which both cattle and sheep had been fed on the same hays, were examined. The study was restricted to low-quality hays of less than 60% apparent digestibility of dry matter. On average, digestibility was higher for cattle than for sheep, and the difference was greatest with the samples of lowest digestibility. A linear equation found to describe best the relationship between the digestibility of hays by cattle and sheep was:

$\text{y = 0.673 x + 20.3}$

where y = digestibility of hay to cattle (%) and x = digestibility to sheep (%) (r = 0.843; S_y.x, ± 3.41; standard error of the slope, ± 0.111). This equation may be used to correct apparent digestibility values of 60% or less measured with sheep, or estimated in vitro with the use of sheep standards, if the digestibility data are to be applied to cattle.     

The behaviour of coupled biochemical oscillators as a model of contact inhibition of cellular division

Intercellular communication of molecules between normal cells by tight junctions, and lack of this in some cancer cells (Loewenstein), can explain contact inhibition of cellular division in tissues. A general theory has been based on assuming the continual rise and fall (intrinsic oscillation) of a key substance x in each cell, with the period of the cell cycle. Periods are asynchronous in different cells, and x is exchanged between cells in contact by diffusion. A reduction in the resultant amplitude of fluctuation of x results, so that it does not reach the threshold x_t required for division to ensue; hence contact inhibition.The mathematical model is defined in its simplest form, and the sets of differential equations for arrays of cells are solved, from the isolated cell to the cell in an infinite sheet. The relative probability of division, P, is computed by numerical analysis from the area of resultant curves of x that lies above the threshold x_t. P depends on four dimensionless parameters, the order of coupling n (the number of cells directly communicating with a given cell), the total number of cells N in the aggregate, the communication constant K, and x_t, as a fraction of the amplitude of the intrinsic oscillation. The degree of synchrony, measured by the coefficient of variation σ of the periods, is important. If σ < ± 4%, contact inhibition is much reduced. The theory predicts that a paradoxical “contact-facilitation” is possible for very small aggregates of cells. For a cell in an infinite sheet, the amplitude of oscillation of x is reduced approximately by the factor $\text{1}\text{nK}$. For normal cells K is probably > 1, for cancer cells that lack communication, K is probably «< 1. However, two other basic causes for lack of regulation of tissue growth (cancer) could be excessive intrinsic oscillation of x, cf. x_t, and partial or complete synchronization of groups of cells by some unknown mechanism.     

Transient state isoelectric focusing: computational procedures for digital data processing with a desk-top programmable calculator

Computational procedures (applicable to desk-top programmable calculators) have been described which are useful in the processing of data generated by transient state isoelectric focusing (TRANSIF) experiments. The system allows the calculation of methological parameters such as peak position, peak separation, segmental pH gradient, resolving power, and resolution. “Apparent” physical constants such as the isoelectric point (pI), diffusion coefficient (D), and the slope of the pH-mobility curve ($\text{du}\text{d}\text{(}\text{pH}\text{)}$) of amphoteric molecules can be calculated either by linear interpolation (pI) or by linear regression analysis (D and $\text{du}\text{d}\text{(}\text{pH}\text{)}$). The computing programs are also capable of performing operations such as smoothing of raw data, conversion to absorbance, baseline detection and correction, and calculation of the higher statistical moments of Gaussian, bi-Gaussian, and asymmetric peak shapes. Moment analysis is carried out by direct statistical procedures, curve simulation, and slope detection methods. The above computational procedures provide a useful and necessary adjunct to the scanning isoelectric focusing assembly which is coupled to a digital data acquisition and recording system.