A functional differential equation for the steady size distribution of a population is derived from the usual partial differential equation governing the size distribution, in the particular case where birth occurs by one individual of size x dividing into a new individuals of size x/a. This leads, in the case of constant growth and birth rate functions, to the functional differential equation y'{x) = -ay(x) + aay(ax) together with the integral condition J°° y(x) dx = 1. We first look at a number of properties that any solution of this equation and boundary condition must have, and then proceed to find the unique solution by the method of Laplace transforms. Results from number theory on the infinite product found in the solution are presented, and it is shown that y(x) tends to a normal distribution a s o -» l + .

Allele frequencies have long been studied by biologists interested in evolution and speciation. More recently, with the application of molecular markers in human DNA profiling we have also seen the need for reliable population allele frequency estimates for making probabilistic inferences. There is now interest in applying the same DNA profiling technology to identification of plant varieties. HortResearch maintains a large germplasm of horticultural plant species. It is becoming evident that accurate identification of these accessions through DNA fingerprinting is essential for effective utilisation and maintenance of this germplasm. Microsatellites are the markers of choice for this fingerprinting. However, such markers do not reveal the dosage of alleles in a polyploid. Polyploidy is common amongst horticultural plants. Estimating allele frequencies in a polyploid population is, therefore, complicated because of some marker genotypes being phenotypically indistinguishable. For example, in a tetraploid, with four alleles at a locus showing polysomic inheritance, although 35 genotypes are possible, these will fall into only 15 marker phenotypic classes. Furthermore 'null' individuals are rarely detected in polyploids. Furthermore, some polyploids can be cryptic exhibiting disomy, instead of the polysomic inheritance. We will discuss the implications of these factors and present an EM-type algorithm for estimating allele frequencies of a polyploid population under certain patterns of inheritance. The method will be demonstrated on simulated data. We also discuss the nature of some of the additional problems that may be encountered with estimating allele frequencies in polyploids for which other solutions still need to be developed. Heredity (2005) 95, 327-334.

A population of cells growing and dividing often goes through a phase of exponential growth of numbers, during which the size distribution remains steady. In this paper we study the function differential equation governing this steady size distribution in the particular case where the individual cells themselves are growing exponentially in size. A series solution is obtained for the case where the probability of cell division is proportional to any positive power of the cell size, and a method for finding closed-form solutions for a more general class of cell division functions is developed.

The 1-aminocyclopropane-1-carboxylic acid synthase (ACS) gene is a member of the ACS gene family that is involved in apple ( Malus x domestica Borkh.) fruit ripening. Presence of an allele ( Md-ACS1-2) of this gene is associated with low internal ethylene concentration in some apple cultivars. In this study, inheritance of Md-ACS1 was determined for 50 apple cultivars/advanced selections and 101 F(1) seedlings from five populations. Following this, the softening pattern of apples stored at 20 degrees C for up to 40 days was examined using 35 fruiting cultivars/selections of defined Md-ACS1 status. Md-ACS1 is inherited in a Mendelian fashion and was found to be linked to fruit softening. Maturity season of genotypes also significantly affected fruit softening. Late-season genotypes in the Md- ACS1-2/2 class had the slowest rate of softening, while early-season Md- ACS1-1/1 genotypes had the most rapid softening rate. The implications of these results are discussed in relation to parental selection and breeding for storage ability in apple.

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