Dihydrofolate reductase inhibition by 2,4-diaminotriazines: A structure-activity study.

Quantitative structure-activity relationships have been formulated for the inhibition of dihydrofolate reductase from bovine and rat liver by 4,6-diamino-1,2-dihydro-2,2-dimethyl-1-(3-X-phenyl)-s-triazines. The best correlations are: bovine, for 28 congeners; rat, for 18 congeners. In these expressons C is the molar concentration of inhibitor producing 50% inhibition of the enzyme, π₃ is the octanol/water hydrophobic parameter for the 3-X-phenyl substituent, β is an iteratively derived coefficient, and r is the multiple least squares correlation coefficient. The implications of these bilinear models are discussed and compared with earlier work by B. R. Baker.     

Quantitative structure-activity relationships of chymotrypsin-ligand interactions: An analysis of interactions in p3 space.

Fourteen derivatives of l-alanine of the type CH₃CH(NHCO-3-C₅H₄N)COOR₃ have been synthesized and their hydrolysis by chymotrypsin was studied with the object of characterizing enzymic space (?₃) to which R₃ binds. The binding of R₃ ($\text{log}\text{1}\text{K}{}_{\mathrm{m}}$) was shown via correlation analysis to correlate with molar refractivity (MR) of R₃ rather than hydrophobicity (π). The results confirmed our earlier predictions. A correlation equation for the hydrolysis of 77 acyl-amino acid esters of the general formula R₂CH(NHCOR₁)COOR₃ relating $\text{log}\text{(k}{}_{\mathrm{<mtext>cat</mtext>}}\text{k}{}_{\mathrm{m}}\text{)}$ to molar refractivity of R₁, R₂, and R₃ and to $\text{σ}{}^1$ (Taft's polar parameter) of R₃ was formulated. The general picture of ligand interactions with chymotrypsin as seen with correlation analysis is discussed.     

The use of triazine inhibitors in mapping the active site region of Lactobacillus casei dihydrofolate reductase.

A quantitative structure-activity relationship for the inhibition of Lactobacillus casei dihydrofolate reductase by 4,6-diamino-1,2-dihydro-2,2-dimethyl-1-(3-X-phenyl)-s-triazines has been derived: for 28 congeners. In this expression C is the molar concentration of triazine causing 50% inhibition, π₃ is the hydrophobic constant for the 3-X-phenyl substituent, MR′ is the molar refractivity of certain substituents, β is an iteratively derived coefficient, and r is the multiple least squares correlation coefficient. This correlation is quite different from those found with the same type of inhibitors acting on bovine and rat liver dihydrofolate reductase. The differences are discussed. The correlation equations for the triazines acting on purified enzymes are compared with equations correlating triazines inhibiting the growth of Staphylococcus aureus and Escherichia coli.     

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 covariance of relatives derived from a random mating population.

Attention is drawn to errors common in the derivation of forms for the genotypic covariance of noninbred relatives from a Hardy-Weinberg population of diploids. A synthesis of Fisher's least-squares method of partitioning the genotypic variance and Malécot's probability method of expressing kinship, yields a general form. For one locus, the form is $\text{(P}{}_{\mathrm{ss}}\text{+ P}{}_{\mathrm{sd}}\text{+ P}{}_{\mathrm{ds}}\text{+ P}{}_{\mathrm{dd}}\text{)}\text{1}\text{2}\text{σ}{}_{\mathrm{a}}{}^2\text{+ (P}{}_{\mathrm{ss}}\text{P}{}_{\mathrm{dd}}\text{+ P}{}_{\mathrm{sd}}\text{P}{}_{\mathrm{ds}}\text{) σa}{}_{\mathrm{d}}{}^2$, where σ_a² is the additive genetic variance, α_d² is the variance of dominance deviations, p_ij is the probability that parental gamete i is identical by descent to parental gamete j, i = s, d indexes the parents of one relative, and j = s, d indexes those of the other. The form provides a framework for obtaining the covariance of relatives from an equilibrium population with linkage.     

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.     

