Cyclic blockade of initiation sites by streptomycin-damaged ribosomes in Escherichia coli: an explanation for dominance of sensitivity

In bacterial extracts streptomycin is known not only to inhibit ribosomal activity but also to cause gradual release of ribosomes from polysomes. Nevertheless, we now find that after streptomycin has virtually halted protein synthesis in cells of Escherichia coli K12 a substantial (though reduced) level of polysomes persists. These polysomes are evidently maintained by turnover rather than by static blockade, for in streptomycin-treated cells [³H]uracil pulses are rapidly incorporated in the polysomal messenger RNA; moreover, if the synthesis of RNA or the formylation of methionyl-transfer RNA is blocked the polysome level decreases rapidly. Streptomycin thus appears to cause a cycle of ribosomal initiation, blockage of chain extension, gradual release, and reinitiation.The resulting cyclic blockade of initiation sites can account for the dominance of streptomycin sensitivity over resistance in $\text{str}{}^{\mathrm{s}}\text{str}{}^{\mathrm{r}}$² heterozygotes. In confirmation of this model, the inactive resistant ribosomes in treated heterozygotes were found to resume activity if the cells were lysed and excess messenger was provided. These findings further suggest that in sensitive cells damage to only a fraction of the ribosomal population by streptomycin may be sufficient to block protein synthesis.     

Determination of the distribution of protein and nucleic acid in the 70 S ribosomes of Escherichia coli and their 30 S subunits by neutron scattering.

Neutron small-angle scattering of the 70 S Escherichia coli ribosomes and of its smaller 30 S subunit has been measured in $\text{H}{}_2\text{O}{}^2\text{H}{}_2\text{O}$ mixtures. A linear dependence of the square of the radius of gyration on the reciprocal of the contrast is found, which is qualitatively similar to the results from contrast variation with the larger 50 S subunit. The slope α in this plot is a measure of radial segregation of RNA and proteins. It is most pronounced with the 50 S subunit. The 30 S particle appears to be more homogeneous, whereas the 70 S ribosome assumes an intermediate value of α. Neither the 30 S and 50 S subunits nor the 70 S ribosome show a significant separation of the centres of mass of their RNA part and proteins. A quantitative comparison of the parameters obtained suggest that the interaction between the two subunits and the 70 S ribosomes does not involve any major change in the latter.     

Binding of the structural protein soc to the head shell of bacteriophage T4

Qβ plus strands with a 70 S ribosome bound to the coat cistron initiation site were used as template for Qβ replicase. Minus strand synthesis proceeded until the replicase reached the ribosome. The ribosome was removed and elongation was continued in a substrate-controlled, stepwise fashion. The nucleotide analog N⁴-hydroxyCMP was introduced into the positions complementary to the third and fourth nucleotides of the coat cistron. The minus strands were elongated to completion, purified and used as template for Qβ replicase. The final plus strand preparation consisted of four species, with the sequences -A-U-G-G- (wild type), -A-U-A-G- (mutant C₃), -A-U-G-A- (mutant C₄) and -A-U-A-A- (mutant $\text{C}{}_3\text{C}{}_4$) at the coat initiation site. The ribosome binding capacity of the mutant RNAs relative to wild type was <0.1 (C₃), 3.2 (C₄) and 0.3 ($\text{C}{}_3\text{C}{}_4$). The finding that mutant C₃ no longer formed an initiation complex suggests that the interaction of the ribosome binding site with fMet-tRNA plays an essential role in the formation of the 70 S initiation complex. The fact that mutant C₄ RNA bound more efficiently than wild type, and that mutant $\text{C}{}_3\text{C}{}_4$ RNA showed substantial ribosome binding capacity whereas the single mutant C₃ did not, can be explained by assuming that an A residue following the A-U-G triplet interacts with a complementary U residue in the anticodon loop sequence. In the case of $\text{C}{}_3\text{C}{}_4$ this additional base-pair may offset the reduced codon-anticodon interaction resulting from the modification of the A-U-G codon.     

