Contribution to the linkage theory of autopolyploids: II

In the first part of this paper, hereinafter referred to as I, a general segregation, distribution was introduced for autopolyploids (2s-ploids) withm, loci, andr alleles. Random mating, chromosome segragation, distinct generations, and equal segregation distributions for males and females were assumed. In this second part, using this segregation distribution as a basis, a recurrence formula is established which enables us to compute the distribution of gametes for any generation, if this distribution is known for an initial generation. This initial distribution of gametes is derived from the initial distribution of genotypes. The limit behavior of these distributions is completely described in a general limit theorem which contains as particular cases the limit theorems for diploids withm loci, and for 2s-ploids with one locus (Geiringer, 1944, 1945, 1948).     

Contribution to the mathematical theory of mass behavior: I. The propagation of single acts

The propagation of a single act in a large population is supposed to depend on some external circumstance and on an “imitation component”, where encounters with individuals who are performing or have already performed the act contribute to the tendency of an individual to perform it. The “tendency” to perform is supposed to be measured by the average frequency of stimuli, randomly distributed in time, impinging on the individual. The deduced equation is a relation between the fraction of the population who have performed the act and time, provided the time course of the “external circumstance” and the way in which the imitation component contributes are known. Several special cases are studied, in particular, cases without the imitation component, cases with imitation only, and various mixed cases. Examples are given of social situations in which such factors may operate and general suggestions are made for the systematization of observations and/or experiments to test the assumptions of the theory.     

Contribution to the physically anchored linkage map of the pig

Thirty-three microsatellites have been mapped on the PiGMaP porcine genetic map. By comparison with the previously published PiGMaP maps, the maps of chromosome 2 (140 cM/70 cM) and chromosome 3 (180 cM/110 cM) were extended and new markers were mapped on the p-arm extremity of chromosome 7 and on the centromeric extremity of chromosome 15. New orders are proposed for markers on chromosomes 3 and 17. Six microsatellites isolated from cosmids were also localized on the cytogenetic map by fluorescent in situ hybridization. We tested the subcloning ligation mixture–polymerase chain reaction (SLiM-PCR) method for isolating microsatellites from cosmids. Subcloning is more effective when the cosmid harbours several microsatellites whereas SLiM-PCR is more straightforward when the cosmid contains a single microsatellite. Fifteen anonymous microsatellites were regionally assigned by using a hybrid cell panel. For map integration, the determination of a regional assignment of anonymous microsatellites by using a hybrid cell panel offers an alternative to microsatellite isolation from cosmids and their localizations by in situ hybridization.     

Contribution to the theory of discrimination learning

In the first two sections, which deal, respectively, with simple and double (or successive) discrimination, a comparison is made between the theory presented and certain experiments on time discrimination. Section III sets forth a possible theoretical approach to multiple choice discrimination.     

Contribution to the probabilistic theory of neural nets: I. Randomization of refractory periods and of stimulus intervals

Aggregates of neurons are considered in which the frequency of occurrence of neurons with a specified value of the refractory period follows certain probability distributions. Input-output functions are derived for such aggregates. In particular, if input and output intensities are defined in terms of stimulus frequencies and firing frequencies per neuron respectively, it is shown that a rectangular distribution of refractory periods leads to a logarithmic input-output curve. If input and output are defined in terms of the total number of stimuli and firings in the aggregate, it is shown how the “mobilization” picture leads to the logarithmic input-output curve. By randomizing the intervals between stimuli received by a single neuron and by introducing an inhibitory neuron a very simple “filter net” can be constructed whose output will be sensitive to a particular range of the input, and this range can be made arbitrarily small.     

Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population

An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.     

Contribution to the theory of random and biased nets

The probabilistic theory of random and biased nets is further developed by the “tracing” method treated previously. A number of biases expected to be operating in nets, particularly in sociograms, is described. Distribution of closed chain lengths is derived for random nets and for nets with a simple “reflexive” bias. The “island model” bias is treated for the case of two islands and a single axon tracing, resulting in a pair of linear difference equations with two indices. The reflexive bias is extended to multiple-axon tracing by an approximate method resulting in a modification of the random net recursion formula. Results previously obtained are compared with empirical findings and attempts are made to account for observed discrepancies.     

Contribution to the theory of organismic sets,III

In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern>1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.     

Contribution to the logistic theory of growth: temporal structure and growth potential

The analysis of a structured population according to three (juvenile, mature and senescent) cellular states is carried out within the framework of Delattre's transformation systems theory. Growth in number, with the dissymmetry of cell divisions, is determined by an autocatalysis process under the constraint of the availability of a source. Two models are presented: their dynamics results in a growth of the exponential type or of the sigmoidal type, respectively. In the sigmoidal case, the logistic equation (Richards-Nelder's function with adjunction of a lower asymptote Y not equal to 0) fits satisfactorily the simulated data of the total cell number Y. The growth potential is defined as the instantaneous capacity of autocatalysis, which is expressed in relation to the present 'mitotic resources' (source + non-senescing mature cells). The acceleration variations d2Y/dt2 are in close agreement with the growth potential gradient. The analysis is then generalized to other population structuring. As a result, the logistic equation can be interpreted in terms of a formal model of growth of a structured population submitted to autocatalysis and competition.     

