The equilibrium between cytochrome oxidase and carbon monoxide

An evolution argument which attempted to trace the development of hemoglobins from such respiratory pigments as cytochrome oxidase presupposed that the latter possesses, in addition to its high affinity for oxygen, an approximately hyperbolic equilibrium function, and little if any Bohr effect (decline in affinity for oxygen with rise in acidity). Since cytochrome oxidase, unlike hemoglobin, is irreversibly oxidized by oxygen, the present experiments examine its combination with carbon monoxide, with which, like hemoglobin, it yields a true equilibrium. In all known hemoglobins the form of the equilibrium function and the vigor of the Bohr effect are similar with carbon monoxide and with oxygen, so that observations involving the former gas are relevant to the relations of the latter. The equilibrium function of cytochrome oxidase with carbon monoxide—percentage saturation vs. partial pressure of CO—is slightly inflected (in the Hill equation n = 1.26; for a hyperbola, n = 1). No Bohr effect is present in the range of pH 7–8. The pressure of carbon monoxide at which half-saturation occurs (p₅₀) is about 0.17 mm. at 10–13°C. The affinity for carbon monoxide is therefore higher than commonly supposed. These properties are consistent with the evolution argument. They are important also for the physiological functioning of cytochrome oxidase, the nearly hyperbolic equilibrium function facilitating a high degree of saturation, and the lack of Bohr effect making this enzyme impervious to hyperacidity. The slight inflection of the equilibrium function shows that the Fe-porphyrin units of cytochrome oxidase interact to a degree, hence that the enzyme must contain more than one such unit per molecule. It is suggested that in cytochrome oxidase two Fe-porphyrin groups may unite with one oxygen in the manner Fe⁺⁺-O₂-Fe⁺⁺; and that the evolution of hemoglobins proceeded over a first stage in which the hemes were separated so that each combines with only one molecule of oxygen, so tending to remain reduced; to a further stage in which the separated hemes interact through the protein to facilitate one another in combining with oxygen.     

On the oscillatory transient stage structure of yeast population

We say that a cell of budding yeast, Saccharomyces cerevisiae, is of stage n when it has n bud scars and hence it has produced n daughters. We then divide the total population X into stage classes according to X = Σ_nX_n where X_n is the number of cells which have produced n daughters. The present work deals with the fractions $\text{X}{}_{\mathrm{n}}\text{X}$ in the transient phase before the demographic equilibrium, where the fractions attain stationary values, is reached. It is found that the oscillatory (in time) stage structure is possible provided that (i) the generation time of a virgin (n = 0) cell differs from those of matured (n ≥ 1) cells by more than the sum of respective standard deviations, and (ii) both the initial stage structure and the initial distribution over the cell cycle phase differ appreciably from those prevailing in the demographic equilibrium.     

A New Interpretation of the Dynamic Changes of the Potassium Conductance in the Squid Giant Axon

The solutions, n(t), of the differential equation dn/dt = α (1 - n) n (4 - 6n + 4n² - n³) - βn² (4 - 6n + 4n² - n³) in which α and β are instantaneous functions of membrane potential, are shown to fit with good accuracy the time courses of the rise of potassium conductance during depolarizing steps in clamp potential, found experimentally by Hodgkin and Huxley and by Cole and Moore. The equation is derived by analysing the dynamic behaviour of a system consisting of a square array of interacting pores. The possible role of Ca⁺⁺ ions in this system is discussed.     

An analogue of the binding polynomial for the case of ligands binding to an aggregating macromolecule

It is known that when a macromolecule exists as a mixture of forms with different molecular weights in equilibrium in dilute solution with ligands then the saturation functions cannot be derived by differentiating a binding polynomial. However, we demonstrate that in such circumstances a saturation function can be expressed in terms of the binding polynomials for each of the polymeric forms of macromolecule and integration of this using the conservation equation affords a function, Q, which is the analogue of the binding polynomial for a non-aggregating system. The existence of Q leads to heterotropic linkage relationships.     

Cooperativity in viral fusion

A simple model for membrane fusion mediated by vial spike glycoproteins is presented. The viral proteins are considered to be allosteric proteins that undergo concerted conformational transitions when they bind the ligand. The ligand in this case is H⁺. The effect of the conformational transition is to bring membranes together and induce their fusion. An equation is derived for the dependence of fusion rates on ligand concentration, for a given dissociation constant (K _d), equilibrium constant for the conformational change (L), and number of cooperating subunits (n). Curves generated by this equation provide a reasonable fit to data on the rates of fusion of Vesicular Stomatitis virus with cells for a pK _d of 6.3,L=1000 andn=6.     

