Reducing Epistasis and Pleiotropy Can Avoid the Survival of the Flattest Tragedy
1. Introduction
Evolution optimizes the fitness of organisms within a population over time. We often think of this process as climbing to the peak of an imaginary fitness landscape, and while doing so, organisms move from simpler to more complex forms of life [1]. Although, over time, more and more information related to adaptation to the environment accumulates in genomes [2]. In biology, the term complexity can pertain to different concepts, such as genotypic complexity [3,4], phenotypic complexity [5], developmental complexity [6], or the genotype–phenotype complexity [7,8,9], for example. However, complexity, in general, remains an elusive term and a difficult-to-measure concept [10,11].
It has been argued that changes in complexity mostly happened at the macroevolutionary level [12], which does not exclude that the complexity of organisms within a population can also change in complexity [13] and give rise to macroevolutionary changes later. Here, we focus on the complexity of the genotype–phenotype map and use the degree to which genes interact with the aspect of complexity we study. Consequently, we are addressing changes within a single population given a static fitness landscape. Therefore, we will focus on the complexity of interactions and use the change in these interactions as a proxy to describe the changes in complexity at a microevolutionary level. However, given the right evolutionary circumstances, evolution can also reduce the number of genetic interactions and result in what one might call simpler solutions [14,15]. The question is now about which circumstances give rise to which outcomes. However, before we get to this question, we need to first elaborate on the interactions in the form of epistasis and pleiotropy, explain how robustness can be affected by either, and then explore the role of the fitness landscape.
Mutations introduce a range of genetic variations that can lead to changes in traits known as phenotypic changes. These changes are caused by gains, losses, or modifications to genes [16]. However, these genes do not act in isolation but instead interact with each other [17]. Thus, these interactions, known as epistasis () [18], play a crucial role in shaping the ultimate effect of any given mutation. Epistatic interactions can also provide information on how proteins, pathways, and components within cells function together [19]. These interactions explain the complexity of some drug resistance mechanisms [20], improve the understanding of synthetic lethality in cancer treatment [21], and provide insight into the inheritance patterns of complex traits [22].
Positive epistasis plays a crucial role in evolutionary adaptation by facilitating synergistic gene interactions that enhance the organism’s fitness beyond the additive effects of individual mutations. This form of epistasis can significantly contribute to the rapid adaptation of organisms to changing environmental conditions, as it allows for the emergence of complex traits that are highly beneficial for survival [23]. Conversely, negative epistasis introduces a layer of genetic constraint that can limit evolutionary potential. This occurs when the interaction between genes results in a phenotypic outcome that is less advantageous than expected based on the individual effects of the mutations involved. Such antagonistic interactions can hinder the accumulation of beneficial mutations, potentially slowing the pace of adaptation [24].
Unlike epistasis, which involves interactions between different genes, a single gene can also influence multiple traits. This is known as pleiotropy () [25]. Pleiotropy demonstrates how a single genetic variation can have wide-ranging effects on an organism’s phenotype, potentially affecting fitness, adaptation, and disease susceptibility. For example, pleiotropic genes have been identified as critical factors in human diseases, where a single gene mutation can lead to multiple symptoms or conditions [26]. Studies on pleiotropy also contribute to the understanding of biological systems, revealing how genes regulate multiple developmental pathways to shape the form and function of an organism [27].
Robustness in biological systems refers to the ability of organisms to maintain the same phenotypic traits despite genetic mutations or variations and environmental challenges [28]. Functional stability is essential to ensure survival and facilitate adaptation. Robustness is achieved through various genetic mechanisms, among which epistasis and pleiotropy play significant roles. For comparison, when a single gene is responsible for a phenotypic trait, when it is hit by the right mutation, that trait disappears, as in the case of antibiotic resistance [29]. However, the effect of such a mutation could be augmented if the same gene is pleiotropic in the sense that it plays multiple roles and thus contributes to multiple traits. A single mutation can have multiple negative effects [25]. Interestingly, this can also go in the other direction; when, for example, a mutation enhances expression, it could benefit multiple traits [25,30]. Mixed situations can be imagined for pleiotropic genes, where a single mutation improves some traits while also preventing others, making pleiotropic genes notoriously hard to study [31].
