Mathematical study of pattern formation accompanied by heterocyst differentiation in multicellular cyanobacterium

Abstract

The filamentous cyanobacterium, Anabaena sp. PCC 7120, is one of the simplest models of a multicellular system showing cellular differentiation. In nitrogen-deprived culture, undifferentiated vegetative cells differentiate into heterocysts at ~10-cell intervals along the cellular filament. As undifferentiated cells divide, the number of cells between heterocysts (segment length) increases, and a new heterocyst appears in the intermediate region. To understand how the heterocyst pattern is formed and maintained, we constructed a one-dimensional cellular automaton (CA) model of the heterocyst pattern formation. The dynamics of vegetative cells is modeled by a stochastic transition process including cell division, differentiation and increase of cell age (maturation). Cell division and differentiation depend on the time elapsed after the last cell division, the "cell age". The model dynamics was mathematically analyzed by a two-step Markov approximation. In the first step, we determined steady state of cell age distribution among vegetative cell population. In the second step, we determined steady state distribution of segment length among segment population. The analytical solution was consistent with the results of numerical simulations. We then compared the analytical solution with the experimental data, and quantitatively estimated the immeasurable intercellular kinetics. We found that differentiation is initially independent of cellular maturation, but becomes dependent on maturation as the pattern formation evolves. Our mathematical model and analysis enabled us to quantify the internal cellular dynamics at various stages of the heterocyst pattern formation.

The multicellular nature of filamentous heterocyst-forming cyanobacteria

“…Not all models of developmental pattern formation in filamentous cyanobacteria have been based on reaction-diffusion schemes (Hammel and Prusinkewicz 1996;Coen et al 2004;Allard, Hill and Rutenberg 2007;Gerdtzen et al 2009;Rutenberg 2012a,b, 2014;Ishihara et al 2015). In a series of works, Rutenberg and coworkers have emphasized fixed nitrogen as a dynamical variable and its transport along filaments, but neglected proteins or peptides in the developmental network such as HetR, PatS and HetN, to model fixed nitrogen and heterocyst spacing distributions (Allard, Hill and Rutenberg 2007;Brown and Rutenberg 2012b).…”

“…No cell growth or cell-cell variability was considered, except for randomness in the initial values of the concentrations of the three proteins. Ishihara et al (2015) implemented a cellular automata scheme involving stochastic cell division, differentiation and increase of cell age in vegetative cells, as well as lateral inhibition from heterocysts flanking a given segment of vegetative cells. This scheme was developed in order to describe two stages in heterocyst pattern formation that they distinguished in their experimental cell lineage data.…”

Morphological and ecological characteristics of potentially toxic invasive cyanobacterium Sphaerospermopsis aphanizomenoides (Forti) Zapomelová, Jezberová, Hrouzek, Hisem, Reháková & Komárková (Nostocales, Cyanobacteria) in Serbia

“…The number of heterocysts alters with a change of nitrogen salts concentration in water (Ogawa and Carr 1969;Zhang et al 2006;Haselkorn 2010;Ishihara et al 2015). Thus, heterocystous cyanobacteria express the genes for heterocyst development and N 2 fixation, only when a suitable source of combined N becomes limited.…”

“…In the presence of ammonium they are unable to assimilate alternative sources of N, such as nitrate, N 2 , or ammonium itself (Adams and Duggan 1999). According to some studies of the strain called Anabaena PCC 7120, differentiation of heterocyst usually occurs at an interval of approximately 10-cell along the filament (Ishihara et al 2015). According to Ogawa and Carr (1969), their number is lowest when amines (NH 4-) are used as a nitrogen source, and they are most abundant when atmospheric nitrogen (N 2 ) is used.…”

Modelling of activator-inhibitor dynamics via nonlocal integral operators

“…Theoretical and mathematical models have been developed using different tools, e.g. cellular automata [6], genetic networks [4,10]. In this paper we model the spatial interaction of the processes of self-activation and lateral inhibition, which produces the well-known pattern of vegetative cells and heterocysts.…”