Respiration-driven proton translocation with nitrite and nitrous oxide in Paracoccus denitrificans

(1) $\text{H}$⁺/electron acceptor ratios have been determined with the oxidant pulse method for cells of denitrifying Paracoccus denitrificans oxidizing endogenous substrates during reduction of O₂, NO^?₂ or N₂O. Under optimal H⁺-translocation conditions, the ratios $\text{H}{}^{\mathrm{+}}\text{O}$, $\text{H}{}^{\mathrm{+}}\text{N}{}_2\text{O}$, $\text{H}{}^{\mathrm{+}}\text{NO}{}^{\mathrm{?}}{}_2$ for reduction to N₂ and $\text{H}{}^{\mathrm{+}}\text{NO}{}^{\mathrm{?}}{}_2$ for reduction to N₂O were 6.0–6.3, 4.02, 5.79 and 3.37, respectively. (2) With ascorbate/N,N,N′,N′-tetramethyl-p-phenylenediamine as exogenous substrate, addition of NO^?₂ or N₂O to an anaerobic cell suspension resulted in rapid alkalinization of the outer bulk medium. $\text{H}{}^{\mathrm{+}}\text{N}{}_2\text{O}$, $\text{H}{}^{\mathrm{+}}\text{NO}{}^{\mathrm{?}}{}_2$ for reduction to N₂ and $\text{H}{}^{\mathrm{+}}\text{NO}{}^{\mathrm{?}}{}_2$ for reduction to N₂O were ?0.84, ?2.33 and ?1.90, respectively. (3) The $\text{H}{}^{\mathrm{+}}\text{oxidant}$ ratios, mentioned in item 2, were not altered in the presence of $\text{valinomycin}\text{K}{}^{\mathrm{+}}$ and the triphenylmethylphosphonium cation. (4) A simplified scheme of electron transport to O₂, NO^?₂ and N₂O is presented which shows a periplasmic orientation of the nitrite reductase as well as the nitrous oxide reductase. Electrons destined for NO^?₂, N₂O or O₂ pass two H⁺-translocating sites. The $\text{H}{}^{\mathrm{+}}\text{electron}$ acceptor ratios predicted by this scheme are in good agreement with the experimental values.     

Elasnin,a new human granulocyte elastase inhibitor produced by a strain of Streptomyces

Elasnin, a new human granulocyte elastase inhibitor, has been isolated from $\text{Streptomyces}\text{noboritoensis}$ KM-2753. Elasnin is a neutral, lipophilic colorless and viscous oil ($\text{n}{}_{\mathrm{D}}{}^{17}\text{=1.4983}$, $\text{[α]}{}_{\mathrm{D}}{}^{18}\text{?0.9°}$, λ_max^EtOH 291 nm (ε, 7760)). The molecular formula was C₂₄H₄₀O₄ (M.W.: 392) as determined by its elemental analysis and mass spectrum. Elasnin inhibits markedly human granulocyte elastase, but is almost ineffective for pancreatic elastase, trypsin, chymotrypsin, thermolysin and papain.     

Sulphate transport by rat ileum. Effect of molybdate and other anions

Kinetic constants for SO₄^2? transport by upper and lower rat ileum in vitro have been determined by computer fitting of rate vs concentration data obtained using the everted sac technique. MoO₄^2? inhibition of this transport is competitive, and kinetic constants for the inhibition were similarly determined. Transport is also inhibited by the anions WO₄^2?, S₂O₃^2? and SeO₄^2?, in the order $\text{S}{}_2\text{O}{}_3{}^{\mathrm{2?}}\text{> SeO}{}_4{}^{\mathrm{2?}}\text{≧ MoO}{}_4{}^{\mathrm{2?}}\text{> WO}{}_4{}^{\mathrm{2?}}$. These anions have no effect on the transport of l-valine. Low SO₄^2? transport rates were observed in sacs from animals fed a high-molybdenum diet. The significance of the results with respect to the problem of molybdate toxicity in animals is discussed, and related to the known protective effect of SO₄^2?.     

Equilibrium constants under physiological conditions for the reactions of L-phosphoserine phosphatase and pyrophosphate: L-serine phosphotransferase