Conversely, epistasis can also modulate the effect of mutations on single genes. Positive epistasis can lower the effect of a deleterious mutation [23,32], while negative epistasis could enhance the effect. In the extreme case, where many genes have a high degree of epistasis and are pleiotropic, the effects of mutations might become even less predictable and more multifaceted.
Consequently, and play a crucial role during evolutionary adaptation but are also the consequence of evolution. However, whether or not an increase or decrease in or makes organisms more or less robust is still highly debated [33,34,35,36,37,38]. What likely made answering this question hard is not only the difficulty one faces when doing evolutionary experiments with natural organisms but also the way how computational models were made. In most previous cases related to this matter, evolution occurred at the phenotypic level or did not distinguish between the genotype and the phenotype [39,40]. This is especially curious, as and are phenomena that happen in translating a genotype to phenotype. Genes affecting multiple traits or interacting to form a trait can only occur within a model with a genotype and phenotype. This translation from a genotype to phenotype is called encoding within the evolutionary computation community [41] and, of course, the subject of biological research, as we want to know how one can make an organism’s phenotype from its genes.
We previously extended the NK fitness model with such encoding [42]. In this model, a genotype experiences mutations that affect not only the expression of genes but also how they interact epistatically and what pleiotropy a gene has while also resulting in a phenotype with multiple traits. This model uses a simple matrix to determine which gene influences which trait. Natural protein or transcription factor networks are likely more complicated, and thus, other indirect encodings have been used [43,44,45]. However, including further complications from more sophisticated models might induce other confounding findings, and we thus use a very simple method to model encoding. Indirect encodings in nature come from the genotype to phenotype mapping [46], which can be very complicated and thus result in different notions of complexity, such as genotypic, phenotypic, or developmental complexity [47,48], and while interesting, we focus on the relation between the components and how they map to traits in the simplest form and thus are, as described above, considering the number of interactions as a proxy for complexity (interaction complexity), but be aware that the notion of complexity is more nuanced in nature. The question now is in which environment can we test whether and increase or decrease to facilitate mutational robustness?
One of the more interesting examples of mutational robustness and how it influences evolution is the survival of the flattest (SoF) phenomenon. Here, populations with wider genetic variability and flatter fitness landscapes have a greater chance of enduring and adapting in variable environments than those perched on high fitness peaks [39]. Adami et al., 2006 [40] demonstrated that a population might not remain on a high fitness peak, instead preferring a lower one, as long as the lower peak is flatter. In this context, a peak refers to a phenotype with higher fitness than all its mutational neighbors, specifically those with a Hamming distance of 1. Flatness relates to a peak’s average fitness, or neutral network, and describes the mean fitness of these mutational neighbors (see Figure 1). If the mutation rate is high enough [49], populations are small [39], or the ratio between the primary and secondary peaks is closer to [39,49], populations will prefer the secondary peak.
This SoF landscape also presents an interesting challenge to adaptation. The higher and narrower peak can only be reached or remained on by a population with higher mutational robustness against the mutation rate. Previous studies clearly showed that reducing the mutation rate is one way un which the population can stay at a higher peak. Alternatively, making the landscape around the higher peak wider or the flatter peak smaller was modeling mutational robustness, and both could control the experiment’s outcome. Previous studies considered only computational models in which there was no distinction between genotype and phenotype. Thus, a change in the fitness landscape did not change the way the genotype translates to the phenotype in any way, as both were the same in these models. Here, instead, we use a model with a genotype-to-phenotype translation and the ability to evolve and , thus allowing us to investigate how both can facilitate or hamper mutational robustness.
