Simpson's Paradox in a Synthetic Microbial System

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Science  09 Jan 2009:
Vol. 323, Issue 5911, pp. 272-275
DOI: 10.1126/science.1166739


The maintenance of “public” or “common good” producers is a major question in the evolution of cooperation. Because nonproducers benefit from the shared resource without bearing its cost of production, they may proliferate faster than producers. We established a synthetic microbial system consisting of two Escherichia coli strains of common-good producers and nonproducers. Depending on the population structure, which was varied by forming groups with different initial compositions, an apparently paradoxical situation could be attained in which nonproducers grew faster within each group, yet producers increased overall. We show that a simple way to generate the variance required for this effect is through stochastic fluctuations via population bottlenecks. The synthetic approach described here thus provides a way to study generic mechanisms of natural selection.

A simple general principle has emerged from theoretical and experimental studies of common-good producer–nonproducer interactions: For producers to be selected and maintained, they have to be the privileged recipients of the common good (118). Producers may become privileged recipients by virtue of kinship or spatial proximity or discrimination through reciprocity or some distinctive feature (15, 9, 10, 14).

In this work, we considered the scenario in which producers and nonproducers are distributed heterogeneously into subpopulations of varying composition, some starting with higher and some with lower fractions of producers. As each of these subpopulations grows, its fraction of producers falls. Yet, it is possible for the overall fraction of producers to rise. This phenomenon, usually analyzed in terms of kin selection or group selection (3), can be distilled into an elementary mathematical feature known as Simpson's paradox (5, 19, 20) (Fig. 1). Because the global producer proportion is a weighted sum of the subpopulation proportions, when there is sufficient covariance between growth rate of a subpopulation and its fraction of producers, a counter-intuitive behavior of the whole population can be observed: Producers increase overall.

Fig. 1.

Principle of Simpson's paradox. Simpson's paradox (5, 19, 20) refers to a situation in which several groups, composed of two types of elements, P and NP, evolve so that the proportion of P elements decreases within each group but nevertheless increases in average overall. The right side shows the evolution of three hypothetical subpopulations represented by pie charts of P (green) and NP (white) slices; the initial and final subpopulations are connected by solid black arrows. The left side shows the corresponding composition of the initial and final global population formed by these three subpopulations (dotted lines). As a whole, the figure illustrates the paradox of P decreasing in each subpopulation (Δpi < 0 for every group i) but increasing overall (Δp > 0). This “paradox” is a purely statistical effect, based on the fact that the global proportion of P is a group size–weighted average that differs from the nonweighted average. In general, if we start from n groups of equal sizes and proportions pi of P (i = 1,..., n), whether Simpson's paradox is observed depends on the changes in proportion of P within each group, Δpi, and on the overall growth of each group, wi. The global variation of the proportion of P, Δp, satisfies (29) 〈wi〉Δp = cov (wi, pi) + 〈wiΔpi〉, where 〈.〉 denotes a nonweighted average over the groups, 〈xi〉 = (Σxi)/n, and cov (wi, pi) = 〈wipi〉 – 〈wi〉〈pi〉 represents the covariance between wi and pi. Simpson's paradox corresponds to having Δpi < 0 for all groups i, but Δp > 0 globally, and requires sufficient correlation between wi and pi, viz, cov (wi, pi) > –〈wiΔpi〉.

A similar situation may arise in natural microbial systems (7, 8, 13), but the effects associated with population heterogeneities are difficult there to assess quantitatively. In addition, in natural systems, evolutionary pressures may have selected for genetic backgrounds that limit the spread of nonproducer mutants and ensure that producers have the growth advantage (11). We have taken a synthetic approach (2125), which circumvents the difficulties of quantifying the interactions in natural populations. Having no evolutionary history, a synthetic system can furthermore be engineered to minimize pleiotropic effects, enabling measurement and control of other parameters.