Contribution to the probabilistic theory of neural nets: III. Specific inhibition

The input-output formula is derived for a neuron upon which converge the axones of two other neurons (one excitatory, the other inhibitory) which are themselves subjected to a “Poisson shower” of excitatory stimuli. If the period of latent inhibition, σ, does not exceed one half the refractory period, δ, the input-output curve has no maximum. If, however, σ>δ/2, a maximum exists in the input-output curve. As the outside frequencyx increases without bound, the output frequencyx ₃ approaches an asymptotic value which ranges from 1/δ to 0, depending on the ratio σ/δ. The maximum output (if it exists) is also derived as a function of σ and δ.     

Renewal process approach to the theory of genetic linkage: case of no chromatid interference

Models are presented in which the distribution of crossovers at a four-four-strand stage of meiosis results from a renewal process. Probability distributions are obtained for the number of crossover events on a meiotic bivalent and for the number of exchange points in random meiotic products. These distributions are found to fit the observed distribution of these variables reasonably well. Using these distributions and assuming no chromatid interference, relations between map distance and the recombination fraction are obtained. These relations give either better or equivalent fit to data when compared to relations that are designed to account for both chromatid and chiasma interference.     

Contribution of genetic effects to genetic variance components with epistasis and linkage disequilibrium

Background

Cockerham genetic models are commonly used in quantitative trait loci (QTL) analysis with a special feature of partitioning genotypic variances into various genetic variance components, while the F_∞ genetic models are widely used in genetic association studies. Over years, there have been some confusion about the relationship between these two type of models. A link between the additive, dominance and epistatic effects in an F_∞ model and the additive, dominance and epistatic variance components in a Cockerham model has not been well established, especially when there are multiple QTL in presence of epistasis and linkage disequilibrium (LD).

Results

In this paper, we further explore the differences and links between the F_∞ and Cockerham models. First, we show that the Cockerham type models are allelic based models with a special modification to correct a confounding problem. Several important moment functions, which are useful for partition of variance components in Cockerham models, are also derived. Next, we discuss properties of the F_∞ models in partition of genotypic variances. Its difference from that of the Cockerham models is addressed. Finally, for a two-locus biallelic QTL model with epistasis and LD between the loci, we present detailed formulas for calculation of the genetic variance components in terms of the additive, dominant and epistatic effects in an F_∞ model. A new way of linking the Cockerham and F_∞ model parameters through their coding variables of genotypes is also proposed, which is especially useful when reduced F_∞ models are applied.

Conclusion

The Cockerham type models are allele-based models with a focus on partition of genotypic variances into various genetic variance components, which are contributed by allelic effects and their interactions. By contrast, the F_∞ regression models are genotype-based models focusing on modeling and testing of within-locus genotypic effects and locus-by-locus genotypic interactions. When there is no need to distinguish the paternal and maternal allelic effects, these two types of models are transferable. Transformation between an F_∞ model's parameters and its corresponding Cockerham model's parameters can be established through a relationship between their coding variables of genotypes. Genetic variance components in terms of the additive, dominance and epistatic genetic effects in an F_∞ model can then be calculated by translating formulas derived for the Cockerham models.

     

PKU locus: Genetic linkage with human amylase (Amy) loci and assignment to linkage group I

Summary The linked alpha-amylase loci Amy 1 and Amy 2 were evaluated for their linkage relationship to the PKU locus using data collected from two (one Czech and one Polish) groups of families. The five sibships informative for Amy 1: PKU give a score of 1.505 at =0.00 and the eight sibships informative for Amy 2: PKU give a score of 2.709 at =0.00. Due to the tandem position of Amy 1 and Amy 2 loci, these data could be combined, and linkage between Amy and PKU loci established with a score 4.214 at =0.00. The practical significance of the linkage, especially for identifying PKU allele carriers, is emphasized.     

Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

Contribution to the probabilistic theory of neural nets: II. Facilitation and threshold phenomena

The output curve of a single neuron with a threshold of response with respect to the frequency of the stimuli is derived. If the stimuli are regularly spaced in time, the output curve has discontinuities. If the threshold and/or refractory period are sufficiently large, the output curve approaches the “all-or-none” curve. In the case of completely randomized stimuli, the output curve is sigmoid. The equation of this curve is derived and some properties are studied. Threshold and “all-or-none” effects can be achieved by “pyramiding” neurons of this type to converge on neurons of higher order.     

Contribution of proton linkage to the thermodynamic stability of the major cold-shock protein of Escherichia coli CspA

The stability of protein is defined not only by the hydrogen bonding, hydrophobic effect, van der Waals interactions, and salt bridges. Additional, much more subtle contributions to protein stability can arise from surface residues that change their properties upon unfolding. The recombinant major cold shock protein of Escherichia coli CspA an all-beta protein unfolds reversible in a two-state manner, and behaves in all other respects as typical globular protein. However, the enthalpy of CspA unfolding strongly depends on the pH and buffer composition. Detailed analysis of the unfolding enthalpies as a function of pH and buffers with different heats of ionization shows that CspA unfolding in the pH range 5.5-9.0 is linked to protonation of an amino group. This amino group appears to be the N-terminal alpha-amino group of the CspA molecule. It undergoes a 1.6 U shift in pKa values between native and unfolded states. Although this shift in pKa is expected to contribute approximately 5 kJ/mol to CspA stabilization energy the experimentally observed stabilization is only approximately 1 kJ/mol. This discrepancy is related to a strong enthalpy-entropy compensation due, most likely, to the differences in hydration of the protonated and deprotonated forms of the alpha-amino group.     

Contribution to the Embryology of Indian Euphorbiaceae: I. Euphorbia rothiana Spreng.

The paper presents an account of the embryology of Euphorbiarothiana Spreng.