The observed equilibrium constants (K_obs) for the l-phosphoserine phosphatase reaction [EC 3.1.3.3] have been determined under physiological conditions of temperature (38 °C) and ionic strength (0.25 m) and physiological ranges of pH and free [Mg²⁺]. Using Σ and square brackets to indicate total concentrations $\text{K}{}_{\mathrm{<mtext>obs</mtext>}}\text{=}\text{Σ L-serine][Σ P}{}_{\mathrm{i}}\text{]}\text{Σ L-phosphoserine]H}{}_2\text{O]}\text{, K =}\text{L-H · serine}{}^{\mathrm{\pm }}\text{]HPO}{}_4{}^{\mathrm{2?}}\text{]}\text{[L-H · phosphoserine}{}^{\mathrm{2?}}\text{]H}{}_2\text{O]}$. The value of K_obs has been found to be relatively sensitive to pH. At 38 °C, K⁺] = 0.2 m and free [Mg²⁺] = 0; K_obs = 80.6 m at pH 6.5, 52.7 m at pH 7.0 [ΔG_obs⁰ = ?10.2 kJ/mol (?2.45 kcal/mol)], and 44.0 m at pH 8.0 ([H₂O] = 1). The effect of the free [Mg²⁺] on K_obs was relatively slight; at pH 7.0 ([K⁺] = 0.2 m) K_obs = 52.0 m at free [Mg²⁺] = 10^?3, m and 47.8 m at free [Mg²⁺] = 10^?2, m. K_obs was insignificantly affected by variations in ionic strength (0.12–1.0 m) or temperature (4–43 °C) at pH 7.0. The value of K at 38 °C and I = 0.25 m has been calculated to be 34.2 ± 0.5 m [ΔG_obs⁰ = ?9.12 kJ/mol (?2.18 kcal/ mol)]([H₂O] = 1). The K for the phosphoserine phosphatase reaction has been combined with the K for the reaction of inorganic pyrophosphatase [EC 3.6.1.1] previously estimated under the same physiological conditions to calculate a value of 2.04 × 10⁴, m [ΔG_obs⁰ = ?28.0 kJ/mol (?6.69 kcal/mol)] for the K of the pyrophosphate:l-serine phosphotransferase [EC 2.7.1.80] reaction. $\text{K}{}_{\mathrm{<mtext>obs</mtext>}}\text{=}\text{[}\text{Σ L-serine}\text{][}\text{Σ}\text{P}{}_{\mathrm{<mtext>i</mtext>}}\text{]}\text{[}\text{Σ L-phosphoserine}\text{][}\text{H}{}_2\text{O}\text{]}\text{, K =}\text{[L-H · serine}{}^{\mathrm{\pm }}\text{]HPO}{}_4{}^{\mathrm{2?}}\text{]}\text{[L-H · phosphoserine}{}^{\mathrm{2?}}\text{]H}{}_2\text{O}$. Values of K_obs for this reaction at 38 °C, pH 7.0, and I = 0.25 m are very sensitive to the free [Mg²⁺], being calculated to be 668 [ΔG_obs⁰ = ?16.8 kJ/mol (?4.02 kcal/mol)] at free [Mg²⁺] = 0; 111 [ΔG_obs⁰ = ?12.2 kJ/mol (?2.91 kcal/mol)] at free [Mg²⁺] = 10^?3, m; and 9.1 [ΔG_obs⁰ = ?5.7 kJ/mol (?1.4 kcal/mol) at free [Mg²⁺] = 10^?2, m). K_obs for this reaction is also sensitive to pH. At pH 8.0 the corresponding values of K_obs are 4000 [ΔG_obs⁰ = ?21.4 kJ/mol (?5.12 kcal/mol)] at free [Mg²⁺] = 0; and 97.4 [ΔG_obs⁰ = ?11.8 kJ/ mol (?2.83 kcal/mol)] at free [Mg²⁺] = 10^?3, m. Combining K_obs for the l-phosphoserine phosphatase reaction with K_obs for the reactions of d-3-phosphoglycerate dehydrogenase [EC 1.1.1.95] and l-phosphoserine aminotransferase [EC 2.6.1.52] previously determined under the same physiological conditions has allowed the calculation of K_obs for the overall biosynthesis of l-serine from d-3-phosphoglycerate. $\text{K}{}_{\mathrm{<mtext>obs</mtext>}}\text{=}\text{[}\text{Σ L-serine}\text{][}\text{Σ NADH}\text{][}\text{Σ}\text{P}{}_{\mathrm{<mtext>i</mtext>}}\text{][}\text{Σ}\text{α-}\text{ketoglutarate}\text{]}\text{[Σ d-3-}\text{phosphoglycerate}\text{][Σ NAD}{}^{\mathrm{+}}\text{][Σ L-glutamat0]}$ The value of K_obs for these combined reactions at 38 °C, pH 7.0, and I = 0.25 m (K⁺ as the monovalent cation) is 1.34 × 10^?2, m at free [Mg²⁺] = 0 and 1.27 × 10^?2, m at free [Mg²⁺] = 10^?3, m.     

Some dilute-solution parameters of the levan of streptococcus salivarius in various solvents

The intrinsic viscosities, weight-average molecular weights (M?_w), and radii of gyration [($\text{R}{}^2{}_{\mathrm{g}}$)^{$\text{1}\text{2}$}≈] of Streptococcus salivarius levan in various solvents were respectively obtained from viscosity and light-scattering measurements. The data showed that the levan in water is not aggregated by hydrogen bonds, and that the values of both the refractive index and ($\text{R}{}^2{}_{\mathrm{g}}$)^{$\text{1}\text{2}$} are lower in water than in aqueous solutions of urea. Urea may break intramolecular hydrogen-bonds, e.g., between branches, allowing the molecule to expand.     