We will show that computational model organisms having an indirect encoding from genotype to phenotype experience the same survival as the flattest phenomenon as directly encoded ones. However, in indirectly encoded organisms, mutational robustness is determined by the effect that mutations have on the individual genes and how they interact, specifically their degree of epistasis and pleiotropy. The question we answer here, which has been debated before, is whether an increase or decrease in epistasis or pleiotropy increases or decreases mutational robustness. We show that a reduction in both epistasis and pleiotropy makes organisms more mutationally robust and thus allows them to avoid the survival of the flattest tragedy of not being able to remain at a higher, narrower peak. It is essential to note that the survival of the flattest phenomenon is about the ability of a population to remain at a higher peak or about two populations in competition ending with the slower replicating one outcompeting the one at the higher peak due to mutational robustness. Therefore, we focus on epistasis and pleiotropy’s effect on mutational robustness and a population’s ability to maintain their position at the higher peak, and while that does not exclude a way for a population to climb to a higher peak by reducing its mutational robustness, our work does not seek to find evidence for the latter. We, however, argue that such a reduction in epistasis and pleiotropy reduces the number of interactions, makes organisms less genetically complex, and thus presents another example where evolution prefers a simpler over a more complicated solution.
3. Results
The two-peak model has previously been tested only on directly encoded organisms [50,51]. It demonstrated that, in the absence of the capacity to evolve mutational robustness, the probability of staying at the higher peak given the mutation rate depends on three factors: the height of the second peak, population size, and the flatness of the first peak. Consequently, we expect the same parameters to control the adaptation of indirectly encoded organisms. Namely, lower mutation rates or lower secondary peak heights allow populations to stay on the higher peak, and smaller population sizes also cause populations to leave earlier.
Upon varying the height of the second peak (, , , and ) or the population size (ranging from 100 to 1000), we find our expectations corroborated (see Figure 4). The critical mutation rate, at which a population can no longer remain at the highest peak, decreases as the secondary peak becomes taller. Likewise, larger populations can withstand a higher critical mutation rate before they cannot stay at the higher peak.
The critical mutation rate for a population to leave a peak is also influenced by the direct mutational environment of the higher peak. The higher the fitness of the mutants surrounding the highest peak, the greater the critical mutation rate must be for the population to vacate the peak. This effect is corroborated in the two-peak model using indirect encoding (see Figure 5).
These results confirm that applying the two-peak model to indirectly encoded organisms, which can potentially evolve mutational robustness and vary in their and , produces the same observations regarding the survival of the flattest phenomenon.
We observed that organisms adapt to more rugged adaptive landscapes (where genotype-to-phenotype relationships are more complex) by reducing and [42]. We hypothesize that the reduction in and causes organisms to experience the survival of the flattest phenomenon less frequently. We propose that to have a chance of staying at a higher peak, organisms increase their mutational robustness by decreasing and . It is known that is correlated with mutational robustness [49].
Consequently, we predict that organisms should experience survival of the flattest frequently with a higher degree of . We believe that plays a similar role, with a gene having more interactions being less robust to mutations. Therefore, a higher degree of should also lead to organisms experiencing the survival of the flattest effect more often.
To test these hypotheses, we took randomly generated organisms with indirect encoding. We determined the critical mutation rate at which they could not remain on the primary peak in the two-peak landscape given the height of the secondary peak (see Figure 6 black line). As expected, the higher the secondary peak, the lower the critical mutation rate at which populations vacate the primary peak.
