Theoretical considerations of the Tsou plot

A graphical method was proposed by Tsou in 1962 for interpreting the data obtained by chemical modification of proteins and determining the number of essential groups involved. A strict mathematical proof and the estimation of deviation probability for this method are presented here. From the well-known Chebyshev's inequality, it has been shown that the possible error which could be derived from the Tsou plot will be much smaller than the usual experimental error obtainable. Some problems related to experiment and to application of Tsou plot in oligomeric enzymes have been discussed. Some analysis of experimental data taken from the literature are presented.     

Theoretical considerations on cell ensembles

Structure-function interdependency must apply at the supracellular level as well as at cellular and molecular levels. The overall structure and the intercellular geometry of the multicellular erythroblastic island, although strongly dependent upon molecular interactions (which determine or modulate recognition, adhesion, and the like), also carry consequences at a more general physical-chemical level. These relate to such factors as local concentrations and gradients, the nature of the forces and the phase-state of the material in the intercellular region, the generalized intercellular transport processes (diffusional, osmotic, electrokinetic, hydrodynamic, etc.), and the dynamic coupling of flows. Such "other-level" phenomena must play a significant role, and therefore must be taken into account, in formulating a comprehensive picture of the ways in which the microenvironment acts in the management and control of the processes of hematopoietic differentiation and development.     

Theoretical considerations of the sensitivity of quantitative subunit hybridization

Summary The theoretical basis of the quantitative subunit hybridization technique and its ability to measure evolutionary amino acid substitutions is examined. Homospecific:heterospecific enzyme ratios found after subunit reassociation depend upon K_1.2, the equilibrium constant for the dissociation of the heterospecific enzyme. It is shown that if this constant is near the geometric mean of the two homospecific enzyme dissociation constants, as it should be in enzymes whose subunits pair isologously, the quantitative subunit hybridization method will not detect most changes in the subunit contact regions of homologous proteins.     

Theoretical considerations on myofibril stiffness.

A discrete model of the interaction between individual myofilaments was developed to study the stiffness of a sarcomere for the case in which filament compliance is not negligible. Our model retains, in the limit, the characteristics of the previously published model by Ford et al. (Ford, L. E., A. F. Huxley, and R. M. Simmons. 1981. The relation between stiffness and filament overlap in stimulated frog muscle fibres. J. Physiol. 311:219-249). In addition, the model is able to model the interaction in cases in which few cross-bridges are attached, or when the distribution of attached cross-bridges is not uniform. Our results confirm previous indications that it might be impossible to calculate the number of attached cross-bridges by using only stiffness measurements in quick-stretch (or release) experiments.     

Foraging behavior of ants: Theoretical considerations

This paper describes models of behavior for two ant species recruiting to very different sorts of resource. One species, Solenopsis geminata, recruits to sugar solution. The other, Pogonomyrmex occidentalis, recruits to patches of seeds. The models clarify the assumptions entering into the theoretical analysis of these behaviors and point to measurements that should be made for their experimental analysis.Both models assume that the colony gains energy when a forager gets a load of resource. Energy is lost during the trip to a patch of resource in a manner dependent upon the current physical conditions of the environment. The only factor counterbalancing an ever increasing net energy gain when the recruitment rates are increased is interference among workers at a patch. As the rate of recruitment is elevated this interference decreases the efficiency with which the resource can be used and, thus, sets an optimum at some intermediate rate of recruitment.The limitation of considerations in this paper to energetic efficiency is discussed and justified in terms of components of fitness and ant physiology. It is found that several qualitative predictions can be experimentally tested, but that quantitative predictions require fine measurement of metabolic costs to individual foragers and of complicated gains to the colony as a whole. Both models can be differentiated experimentally from corresponding time minimization models.     

