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For positive recoverable counts $N_0,N_1$ bracketing one growth interval, define its net cell line population doublings byThe formula counts how many factors of two produce the observed net factor. For exactly two divisions per original biological cell, with all daughters recovered, $N_1=2^2{N}_0$, giving $d=2$. Death or loss of biological cells makes this a net estimate of division.^[ref]

Under constant-rate exponential growth, $N(t)=N_0{e}^{rt}$, a growth interval of duration $\Delta t$ givesFor the constructed fourfold increase over $48\mathrm{h}$, $d=2$ and the model doubling time is $24\mathrm{h}$. The proof follows by taking the natural logarithm and solving $e^{r{t}_2}=2$. This is a fitted property of that interval and model: it is not a universal HeLa doubling time, nor the interval between two divisions of each individual biological cell. Across successive growth intervals, add their $d$ values; keep transfer losses separate as in cell culture passage accounting.

Because the hazard functions are constant, normalization gives ${\Lambda }_1=2\alpha$ and ${\Lambda }_2=3\alpha$. The population characteristic polynomial factors asThusThese exponential distributions describe ages among currently present biological cells. Their rates differ from the phase-exit rates because exponential growth overrepresents younger entrants. The stable phase ratio is $N_2/N_1=1/2$, so the G1 phase and G2 phase fractions approach $2/3$ and $1/3$.

Increasing $\alpha$ rescales all times by $1/\alpha$: the population undergoes faster exponential growth, its doubling time is $\boxed{(\log 2)/\alpha }$, and the profile means $1/(3\alpha )$ and $1/(4\alpha )$ decrease. The phase fractions do not depend on $\alpha$. The population matrix has exact solutionso population-ratio transients decay relative to the growing term at rate $7\alpha$. This exact population formula does not require initially stationary cell-cycle age densities.