Steady size distributions for cells in one-dimensional plant tissues

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.

Solutions to an advanced functional partial differential equation of the pantograph type

“…In general, for n ≥ 0, it can be shown that 4) and that G n (0) = 0 for n ≥ 1. Substituting u = w n,n = α n x − t into equation (4.4) yields…”

“…The cell division problem was studied by Hall & Wake [3]. The motivation for the study came from experimental results for certain plant cells [4] that suggested solutions of the type n(x, t) = w(t)y(x), (1.4) at least as a long-term approximation. Here, y is a probability density function.…”

Holomorphic solutions to functional differential equations

“…More recently, this equation has found applications in a cell growth model [6,8], where the solution represents a probability density function for the size of a cell (as measured by DNA content). A detailed analysis of the pantograph equation on the real line including the asymptotics of solutions as x → ∞ is given in [12] and generalized in [11].…”

A functional partial differential equation arising in a cell growth model with dispersion

“…Most of the research on this model focussed on the long time asymptotic solution to the problem. Hall and Wake referred to such solutions as steady size distributions (SSDs), and for the case of constant G and B they proposed an SSD solution based on the separable solution n ( x , t )= N ( t ) y ( x ) to the pde (see also Hall and Wake). They were able to solve this problem analytically and showed that y satisfied the pantograph‐type equation: where ν = B α / G .…”