Reproductive biology of the Pteridophyta. II. Theoretical considerations

Because of the homothallic nature of many pteridophytes, two categories of mating are possible: intragametophytic selling (the origin of both gametes from a single gametophyte) and inter-gametophytic mating (the origin of each gamete from a different gametophyte).Various morphological and genetical criteria (placement of the gametangia on the thallus, their sequence of ontogeny, the capacity for simple polyembryony and genetic self-incompatibility) can be used to indicate the relative probability of intragametophytic selfing or intergametophytic mating. Only the former has genetic significance (i.e.complete homozygosity); if the latter is evidenced, then detailed studies of population variability are required to ascertain the breeding system. Three types of reproductive systems involve the gametophyte generation: intragametophytic selfing, intergametophytic mating and apogamy. Apogamy generally offers the shortest gametophyte generation and the least evolutionary potential, intergametophytic mating systems generally have the longest gametophyte generation and the greatest evolutionary potential, and intragametophytic mating systems are intermediate in both respectS. It is envisioned that the interaction between gametophyte ecology and evolutionary potential is important in the evolution of a taxon's reproductive system.     

Theoretical considerations on potentiation in drug interaction

To account for some of the more important aspects of drug interaction we shall consider a model which can also account for certain general properties of the action of a single drug. A simple model in which there may be enzymatic detoxification of a drug is studied theoretically. The relation between time for appearance of an effect due to the drug and the size of the dose is found to contain the same parameters as the relation between the effectiveness of paired doses and the interval of time between doses. A similar situation holds when the drug is given at a constant rate. When two drugs are administered together, their effect will depend on the manner of interaction, how much of each drug is given, which is given first, and on the interval of time between each administration. A number of plausible types of interaction is considered theoretically in terms of the model, analytical expressions being given for a number of cases. The interaction may be synergistic or antagonistic. In the former case the potentiation may be more than or less than additive depending on the order of delivery and on the time between injections. Methods for the estimation of the parameters from data are discussed.     

Analysis of derivative binding isotherms. Theoretical considerations

The basic theoretical groundwork for the use of derivative binding isotherms in the analysis of ligand binding is presented. The derivative binding isotherm is defined as Γ (Y) = df/dy where f = fractional degree of saturation and y = natural logarithm of the free ligand concentration. Since Γ (y) is a positive function, which goes to zero as y → ±∞, the mean value of y, 〈y〉, and the second and third moments, μ₂ and μ₃ about 〈y〉 are well defined. For a macromolecular system consisting of N equivalent and independent binding sites, Γ (y) is a symmetrical bell-shaped function with one maximum. The maximum occurs when y = ?ln K_assoc; μ₂ = π²/3, and μ₃ = 0. For multiple sets of independent binding sites, Γ (y) is a superposition of Γ-type functions. If the sets are sufficiently well separated in binding free energy, multiple extrema may be seen at positions corresponding to the logarithms of the dissociation constants for the individual sets. In any case, 〈y〉 is equal to the mean value of the logarithms of the dissociation constants for the sets; μ₂ > π²/3 and equal to π²/3 plus the variance of the logarithms of the dissociation constants about their mean value; and μ₃ is, except by coincidence, not equal to zero and equals the third moment of the distribution of logarithms of the dissociation constants about their mean value. Analysis of Γ(y) for the case of cooperative interactions within a set of binding sites was investigated by examining (1) the Hill model (whose mathematical representation is equivalent to that used to describe antibody heterogeneity except that in the latter case the parameter a, the Sips, constant, is constrained (0 < a ≤1);(2) a common model for cooperativity in which the cooperative free energy is a linear function of the fraction bound; and (3) a general representation of cooperative interactions within a set of sites in terms of ?(f), a smooth function that gives the interaction free energy in units of RT. For the Hill model (or Sips model) Γ(y) is a symmetrical function with one maximum at y = (?1)/a)lnK, μ₂ = π²/3a²; and μ₃ = 0. For the case in which the cooperative free energy is a linear function of f [?(f) = cf], 〈y〉 = ?ln K₀ + (c/2); μ₂ = (π²/3) + c[(c/12) + 1] where c > ?4; and μ₃ = 0. General expressions for the moments in terms of ?(f) are derived. In general, μ₂ < (π²/3) for positive cooperativity and μ₂ > (π₂/3) for negative for negative cooperativity. Γ(y) will be symmetrical if and only if the cooperative free energy is introduced symmetrically about f = 0.5.     