Kinetic analysis of biphasic protein modification reactions cooperative effects

A mathematical treatment of protein modification reactions is presented, and it is shown thai in these cases protein modification is described by a summation of exponential functions of reaction time, the number of exponentials being equal to the number of modified protein species. It is shown that in cases of protein modification cooperativity, there is a strict dependence of the coefficients of the multiexponential modification equation on the constants of the same equation. The conditions necessary for a reduction of a multiexponential protein modification equation to one of a summation of two exponentials only are examined. The possible formulae for the coefficients of a two-exponential-summation equation, used to describe the modification of protein models with two, three or four modifiable residues (as well as some aspects of models with five and six modifiable residues) per protein molecule are derived. It is seen that the number of such coefficients is severely limited. The most frequently obtained formula for the lower stoichiomelric coefficient of a 'wo-exponential-summation equation is Ak_a/(k_a-k_b). where k_b and k_b are the constants of the two exponentials of the equation, and A is a constant. The value most frequently arrived at for A is (n?1)/n, where n is the number of modifiable residues per protein molecule, while values such as 1/n, or a/n (where a is an integer, and also where a < n) are also possible. In most of the cooperative protein modification models worked out, k_a is identical with k_n, viz., k_a is identical with the rate constant for the first stoichiometric protein modification.     

The equilibrium and stability for n alleles under the density-dependent selection

The problem of the equilibrium under the density-dependent selection for n-alleles belonging to one locus is considered. The new “measure of quality” of the population φ is introduced and it is shown that the equilibrium points are the constrained stationary points of the function φ and all locally stable equilibriums are its local maximum points. The analoque of the Fisher theorem for density-dependent selection is considered.     

Cardiac chemical constant. Part I: Derivation of the characteristic phenomenological equation

A physiochemical parameter is derived and defined as the cardiac chemical equilibrium dissociation constant (K_D), K_D is based upon a phenomenological model in which the cardiac muscle chemical reaction kinetics describe the interconversion between long and short unils (i.e. the individual sarcomere is fully extended or fully contracted). K_D is defined as the ratio of the number of units in the long state to the number of units in the short state. The mathematical development proceeds through four stages: derivation of the governing differential equation during cardiac systole; simplification of the differential equation to describe the cardiac model; determination of the upper and lower limits and average value of Nt (the total number of units in a hypothetical mid-wall circumferential fibre); definition and calculation of the cardiac chemical constant (K_D). K_D is shown to describe a series of equilibrium points throughout cardiac systole. This requires that each mechanical equilibrium state (a series of static, steady-state intervals over time) is also associated with its own specific chemical equilibrium state.     

Coordination of poly(1-vinylimidazole) by Hemin

It is shown that when hemin (Fe) coordinates imidazole groups (L) attached to a polymer such as poly(1-vinylimidazole) (PVI), then the experimentally observed equilibrium will correspond to the apparent equilibrium constant K_x = [FeL_x]/[Fe][L] and the plot of log ([FeL_x]/[Fe]) against log [L] will give a slope (n) of unity both when x = 1 and, provided both imidazole groups are attached to the same polymer molecule, when x = 2. In agreement with this, hemin reacts with PVI in dimethylsulfoxide at 25°C to give a single product that has a spectrum identical to that of the bisimidazole adduct and the equilibrium corresponds to n = 1.     

Cole-Moore superposition and kinetic models for the potassium conductance of nerve fiber membranes

Restrictions imposed by “Cole-Moore superposition” on a class of models for the potassium conductance of nerve fiber membranes are studied. The models studied assume that: (1) different potassium transfer sites on the membrane are not coupled; (2) each site is characterized by a set of states |α>, α = 0, …, N, each of energy E_α, and degeneracy, r_α. (3) The time dependence is described by kinetic equations.It is rigorously shown that: (a) there is superposition if, and only if, in going from one equilibrium state to another, the system passes only through equilibrium states. (b) Condition (a) is satisfied only if: (i) the energy levels are equally spaced, (ii) the state |n> is [N!/n!(N-n)!]-fold degenerate, and (iii) only transitions between nearest neighbor levels are allowed.     

