Large Deviation Principle Linking Lineage Statistics to Fitness in Microbial Populations

In exponentially proliferating populations of microbes, the population 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 nonmonotonic convergence of the estimate, which we verify in both synthetic and experimental data sets.

Figure 1

(a) A population tree starting from a single ancestor. The distinction is made between single lineages (highlighted) and a population (black). Lineages can be sampled by traveling down the tree and randomly selecting a daughter cell at the end of each branch. The probability of selecting any specific lineage with $n$ divisions is $2^{-n}$. (b) $M$ independent lineages of length $T$. For the $i$th lineage, $n_i$ is the number of cell divisions along that lineage. For each lineage we have shown the cell size, which typically increases exponentially between divisions as a function of time. The lineage division distribution can be approximated from these independent lineages by recording the division events and using the highlighted formula. Here, ${\delta }_{n,m}=1$ if $m=n$ and 0 otherwise. (c) A growing population of cells from which one can compute the fitness directly by counting the number of cells as a function of time [or utilizing Eq. (1)]. Using the lineage distribution of divisions we can obtain the fitness from independent lineages.

Figure 2

Comparison of the analytical formula for the population growth rate [Eq. (6)] shown as solid lines with the lineage representation as a function of the mother-daughter correlations shown as open circles, $c$. Simulations were carried out by drawing the generation times of each daughter cell from a normal distribution with mean ${\tau }_0(1-c)+{\tau }^{\prime }c$ and variance ${\sigma }_{\xi }^2$. The parameters were ${\tau }_0=1$, and ${\sigma }_{\tau }^2=0.2$, 0.15, and 0.1 with ${\sigma }_{\tau }^2={\sigma }_{\xi }^2/(1-c^2)$. Error bars are smaller than the symbols. Desired accuracy was achieved with $M={10}^3–{10}^6$ lineages and $T={10}^3$ time duration.

Figure 3

Convergence of the error from the lineage representation as a function of the lineage durations, $T$, for different numbers of lineages, $M$. Data was generated from lineage simulations of the Langevin model with $⟨\tau ⟩=1$, $c=0.2$, and ${\sigma }_{\tau }=0.2$. Here it can clearly be seen that the error initially scales as $1/T$, eventually increasing to approach the limit imposed on the sampling error by the large deviation rate function [see Eq. (14)]. The inset shows the lineage representation ${\widehat{\mathrm{\Lambda }}}_{\mathrm{lin}}$ as a function of $T$ using $M=80$ lineages. This plot is noisy because only a single ensemble of lineages is used, in contrast the error in the main plot is computed by averaging over many ensembles of lineages.