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The NK model is designed to study evolutionary adaptation in rugged fitness landscapes. The factor K determines the number of interacting genes, their degree of pleiotropy and epistasis, and consequently the ruggedness of the fitness landscape. However, in natural organisms, the degree of epistatic interactions and the number of functions a gene can have are to a certain degree determining the ruggedness of the landscape. Still, pleiotropy and epistasis can evolve independently from each other, and are to some degree independent of the ruggedness of the landscape. Here, we propose an extension to the standard NK model to investigate these factors independently of each other. Over the course of evolution the computational model organisms can now change how their genes interact and how they control phenotypic traits. Further, the degree of epistasis and pleiotropy is affected by the ruggedness of the landscape and becomes reduced with increasing ruggedness. While this proves that the extension of the model performs as expected, the adaptations are minor, presumably because only relatively short periods of adaptations with few mutations can be studied. © 2022 IEEE.

The study of gene regulatory networks (GRNs) is fundamental to the understanding of evolutionary dynamics and artificial life modeling. This paper presents an integration of a GRN into the NK-fitness landscape model and explores the impact of sparsity on epistasis and pleiotropy. As sparsity augments, gene interactions diminish, expectedly leading to a reduction in both epistasis and pleiotropy. Our findings corroborate the model’s response to such perturbations, demonstrating its potential for investigating a range of GRN adaptations within the NK-fitness landscape framework.

This study investigates whether reducing epistasis and pleiotropy enhances mutational robustness in evolutionary adaptation, utilizing an indirect encoded model within the “survival of the flattest” (SoF) fitness landscape. By simulating genetic variations and their phenotypic consequences, we explore organisms’ adaptive mechanisms to maintain positions on higher, narrower evolutionary peaks amidst environmental and genetic pressures. Our results reveal that organisms can indeed sustain their advantageous positions by minimizing the complexity of genetic interactions—specifically, by reducing the levels of epistasis and pleiotropy. This finding suggests a counterintuitive strategy for evolutionary stability: simpler genetic architectures, characterized by fewer gene interactions and multifunctional genes, confer a survival advantage by enhancing mutational robustness. This study contributes to our understanding of the genetic underpinnings of adaptability and robustness, challenging traditional views that equate complexity with fitness in dynamic environments. © 2024 by the authors.

Understanding the balance between robustness and evolvability is crucial in evolutionary dynamics. This study aims to determine how varying mutation rates and valley depths affect this interplay during adaptation. Using a two-peak fitness landscape model requiring populations to cross a fitness valley to reach a higher peak, we investigate how mutation rates and valley depths influence both evolvability—the capacity to generate beneficial mutations—and mutational robustness, which stabilizes populations at the highest peak. Our experiments reveal that at low mutation rates, populations struggle to cross fitness valleys, reducing the occurrence of pioneers. As mutation rates increase, valley crossing becomes more frequent, but organisms forming a majority at the highest peak are less common and tend to arise at intermediate mutation rates. Although pioneers reach the highest peak, they are often replaced by more mutationally robust organisms that later form a majority. This suggests that while evolvability aids in valley crossing, long-term stability at the highest peak requires greater mutational robustness. Our findings highlight that adaptations in epistasis and pleiotropy facilitate the trade-off between evolvability and robustness, providing insights into how organisms navigate complex fitness landscapes. These results can also inform the design of genetic algorithms that balance evolvability with robustness to optimize outcomes. © 2024 by the authors.

Understanding the balance between evolvability and mutational robustness is crucial for exploring adaptation in complexfitness landscapes. This study examines how heterogeneous populations adapt under varying mutation rates (μ) andfitness landscape ruggedness (K), emphasizing their distinct starting conditions in valleys, slopes (low prominence), orpeaks (high prominence). Using an extended NK model, we simulate populations capable of initiating anywhere in thelandscape. Our findings reveal that starting positions strongly influence whether robustness or evolvability is advantageous.Populations beginning in low-prominence regions (valleys and slopes) exhibited high levels of epistasis (ϵ) and pleiotropy(π), enhancing evolvability and enabling exploration of the fitness landscape. In contrast, populations starting in highprominenceregions (peaks) reduced ϵ and π, prioritizing robustness to maintain stability against mutations. This studyhighlights the role of starting conditions in shaping evolutionary trajectories, offering insights into the interplay betweenevolvability and robustness.

The NK fitness landscape is a well-known model with which to study evolutionary dynamics in landscapes of different ruggedness. However, the model is static, and genomes are typically small, allowing observations over only a short adaptive period. Here we introduce an extension to the model that allows the experimenter to set the velocity at which the landscape changes independently from other parameters, such as the ruggedness or the mutation rate. We find that, similar to the previously observed complexity catastrophe, where evolution comes to a halt when environments become too complex due to overly high degrees of epistasis, here the same phenomenon occurs when changes happen too rapidly. Our expanded model also preserves essential properties of the static NK landscape, allowing for proper comparisons between static and dynamic landscapes.

The genotype-phenotype map determines how genetic variation translates into traits, influencing evolutionary adaptability. While previous models often assume a static relationship, genetic architectures evolve dynamically in response to selective pressures. In this study, we investigate how epistasis and pleiotropy adapt under varying fitness landscape ruggedness (K) and environmental variability (V ) together, shaping genetic robustness and evolvability. Using the NK treadmill model, we systematically explore the independent effects of K and V on genetic complexity. Our findings reveal that increased ruggedness (K) reduces genetic interdependencies, favoring modular architectures that enhance mutational robustness. Conversely, higher environmental variability (V ) promotes interconnected genetic networks, increasing evolvability. Empirical validation using bacterial genomes supports these results, showing strong correlations between genetic complexity measures and mutational robustness, reinforcing the role of environmental pressures in shaping genetic architectures.