Graphic method for determination of allosteric constants

The maximum slope of the plot, appearing in the paper of Watari & Isogai (1976), was derived algebraically as a function of allosteric constants c and α_mor β_m (= cα_m), and the relation between L, c, and α_mor β_m, was also obtained, where $\text{L =}\text{T}{}_{\mathrm{o}}\text{R}{}_{\mathrm{o}}\text{, c =}\text{K}{}_{\mathrm{R}}\text{K}{}_{\mathrm{T}}\text{, α}{}_{\mathrm{m}}\text{=}\text{F}{}_{\mathrm{m}}\text{K}{}_{\mathrm{R}}\text{, β}{}_{\mathrm{m}}\text{=}\text{F}{}_{\mathrm{m}}\text{K}{}_{\mathrm{T}}\text{, R}{}_{\mathrm{o}}\text{and}\text{T}{}_{\mathrm{o}}$ are concentrations of unligated R and T states respectively, K_Rand K_T are microscopic dissociation constants, and F_m is the ligand concentration at the maximum slope of the plot. When the maximum slope is increased by one, the value becomes Hill constant, n. Nomographs which enable easier estimation of allosteric constants, L and c, were constructed from the two given values, the maximum slope of the plot, n ? 1, and α_mor β_m, in the cases where the maximum number of ligands, N, was 2 and 4. In the nomograph, log c is plotted against log L²c^N keeping the value of the maximum slope of the plot and that of α_mor β_m constant. These nomographs show that the representation is symmetrical in the cases of L²c^N > 1 and L²c^N < 1.     

Optimizing enzyme assays with one or two coupling enzymes

The cost of assays using one or two coupling enzymes is optimized by using equations to calculate the minimum amount(s) of enzyme(s) which should be used to obtain a given time (t₉₉) in which 99% of the rate V₀ of the first reaction is obtained. Using two coupling enzymes and given a value of t₉₉, the induction period L = L₁ + L₂ fulfills the requirement $\text{t}{}_{99}\text{2}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{4.6}\text{≥ L ≥}\text{t}{}_{99}\text{4.6}$, allowing one to choose a cost lower than that derived from the until-now generally applied assumption of t₉₉ = 4.6L. Being $\text{α =}\text{L}{}_1\text{L}{}_2$, in optimized assays the values α, t₉₉, and L are related by $\text{T}{}_{99}\text{=4.6}\text{(1+α)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{1+α}\text{L}$, thus allowing (graphical) calculation of the amounts of coupling enzymes which will minimize the cost for every chosen t₉₉ or L. Maximum practical rates, allowed in some supposed interesting cases, have been calculated.     

Half-lives and relative amounts of stored and polysomal ribosomes and poly(A)+ RNA in mouse oocytes

Growing mouse oocytes were labeled in vitro with [³H]uridine and chased for 2 or for 7 days to estimate the relative amounts of RNA appearing in different fractions and to follow their turnover. Oocytes were lysed and thoroughly dispersed in the presence of 1% DOC, and centrifuged on sucrose gradients to separate polysomes from smaller components not engaged in translation. After the short chase, one-third of the labeled ribosomes appeared in EDTA-sensitive polysomes. The proportion of ribosomes in both fractions remained stable during the long chase, demonstrating no net flow from one fraction to the other. When gradient fractions were analyzed by poly(U) Sepharose chromatography, it was found that about 20% of the labeled poly(A)+ RNA appeared in polysomes after the short chase. The half-lives of stored and translated mRNA were followed relative to stable rRNA during the long chase. Stored mRNA was completely stable, but translated mRNA turned over with a $\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of about 6 days. Other methods for separating stored from translated components were not successful, including sedimentation of putative large complexes (fibrillar lattices) containing stored components, or chromatography of lysates on oligo(dT)-cellulose. Results presented here combined with our previous results demonstrate that, during meiotic maturation, the percent of labeled stable RNA which is polyadenylated declines from 19 to 10%, suggesting deadenylation or degradation of half of the accumulated maternal mRNA.     

