Asymmetrical Damage Partitioning in Bacteria: A Model for the Evolution of Stochasticity, Determinism, and Genetic Assimilation

Abstract

Non-genetic phenotypic variation is common in biological organisms. The variation is potentially beneficial if the environment is changing. If the benefit is large, selection can favor the evolution of genetic assimilation, the process by which the expression of a trait is transferred from environmental to genetic control. Genetic assimilation is an important evolutionary transition, but it is poorly understood because the fitness costs and benefits of variation are often unknown. Here we show that the partitioning of damage by a mother bacterium to its two daughters can evolve through genetic assimilation. Bacterial phenotypes are also highly variable. Because gene-regulating elements can have low copy numbers, the variation is attributed to stochastic sampling. Extant Escherichia coli partition asymmetrically and deterministically more damage to the old daughter, the one receiving the mother's old pole. By modeling in silico damage partitioning in a population, we show that deterministic asymmetry is advantageous because it increases fitness variance and hence the efficiency of natural selection. However, we find that symmetrical but stochastic partitioning can be similarly beneficial. To examine why bacteria evolved deterministic asymmetry, we modeled the effect of damage anchored to the mother's old pole. While anchored damage strengthens selection for asymmetry by creating additional fitness variance, it has the opposite effect on symmetry. The difference results because anchored damage reinforces the polarization of partitioning in asymmetric bacteria. In symmetric bacteria, it dilutes the polarization. Thus, stochasticity alone may have protected early bacteria from damage, but deterministic asymmetry has evolved to be equally important in extant bacteria. We estimate that 47% of damage partitioning is deterministic in E. coli. We suggest that the evolution of deterministic asymmetry from stochasticity offers an example of Waddington's genetic assimilation. Our model is able to quantify the evolution of the assimilation because it characterizes the fitness consequences of variation.

Fig 1. Evolution of genetic assimilation.

An activation factor is assumed to be needed to express a phenotype such as crossveinless (CVL). The factor is produced stochastically and its concentration varies between individuals within a population. For CVL to be expressed the concentration needs to exceed a threshold. (A) Under Control conditions the threshold has a value C, and CVL is not expressed because no fly in a pre-selection wild type population exceeds the threshold. The effect of subjecting a pre-selection fly pupa to a heat shock is to lower the threshold to a value H, in which case some flies become CVL (yellow fraction). (B) After selection for CVL following heat shock, the selected flies evolved to produce the activation factor with a distribution that has a higher mean. Under Control conditions, more selected flies are CVL after heat shock (yellow and red fraction), but some flies are able to express CVL even under Control conditions (red fraction). (C) Alternatively, selected flies may have evolved an activation factor with a distribution that has a larger variance but the same mean as before selection. CVL is expressed under both Control conditions (red) and after heat shock (yellow and red).

Fig 2. Cell polarity in E. coli cells.

The cell polarity can be determined by tracking a lineage. Because division cleaves the short axis (-—-) of the cell, poles formed at the cleavage are new (N) and distal poles are old (O). After the next division, the daughter receiving the mother’s new pole is the new daughter and the other is the old daughter.

Fig 3. Distributions of the asymmetry coefficient a.

