Hierarchical population model with a carrying capacity distribution for bacterial biofilms

Abstract

A time-and space-discrete model for the growth of a rapidly saturating local biological population N (x, t) is derived from a hierarchical random deposition process previously studied in statistical physics. Two biologically relevant parameters, the probabilities of birth, B, and of death, D, determine the carrying capacity K. Due to the randomness the population depends strongly on position, x, and there is a distribution of carrying capacities, Π(K). This distribution has selfsimilar character owing to the imposed hierarchy. The most probable carrying capacity and its probability are studied as a function of B and D. The effective growth rate decreases with time, roughly as in a Verhulst process. The model is possibly applicable, for example, to bacteria forming a "towering pillar" biofilm. The bacteria divide on randomly distributed nutrient-rich regions and are exposed to random local bactericidal agent (antibiotic spray). A gradual overall temperature change away from optimal growth conditions, for instance, reduces bacterial reproduction, while biofilm development degrades antimicrobial susceptibility, causing stagnation into a stationary state.

Cellular automaton for bacterial towers

“…After a short transient, the populations saturate exponentially rapidly to their carrying capacity. For long times, self-similar carrying capacity distributions result [6], which bear a resemblance to that of Fig.7, except in the details of the fractal geometry. We now turn to the important question of whether we can provide a microscopic basis for the rescaling factor λ. Hierarchical-model population curves for 9 samples generated with P = 0.33.…”

“…This convergence rate applies not only to the average but also to each curve of Fig.10 individually, since the process is self-averaging for long times [6]. From (6) we learn that a constant microscopic ("cellular") growth probability G combined with a finite nutrient supply, fixing K, leads to an exponential saturation, justifying the use of and providing a microscopic interpretation for the constant rescaling factor λ in the hierarchical population model.…”

“…This quantity is proportional to the birth probability B (in the present context the probability of death, D, equals zero, because no antibiotic or other bactericidal agent is considered) [6]. Therefore,K ∝ B.…”

“…In the exactly soluble random hierarchical population model the average excess population of bacteriaN(t) obeys the simple time evolution in discrete time t = n [6],…”

“…The random hierarchical population model [6] generates bacterial towers through a more macroscopic construction algorithm. A typical local population N(x, y; t) grown over a substrate in the xy-plane, for t = 4 (after four generations) and for P = 0.33 is shown in Fig.8.…”