Structural dynamics of bacterial ribosomes: V. Magnesium-dependent dissociation of tight couples into subunits: Measurements of dissociation constants and exchange rates

The magnesium ion-dependent equilibrium of vacant ribosome couples with their subunits

$\text{70}\text{S}\text{?}\text{k}{}_{\mathrm{?1}}\text{k}{}_1\text{50}\text{S}\text{+30}\text{S}$

has been studied quantitatively with a novel equilibrium displacement labeling method which is more sensitive and precise than light-scattering. At a concentration of 10^?7m, tight couples (ribosomes most active in protein synthesis) dissociate between 1 and 3 mm-Mg²⁺ at 37 °C with a 50% point at 1.9 mm. The corresponding association constants K_a′ are 5.1 × 10⁵m^?1 (1 mm-Mg²⁺), 3.5 × 10⁷m^?1 (2 mm), and 1.2 × 10⁹m^?1 (3 mm), about five orders of magnitude higher than the K_a′ value of loose couples studied by Spirin et al. (1971) and Zitomer & Flaks (1972).In this range of Mg²⁺ concentrations ($\text{37 °C, 50 mm-}\text{NH}{}_4{}^{\mathrm{+}}$) the rate constants depend exponentially and in opposite ways on the Mg²⁺ concentration: k₁ = 2.2 × 10^?3s^?1, k_?1 = 7.7 × 10⁴m^?1s^?1 (2mm-Mg²⁺); k₁ = 1.5 × 10^?4s^?1, k_?1 = 1.7 × 10⁷m^?1s^?1 (5 mm-Mg²⁺). Under physiological conditions (Mg²⁺ ～- 4 mm, ribosome concn ～- 10^?7m), the equilibrium strongly favors association and the rate of exchange is slow ($\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{～- 10}\text{min}$). In the range of dissociation (2 mm-Mg²⁺), association of subunits proceeds without measurable entropy change and hence ΔG^O = ΔH^O. The negative enthalpy change of ΔH^O = ? 10 kcal suggests that association of subunits involves a shape change.Below a critical Mg²⁺ concentration (～- 2 mm), the 50 S subunits are converted irreversibly into the b-form responsible for the transition to loose couples. The results are compatible with two classes of binding sites, one class binding Mg²⁺ non-co-operatively and contributing to the free energy of association by reduction of electrostatic repulsion, and another class probably consisting of hydrogen bonds between components at opposite interfaces whose critical spatial alignment rapidly denatures in the absence of stabilizing magnesium ions.     

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.     

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?.     

14N-ENDOR evidence for imidazole coordination in copper proteins

¹⁴N-ENDOR studies of simple nitrogen-coordinated copper(II) complexes in frozen aqueous solutions show that the nitrogen hyperfine constants, A_″ and A_⊥, of imidazole are much more isotropic ($\text{R =}\text{A}{}_{\mathrm{″}}\text{A}{}_{\mathrm{\perp }}\text{= 1.05}$) than those of the other biologically-related ligand nitrogens. From this result, combined with ¹⁴N-ENDOR results of some copper proteins containing imidazoles as ligands, it is concluded that R < 1.10 for nitrogen hyperfine constants can be employed as an empirical criterion for demonstration of the existence of imidazole coordination in copper proteins.     

Characterization of sodium and proton flows in sub-bacterial particles of Halobacterium halobium in terms of nonequilibrium thermodynamics