Preferential solvation of methylmercurated calf thymus deoxyribonucleic acid.

The changes in polymer-solvent interactions that occur when native calf thymus DNA is dialyzed against Na₂SO₄ solutions of a given ionic strength and buffer concentration but of varying concentrations in methylmercuric hydroxide have been investigated with the help of solution density measurements at 25 °C and pH 6.8–7.0. From measurements executed under equilibrium dialysis conditions at the three salt levels 5 mm, 0.05 m, and 0.5 m Na₂SO₄ (m refers to molality) and in the presence of 5 mm cacodylic acid buffer, the density increments $\text{(}\text{??}\text{?c}{}_2\text{)}{}_{\mathrm{\mu }}{}^0$ for native calf thymus DNA were determined as a function of CH₃HgOH concentration. $\text{(}\text{??}\text{?c}{}_2\text{)}{}_{\mathrm{\mu }}{}^0$ was found not to vary with organomercurial concentration, irrespective of the concentration of supporting electrolyte, until a certain CH₃HgOH concentration level has been reached, viz., , beyond which $\text{(}\text{??}\text{?c}{}_2\text{)}{}_{\mathrm{\mu }}{}^0$ increases strongly with increasing concentration of CH₃HgOH. As is shown by optical melting, $\text{(}\text{??}\text{?c}{}_2\text{)}{}_{\mathrm{\mu }}{}^0$ becomes a function of organomercurial concentration the moment DNA undergoes denaturation brought about by the complexing of CH₃HgOH with the various N-binding sites of the base residues in the DNA double helix.Polymer-solvent interactions, expressed in terms of preferential water interactions (“net hydration”) and preferential salt interactions (“salt solvation”), were derived from the $\text{(}\text{??}\text{?c}{}_2\text{)}{}_{\mathrm{\mu }}{}^0$ data in combination with data obtained on the preferential interaction of CH₃HgOH with denatured DNA and data on the partial specific volumes of all major solution components, gathered from density measurements on solutions with fixed concentrations of diffusible components. Evidence is presented which shows that denaturation in general decreases the net hydration while salt becomes preferentially associated with the polyelectrolyte. This process is further amplified by the interaction of CH₃HgOH with denatured DNA: Methylmercurated DNA alters the redistribution of diffusible components at dialysis equilibrium to such an extent that in a formal sense large amounts of water are rejected from the immediate vicinity of the polymer. The molecular implications of these findings are explored. The results are further discussed in the light of previous findings where the methylmercury-induced denaturation of DNA had been studied with the help of buoyant density measurements in a Cs₂SO₄ density gradient and by velocity-sedimentation in a variety of sulfate media.     

Evidence for general catalysis and formation of nitrobenzene in the oxidation of phenylhydroxylamine in aqueous phosphate buffer

N-Phenylhydroxylamine is oxidized in aqueous phosphate buffer to nitrosobenzene, nitrobenzene, and azoxybenzene. Degradation is O₂ dependent and shows general catalysis by $\text{H}{}_2\text{PO}{}_4{}^{\mathrm{?}}$ (k₁ = 2.3 M^?2 sec^?1) and $\text{PO}{}_4{}^{\mathrm{?3}}$ (k₂ = 2.3 × 10⁵M^?2 sec^?1) or kinetically equivalent terms. Evidence is presented suggesting the intermediacy of a highly reactive species leading to these products.     

Oxidative phosphorylation by membrane vesicles from Bacillus alcalophilus

ADP and P_i-loaded membrane vesicles from l-malate-grown Bacillus alcalophilus synthesized ATP upon energization with $\text{ascorbate}\text{N,N,N′,N′-}\text{tetramethyl}\text{-p-}\text{phenylenediamine}$. ATP synthesis occurred over a range of external pH from 6.0 to 11.0, under conditions in which the total protonmotive force was as low as ?30 mV. The phosphate potentials (ΔG_p) were calculated to be 11 and 12 kcal/mol at pH 10.5 and 9.0, respectively, whereas the values in vesicles at these two pH values were quite different (?40 ± 20 mV at pH 10.5 and ?125 ± 20 mV at pH 9.0). ATP synthesis was inhibited by KCN, gramicidin, and by N,N′-dicyclohexylcarbodiimide. Inward translocation of protons, concomitant with ATP synthesis, was demonstrated using direct pH monitoring and fluorescence methods. No dependence upon the presence of Na⁺ or K⁺ was found. Thus, ATP synthesis in B. alcalophilus appears to involve a proton-translocating ATPase which functions at low .     

