No abstract

No abstract

We consider a population of cells growing and dividing steadily without mortality, so that the total cell population is increasing, but the proportion of cells in any size class remains constant. The cell division process is non-deterministic in the sense that both the size at which a cell divides, and the proportions into which it divides, are described by probability density functions. We derive expressions for the steady size/birth-size distribution (and the corresponding size/age distribution) in terms of the cell birth-size distribution, in the particular case of one-dimensional growth in plant organs, where the relative growth rate is the same for all cells but may vary with time. This birth-size distribution is shown to be the principal eigenfunction of a Fredholm integral operator. Some special cases of the cell birth-size distribution are then solved using analytical techniques, and in more realistic examples, the eigenfunction is found using a simple, generally applicable numerical iteration. Key words" Cell population g r o w t h -Exponential g r o w t h -Size/birth-size distributions -Fission into unequal parts -Fredholm integral equationIn most studies of steady size and size/age distributions of cells, it is assumed that the cell population is "well-mixed", and that cells are physically independent entities (for example, see Collins and Richmond [2], Bell and Anderson [1], or Sinko and Streifer [13]). In this paper we consider a population of cells constrained to grow in one dimension only, within a structured plant tissue. While this places no particular restrictions on the manner in which cells may divide, it does allow a priori specification of the form of the cell growth rate function, and this leads to considerable theoretical simplification.In Part I, we first discuss the case where cells grow without structural constraints, then show how the theory of this general case can be simplified and developed when the cells are constrained to grow in one dimension within structured tissues. Steady size/age and size/birth-size distributions of cells are expressed in terms of the steady cell birth-size distribution, which is shown to be the principal eigenfunction of a Fredholm integral operator. In Part II, we solve some special cases analytically and present a simple numerical method which can be used in analytically intractible cases.

No abstract