Practice question

Exponential increase in bacterial population when resources unlimited follows first order kinetics proportional to current population. Mathematically rate change dN over dt equals mu N, where N cell number, mu specific growth rate inverse time. Factor mu reflects nutritional quality, temperature, aeration, strain genotype. Integration yields N(t) equals N0 e to mu t, equivalently N0 2 to power t over g, where g equals ln2 over mu generation time. Taking natural logarithm ln N versus time gives straight line slope mu, confirming balanced growth. Monod equation refines mu as function substrate concentration mu equals mu max S over Ks plus S, saturation constant Ks. This simple model assumes no death, constant cell size, asynchronous divisions averaging. It underpins predictions biomass yield, productivity in fermentors, design of fed batch feeds, modeling infection expansion, and estimating time to reach detectable thresholds in diagnostics. Deviations when substrates limit or toxins accumulate modeled by logistic or inhibition terms. Understanding dn/dt equals mu N is cornerstone quantitative microbiology for both laboratory and industrial contexts.