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.     

A dynamical behaviour of active regions in randomly connected neural networks

The field of the randomly connected neural network is approximately formulated by Griffith's equation, regarding the network as being continuous. An integral representation of Griffith's equation is derived. If a relative refractory period can be ignored, it is $\text{X(x,t)}\text{=1}\text{∫}\text{o}\text{∞}\text{ds}\text{∫}\text{?vs}\text{vs}\text{dn}\text{kv}\text{2t}\text{e}{}^{\mathrm{?avs}}\text{X(x?n, t?s) ? θ}$ where X(x, t) corresponds to the firing rate and θ means the threshold of the neural firing, τ the absolute refractory period and v the velocity for the spike potential travelling down the axon. The above equation is formally analogous to Caianiello's equation, but the former describes the more macroscopic behaviour of the neural network than the latter. With the aid of computer simulation, appropriate solutions are successfully obtained.In regions where X = 1, neurones are firing at a high constant rate of $\text{1}\text{τ}$ (active regions). In regions where X = 0, there is no firing of neurones (resting regions). In the neural net for which $\text{0 <}\text{a}{}^2\text{τθ}\text{k}\text{< 1}$, the net is generally a mixture of the active regions and of the resting regions. In the case that a large active region is in contact with a large resting region, the propagation velocity of boundary between the two regions tends to the velocity u given by $\text{u = (1 ?}\text{2a}{}^2\text{τθ}\text{k}\text{)v}$. This expression of velocity u was deduced from the fact that there exists a solution of the type X(x, t) = 1 (ut ? x) for equation (A). In the case of $\text{0 <}\text{a}{}^2\text{τθ}\text{k}\text{< 0 · 5}$, the active region grows and in the case of $\text{0 · 5 <}\text{a}{}^2\text{τθ}\text{k}\text{< 1}$, the resting region grows. A fatigue effect is introduced, for which it is hard for neurones to maintain firing states. In this case an active region of definite width L propagates with constant velocity u′. The dependence of L and u′ on characteristics of neural network and on the fatigue effect is investigated.     

Study on models of single populations: an expansion of the logistic and exponential equations

Using the adsorption theory of chemical kinetics, a new equation concerning the growth of single populations is presented:

$\text{d}\text{X}\text{d}\text{t}\text{=}\text{μ}{}_{\mathrm{c}}\text{X(1 ?)}\text{X}\text{X}{}_{\mathrm{m}}\text{1?}\text{X}\text{X}{}_{\mathrm{m}}{}^{\mathrm{\prime }}$

or in its integral form:

$\text{ln}\text{X}\text{X}{}_{\mathrm{o}}\text{?}\text{ln}\text{X}{}_{\mathrm{m}}\text{?X}\text{X}{}_{\mathrm{m}}\text{?X}{}_{\mathrm{o}}\text{+}\text{X}{}_{\mathrm{m}}\text{X}{}^{\mathrm{\prime }}{}_{\mathrm{m}}\text{X}{}_{\mathrm{m}}\text{?X}\text{X}{}_{\mathrm{m}}\text{?X}{}_{\mathrm{o}}\text{=μ}{}_{\mathrm{c}}\text{(t?t}{}_{\mathrm{o}}\text{)}$

This equation attempts to explain the relationship between population increment and limiting resources. It can be reduced to either the logistic or exponential equation under two extreme conditions. The new equation has three parameters, X_m, X′_m and μ_c, each of which has ecological significance. $\text{X}{}_{\mathrm{m}}\text{X′}{}_{\mathrm{m}}$ concerns the efficiency of nutrient utilization by an organism. Its value is between zero and one. With ratios approaching unity, the efficiency is high; lower ratios indicate that population increment is quickly restricted by limiting resources. μ_c, is a velocity parameter lying between μ_e, (exponential growth) and μ_L (logistic growth), and is dependent on the value of $\text{solX}{}_{\mathrm{m}}\text{X′}{}_{\mathrm{m}}$. From μ_c we can predict the time course of population incremental velocity ($\text{d}\text{X}\text{d}\text{t}$), and can observe that it is not symmetrical, unlike that derived from the logistic equation. At $\text{X}{}_{\mathrm{m}}\text{X′}{}_{\mathrm{m}}\text{= 1}$ the maximum velocity of the population increment predicted from the new equation is twice that of the logistic equation.Population growth in nature seems to support the new equation rather than the logistic equation, and it can be successfully fitted by means of a least square method.     

Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks

In the mathematical modeling of biochemical reaction networks the application of the quasi-steady-state approximation permits a reduction of the number of dynamic variables as well as of the number of parameters. It is shown that the quasi-steady-state approximation represents the zeroth approximate solution of the perturbation problem$\text{dX}\text{dt}\text{= RV(X)+}\text{1}\text{μ}\text{S}\text{W}\text{?}\text{(X)}$ with μ ? 1. The perturbation equation develops by subdivision of the flux rates of the model into the rates $\text{w}{}_{\mathrm{i}}\text{(X) = (1/μ)}\text{w}\text{?}{}_{\mathrm{i}}\text{(X)}$ of fast reactions and the rates v_j(X) of slow reactions. The matrix C=(R?S) denotes the stochiometric matrix of the reaction network. The analysis of this perturbation problem provides conditions for the applicability of the quasi-steady-state approximation in a given network. The paper presents a practical guide for the construction of the approximate solution.     

A simple stochastic gene substitution model

If the fitnesses of n haploid alleles in a finite population are assigned at random and if the alleles can mutate to one another, and if the population is initially fixed for the kth most fit allele, then the mean number of substitutions that will occur before the most fit allele is fixed is shown to be

$\text{1}\text{2}\text{+}\text{1}\text{k}\text{+}\text{∑}\text{i=2}\text{k?1}\text{(i+3)}\text{(2i(i+1))}$

when selection is strong and mutation is weak. This result is independent of the parameters that went into the model. The result is used to provide a partial explanation for the large variance observed in the rates of molecular evolution.     

Steady-state current-voltage characteristics of amino acid transport in rhizoid cells of Riccia fluitans Is the carrier negatively charged?

The amino acid transport across the plasmalemma of Riccia fluitans rhizoid cells has been further characterized by means of current-voltage $\text{I?V}$) analysis. On the basis of two cyclic transport models which include six different carrier states, the question is raised, whether the electrochemical pH-gradient drives a negatively charged carrier or a positively charged alanine-proton-carrier complex across the membrane. $\text{I?V}$ analysis shows that (1) the typical $\text{I?V}$ characteristic of l-alanine transport follows a sigmoid curve, (2) maximal accumulation of l-alanine within the cytoplasm is reached after about 1 hour, (3) the electrically accessible cytoplasmic l-alanine concentration is limited to about 20 mM, and (4) the steady-state saturation current depends directly on external l-alanine concentration. It is concluded that (a) these results are consistent with the predictions of the models for a negatively charged carrier, and (b) that the rate-limiting step involves the translocation of the ternary complex.     

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.     

A modified assay of adenylate cyclase using 3H and 14C

Kinetic data from enzyme-catalyzed reactions have been analyzed traditionally in terms of the Michaelis-Menten equation, which assumes that the maximal velocity (V) and the Michaelis constant (K) are the primary kinetic constants. But what is needed from most kinetic studies today is $\text{V}\text{K}$. A new form of the equation is proposed which assumes that V and $\text{V}\text{K}$ are the primary kinetic constants: $\text{v = (V·S·}\text{V}\text{K}\text{)/(V+S·}\text{V}\text{K}\text{)}$. Computer fittings of both experimental and simulated velocity data to both equations give results favoring the new equation.     

A stochastic model for the economic management of a renewable animal resource

An optimal economic harvesting policy, which maximizes the present value of an animal population, capable of renewing itself, is discussed. It is assumed that, unhindered, the successive population levels, X_n, form a Markov chain, with transitions

$\text{X}{}_{\mathrm{n+1}}\text{=?(X}{}_{\mathrm{n}}\text{) + ?}{}_{\mathrm{n}}\text{?(X}{}_{\mathrm{n}}\text{)}$

, where f is the recruitment function, and {?_n} is an iid sequence of random shocks. When a positive set-up cost is present an optimal policy is of the (S,s) type. The optimal population level is compared with that of an equivalent deterministic model. Bioeconomic conditions, which imply the optimality of conservation, or extinction are investigated.     

Evolution of the three overlapping gene systems in G4 and phi X174.

