Synergistic and antagonistic effects of deterministic and stochastic cell-cell variations

By diversifying, cells in a clonal population can together overcome the limits of individuals. Diversity in single-cell growth rates allows the population to survive environmental stresses, such as antibiotics, and grow faster than the undiversified population. These functional cell-cell variations can arise stochastically, from noise in biochemical reactions, or deterministically, by asymmetrically distributing damaged components. While each of the mechanisms is well understood, the effect of the combined mechanisms is unclear. To evaluate the contribution of the deterministic component we developed a mathematical model by mapping the growing population to the Ising model. To analyze the combined effects of stochastic and deterministic contributions we introduced the analytical results of the Ising-mapping into an Euler–Lotka framework. Model results, confirmed by simulations and experimental data, show that deterministic cell-cell variations increase near-linearly with stress. As a consequence, we predict that the gain in population doubling time from cell-cell variations is primarily stochastic at low stress but may cross over to deterministic at higher stresses. Furthermore, we find that while the deterministic component minimizes population damage, stochastic variations antagonize this effect. Together our results may help identifying stress-tolerant pathogenic cells and thus inspire novel antibiotic strategies.

Figure 1

(a) Schematic illustration of damage redistribution in a population of dividing cells. The doubling time $T$ of the cell is slowed down by the accumulated damage [10, 16] [Eq. (2)], and all cells acquire environmental damage (each red star in the schematic corresponds to $\lambda$ amount of damage). At division, the damage is inherited asymmetrically resulting in different doubling times in the two sisters [Eq. (3)]. Green shading marks the lineage starting from an old cell of age $j$ where cells consequently evade damage, i.e., the lineage $\ell =\mathrm{ONNN}$. (Model parameters in the schematic: $a=1, t_{\min }=0, \mu \lambda =1$). (b), (c) The distribution of ADS doubling times (b) and noise [variations from average doubling time (c)] extracted via noise-filtering [Eq. (1)]. (d) Histogram of noise distribution around ADS distribution based on a grid of $(N_{\mathrm{ADS}},N_{\xi })=(100,100)$ bins using the data from Stewart et al. [10] illustrates the absence of statistical dependence between noise and ADS.

Figure 2

Mapping of population with asymmetric damage segregation to statistical physics. (a) Schematic illustration of the analogy between lineages in a growing population and 1D Ising model [illustrated lineage is the same highlighted by green shade in Fig. 1]. For a lineage, the history of damage inheritance corresponds to a unique configuration of spins on an Ising chain. The damage accumulation/evasion events shown above each cell [their fraction of inheritance of ancestor damage via Eq. (3)] correspond to the values of the spins ($\pm 1$, red arrows) in the analogous Ising chain. (b) Each lineage in a population tree maps to a microstate in an Ising model. Select lineages are marked with the corresponding spin states as in panel (a). (c) Population growth rate as predicted by the theory [lines, Eq. (6)] agrees with numerical simulations (circles), as illustrated for $a=0.3$ (blue, bottom line) and $a=1$ (black, top line). Results for the reported damage-dependent asymmetry, $a\left(D\right)$ ([16], red, large circles) overlap with those for $a=1$. Both $t_{\min }=22$ min and $\mu \lambda$ are in relevant ranges for E. coli under experimental conditions [16]. (d) The population growth gain from ADS increases with stress for low ADS ($a=0.3$, top dashed line) and high ADS ($a=1$, bottom dashed line). The growth gain is quantified by the difference of mean population doubling times $\langle T\rangle$ with and without ADS.