Properties of the binding of copper by bleomycin

The glycopeptide, bleomycin, binds metal ions including Cu²⁺. It is the copper complex of this material that is isolated from Streptomyces verticillus. Both free ligand and copper complex are excellent antitumor agents in animals. The biochemical and pharmacological relationship between these compounds has not been established. The present study begins an analysis of the chemistry and biochemistry of copper-bleomycin with structural and equilibrium properties of the complex. Potentiometric and fluorometric titrations of bleomycin confirm three acidic groups with pK_a values of 7.50, 4.93, and 2.72. The conjugate nitrogen bases of these groups, comprise three of the binding sites for Cu²⁺ according to similar titrations of copper-bleomycin. The fourth is a conjugate base of an acid with a very large pK_a that cannot be measured by these techniques. The participation of a fourth such group is inferred from both proton release studies of the binding of metal and ligand above pH 8 and from several studies of the thermodynamic stability of copper bleomycin. At low pH binding of copper to bleomycin occurs in two steps, as observed by several independent techniques which monitor either the metal or the ligand. Log stability constants for the reactions Cu²⁺ + H_kBlm ? CuH_k-nBlm + nH⁺ and CuH_k-nBlm ? CuH_k-n-rBlm + rH⁺ are 1.32 and ?4.31, respectively, with n of 2.21 in the first equation and r of 2.07 in the second equation. The derived logarithm of the pH independent stability constant for copper bleomycin multiplied by the protonation constant for the unknown fourth ligand in the binding site is 12.16. This agrees closely with values obtained from measurements of conditional formation constants. One of the groups which binds in the second reaction is the substituted pyrimidine.     

The information capacity of hypercycles

Hypercycles are information integration systems which are thought to overcome the information crisis of prebiotic evolution by ensuring the coexistence of several short templates. For imperfect template replication, we derive a simple expression for the maximum number of distinct templates n_m that can coexist in a hypercycle and show that it is a decreasing function of the length L of the templates. In the case of high replication accuracy we find that the product n_mL tends to a constant value, limiting thus the information content of the hypercycle. Template coexistence is achieved either as a stationary equilibrium (stable fixed point) or a stable periodic orbit in which the total concentration of functional templates is nonzero. For the hypercycle system studied here we find numerical evidence that the existence of an unstable fixed point is a necessary condition for the presence of periodic orbits.     

A model of the complex response of Staphylococcus aureus to methicillin

It is widely accepted that β-lactam antimicrobials cause cell death through a mechanism that interferes with cell wall synthesis. Later studies have also revealed that β-lactams modify the autolysis function (the natural process of self-exfoliation of the cell wall) of cells. The dynamic equilibrium between growth and autolysis is perturbed by the presence of the antimicrobial. Studies with Staphylococcus aureus to determine the minimum inhibitory concentration (MIC) have revealed complex responses to methicillin exposure. The organism exhibits four qualitatively different responses: homogeneous sensitivity, homogeneous resistance, heterogeneous resistance and the so-called ‘Eagle-effect’. A mathematical model is presented that links antimicrobial action on the molecular level with the overall response of the cell population to antimicrobial exposure. The cell population is modeled as a probability density function F(x,t) that depends on cell wall thickness x and time t. The function F(x,t) is the solution to a Fokker-Planck equation. The fixed point solutions are perturbed by the antimicrobial load and the advection of F(x,t) depends on the rates of cell wall synthesis, autolysis and the antimicrobial concentration. Solutions of the Fokker-Planck model are presented for all four qualitative responses of S. aureus to methicillin exposure.     

Derivation of the equations that describe the effects of unstirred water layers on the kinetic parameters of active transport processes in the intestine

Unidirectional flux of solutes into the intestinal mucosal cells is determined by the rate of movement of these molecules across both an unstirred water layer and the microvillus membrane of the epithelial cell. Therefore, an equation is derived in this paper that describes the velocity of active transport as a function of the characteristics of both the transport carrier in the membrane and the resistance of the overlying unstirred water layer. Using this equation a series of curves are presented that depict the effect on the kinetics of active transport of varying the thickness (d) or surface area (S_w) of the unstirred water layer, the free diffusion coefficient (D) of the solute, the distribution of active transport sites along the villus ($\text{?}{}_{\mathrm{<mtext>n</mtext>}}$), the maximal transport velocity (J^m_d) and the true Michaelis constant (K_m). These theoretical curves illustrate the serious limitations inherent in interpretation of previously published data dealing with active transport processes in the intestine.     