Bacteriophage G4 has the same $\text{A}\text{B}$ and $\text{D}\text{E}$ overlapping gene systems as φX174 and together with the A and $\text{C}\text{K}$ overlapping gene system (Shaw et al., 1978), 7 of the 11 G4 and φX174 genes are involved in overlaps. The nucleotide differences between G4 and φX174 in the overlapping portions of the A, C and D genes are 23%, 27% and 21%, respectively, compared with 32%, 36% and 34% in the non-overlapping portions of the same genes. The amino acid differences between the G4 and φX174 overlapping B, K and E proteins, are 44%, 39% and 44%, respectively, compared with 28%, 26% and 16% in the regions of genes A, A and C, and D which contain genes B, K and E. These results suggest that the nucleotide sequences of overlapping genes evolve at almost the same rate as in non-overlapping genes, and that this is made possible by a lower amino acid sequence stringency of one of the pairs of proteins. The overlapping $\text{D}\text{E}$ and A and $\text{C}\text{K}$ gene systems may have originated by taking advantage of a high incidence of T nucleotides in the second codon position to produce a hydrophobic protein and the $\text{A}\text{B}$ gene system may have evolved by read-through of the A gene into the B gene. From the nucleotide sequences, other overlapping genes appear to be possible in these bacteriophages.     

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.     

Conformations and CD curves of cyclo(L- or D-Phe-L-Pro-Aca): cyclized models for specific types of β-bends

Cyclic tripeptides cyclo(L-Phe-L-Pro-Aca) (molecule $\text{3}$) (Aca, ?-aminocaproic acid) and cyclo(-D-Phe-L-Pro-Aca) (molecule $\text{4}$) are designed as models of specific types of β-bend. Energy calculation and ¹H and ¹³C NMR studies have indicated that peptides $\text{3}$ and $\text{4}$ form β-bend types VI and II', respectively. Circular dichroism spectra of $\text{4}$ have a double minimum negative band at the region of 200–230 nm like those of gramicidin S. The spectra of $\text{3}$, forming the cis peptide bond just before Pro, have a negative extremum at the 210–213 nm region. The spectra are used to estimate the contribution of various bend types in peptides.β-BendCD MeasurementConformational energy calculationCyclic peptideGramicidin SNMR measurement     

Ultracentrifugation studies on early stage polymerization of tobacco mosaic virus protein

The effects of absolute temperature (T), ionic strength (μ), and pH on the polymerization of tobacco mosaic virus protein from the 4 S form (A) to the 20 S form (D) were investigated by the method of sedimentation velocity. The loading concentration in grams per liter (C) was determined at which a just-detectable concentration (β) of 20 S material appeared. It was demonstrated experimentally that under the conditions employed herein, an equilibrium concentration of 20 S material was achieved in 3 h at the temperature of the experiment and that 20 S material dissociated again in 4 h or less to 4 S material either upon lowering the temperature or upon dilution. Thus, the use of thermodynamic equations for equilibrium processes was shown to be valid. The equation used to interpret the results, log ($\text{C?β) =}\text{constant}\text{+ (}\text{ΔH}{}^1\text{2.3RT}$) + ($\text{Δ}\text{W}{}^1{}_{\mathrm{<mtext>el</mtext>}}\text{2.3RT}$) ? K′_sμ + ζpH, was derived from three separate models of the process, the only difference being in the anatomy of the constant; thus, the method of analysis is essentially independent of the model. $\text{ΔH}{}^1$ and ΔW¹_el are the enthalpy and the change in electrical work per mole of A protein (the trimer of the polypeptide chain), K^′_s is the salting-out constant on the ionic strength basis, ζ is the number of moles of hydrogen ion bound per mole of A protein in the polymerization, and R is the gas constant. The three models leading to this equation are: a simple 11th-order equilibrium between A₁ (the trimer of the polypeptide chain) and D, either the double disk or the double spiral of approximately the same molecular weight, designated model A; a second model, designated B, in which A₁ was assumed to be in equilibrium with D at the same time that it is in equilibrium with A₂, A₃, etc., dimers and trimers, etc., of A₁ in an isodesmic system; and a phase-separation model, designated model C, in which A protein is treated as a soluble material in equilibrium with D, considered as an insoluble phase. From electrical work theory, $\text{Δ}\text{W}{}_{\mathrm{el}}{}^1\text{/T}$ was shown to be essentially independent of T; therefore, in experiments at constant μ and constant pH the equation of log (C ? β) versus 1/T is linear with a slope of $\text{ΔH}{}^1\text{/2.3R}$. The results fit such an equation over nearly a 20 °C-temperature range with a single value of $\text{Δ}\text{H}{}^1$ of +32 kcal/mol A₁. Results obtained when T and pH were held constant but μ was varied did not fit a straight line, which shows that more than simple salting-out is involved. When the effect of ionic strength on the electrical work contribution was considered in addition to salting-out, the data were interpreted to indicate a value of $\text{Δ}\text{W}{}^1{}_{\mathrm{el}}$ of 1.22 kcal/mol A₁ at pH 6.7 and a value of 4.93 for K_s^′. When μ and T were held constant but pH was varied, and when allowance was made for the effect of pH changes on the electrical work contribution, a value of 1.1 was found for ζ. This means that something like 1.1 mol of hydrogen ion must be bound per mole of A₁ protein in the formation of D. When this is added to the small amount of hydrogen ion bound per A₁ before polymerization, at the pH values used, it turned out that for D to be formed, 1.5 H⁺ ions must be bound per A₁ or 0.5 per protein polypeptide chain. This amounts to 1 H⁺ ion per polypeptide chain for half of the protein units, presumably those in one but not the other layer of the double disk or turn of the double spiral. When polymerization goes beyond the D stage, as shown by previously published data, additional H⁺ ions are bound. Simultaneous osmotic pressure studies and sedimentation studies were carried out, in both cases as a function of loading concentration C. These results were in complete disagreement with models A and C but agreed reasonably well with model B. The sedimentation studies permitted evaluation of the constant, β, to be 0.33 g/liter.     

