  In exponentially proliferating populations of microbes, the population
typically doubles at a rate less than the average doubling time of a
single-cell due to variability at the single-cell level. It is known that the
distribution of generation times obtained from a single lineage is, in general,
insufficient to determine a population's growth rate. Is there an explicit
relationship between observables obtained from a single lineage and the
population growth rate? We show that a population's growth rate can be
represented in terms of averages over isolated lineages. This lineage
representation is related to a large deviation principle that is a generic
feature of exponentially proliferating populations. Due to the large deviation
structure of growing populations, the number of lineages needed to obtain an
accurate estimate of the growth rate depends exponentially on the duration of
the lineages, leading to a non-monotonic convergence of the estimate, which we
verify in both synthetic and experimental data sets.