Competitive Model on Denaturant-Mediated Protein Unfolding

A denaturant-mediated protein unfolding model, which is different from already existing ones based on the assumption that denaturant competes for water molecules to interact and thus reduces water–protein interactions, which leads to unfolding phenomenon, has been developed with a detailed mathematical justification. Theoretical results suggested that the parameter (m_u) obtained from the usual linear extrapolation model must be a linear function of the number of bound water molecules (n) on protein with a zero intercept. However, application of this theory to a set of proteins for which m_u values for urea denaturation are already known showed that m_u was a linear function of n but with a nonzero intercept. Finally this nonzero intercept was attributed to binding of denaturant to protein at n = 0. Detailed investigation of this factor showed that average equilibrium constant for binding of urea with aromatic side chains (generally nonpolar side chains) was k_b ≈ 0.65 ± 0.45 mol⁻¹, which agreed well with earlier experimental estimations, and also suggested that an integrated approach was necessary to avoid discrepancy in ΔG^H₂O estimated from different models.     

Population dynamics of sex-determining alleles in honey bees and self-incompatibility alleles in plants

Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is 1/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from 1/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.     

Surface active site model for Ni2+ adsorption of the surface imprinted adsorbent

In this work, a new surface active site (SAS) adsorption equilibrium model was presented, which explicitly accounted for the H⁺ competitive adsorption with Ni²⁺ in adsorption equilibrium. Static adsorption experiments with Ni²⁺ as a model metal ion were carried out to determine the model parameters, those were, equilibrium constant for Ni²⁺ (K_a), for H⁺ (K_s), characteristic number of binding sites for Ni²⁺ (n), for H⁺ (a), and the non-imprinted factor (σ). It was found that those model parameters n and a were all constant, and that they all expressed that one active site bound two Ni²⁺ or two H⁺, while the non-imprinted factor, σ, was effected by Ni²⁺ concentration, H⁺ concentration in solution and imprinted Ni²⁺ concentration in the preparation. Simulated result was compared with experimental data of the adsorption for Ni²⁺. It was showed that this model could be well used to predict the adsorption equilibrium for Ni²⁺ on the surface imprinted adsorbent. And it was demonstrated that the efficacy of the active sites formalism could be used in describing adsorption behavior for Ni²⁺ on the surface imprinted adsorbent.     

The Deterministic Behavior of Self-Incompatibility Alleles

For a system of n self-incompatibility alleles, neglecting mutation and random drift, it is shown that the completely symmetric equilibrium is locally stable, and any allelic frequency less than q = 1+a-(see PDF), where a = [2(n - 1)]^-1, will increase. For all n, q > (2n)^-1, but if n > > 1, q ≈ (2n)^-1.     

Combinatorics of Contacts in Protein Contact Maps

Contacts play a fundamental role in the study of protein structure and folding problems. The contact map of a protein can be represented by arranging its amino acids on a horizontal line and drawing an arc between two residues if they form a contact. In this paper, we are mainly concerned with the combinatorial enumeration of the arcs in m-regular linear stack, an elementary structure of the protein contact map, which was introduced by Chen et al. (J Comput Biol 21(12):915–935, 2014). We modify the generating function for m-regular linear stacks by introducing a new variable y regarding to the number of arcs and obtain an equation satisfied by the generating function for m-regular linear stacks with n vertices and k arcs. Consequently, we also derive an equation satisfied by the generating function of the overall number of arcs in m-regular linear stacks with n vertices.     

Binding Characteristics of N2-Fixing Bacteria to Cereal Roots

The attachment of Rhizobium japonicum 61A89 and Rhizobium spp. 32H1 to the roots of wheat and rice seedlings is analyzed in terms of an equilibrium model. A Langmuir adsorption isotherm describes the binding. Strain 61A89 binds to a greater extent than does strain 32H1, and the equilibrium constants for each strain binding to wheat are strongly temperature dependent. Both time-dependent dissociation and association, predicted by an equilibrium model, have been found. The dissociation rate constant for 32H1 is approximately twice that of 61A89, and each is weakly temperature dependent. The rate equation for the binding of exponentially growing 61A89 to wheat roots has been solved as a function of time. Theory and experiment both indicate that the binding at very short times is much less than the equilibrium values. The binding of Azotobacter vinelandii 12837 to wheat roots has also been measured. Root-associated Azotobacter fixes nitrogen, whereas under aerobic growth conditions, root-associated 61A89 and 32H1 do not. The effect of metabolic inhibitors and antibiotics on the binding of Rhizobia and Azotobacter was examined.