A general stochastic model for nucleation and linear growth

Abstract

The model considered here has arisen in a number of completely separate contexts: release of neurotransmitter at neuromuscular synapses, unravelling of strands of DNA, differentiation of cells into heterocysts in algae and growth of crystals. After a shear transformation the model becomes a Markov process, based on a Poisson process on the upper half plane, homogeneous in the horizontal (time) direction, which increases at unit rate except for occasional "drops." By considering the process separately when it is above or below a given "level," for any interval on the time axis, we obtain in particular exact moment results and prove asymptotic normality for long time intervals for the number of downcrossings in the interval, the total time in the interval when the process is below the specified level and the number of drops in the interval. Limit distributions as the length of interval tends to infinity are obtained for the level at which the interval is "covered." It is shown that several problems considered in the literature have analytic solutions as special cases of the general model. The numerical results from one special case are compared to statistics obtained from experimental data from neurobiology.

A Regularity Condition and Strong Limit Theorems for Linear Birth–Growth Processes

“…We use the approach of [11] and [18]. Assuming, without loss of generality, that v = 1/2, we take a shear transformation (x, t) → (x + t/2, t) for the spatio-temporal path of the moving frontiers of the bidirectional growth process.…”

“…Thus, the x i 's are the location points where {τ x } crosses the time level a. Following the terminology used in [11], we say that the path of {τ x } has been separated by {x 1 , x 2 , . .…”

“…Note that Y 2 = 0 if M n = 1. Because of locational stationarity, the distribution is unchanged under the shear transformation (see [11], p. 904 and [18], p. 302) we have Y 1 } and {β 1 , α 2 , . .…”

“…The resultant structure is known as a one-dimensional Johnson-Mehl tessellation [12], [13] and [16]. Such a linear birth-growth process was studied in details by [4], [5], [6], [8], [9], [11], [15] and [17]. Applications can be found in various biological context [1], [7], [18] and [19].…”

“…Chiu [4], Holst et al [11] and Quine and Robinson [17] established the asymptotic normality of the total number of seeds born on a very long interval. Chiu and Quine [5] considered a d-dimensional birth-growth process, d ≥ 1, and obtained the asymptotic normality of the number of seeds born in a very large cube.…”