Theoretical considerations in the equilibrium binding of myosin fragments on F-actin

In a previous paper, equilibrium constants for the binding of myosin fragments onto F-actin were assumed known and the statistical problems encountered when the actin sites are occupied to an arbitrary fractional extent were analyzed. The object of the present paper is to attempt to understand the observed order of magnitude of these equilibrium constants in terms of the statistical mechanical degrees of freedom involved. That is, we examine here the equilibrium constants them- selves rather than the statistical consequences of the equilibrium constants. The treatment given amounts to a semi-quantitative sketch or outline of the problem. Structural details are much too uncertain to warrant a careful and rigorous treatment at this time. But the discussion suffices to establish the essential qualitative features of the problem. The procedure used is to exa- mine the important equilibrium constants, one at a time, in terms of the factors (partition functions) that contribute to each constant, together with numerical estimates for these factors.     

Equilibrium muscle cross-bridge behavior. Theoretical considerations.

We have developed a model for the equilibrium attachment and detachment of myosin cross-bridges to actin that takes into account the possibility that a given cross-bridge can bind to one of a number of actin monomers, as seems likely, rather than to a site on only a single actin monomer, as is often assumed. The behavior of this multiple site model in response to constant velocity, as well as instantaneous stretches, was studied and the influence of system parameters on the force response explored. It was found that in the multiple site model the detachment rate constant has considerably greater influence on the mechanical response than the attachment rate constant. It is shown that one can obtain information about the detachment rate constants either by examining the relationship between the apparent stiffness and duration of stretch for constant velocity stretches or by examining the force-decay rate constants following an instantaneous stretch. The main effect of the attachment rate constant is to scale the mechanical response by influencing the number of attached cross-bridges. The significance of the modeling for the interpretation of experimental results is discussed.     

Theoretical considerations of the role of antigen structure in B cell activation

Thymus-independent antigens generally are polymeric molecules with repeating arrays of antigenic determinants. Immunological studies of the activity of haptenated thymus-independent antigens have shown that small changes in hapten density can transform a polymeric antigen from nonimmunogenic to immunogenic, and from immunogenic to tolerogenic. In this paper we compute the equilibrium configuration of a linear flexible, haptenated polymer absorbed to a B cell surface, and correlate configurational features of the molecule with its immunological functioning. A polymeric molecule bound to a cell generally will not lie entirely on the surface; rather there will be sections that form loops extending into solution, separated by tightly bound sections, or trains. Trains link antibody receptors on the B cell surface in a fashion that restricts their mobility. Thus trains cause restrictive cross-linking. Our computations show that there is a critical hapten density below which the polymer does not bind to the surface. At hapten densities slightly above the critical density, the polymer binds weakly to the surface with a configuration dominated by a few, rather long loops. These loops cross-link receptors, but do so without bringing the cross-linked receptors into close proximity and without substantially restricting their motion. Long loops thus cause unrestrictive cross-linking. As the hapten density increases, the average loop length decreases and the average train length increases. Thus cross-linking becomes restrictive. In this density range, immune stimulation is observed. At high hapten densities long trains form, separated by few, very short loops and almost all receptors are cross-linked. Consequently cross-linking may be overly restrictive, freezing receptors into place and generating an abundance of cross-linking or other signals that induce a state of immunological tolerance.     

Theoretical considerations and probability models for the somatic development of the T-cell repertoire

A theory about the somatic development of the T-cell repertoire is described which explains the high proportion of alloreactive T lymphocytes as a side effect resulting from two biologically important selection processes. It is assumed that the T-cell receptors contain two binding sites X and Y similar to the two V regions of the B-cell receptors and immunoglobulin molecules, but with a different specificity repertoire as a result of the two selection processes. The first selection process is mediated by T-lineage inducer cells, which have already been postulated by Zinkernagel. They are believed to induce only cells with a binding site Y that can bind to major histocompatibility antigens (MHA) on the inducer cells. This selection forces the T-cell system to recognize preferentially cell-bound antigens; and it introduces under certain circumstances a high degree of self-MHA restriction and self-reactivity, and also a low degree of alloreactivity on the basis of cross-reactivity. The subsequent second selection process serves the purpose of self-tolerance induction. This process is expected under certain circumstances to result in a loss self-reactivity combined with a partial loss of self-restriction and selective enrichment of alloreactive T cells. Probability models are presented to illustrate how the two selection processes shape the T-cell repertoire and how the proportion of alloreactive cells is increased under special circumstances. It is possible that some T-cell subclasses are not affected by the second selection and maintain the primary repertoire. The theory is, in contrast to major competing theories, compatible with the assumption that the original V gene repertoires for the T-cell receptors are random.