Dynamics of the adder model of cell division using master equation

Bacterial cells achieve size homeostasis through the adder mechanism, coordinating stochastic protein dynamics to add constant volume between divisions. We develop a master equation framework coupling exponentially replicating growth proteins to threshold-triggered division proteins produced burstily through the geometric distribution. We derive exact probability and inter-division time distributions along with tractable approximations. Key results are that (1) The added size(biomass) follows a convolution of the negative binomial distribution with the burst distribution at division in the large protein approximation (2) Division timing variability stems from coupling of growth noise and approximate Poissonian trigger accumulation convolved with the geometric distribution (3) The expression for the Mean inter-division time provides the relationship between growth rates (α), trigger synthesis (β), initial size (i0) and threshold number (N). The results for the distributions and their properties show universal parameter dependencies. The α/β ratio governs a precision-metabolism trade-off, where suppressing timing variability requires costly protein overproduction. The discrete Master equation methodology unifies discrete molecular-scale stochastic kinetics with cellular homeostasis, showing how noise propagation restricts division control.