Figure 3

The interplay of stochastic and deterministic sources of diversity under stress. (a) Analytical predictions [Eq. (6), dashed lines] of a linear increase of the effects of ADS with stress confirmed by simulations (circles), as illustrated by ${\langle T\rangle }_{\mathrm{ADS}}-t_{\min }$ (top dashed line) and ${\sigma }_{T,\mathrm{ADS}}$ (bottom line). Errorbars show sample error of the means from the simulations (Appendix pp3). The same trends hold for damage-dependent asymmetry, although exact linearity is not satisfied, see Supplemental Material [44] Fig. S2. (b) Comparison of model predictions for ${\sigma }_{T,\mathrm{ADS}}$ with experimental results across all stresses. Squares mark heat-stress, triangles antibiotic (kanamycin) stress and circle marks low UV stress (Stewart et al. [10]). Perceived experimental stress levels $\mu \lambda$ are inferred by fitting the model to the ADS-filtered data (Methods). Each mark represents the average over all cells with those particular experimental conditions, while the red line is the simulation results from 100 repeat simulations of $a\left(D\right)$-model (line indicates 50th percentile, with the shaded region indicating $\text{25th}-\text{75th}$ percentile). C Synergistic contributions of noise and ADS to the population growth rate. Points mark simulation results, while lines are the analytical results [Eq. (7)]. The gain in population growth from noise alone, $\mathrm{\Delta }G(a=0,\xi )$, is in green [Eq. (E8); bottom line at small $\mu \lambda ]$; from ADS alone, $\mathrm{\Delta }G(a=1,\xi =0)$, is in red [Eq. (6); decreasing middle line] and from combined noise and ADS, $\mathrm{\Delta }G(a=1,\xi )$, is in blue [Eq. (7); top line]. Errorbars mark 95% confidence interval on fitting exponential growth to the number of cells across 10 repeat simulations (Appendix pp3pp3-s2). (d), (e) Benefits of doubling time, $B_{\langle T\rangle }$, and damage, $B_{\langle D\rangle }$, defined in Eq. (8), under various combinations of noise and ADS in a case of low (d) and medium (e) damage induced by heat stress [16]. While the population benefits from both noise and ADS in reducing population doubling time (compare results for $B_{\langle T\rangle }$), noise antagonizes the beneficial effects of ADS in decreasing population damage (compare results for $B_{\langle D\rangle }$). For simulations with noise, we show the mean across 10 realizations. Here the asymmetry is set to be damage-dependent, $a\left(D\right)$; however, results are similar for $a=1$. Asterisks indicate statistically significant differences with $p<5.0\times {10}^{-8}$ (bootstrap, $t$-test, see Appendix pp1pp1-s3).

Figure 4

Illustration of the three ensembles from Appendix pp1-s1 for sampling data from experiments or simulations, with the data in the figure coming from simulations using $(\lambda ,\mu ,t_{\min })=(2,1,18\min )$ corresponding to the parameters of the experimental conditions under highest stress (mutant $\text{MC4100}\mathrm{\Delta }\mathrm{clpB}$ strain exposed to antibiotic stress $[0.05\mu \mathrm{G}$/ml kanamycin)]. Using either ensembles 1 (blue) or 2 (green) to calculate the moments replicates the results of our theory (dotted lines) presented in Eq. (6), but ensemble 3 (red) gives the results without ADS. (a) Time evolution of the instantaneous values $\overline{D}$ of the mean-population damage computed based on all cells dividing exactly at the sampling time $t$ using $a=0.8$, with the number of dividing cells indicated for every fifth data point. $\overline{D}$ (black) converges towards the theoretical population mean $\langle D\rangle$ from Eq. (6) (gray line) and settles after about 150–200 min, although some fluctuations persist beyond this, for both the case where the initial cell in the population has 0 damage (top panel) and when the population starts from a cell with the average damage (bottom panel). (b), (c) Results for the population mean damage $\langle D\rangle$ and standard deviation of damage ${\sigma }_{\langle D\rangle }$ for each of the three ensembles (colors) and compared to theoretical results (black dotted lines). At both investigation times $T_0=210$ min (b) and 300 min (c) we find good agreement between theoretical predictions from Eq. (6) for infinite population (black dotted line) and the results from the finite simulations using either ensembles 1 (blue) or 2 (green) for both initial conditions for the simulations (compare dashed and full colored lines).

Figure 5

Adding cell-size control to the system does not influence the conclusions of this study. For either model, the decrease in $\langle T\rangle$ (a), (d) predicted theoretically (dashed line) is still found both with (circles) or without (triangles) stochasticity in cell length at division. There is also little difference between the average pure ADS lifetime ($T^{\mathrm{ADS}}=ln2/\gamma$, blue) and the full average single-cell lifetime [including effects of ADS as well as cell-size control and given by Eq. (C5), red] for all studied cases. Examples of distributions of $\mathrm{\Delta }T$, the variations between pure ADS lifetimes and full single-cell lifetimes, have also been included for $a=0.5$ (b), (c), (e), (f). These illustrate that the difference $\mathrm{\Delta }T$ between the full single-cell lifetime and the pure ADS lifetime [see Eq. (D1)] is very small when cells divide deterministically exactly at the middle (panels B and E) because cell-size control ensures that any variations in cell length are quickly made negligible over few generations. However, the variations are noticeable when the cells divide stochastically (c), (f), but these are evenly distributed around the ADS lifetime, so these also do not impact the average population behavior because their contributions average to 0. Results calculated with $\lambda =2, t_{\min }=18$ min, $\mu =1, N_{\mathrm{gens}}=20$ and interrogated in a 20 min interval starting at 290 min; the first cell in the population has a random cell length drawn from a normal distribution with mean 1 $\mu \mathrm{m}$ and standard deviation of 0.25 $\mu \mathrm{m}$, for “sizer” the cells divide when the reach a length of 3 $\mu \mathrm{m}$ and for “adder” they divide after adding 2 $\mu \mathrm{m}$ to their length at birth.