The ability of rare types to invade populations is important for the maintenance of diversity and spread of beneficial variants. Spatial structure promotes strategies of interference competition by limiting diffusion of interference toxins and resources, often allowing interference competitors to invade when rare. Consistent with previous results in other microbial systems, toxin production by Saccharomyces cerevisiae is advantageous in spatially structured, high-density environments, but not in unstructured environments. However, at low density and at low frequency, rare toxin producers cannot invade populations of common, sensitive, toxin nonproducers. This is because the likelihood of interaction between toxin producers and sensitives depends upon the density and frequency of both competitors.

Strategies for resource competition involve direct acquisition of a resource, with no interaction with competitors beyond depriving them of that resource (Schoener 1983). The success of a strategy for resource competition does not depend directly on the state of a competitor, only on the state of the resource. Resource competition can maintain diversity if availability varies in time or space (Chesson 2000) and if trade-offs exist in the ability to acquire the resource when it is abundant or rare (Stewart and Levin 1973). In contrast, strategies for interference competition target competitors themselves, rather than a resource (Park 1962). Interference competition thus often requires additional activities to those needed for resource competition and the success of a strategy for interference competition depends on the state of the competitor (Amarasekare 2002). Allelopathy is a type of interference competition in which toxic compounds are produced that kill or suppress the growth of competitors. Numerous instances of allelopathy have been observed across diverse taxonomic groups including plants (Callaway and Aschehoug 2000), marine invertebrates (Jackson and Buss 1975), bacteria (Adams et al. 1979), and yeast (Starmer et al. 1987).

The success or failure of allelopathic strategies is thought to depend on the frequency of toxin producers, environmental structure, the costs associated with toxin production, the effect of a toxin on competitor growth, and the relative importance of interference competition and resource competition (Frank 1994). These first two factors have been investigated experimentally by Chao and Levin (1981), who showed that the fate of an interfering competitor can depend critically on its frequency. Interfering bacteria that produced an anticompetitor toxin were allowed to compete against bacteria that were sensitive to the toxin and did not produce it. Toxin producers grew more slowly than sensitive strains because of the costs associated with toxin production, but in certain conditions the interfering strain could invade. Chao and Levin (1981) found that in an unstructured environment—a well-mixed liquid culture—the toxin producers were eliminated from the population after a few generations if they were introduced at low frequency, but if they started at a high enough initial frequency they rapidly became fixed by killing all the sensitive individuals (dynamically, this community exhibits bistability). In a well-mixed environment, the death of a competitor is of equal benefit to all remaining individuals, whether they produce toxin or not. When rare, the toxin producers lower the growth rate of nonproducers by such a small amount that nonproducers can grow faster than producers, whose growth is slowed by the cost of toxin production. Thus toxin producers were unable to invade a liquid culture when initially rare. But at sufficiently high initial frequency of producers there is enough toxin to slow the growth rate of nonproducers below that of producers, so toxin producers increase in frequency (see also Adams et al. 1979). In contrast, in a structured environment—the surface of an agar plate—cells that produced toxin had an advantage over sensitive cells regardless of their initial frequency. This is because spatial structure allows local frequency of toxin producers to be much higher than their global frequency and ensures that toxin producers benefit disproportionately from local sensitive competitor death by enjoying increased local resources (Chao and Levin 1981; Wiener 2000). Hence toxin producers are able to invade in a structured environment, even when initially very rare. The potential impact of spatial structure on the success of interference competitors has been observed in both plant and insect species (Amarasekare 2003).

The importance of frequency and spatial structure suggests that density is likely to affect the costs and benefits of allelopathy. Because interference interactions target competitors, the potential for competitor density to affect the fitness of interference competitors is readily apparent. Ecological interactions are often density dependent (Levin 1988), and allelopathic density dependence has been observed previously (Weidenhamer et al. 1989; Brown et al. 2006).

