A mathematical model of pollen-tube penetration in apple styles

Cross-compatible pollen tubes were cultured in detached flowers of apple (Malus domestica Borkh. cv. 'Cox's Orange Pippin'). Although the culture method was simple, the detached flowers proved to be adequate for our requirements, and after pollination they were incubated at 12 constant temperatures from 3·5° C to 33·5° C and sampled at intervals for microscopy. S-shaped growth curves (Gompertz function) were fitted to measurements of pollen-tube penetration within the styles, and each of the Gompertz parameters was expressed as a function of temperature. The model was defective at temperatures below 6° C, but a modification is described which improves this deficiency and yields a ten-parameter empirical model with temperature and incubation time as inputs, and pollen penetration as its output. An application of the model is described in which it provides a numerically integrated output of pollentube penetration from a series of temperature records. Potential uses for the model include predictions of pollen penetration for experimental use and testing for small differences in pollen-tube response to temperature, between cultivars, or between experimental treatments.