To assess the effect of and , we took the same randomly generated organisms. Still, before exposing them to the two-peak landscape, we artificially increased or decreased their levels of and using a hill climber (see Materials and Methods). Observe that the hill climber is not supposed to model an evolutionary process; it is only used to alter and so we can test those organisms in the SOF landscape. We found that, as predicted, organisms with increased or have a lower critical mutation rate, while decreased and result in a higher critical mutation rate (see Figure 6). Moreover, the critical mutation rate also depends on how much and were modified. Finally, the effect of altering and together, or each individually, is nearly indistinguishable, suggesting that a change in one causes a change in the other in the same direction. Such behavior might be explained by the relationship between genes and traits. If we consider a functional matrix that maps a column of components to a row of functions, we can define epistasis for a trait as the sum of a column and pleiotropy as the sum of a row. If we were to alter that matrix, such as by removing a connection between a gene and a trait, then both and would be reduced. This concept applies to any functional mapping matrix and is not reliant on a specific model. We consequently believe that this constraint affects and in the same manner.
These findings suggest that organisms could endure the survival of the fittest phenomenon for more extended periods if they decreased and .
4. Discussion
Incorporating indirect encoding into our evolutionary model maintains phenotypic selection pressures while introducing a new dimension to the fitness landscape through genotypic mutations. Unlike indirect encoding, where genotypic alterations directly translate to phenotypic changes, the indirect approach introduces a broader array of potential genotypic variations, especially epistatic and pleiotropic effects. This change in the interaction complexity alters the fitness landscape, yet our findings reveal that the SoF principle can still be observed. Indirectly encoded organisms face challenges in sustaining themselves on the evolutionary peak, especially as mutation rates climb, population sizes diminish, or the secondary peak’s prominence increases. Thus, mutational robustness is still present, and therefore, the SoF landscape allows us to investigate how epistasis and pleiotropy facilitate said robustness.
Our analysis demonstrates that organisms exhibiting lower levels of epistasis and pleiotropy exhibit enhanced mutational robustness, stabilizing their position at the highest peak. This insight underscores the intricate balance between genetic architecture and evolutionary adaptability, highlighting how the configuration of genotypic to phenotypic mappings can significantly influence an organism’s evolutionary trajectory.
While our computational model and the concept of indirect encoding serve as abstract representations of genetic interactions, they offer valuable insights that may align with biological realities. The model suggests that organisms characterized by lower levels of and exhibit greater mutational robustness—a hypothesis that merits empirical investigation in natural settings. Similarly, more complicated models incorporating indirect encoding, such as gene regulatory networks, should be tested for the same effect. While we expect the same phenomenon to occur qualitatively, we also expect quantitative differences. However, our proposition aligns with the broader scientific inquiry into genetic robustness and its evolutionary implications, suggesting that similar principles could underpin both computational and biological systems.
Moreover, our findings contribute to the ongoing debate about complexity in biological systems, specifically the complexity of interactions between genes and traits. Contrary to the prevailing notion that complex environments necessitate complex genetic architectures, our model presents a counter-narrative. It illustrates that simplicity, regarding reduced genetic interactions and functionalities, can confer a survival advantage in certain evolutionary landscapes.
While our results show that organisms can remain at high and narrow peaks by reducing epistatic interactions and pleiotropy of genes to increase mutational robustness, it falls short of elucidating the evolutionary journey from a lower to a higher peak. The SoF landscape is characterized by a vast fitness valley between the lower (ancestral) peak and the higher one. Valley crossing mutations are required to traverse this landscape to transition from a lower, broader peak to a higher, narrower one. However, we have not studied this phenomenon here.
Still, we can imagine a dynamic landscape where peaks evolve—growing higher and narrower—suggesting that organisms must adapt by honing their mutational robustness to secure their position atop the increasingly precarious peak, thereby avoiding descent into less optimal fitness zones. Our study demonstrates that adaptation, in terms of retaining a position on the higher peak, can be achieved by reducing and . However, this mechanism’s role in the ascent of the fitness landscape needs to be verified using alternate fitness models. This exploration underscores the interplay between genetic architecture and evolutionary dynamics. Future research, perhaps utilizing models with temporally varying landscapes, is crucial to unravel these dynamics further.