Here we perform similar experiments to Chao and Levin (1981) bacterial studies but using a different system: the killer toxin system of yeast strains infected with toxin-encoding viruses. Yeast-killer viruses are dsRNA viruses that cause permanent infections in Saccharomyces species (Magliani et al. 1997). There is no known extracellular transmission: cells are infected from virus in their parents' cytoplasm, either asexually during budding of daughter cells, or sexually by fusion of an infected haploid gamete with an uninfected gamete. Having no extracellular transmission, the fate of the virus is tied to the fate of its host cell. Thus the encoding of host beneficial traits, such as allelopathic toxins, by killer viruses is not surprising. We studied the effects of the K1 killer virus, whose toxins kill uninfected yeast cells by binding to cell wall receptors and forming transmembrane channels that cause ion leakage and subsequent cell death (Martinac et al. 1990). Killer toxin is secreted by yeast, in contrast with colicins that are released only when E. coli cells burst, so infected cells do not commit suicide to release the toxin. K1 toxicity is due to the secreted toxin-forming transmembrane channels in sensitive cells causing ion leakage and cell death. Self-immunity is provided by the action of the toxin precursor within the toxin-producing cell. The toxin precursor within the cell inhibits formation of channels by the mature toxin acting from outside the cell (Magliani et al. 1997). Thus the mechanisms of toxin production and self-immunity are overlapping. An infected cell should benefit by the killing of competitors (and thereby gaining their resources), but will pay the costs of producing viral products (including killer toxin) and pleiotropic interactions of viral products with host cell components.

We extend the work of Chao and Levin (1983) by examining the effects of density rather than simply frequency. In a well-mixed environment, the killing of sensitive yeast cells is a function of toxin concentration in the culture medium (Kurzweilova and Sigler 1994), which is in turn a function of the density of toxin-producing individuals. In a well-mixed environment containing a low density of toxin producers, the effects of toxin production will be small even if toxin producers make up a high proportion of the population (i.e., they are relatively frequent), because their absolute numbers are low (i.e., they are rare).

In a spatially structured environment, the probability that a sensitive neighbor will be close enough that a toxin producer kills it, depends on the number of toxin producing and sensitive individuals. At low densities the number of such killing interactions is likely to be small and may not cover the potential cost of toxin production, reducing the likelihood of invasion of the toxin producer. Here we present an experiment in which both frequency and density of toxin producers is manipulated and the relative fitness of the toxin producers is measured.

Materials and Methods


The K1 killer virus was transduced from its native strain M484, a gift from Mike Leibowitz, into the laboratory strain Y55 (McCusker and Haber 1988). In the experiments described here, the toxin-producing strain is YDG 963 MATalpha, lys2::KanMX4 and the toxin-sensitive strain is YDG 571 MATalpha, lys2. Strains were grown in buffered pH 4.5 YEPD-4.5 (yeast extract 1%, peptone 2%, glucose 2%, citric acid 2.1%, potassium phosphate dibasic 2.8%, methylene blue 0.0003%), with the addition of 2.5% agar for plates when required. Liquid medium was used in 5 mL volumes in 30 mm wide capped test tubes, shaken at 250 revolutions per minute. Solid medium was used in 25 mL volumes in 90 mm wide plastic petri dishes. For determining cell numbers, cultures were serially diluted and plated on normal YEPD (yeast extract 1%, peptone 2%, glucose 2%, agar 2.5%), allowed to form colonies, and then replica-plated to YEPD plates supplemented with 0.03% of the drug G418, which only allows the resistant cells (those containing the KanMX4 gene—the toxin producers) to grow. By counting the colonies formed on YEPD, and those formed on YEPD-G418, the number of killer and sensitive cells in the sampled cultures, at the beginning and at the end of the experiment, can be determined. The fitness of the toxin-sensitive strain relative to the toxin-producing strain was calculated as the ratio of Malthusian parameters (Lenski et al. 1991). A value of 1 indicates equal fitness of both competitors. Fitness assays were corrected for differences in the number of generations between the two environments caused by the ∼30.9% greater final density in the spatial plate environment (data not shown).


Both strains were grown to a stationary phase in liquid, mixed in equal volumes, and concentrated 10-fold by spinning the cells down and resuspending them in sterile water. This 10-fold concentration was performed to allow spreading of high density (10−1 dilution) cells on plates, which would otherwise be unable to absorb the high volume of liquid medium. The number of cells of each type was determined using YEPD-G418 plates as described above. The mixture was then serially diluted 10-fold in sterile water five times, making six suspensions of cells at different densities ranging from 10× to 0.0001× stationary phase density. Fifty microliter of each suspension was used to inoculate 5 mL liquid cultures and 250 μl of each suspension was plated onto 25 mL agar plates (i.e., a 100-fold dilution, resulting in initial cell densities ranging from 10−1 to 10−6 of the unconcentrated stationary phase densities). There were four replicates per treatment per initial density. After incubation at 30°C for three days, the total number of each cell type in each culture was again determined using the KanMX4 marker and the fitness of the toxin-producing strain in each treatment was calculated. Note that the fitness measurements depend on the change in the measured frequency of toxin-producing cells, so any difference in stationary phase density between the two strains prior to mixing would not affect the fitness results. The experiment was replicated three times. The experimental design included two fixed factors (spatial structure and initial density) and a single random factor (replicate), and all fitness estimates of the toxin producer were log10(Wks+ 1) transformed to equalize variances.