We constructed two Escherichia coli strains that recapitulate the interaction of producers and nonproducers (26) (fig. S1). The common good in this system is a membrane-permeable (27) Rhl autoinducer molecule (fig. S1), rewired to activate antibiotic (chloramphenicol; Cm) resistance gene expression. Otherwise isogenic, green fluorescent protein (GFP)–marked producers synthesize the Rhl autoinducer constitutively, whereas nonfluorescent nonproducers do not. The system, denoted rhl-catLVA (fig. S1), exhibited the expected properties for public-good producers and nonproducers. First, in antibiotic-containing media, producers grew in a density-dependent manner (fig. S2, left) that was abolished when a synthetic autoinducer was exogenously supplied (fig. S2, right), indicating that autoinducer production was limiting. Second, when started from the same initial density, pure cultures of nonproducers grew slower than pure cultures of producers in antibiotic (fig. S3). However, addition of either synthetic autoinducer or cell-free conditioned medium (containing autoinducer made by producers) increased nonproducer growth in antibiotic-containing media (fig. S3).

When cultures containing preincubated mixtures of producers and nonproducers were diluted into antibiotic-containing medium, both the relative growth of the two strains within a mixture and the growth of the whole mixture depended on the initial proportion of producers (Fig. 2A). In our system, nonproducers proliferated faster than producers within each mixture, independently of the initial composition, as verified by flow cytometry (Fig. 2A). The slower growth of producers can be attributed to the cost of production of the autoinducer, together with the linked GFP reporter. Furthermore, cultures with a higher proportion of producers grew to larger population sizes during the course of the selection phase (12 to 13 hours, 30°C) in antibiotic, and ending subpopulation size was strongly correlated with the initial producer proportion (r = 0.988 in 21 trials).

Fig. 2.

Simpson's paradox observed in a synthetic microbial system. (A) Growth and competition of rhl-catLVA producers and nonproducers in antibiotic-containing media (6.25 μg/ml of Cm). Different colors represent 21 different trials. In each trial, subpopulations containing producers and nonproducers mixed in 12 different proportions [about 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, and 100% (vol/vol) producers] were first grown in the absence of Cm and then diluted 1/100 into Cm at time 0. (Top) Before dilution, initial producer proportions (pi) for each subpopulation i were determined. After 12 to 13 hours at 30°C, the normalized growth wi (y axis) and the final producer proportion (pi′) of each mixture were measured. The x component of each vector represents the change from pi to pi′. (Bottom) The same data are plotted to show the change in producer proportion (Δpi) as a function of pi. Note that Δpi is close to zero when pi is very small because of the low overall growth of those subpopulations. Also, Δpi is zero when pi = 1 because those cultures are pure producer populations. The curved arrows joining purple points and individual test tubes depict how the 10 nonpure subpopulations from each trial were pooled to form the global populations (10-well pools) in (B). (B) Simpson's paradox satisfied by rhl-catLVA producers and nonproducers. Initial (p) and final (p′) global proportions of producers were determined in 19 of the 21 trials from (A) (the first two trials were not measured at a global level). Each vector plots the change from p to p′ (mean change = 0.105, SD = 0.011, n = 19). Colors indicate the correspondence between each trial of subpopulations in (A) (leftward vectors) and each global pool in (B) (rightward vectors). Square symbols show predicted p′ values calculated by using the data from individual subpopulations of (A).

The relative growth advantage of nonproducers versus producers within a mixture, combined with the observed correlation between the overall growth of a mixture and the initial proportion of producers, implied that a situation described by Simpson's paradox could be observed (Fig. 1). To test this, we measured the global proportion of producers in the 10 nonpure cultures with different initial proportions of producers by mixing equal volumes of these cultures and then determining the composition with flow cytometry. As predicted by straightforward calculations (Fig. 1) based on the data obtained for the growth of independent mixtures (Fig. 2A), the global proportion of producers increased relative to its initial value (Fig. 2B), despite the fact that within each mixture, producers did not increase in proportion (Fig. 2A). These results are an illustration of Simpson's paradox in growing microbial populations.

The question of the maintenance of producers has also been addressed in studies of bacterial quorum-sensing mutants (16) and siderophore nonproducers (12) using simplified laboratory conditions and quantitative measurements. However, in our experiments, there is no direct benefit of being a producer, specifically, no negative frequency dependence (Δpi > 0 for lower values of pi). More importantly, producers and nonproducers are grown here as nonclonal mixtures under selective conditions in which the public good confers an advantage to nonproducers [in contrast, see (18)].