The value of a represents the proportion of damage partitioned by a mother bacterium to its new daughter. Asymmetry requires that a < ½. If a = ½, the partitioning is symmetrical. Distributions are illustrative representations except for (A), which was derived from the experiments of Stewart et al. [9]. (A) Stochastic variation for observed values of a estimated from experimental E. coli data. Distribution mean = .4845, variance σ_S ² = .0004557, and sample size n = 128. (B) Distribution of a when the partitioning of damage is stochastic but symmetrical with a mean of ½. A Gaussian distribution with a variance of σ_S ² = .0004557 is assumed for illustration. (C) Distribution of the proportion of damage allocated to the daughter that gets less damage when partitioning is stochastic but symmetrical. Because symmetrical partitioning is random with respect to whether a daughter is old or new, polarity can be ignored and all the daughters can be re-categorized into ones that get less and ones that get more damage. If only the lesser daughters are considered, the resulting distribution is the half- or folded normal of the Fig 3B distribution. The mean of the half-normal is ½—√(σ_S ² • 2 / 3.141593…), which equals .483 (●). (D) Gaussian distributions representing four populations: a of new daughters (mean = .48; var = σ_S ² = .00046; a of old daughters (mean = 1 –.48 = .52; var = σ_S ² = .00046); a population made by pooling the new and old daughters; and daughters produced by a stochastic but symmetric mother where the variance is increased to σ_S ² + D ²/4 = .00046 + .0004²/4 = .00086 and mean = ½.

Fig 4. Modeling fitness for damage partitioning in bacteria.

Results report relative fitness over time for populations propagated in a computer model as described (Methods). Parameter values of λ = .0095 min^-1 and Π = 18.30 min were used for all simulations. A relative fitness of .5 corresponds to a severely damaged and effectively dead cell that no longer can divide. Because fitness stabilizes after about 1500 min with these parameter values, fitness values between 1500 to 5000 min were used to calculate mean fitness. (A) Relative fitness over time for asymmetrical partitioning with stochasticity (a = .48; var = σ_S ² = .00046); symmetrical partitioning with stochasticity (a = ½; var = σ_S ² = .00046); and symmetrical partitioning with no stochasticity (a = .5; var = 0). (B) Relative fitness over time for asymmetrical partitioning with stochasticity (a = .48; var = σ_S ² = .00046; no anchored damage); symmetrical partitioning with elevated stochasticity (a = ½; var = σ_S ² + D ²/4 = .00046 + .0004²/4 = .00086; no anchored damage); asymmetrical partitioning with stochasticity (a = .48; var = σ_S ² = .00046; with anchored damage C = .05); symmetrical partitioning with elevated stochasticity (a = ½; var = σ_S ² + D ²/4 = .00046 + .0004²/4 = .00086; with anchored damage C = .05). (C) Anchored damage in asymmetrically produced daughters. Because asymmetrical partitioning (gray shading) allocates movable damage to the old daughter and anchored damage (_*) is more likely to appear first in the mother’s older pole, the difference between old and new daughters is magnified. The magnification increases the variance of damage partitioning. (D) Anchored damage in symmetrically produced daughters. If partitioning is symmetric but stochastic, 50% of the time movable damage is allocated to the old daughter as in Fig 4C. However the other 50% of the time it is as depicted here, where movable damage (gray shading) is allocated to the new daughter and anchored damage (_*) is in the old daughter. The old and new daughters are rendered more similar and the variance of damage partitioning is reduced.

Fig 5. Fitness landscape for damage partitioning with anchored damage.

Landscape compares asymmetric and stochastic bacteria (a = .48; var = .00046) with symmetric and stochastic bacteria (a = ½; variance explored over a range of .0046 to .00046). The partitioning variance of asymmetric bacteria was held constant because this value was the estimate obtained from experimental data in E. coli. All reported ratios are for values of asymmetric bacteria divided by values of symmetric bacteria. Contour lines represent the fitness ratio of mean relative fitness determined from simulated populations after values stabilized (see Fig 4). Parameter values of λ = .0095 min^-1 and Π = 18.30 min were used for all simulations. The x-axis represents values of the fraction C of anchored damage. The y-axis represents the ratio of the partitioning variance. Region above contour line 1.0 represent C and variance ratio values for which asymmetric bacteria have higher fitness. The points (■, ●, and ▲) on the surface denote fitness ratios of populations previously presented, respectively, in Fig 3A (variance of symmetric bacteria = .00046; no anchor), Fig 3B (variance of symmetric bacteria = .00086; no anchor), and Fig 3B (variance of symmetric bacteria = .00086; C = .05).