Ouabain receptor interactions in (Na+ + K+)-ATPase preparations. II. Effect of cations and nucleotides on rate constants and dissociation constants

The action of ATP and its analogs as well as the effects of alkali ions were studied in their action on the ouabain receptor. One single ouabain receptor with a dissociation constant ($\text{K}{}_{\mathrm{<mtext>D</mtext>}}$) of 13 nM was found in the presence of ($\text{Mg}{}^{\mathrm{2+}}\text{+ P}{}_{\mathrm{i}}$) and ($\text{Na}{}^{\mathrm{+}}\text{+ Mg}{}^{\mathrm{2+}}\text{+ ATP}$). pH changes below pH 7.4 did not affect the ouabain receptor. Ouabain binding required Mg²⁺, where a curved line in the Scatchard plot appeared. The affinity of the receptor for ouabain was decreased by K⁺ and its congeners, by Na⁺ in the presence of ($\text{Mg}{}^{\mathrm{2+}}\text{+ P}{}_{\mathrm{i}}$), and by ATP analogs (ADP-C-P, ATP-OCH₃). Ca²⁺ antagonized the action of K⁺ on ouabain binding. It was concluded that the ouabain receptor exists in a low affinity (R_α) and a high affinity conformational state (R_β). The equilibrium between both states is influenced by ligands of $\text{(Na}{}^{\mathrm{+}}\text{+ K}{}^{\mathrm{+}}\text{)-ATPase}$. With 3 mM Mg²⁺ a mixture between both conformational states is assumed to exist (curved line in the Scatchard plot).     

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.     

Dipole moments of random coil polypeptides

The Rotational Isomeric States model is applied to calculate dipole moments of polypeptides of the twenty natural α-amino acids in the random coil state. Dipole moments of each repeat unit (μ_i), are evaluated using a quantum mechanics procedure. Dipole moment ratios ($\text{D}{}_{\mathrm{x}}\text{=}\text{〈μ}{}^2\text{〉}\text{x}\text{μ}{}_{\mathrm{i}}{}^2\text{,}\text{x = number of repeat units}$) of homopolypeptides are calculated and extrapolated to x →?. With a few exceptions, D_? = 0.36 ± 0.1. Ten actual proteins and three enzymes are also studied; their dipole ratios (D_x′ =〈μ〉/x) range from 7.34 to 10.57 in 10^?59 C² m² (6.6–9.5 D²). Diffferences in the values of D_x′ are due mainly to the different contributions, μ_i, of the amino acid residues contained in each polymer, whereas the sequence of amino acids has a very minor effect.     

Variance of actual inbreeding

The variances of actual inbreeding and coancestry in terms of their corresponding identities by descent were studied for finite populations. For inbreeding at a single locus, the total variance σ² = F(1 ? F) (F is the inbreeding coefficient) is comprised of a component σ_w² within populations and a component σ_b² between replicate populations. These variances increase in time to a maximum at about 1.1N_e generations for σ_w², about 2.3N_e generations for σ_b², and about 1.4N_e generations for σ², and decrease thereafter (N_e is effective population size). The ratio $\text{σ}{}_{\mathrm{<mtext>b</mtext>}}{}^2\text{σ}{}^2$ is ever increasing to an asymptote in the range 0.4-0.5 depending on N_e and the mating system. For finite populations with variation in pedigree F's, there are contributions σ_wF² within and σ_bF² between populations. The component σ_bF² is insignificant except for very small populations, and σ_wF² is largest in the early generations and then decreases roughly as $\text{(1 ? F)}{}^2\text{KN}{}_{\mathrm{<mtext>e</mtext>}}\text{where}\text{K}$ is formulated in terms of the mating strategy and the degree of avoidance of mating relatives. An additional degree of avoidance increases K by a factor of 4. In a large population at equilibrium with respect to mixed self and random mating, σ_wF² accounts for onehalf to two-thirds of σ_w². Bringing in more loci leads to the decomposition of the total variance into four components whose values are affected by linkages among the loci. The relationships between these components and σ_w², σ_b², σ_wF², and σ_bF², are elaborated in terms of tight and loose linkage. The exact computations of σ_wF² and σ_bF² require the use of two locus descent measures without linkage. The variances of various averages of actual identities by descent, such as the proportions for individuals or populations, are formulated for a sample of individuals.