Understanding microbial cooperation

Abstract

The field of microbial cooperation has grown enormously over the last decade, leading to improved experimental techniques and a growing awareness of collective behavior in microbes. Unfortunately, many of our theoretical tools and concepts for understanding cooperation fail to take into account the peculiarities of the microbial world, namely strong selection strengths, unique population structure, and non-linear dynamics. Worse yet, common verbal arguments are often far removed from the math involved, leading to confusion and mistakes. Here, we review the general mathematical forms of Price's equation, Hamilton's rule, and multilevel selection as they are applied to microbes and provide some intuition on these otherwise abstract formulas. However, these sometimes overly general equations can lack specificity and predictive power, ultimately forcing us to advocate for more direct modeling techniques.

Figure 1

There are four basic, distinct outcomes of a cooperative interaction. (A) Defectors (red) may always have a higher fitness than cooperators (blue), which would lead to the extinction of cooperators. (B) Alternatively, cooperators may have a higher fitness than defectors when they make up a small fraction of the population and a lower fitness at high fractions, leading to coexistence of cooperators and defectors. (C) Also, cooperators may have a higher fitness only when they make up a large fraction of the population and a lower fitness otherwise. This creates an unstable fixed point, above which cooperators fix in the population and below which defectors fix, characteristic of bistability. (D) Finally, cooperators may always have a higher fitness than defectors. Note, the last case is distinct from the first because cooperators increase the fitness of the population and are therefore not arbitrarily labeled. In game theory, the popular names for these interactions (when linear) are (A) prisoner’s dilemma, (B) snowdrift, (C) stag hunt, and (D) mutually beneficial or harmony game.

Figure 2

Non-linear dynamics can lead to coexistence of cooperators and defectors. (A) Cooperators and defectors coexist in the cooperative sucrose metabolism system, which can be seen by their mutual invasability, where cooperators can invade pure defector cultures (bottom) and defectors can invade pure cooperator cultures (top). (B) In a simple model of the cooperative yeast sucrose metabolism system, if the normalized growth rate, γ, is a linear function of the available glucose, then cooperators or defectors always dominate depending on the parameters. Here, we assume that cooperators pay a cost, c, and capture a small fraction, ε, of the glucose made personally, with the rest, 1 − ε, diffusing away to be used by other cells. The amount of available glucose to a cell is then approximately given by ε + p(1 − ε), where p is the proportion of cooperators in the population. Thus cooperators always grow faster than defectors if c < ε and grow slower otherwise. (C) If, however, the growth rate is a non-linear function of the available glucose, then there is a large region where coexistence between cooperators and defectors is measurable (between dark lines). The exponent, α, was experimentally measured to be 0.15 ± 0.01.

Figure 3

Understanding how the Price equation works can aid in understanding evolution. (A) In its simplest form, Price’s equation states that the change in genotype in a population is ΔḠ = Cov(W, G) = β_WGVar(G), where W is relative fitness here. (B) Doubling the slope of the linear regression, β_WG, while keeping everything else constant, doubles ΔḠ. (C) Also note that increasing the spread of genotypes by a factor of 2 reduces β_WG by a factor of $\frac{1}{2}$, but it multiplies the variance by 2² = 4 so ΔḠ still doubles. (D) Effects multiply and combining (B) and (C) leads to a quadrupling of ΔḠ. For the purpose of illustration, the data points all sit perfectly on the regression, but that is not necessary for Price’s equation to work.

Figure 4

Relatedness as a regression coefficient. To apply the general form of Hamilton’s rule, we need to assign a number to each genotype, and if we are working with two strains (e.g. a cooperator and defector) we can arbitrarily set the cooperator genotype to 1 and the defector genotype to 0. The relatedness, r, is then the linear regression coefficient connecting an individual’s genotype with the genotype of its interactants. Note that when everyone interacts with the same number of individuals, the regression passes through the average of each’s interactants (black dots). This allows us to redefine relatedness as r = ℙ(C|C) − ℙ(C|D): the probability for a cooperator to interact with a cooperator minus the probability for a defector to interact with a cooperator. Also note that scaling the x axis would also scale the y axis by the same amount, leaving the slope the same, so arbitrarily setting the genotypes to 0 and 1 does not lose generality.

Figure 5

Hamilton’s rule is a linear regression to generally non-linear data. (A) In the simplest case, where everything is linear, the slope of the fitness function is b and the difference between a defector and cooperator’s fitness with the same frequency of cooperators is c. For a cooperator to have a higher fitness than a defector, it must interact with r more cooperators than defectors do (see figure). Because of the geometry, this results in a critical relatedness, $r_c=\frac{c}{b}$, above which cooperators are favored (i.e. when rb − c > 0). Unfortunately, life, and particularly microbes, is rarely this simple. (B and C) Because linearity is implicitly assumed with Hamilton’s rule, when the fitness is a non-linear function of the fraction of cooperators (light lines), b and c become linear regression coefficients based on the sample points (gray dots). Unfortunately this masks many of the interesting (non-linear) qualities of the system and provides no prediction power because b and c change with the population structure. Also note that this method can lead to negative fitness values (B) and situations where b < c (C), which would incorrectly suggest that cooperation can never evolve. Note that b and c determine the necessary r for cooperation to evolve, but b and c are in turn affected by the population structure and thus r, which makes disentangling fitness effects from population structure often impossible. It is important to note that the cooperator and defector fitnesses are not independently regressed, rather only one regression is done with the slope (b), vertical difference between defectors and cooperators at a fixed cooperator fraction (c), and base fitness (y-intercept) as the three parameters.

Figure 6

Anscombe’s famous quartet illustrates the need of visualizing one’s data before statistically analyzing it. All four datasets have identical summary statistics (e.g., mean, variance, regression coefficients, and covariance), but vastly different graphs, which are more important in interpreting the data. Researchers should graph and interpret their data before blindly applying Price’s equation or Hamilton’s rule without a model. Importantly, if x was the genotype and y the fitness, Price’s equation would predict the same change in average genotype in all four populations, but we would still know nothing about the underlying dynamics if we fail to visualize the data and model the situation. Note that the population variance and covariance are listed here rather than their unbiased sample equivalents because these are the values used in Price’s equation and thus also Hamilton’s rule and multilevel selection. Also note that the variance of x times the regression coefficient is equal to the covariance, as expected.