Practice question

Exponential growth is idealized as first-order autocatalytic process where instantaneous rate of new cell formation is proportional to existing population. Differential equation is dN/dt equals mu times N where N is cell number at time t and mu is specific growth rate per hour reflecting intrinsic replication speed. Separation and integration yields Nt equals N0 times e to power mu t giving linear relationship between ln Nt and time with slope mu during log phase representing balanced growth. During true log phase mu approximates constant as all components increase proportionally. Alternative forms like dN/dt equals K N may use K generically synonymous to mu but precise literature distinguishes mu for biomass versus K for carrying capacity in logistic model. Forms like dN/dt equals Nt lacking constant predict growth proportional to time alone not population, dN/dt equals mu over N predicts decreasing growth with increasing cells opposite to observation. Therefore dn/dt equals mu N accurately captures exponential dependence underpinning fundamental relation mu equals ln 2 divided by generation time g and routinely used to calculate maximal growth rate, biomass yield and productivity in batch and continuous chemostat cultures for bioprocess design.