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

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.