A thermodynamic characterization of the Na⁺-H⁺ exchange system in Halobacterium halobium was carried out by evaluating the relevant phenomenological parameters derived from potential-jump measurements. The experiments were performed with sub-bacterial particles devoid of the purple membrane, in 1 M NaCl, 2 M KCl, and at pH 6.5–7.0. Jumps in either pH or pNa were brought about in the external medium, at zero electric potential difference across the membrane, and the resulting relaxation kinetics of protons and sodium flows were measured. It was found that the relaxation kinetics of the proton flow caused by a pH-jump follow a single exponential decay, and that the relaxation kinetics of both the proton and the sodium flows caused by a pNa-jump also follow single exponential decay patterns. In addition, it was found that the decay constants for the proton flow caused by a pH-jump and a pNa-jump have the same numerical value. The physical meaning of the decay constants has been elucidated in terms of the phenomenological coefficients (mobilities) and the buffering capacities of the system. The phenomenological coefficients for the Na⁺-H⁺ flows were determined as differential quantities. The value obtained for the total proton permeability through the particle membrane via all available channels, $\text{L}{}_{\mathrm{<mtext>H</mtext>}}\text{= (}\text{?J}{}_{\mathrm{<mtext>H\; +</mtext>}}\text{?Δ}\text{pH}\text{)}{}_{\mathrm{<mtext>\Delta \psi ,\Delta </mtext><mtext>pNa</mtext>}}$, was in the range of 850–1150 nmol H⁺·(mg protein)^?1·h^?1·(pH unit)^?1 for four different preparations; for the total Na⁺ permeability, $\text{L}{}_{\mathrm{<mtext>Na</mtext>}}\text{= (}\text{?J}{}_{\mathrm{<mtext>Na</mtext>}}{}^{\mathrm{+}}\text{?Δ}\text{pNa}\text{)}{}_{\mathrm{<mtext>\Delta \psi ,\Delta </mtext><mtext>pH</mtext>}}$, it was 1620–2500 nmol Na⁺·(mg protein)^?1·h^?1·(pNa unit)^?1; and for the proton ‘cross-permeability’, $\text{L}{}_{\mathrm{<mtext>HNa</mtext>}}\text{= (}\text{?J}{}_{\mathrm{<mtext>H</mtext>}}{}^{\mathrm{+}}\text{?Δ}\text{pNa}\text{)}{}_{\mathrm{<mtext>\Delta \psi ,\Delta </mtext><mtext>pH</mtext>}}$, it was 220–580 nmol H⁺·(mg protein)^?1·h^?1·(pNa unit)^?1, for different preparations. From the above phenomenological parameters, the following quantities have been calculated: the degree of coupling (q), the maximal efficiency of Na⁺-H⁺ exchange (η_max), the flow and force efficacies (?) of the above exchange, and the admissible range for the values of the molecular stoichiometry parameter (r). We found q ? 0.4; η_max ? 5%; 0.36 ? r ? 2; ?_JNa⁺ ? 1.3 · 10⁵μmol · (RT unit)^?1 at J_Na = 1 μmolNa⁺ · (mgprotein)^?1 · h^?1; and ?_ΔpNa ? 5 · 10⁴ ΔpNa · (mg protein) · h · (RT unit)^?1 at ΔpNa = 1 unit, for different preparations.     

Saturation transfer electron paramagnetic resonance on membrane bound proteins. II-Absence of rotational diffusion of the cholinergic receptor protein in Torpedo marmorata membrane fragments.

We have applied the technique of saturation transfer electron paramagnetic resonance to study the rotational diffusion of spin labeled membrane bound cholinergic receptors from Torpedo marmorata. Two different spin labels were used: a spin labeled maleimide derivative which binds covalently to proteins and a long chain spin labeled acylcholine which binds reversibly with a high affinity to the receptor protein. The maleimide spin label has a motion whose rotational correlation time is τ₂ > 10^?3 sec. The long chain spin labeled acylcholine indicates slightly more motion ($\text{τ}{}_2\text{? 10}{}^{\mathrm{?4}}\text{sec}$), but the nitroxide in this latter case is probably more loosely bound.     

The reduction of the oxygen-evolving system in chloroplasts by thylakoid components