Both strains were grown to stationary phase in liquid and mixed in equal volumes. The mixture was serially diluted 10-fold four times in a stationary phase liquid culture of the sensitive strain. This produced five mixtures of cells containing the same density of cells, but different frequencies of toxin producers ranging from 50% to 0.005%. The initial number of cells of each type in the five cultures was determined by plating as described above. Then the five mixtures were diluted 1000-fold into 5 ml of fresh liquid medium and onto 25 ml solid plates. Thus the 50% initial frequency treatment corresponds to the 10−3 treatment in the manipulation of density experiment described above. As above, the experimental design included two fixed factors (spatial structure and frequency) and a single random factor (replicate), and all fitness estimates were log10(Wks+ 1) transformed to equalize variances. The fitness of the toxin-producing strain at different frequencies was determined as before, and the experiment was replicated four times.


Five replicate cultures of killer and sensitive strains were grown to stationary phase in liquid in overnight cultures. Each culture was then used to initiate 10 new replicate cultures, 550 μl into 5 mL medium. Optical density at 600 nm (OD600) was used as a measure of cell density. One milliliter of culture from randomly chosen tubes of each set was sampled periodically, diluted into 2 mL fresh medium, and measured with a Biochrom CO8000 cell density meter (Biochrom, Cambridge, UK). OD600 values were log10(OD600 + 1) transformed and were used to obtain rates of growth at each time point. Growth rates for each time segment were estimated by


in which r is the log transformed change in absorbance determined by n estimates over time t, with the zero time density estimate serving as a fixed intercept for all rates calculated from a single replicate. This approach provides independent measures of growth for each time segment (Lenski et al. 1991; Travisano 1997a, b). Cultures were allowed to reach stationary phase after further incubation overnight and five cultures of killer and sensitive strains were sampled, serially diluted, and plated. After incubation, colonies were counted to estimate the stationary phase cell density of killer and sensitive cultures.



The fitness of the toxin producer was highly dependent on the initial density (Fig. 1; analysis of covariance (ANCOVA): F1,24= 283.03, P < 0.00001). As all cultures were grown to stationary phase, the differences between the density treatments are essentially in the initial density and the length of the log phase, rather than in the final density. This suggests that toxin production is neutral or mildly deleterious at low cell densities and producers only gain an advantage in the last division or two before stationary phase. Although this same advantage would exist for producers in all experimental treatments, its effect on overall per-generation fitness would have been less in treatments with low initial densities because producers would have to go through many divisions before reaching high enough density for the toxin to be effective at interference. The advantages of producing toxin were, as expected, more pronounced on plates than in liquid culture (full factorial ANCOVA: F1,24= 35.76, P < 0.00001). The effect of density was far greater on plates than in liquid culture (full factorial ANCOVA: F1,24= 49.90, P < 0.00001). Analysis by ANOVA gave similar results on density (F5,10= 122.45, P < 0.00001), spatial structure (F1,10= 74.01, P < 0.00001), and density by structure interaction (F5,10= 23.29, P < 0.00001).

Figure 1.

Higher density increases the benefit of toxin production. Light and dark bars represent liquid and plate environments, respectively. Error bars represent a one standard deviation range. 50:50 mixed cultures of toxin producers and sensitives were initiated across a density gradient, from a 106 to a 101 dilution of maximal liquid culture density. Environment, density, and environment × density all had statistically significant effects.


We found that the ability of a toxin producer to invade a liquid culture (i.e., to have a relative fitness >1) is frequency dependent (Fig. 2), with toxin producers at lower frequencies having lower fitness than toxin producers at higher frequencies (Table 1), consistent with previous results with other microbial systems (Chao and Levin 1981). However, we found that the benefits of toxin production in a spatially structured environment were also highly frequency dependent (Table 2), unlike in previously reported systems (Chao and Levin 1981). Toxin producers at low frequency cannot easily invade from rare, even when there is spatial structure, because their fitness is little or no higher than their sensitive competitors. There were statistically significant effects on killer fitness due to environment (full factorial ANCOVA: F1,24= 15.62, P= 0.001), initial frequency (ANCOVA: F1,24= 412.31, P < 0.00001), and in the interaction between environment and frequency (ANCOVA: F1,24= 75.79, P < 0.00001). An analysis of variance (ANOVA) provided only marginal support for an environmental effect (F1,12= 3.31, P= 0.094), but did support a frequency effect (F4,12= 22.12, P= 0.0001) and an interaction (F4,12= 4.05, P= 0.026). The effect of frequency is more pronounced in the spatially structured environment, as the benefits are greater at higher densities in the spatially structured environment, but presumably the per capita costs remain the same regardless of the environment of frequency.