The conditions for observing Simpson's paradox are not only based on the properties of the two strains. First, the time scale of the antibiotic selection phase is important; if cultures are allowed to reach stationary phase, those differing greatly in producer proportion (e.g., 20% compared with 80%) will have comparable group sizes, thus preventing Simpson's paradox from occurring. Second, Simpson's paradox also depends on the variance of initial group composition. Thus, long-term maintenance of producers requires an ecological process that enables repeated formation of new groups with a sufficiently large composition variance (1). The degree of dispersal can be controlled by varying the “spatial scale of competition” (6), that is, the degree to which the groups are mixed at each step. From this point of view, Simpson's paradox corresponds to the observation that “global competition” (complete mixing) may have an effect opposite to “local competition” (no mixing).

Notably, a simple possible mechanism for generating sufficiently large composition variance is an extreme dilution of the group populations (1, 2). Such Poisson dilution gave rise to stochastic fluctuations of group compositions with sufficiently large variances to generate Simpson's paradox. We diluted a 50:50 mixture of producers and nonproducers into 288 wells (three 96-well plates), so that the number of founder cells in these groups was Poisson-distributed with a mean λ (estimated by counting the number of wells with no cells). As before, all founding groups of bacteria were first grown in the absence of antibiotic and then diluted into the antibiotic-containing medium for 12 hours at 30°C. The 288 wells were subsequently pooled together, and the overall proportion of producers was measured. Figure 3A shows the change in producer proportion in a single round: For small enough λ, the generated variance was sufficient to reach the Simpson's paradox regime.

Fig. 3.

Poisson dilution conditions can generate sufficient variance to satisfy Simpson's paradox. (A) Effect of Poisson mean on the global change in producer proportion. A 50:50 mixture of producers and nonproducers was strongly diluted into three 96-well plates per treatment and first grown without Cm. Then, at time 0, the mixtures were diluted 1/100 into 6.25 μg/ml of Cm for 12 hours at 30°C. The change (Δp) in producer proportion for the global population (288-well pool) is shown for different dilutions, which resulted in the indicated values of λ, the mean number of founder cells per well. Note that the x axis is not to scale. Red circles, showing the observed change in separate subpools of each of the three individual plates (96-well pools), provide an estimate of the variation arising from pooling smaller numbers of wells. (Insets) Histogram show the distributions of initial (time 0) producer proportions (pi) resulting from Poisson dilution conditions. (B) Iteration of Poisson dilution conditions and growth in Cm. A 10% producer mixture underwent five successive rounds of the dilution and growth described in (A). After each round, the three 96-well plates were pooled (288-well pool), analyzed for global producer proportion (p), strongly rediluted, and then aliquoted to three new 96-well plates for the subsequent round. The Poisson mean λ ranged from 2 to 3 in each round. Red circles are as described in (A). Resulting from stochastic effects, the larger variation observed in earlier rounds (red circles) is expected because p is small. Note that, in general, the variation also depends on λ and the number of pooled wells.

Subsequently, we tested whether this process could be iterated over several rounds, starting with a 10% producer mixture. After each round, the Poisson dilution–antibiotic selection protocol was repeated with use of the pooled population as the new starting mixture for the next round; as a result the overall proportion of producers increased over several consecutive rounds to >95% producers (Fig. 3B). The stochastic fluctuations constitute, therefore, a possible mechanism for reversing the direction of selection in growing populations [as noticed in (1) and (2)]. Although others (15) have demonstrated the effects of small group size on producer dynamics within a single population, in this study small founder group size is shown to indirectly favor producers through its effect on population distributions.

From a biological standpoint, our synthetic system provides a simple example of conflicting levels of selection (28): Producer cells are beneficial in terms of growth at the higher level of populations (whether single groups or a set of groups), but they are nevertheless not necessarily favored by natural selection, which primarily operates at the lower level of individuals (producers cells have thus a disadvantage within any single group). Simpson's paradox corresponds to a situation where this conflict is overcome, and the trait beneficial to the population is selected in spite of its individual cost. The synthetic system presented here is amenable to further analysis. For example, both the cost of being a producer and the benefit of receiving the public good could in principle be experimentally manipulated. Our work thus shows that one can analyze quantitatively how the direction of selection can be influenced by a combination of cellular and ecological parameters.

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Figs. S1 to S3


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