Using thoroughly dark-adapted thylakoids and an unmodulated Joliot-type oxygen electrode, the following results were obtained. (i) At high flash frequency (4 Hz), the oxygen yield at the fourth flash (Y₄) is lower compared to Y₃ than at lower flash frequency. At 4 Hz, the calculated S₀ concentration after thorough dark adaptation is found to approach zero, whereas at 0.5 Hz the apparent $\text{S}{}_0\text{(}\text{S}{}_0\text{+}\text{S}{}_1\text{)}$ ratio increases to about 0.2. This is explained by a relatively fast donation ($\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 1.0–1.5}\text{s}$) of one electron by an electron donor to S₂ and S₃ in 15–25% of the Photosystem II reaction chains. The one-electron donor to S₂ and S₃ appears to be rereduced very slowly, and may be identical to the component that, after oxidation, gives rise to ESR signal II_s. (ii) The probability for the fast one-electron donation to S₂ and S₃ has nearly been the same in triazine-resistant and triazine-susceptible thylakoids. However, most of the slow phase of the S₂ decay becomes 10-fold faster ($\text{t}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 5–6}\text{s}$) in the triazine-resistant ones. In a small part of the Photosystem II reaction chains, the S₂ decay was extremely slow. The S₃ decay in the triazine-resistant thylakoids was not significantly different from that in triazine-susceptible thylakoids. This supports the hypothesis that S₂ is reduced mainly by Q^?_A, whereas S₃ is not. (iii) In the absence of CO₂/HCO^?_A and in the presence of formate, the fast one-electron donation to S₂ and S₃ does not occur. Addition of HCO^?₃ restores the fast decay of part of S₂ and S₃ to almost the same extent as in control thylakoids. The slow phase of S₂ and S₃ decay is not influenced significantly by CO₂/HCO^?₃. The chlorophyll a fluorescence decay kinetics in the presence of DCMU, however, monitoring the Q^?_A oxidation without interference of Q_B, were 2.3-fold slower in the absence of CO₂/HCO^?₃ than in its presence. (iv) An almost 3-fold decrease in decay rate of S₂ is observed upon lowering the pH from 7.6 to 6.0. The kinetics of chlorophyll a fluorescence decay in the presence of DCMU are slightly accelerated by a pH change from 7.6 to 6.0. This indicates that the equilibrium Q^?_A concentration after one flash is decreased (by about a factor of 4) upon changing the pH from 7.6 to 6.0. When direct or indirect protonation of Q^?_B is responsible for this shift of equilibrium Q^?_A concentration, these data would suggest that the pK_a value for Q^?_B protonation is somewhat higher than 7.6, assuming that the protonated form of Q^?_B cannot reduce Q_A.     

Bovine fibrinogens: Reversible polymer formation

Reversible flbrinogen polymer formation was examined at pH 6.6 and Γ/2 0.3. The equilibrium fraction of fibrinogen present as polymer, (P_mf)_e, was determined by gel filtration for fibrinogen concentrations, F_O, from 48 to 166 μm. Using F_O in molarity, the experimental relation is ln [F_O(P_mf)_e] = 3.53 ln[F_O(1 ? (P_mf)_e)] + 23.73. This relation and attendant confidence limits are examined assuming, during filtration, that the original polymer population is either stable or selected polymer species dissociate to monomer. The possibility that all polymers are open is excluded since the calculated microscopic association constant would then increase with F_O. Acceptable models are based on the assumptions that polymers are open, with association constant K_a, until restricted by closure, with association constant K_r, at an integral degree of polymerization, n. Values are selected on the basis that interaction parameters are independent of F_O and that the required molar decrease in free energy is a minimum. Assuming polymer stability, the experimental relation at 273 °K gives $\text{n = 4,}\text{K}{}_{\mathrm{r}}\text{K}{}_{\mathrm{a}}\text{= 1.2 m,}\text{and}\text{K}{}_{\mathrm{a}}$ = 736 m^?1. Temperature dependence gives $\text{ΔH= ?16.9 kcal/mol}\text{and}\text{ΔS}{}^{\mathrm{O}}{}_{\mathrm{a}}$ = ?48.8 e.u. $\text{K}{}_{\mathrm{r}}\text{K}{}_{\mathrm{a}}$ indicates a relation between changes in entropy. The probability is >0.90 that $\text{K}{}_{\mathrm{r}}\text{K}{}_{\mathrm{a}}\text{? 56 m}$, which indicates a greater loss of degrees of freedom on closure than on association. Conclusions are not altered by the assumption that only the closed polymer species is stable. As ionic strength is decreased at pH 6.6, K_a increases. The clotting time of an otherwise constant system decreases as system P_mf is increased.     

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.     

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.