Figure 2.

Low initial frequency reduces the benefit of toxin production. Light and dark bars represent liquid and plate environments, respectively. Error bars represent a one standard deviation range. Mixed cultures of toxin producers and sensitives were initiated across a frequency gradient, from 0.005% to 50% toxin producer at a 104 dilution from maximal density. Environment, frequency, and environment × frequency all had statistically significant effects.

Table 1.  Linear regression of fitness on frequency in the unstructured liquid environment.
Source of variationdfSSMSFsP
Linear regression 11.52×10−21.52×10−253.30.0053
Deviation from regression 38.53×10−42.85×10−4 0.2680.847
Table 2.  Linear regression of fitness on frequency in the structured environment.
Source of VariationdfSSMSFsP
Linear regression 19.48×10−29.48×10−2524.70.00018
Deviation from regression 35.42×10−41.81×10−4  0.3030.821


In the absence of killer versus sensitive competition, production of toxin may reduce growth, which we observed by a more rapid approach to stationary phase by sensitive cells (Fig. 3). The killer genotype achieves maximal growth rate later, ∼332 min, than the otherwise isogenic nonproducer, ∼263 min, (Mann–Whitney on time to maximal growth rate P= 0.01), although without a detectable difference in the maximal rate of growth (F1,8= 2.48, P > 0.1). This “time-shift” is evident in Figure 4, where the nonkiller and killer strains have similar shaped quadratic regressions, but with the killer shifted later. The mean stationary phase density of cells in the five killer strain cultures sampled was 4.72 × 107cells/mL (standard deviation 9.15 × 106 cells/mL) and in the five sensitive cultures was 4.18 × 107cells/mL (standard deviation 9.18 × 106 cells/mL). We determined therefore that the smallest inoculum used in the experiment, the 10−6 liquid treatment in the density experiment, contained around 40 cells and was thus unlikely to be strongly affected by stochastic variation. We found no significant difference between the stationary phase densities of the two strains (ANOVA F1,8= 0.86, P= 0.379).

Figure 3.

Toxin production affects growth, with sensitive cells getting a more rapid start. Squares and circles represent the average sensitive and killer optical densities of five replicates during separate growth in liquid culture, respectively. Error bars indicate 95% confidence intervals based upon n− 1 degrees of freedom. Cultures of sensitive cells are consistently denser during exponential growth.

Figure 4.

Squares and circles represent sensitive and killer growth rate estimates during separate growth in liquid culture, respectively. Five replicates of each are shown. Yeast initially grow slowly upon inoculation into fresh medium, reach a maximal growth rate, and then decline as they saturate their environment. These growth dynamics are well fit by a quadratic regression (adj. r2= 0.80 and 0.86 for sensitive and killer respectively). Quadratic regression also indicates an impact of killer on growth (strain × time F1,8= 8.07, P= 0.006), that is particularly evident by the initially more rapid growth and subsequent earlier decline in growth rate of the sensitive type.


Theory predicts that allelopathic interference competition should occur more readily in a spatially structured environment than in a well-mixed environment (Durrett and Levin 1994; Frank 1994; Iwasa et al. 1998). In spatially structured environments, production of allelopathic compounds (toxins) may reduce local resource competition by the killing of sensitive competitors. In well-mixed environments, the costs of toxin production are borne only by toxin producers but additional resources, freed by the death of sensitive competitors, are available to all individuals whether they are toxin producers or surviving sensitives. Because the density of sensitive and toxin-producing individuals affects the global and local quantities of toxin and competitors, theory also predicts that fitness costs and benefits may vary depending on density (Durrett and Levin 1994; Frank 1994; Iwasa et al. 1998) The potential benefits of toxin production decline with decreasing cell density in a well-mixed environment as toxin concentration is reduced. The results presented here indicate that the potential benefits of toxin production are highly dependent on the densities of both toxin-producing and toxin-sensitive individuals, even in spatially structured environments.

Our results demonstrate that changes in density can have profound effects on the outcome of allelopathic interactions. Previous studies demonstrated that allelopathic competitors in well-mixed environments are competitively superior at high frequencies and inferior at low frequencies, but in spatially structured environments they are generally superior regardless of frequency. In our study, interference competitors are competitively superior at a 50% frequency in a dense well-mixed environment, but that superiority vanishes at lower density. The explanation for this new observation is that the absolute density of toxin producers, and not their frequency, is the determinant of toxin concentration. Because toxin concentration is the major determinant of its effect in a well-mixed environment, low densities of toxin producers, even at high frequencies, provide no benefit that is restricted to toxin producers.