The role of the membrane-bound iron-sulphur centres A and B in the Photosystem I reaction centre of spinach chloroplasts

Photosystem I particles prepared from spinach chloroplast using Triton X-100 were frozen in the dark with the bound iron-sulphur Centre A reduced. Illumination at cryogenic temperatures of such samples demonstrated the photoreduction of the second bound iron-sulphur Centre B. Due to electron spin-electron spin interaction between these two bound iron-sulphur centres, it was not possible to quantify amounts of Centre B relative to the other components of the Photosystem I reaction centre by simulating the line-shape of its EPR spectrum. However, by deleting the free radical signal I from the EPR spectra of reduced Centre A alone or both Centres A plus B reduced, it was possible to double integrate these spectra to demonstrate that Centre B is present in the Photosystem I reaction centre in amounts comparable to those of Centre A and thus also signal I ($\text{P-700}$) and X.Oxidation-reduction potential titrations confirmed that Centre A had $\text{E}{}_{\mathrm{<mtext>m</mtext>}}\text{? ?550}\text{mV}$, Centre B had $\text{E}{}_{\mathrm{<mtext>m</mtext>}}\text{? ?585}\text{mV}$. These results, and those presented for the photoreduction of Centre B, place Centre B before Centre A in the sequence of electron transport in Photosystem I particles at cryogenic temperatures. When both A and B are reduced, $\text{P-700}$ photooxidation is reversible at low temperature and coupled to the reduction of the component X. The change from irreversible to reversible $\text{P-700}$ photooxidation and the photoreduction of X showed the same potential dependence as the reduction of Centre B with $\text{E}{}_{\mathrm{<mtext>m</mtext>}}\text{? ?585}$ mV, substantiating the identification of X as the primary electron acceptor of Photosystem I.     

Investigations on purine and pyrimidine bases stacking associations in aqueous solutions by the fluorescence quenching method: II. Heteroassociation between 2-aminopurine and thymidine

Heteroassociation between A and B compounds in liquid solution was considered. Provided that concentration of A molecules is low, a geneial equation describing fluorescence quantum yield and lifetime of compound A as a function of B molecules concentration was derived. The heteroassociation between 2-aminopurine and thymidine in aqueous solutions was examined within the range of temperatures 0 to 90° C. The equilibrium constants of the first step of association, namely heterodimer formation, were determined and its thermodynamic parameters (ΔH = ?2.76 $\text{kcal}\text{mol}$, ΔS = ?5.9 e.u.) were calculated. The observed changes of the stacking rate constants with temperature confirm the two-step mechanism of the reaction. The activation energy (~2.7 $\text{kcal}\text{mol}$) and the encounter distance (~10.7 A) are only slightly larger than in the case of 2-aminopurine autoassociation, most probably because of a stronger solvation of thymidine molecules.     