The absence of any advantage to interference competitors when rare in spatially structured environments is a striking difference from common expectations. Many models of competitive interactions emphasize the potential for invasion from rare as a key factor. For example, nontransitive interactions between different competitors can maintain diversity in lizards (Sinervo and Lively 1996) and bacteria (Kerr et al. 2002; Kirkup and Riley 2004), but such maintenance depends critically upon the potential to invade when rare. In our system, reduced competitive advantage when rare results in part from the reduced interactions. If both sensitives and toxin producers are sufficiently rare, then the likelihood that toxin production affects growth of sensitives is low. When toxin producers become unable to interfere with the growth of their competitors, they are likely to be at a disadvantage, due to the energetic and pleiotropic costs of toxin production. Such situations are likely during dispersal between patchily distributed habitats, but frequency is less likely to be important if individuals are at higher densities and are more uniformly distributed.

Our conclusions are extensions of those described by Frank (1994), where he delineates the importance of toxin diffusion and competitor migration. Changes in toxin diffusion may increase toxin “producers' colonization of neighboring habitats” and migration “influences the relative strength of competitive advantage” (Frank 1994). The difference in outcomes between our results and those of previous experiments and theory is due to our systematic examination of frequency effects at very low densities where interactions are rare. An additional important factor is the reduction in resource competition when cells are physically distant. In the complete absence of dispersal in our spatially structured treatment and at low population density, the resources that are made available by killing may be too remote to exploit. In natural environments, local dispersal in moderately viscous populations may increase resource competition because neighbors can access the same resource (Gardner and West 2006). Additionally, the effects of density and frequency will vary from system to system, and most microbial studies to date have been with bacteria and not yeast. Compared to bacterial toxins, killer toxin appears to be somewhat less lethal and the production of killer toxin does not require the destruction of a producing cell because the killer toxin is secreted through the membrane of the live cell.

As expected, our results show that toxin producers are favored at high density. But surprisingly, toxin producers appear to be only at a minor disadvantage at low densities, in contrast to what might be expected because of the cost of producing an (ineffective) toxin (Fig. 3). One possible explanation is that the costs associated with toxin production are low. Small costs increase the range of environments in which toxin production is beneficial. This is consistent with the substantial vertical inheritance of killer toxin systems in yeast. It is also consistent with the small decline in fitness observed in Figure 2, as toxin producer frequency declines from 5%. Another possible explanation is that toxin producers primarily release toxin in high-density populations, when it would have the largest impact. The growth rate experiment (Fig. 3) indicates that killer and sensitive strains can both grow rapidly at high density when cultured separately, but that sensitive cells reach maximal, stationary phase, density more rapidly than killer cells. Part of this difference in growth may be due to toxin production, and cells may sense cell density and preferentially produce toxin at higher densities and thus slow their approach to stationary phase. Thus the cost of infection may be small at low densities because toxin production is repressed, and even at the low initial densities in our competition experiment (Fig. 2), the competitive advantage that killer gains when the culture approaches stationary phase (and it produces toxin) equals or outweighs the (low) cost of infection in early stages of growth.

Many microbes can secrete substances that affect others, either deleteriously or advantageously. Both toxic and beneficial secretions create antagonistic interactions between producers and nonproducers. In this article, we investigated the antagonism created by secretion of a toxin, and found that the benefit of interference competition far outweighs the cost of producing an anticompetitor toxin when producers are dense and common and toxin concentration is high, but this advantage is reduced when the toxin concentration is low. Other studies have shown that the cost of interference competition strategies can far exceed their benefit (e.g., Turner and Chao 1999; Travisano and Velicer 2004). The antagonism created by production of an advantageous substance is similar, but inverse, to that created by the production of an anticompetitor toxin. Producers of a useful resource can be exploited by nonproducers that can benefit from the resource without incurring the costs of producing it, but this advantage to nonproducers decreases as producers become less dense or less frequent because the resource becomes less abundant (e.g., Greig and Travisano 2004; Allison 2005). So resource competition may resemble interference competition when a resource is produced by a competitor, even though there are important dynamical differences between the two interactions. Our current results and prior results on resource competition suggest cooperative interactions, cheating strategies, and fitness will be often be highly dependent on both frequency and density.

Associate Editor: H. Kokko


We are grateful to M. Larios, A. Gardner, S. Otto, and two anonymous reviewers for helpful comments. DG is supported by a Royal Society University Research Fellowship. MT is supported by the US National Science Foundation.