Investigations on purine and pyrimidine bases stacking associations in aqueous solutions by the fluorescence quenching method: I. Autoassociation of 2-aminopurine

A general equation was derived, describing fluorescence quantum yield and lifetime of an autoassociating compound in liquid solutions. The autoassociation of 2-aminopurine in aqueous solution was examined within the range from 0 to 90°C. The compound seemed to associate cooperatively. The thermodynamic parameters of polymerization change with temperature, so that its free enthalpy ΔG = ?0.0797 T² + 45.4 T ?7893. The dimerization enthalpy and entropy are approximately temperature-independent (ΔH₂ = ?4.17 $\text{kcal}\text{mol}$, ΔS₂ = ?10.9 e.u.), although the function: ΔG₂ = ?0.0308 T² + 30.3 T - 7213 fits experimental points better. The observed dependences can be explained by the increasing role of the hydrophobic effect with temperature and size of the aggregates. The association rate constants were determined, and a two-step reaction mechanism was demonstrated. The first step is diffusion-controlled. The second is characterized by an activation energy of ~2 $\text{kcal}\text{mol}$ and an encounter distance of ~8.3 Å.     

The joint effects of the release of sterile males and immigration of fertilized females on a density regulated population

The joint effects are studied of the release of sterilized males and immigration of mated females on a population whose discrete generation recursion is, $\text{N′ = N(}\text{RK}\text{K + (R ? 1)N}\text{)}$ where, R ≡ reproductive potential, K ≡ equilibrium. This form of growth is derived from life history considerations so that the impact of the release of sterile males on immatures and on adults can be compared. When the migration parameter and the sterile male release parameter are small, the system has three internal equilibriums (the middle one being unstable). Increase in immigration or in release results in one stable equilibrium. The practical conclusion is that migration must be very small in order for the release of sterile males to be effective on suppressing numbers of adults, while more migration can be tolerated if, as in many agricultural pests, immature stages are the object of concern.     

The concept of high- and low-affinity reactions in bovine cytochrome c oxidase steady-state kinetics

(1) Analysis of the data from steady-state kinetic studies shows that two reactions between cytochrome c and cytochrome c oxidase sufficed to describe the concave Eadie-Hofstee plots ($\text{K}{}_{\mathrm{<mtext>m</mtext>}}\text{? 1 · 10}{}^{\mathrm{?8}}\text{M}$ and $\text{K}{}_{\mathrm{<mtext>m</mtext>}}\text{? 2 · 10}{}^{\mathrm{?5}}\text{M}$). It is not necessary to postulate a third reaction of $\text{K}{}_{\mathrm{<mtext>m</mtext>}}\text{? 10}{}^{\mathrm{?6}}\text{M}$. (2) Change of temperature, type of detergent and type of cytochrome c affected both reactions to the same extent. The presence of only a single catalytic cytochrome c interaction site on the oxidase could explain the kinetic data. (3) Our experiments support the notion that, at least under our conditions (pH 7.8, low-ionic strength), the dissociation of ferricytochrome c from cytochrome c oxidase is the rate-limiting step in the steady-state kinetics. (4) A series of models, proposed to describe the observed steady-state kinetics, is discussed.     

Computer-determined rate constants of hemolysis induced by Prymnesium parvum toxin

Rates of hemolysis of rabbit erythrocyte suspensions induced by P. parvum (prymnesin) have been measured colorimetrically at 25.5°C and pH 5.5. The data have been treated previously as consecutive first-order rate processes associated with the prolytic and lytic periods from which two specific rate constants have been obtained, k′ and kψ, respectively. These constants have been related to those obtained by a computer-generated fit of the rate data (absorbance A_t, as a function of time t) with the rate equation $\text{Y =}\text{D}\text{[1 +}\text{exp}\text{((X ? B)}\text{C)}\text{]}\text{+ E}$. Here Y equals A_t, X = time, t; D is equal to a spread factor, A_i ? A_∞; C is the slope of the curve at the inflection point; B is the midpoint time value, i.e., the time at which $\text{A}{}_{\mathrm{t}}\text{=}\text{D}\text{2}$; E is termed the off-set constant and is equal to A_∞. Of these constants, B is directly related to the length of the prolytic period, and C^?1 is directly related to the specific first-order rate constant for hemolysis, k_ψ.     

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.