Accessibility of Modified NK Fitness Landscapes
Abstract
In this paper we present two modifications of traditional $NK$ fitness landscapes, the $θNK$ and $HNK$ models, and explore these modifications via accessibility and ruggedness. The $θNK$ model introduces a parameter $θ$ to integrate local Rough Mount Fuji-type correlations in subgenotype contributions, simulating more biologically realistic correlated fitness effects. The $HNK$ model incorporates gene regulation effects by introducing a masking mechanism where certain loci modulate the expression of other loci, simulating effects observed in gene regulatory networks without modeling the full network. Through extensive simulations across a wide range of parameters ($N$, $K$, $θ$, and $H$), we analyze the impact of these modifications on landscape accessibility and the number of local optima. We find that increasing $θ$ or the number of masking loci ($H$) generally enhances accessibility, even in landscapes with many local optima, showing that ruggedness doesn't necessarily hinder evolutionary pathways. Additionally, distinct interaction patterns (blocked, adjacent, random) lead to different observations in accessibility and optimum structure. While more complex than traditional $NK$, we believe each model provides a new biologically relevant facet to fitness landscapes and provides insight into how genetic and regulatory structures influence the evolutionary potential of populations.
Summary
This paper introduces two modifications to the traditional NK fitness landscape model, aiming to enhance its biological relevance: the θNK and HNK models. The θNK model incorporates a parameter θ to introduce correlations in subgenotype contributions, mimicking correlated fitness effects observed in biological systems. The HNK model simulates gene regulation by introducing a masking mechanism where certain loci modulate the expression of other loci, without modeling the full gene regulatory network. The research question revolves around understanding how these modifications affect landscape accessibility and ruggedness, specifically the number of local optima. The authors conducted extensive simulations across a range of parameters (N, K, θ, and H) and analyzed the impact of these modifications on landscape accessibility and the number of local optima. They used an exhaustive approach to calculate accessibility, performing a recursive depth-first search starting from the global optimum. The key finding is that increasing θ or H generally enhances accessibility, even in rugged landscapes, challenging the notion that ruggedness necessarily hinders evolutionary pathways. They also found that different interaction patterns (blocked, adjacent, random) lead to different observations in accessibility and optimum structure. The research contributes by providing new biologically relevant facets to fitness landscapes, offering insights into how genetic and regulatory structures influence the evolutionary potential of populations. By demonstrating that increased ruggedness doesn't necessarily impede accessibility, the paper challenges conventional assumptions and provides a more nuanced understanding of evolutionary dynamics in complex systems. The paper also investigates methods for fitting curves to the N=K ridge in the modified landscapes, providing tools for prediction.
Key Insights
- •The θNK model demonstrates that introducing correlations in subgenotype contributions via the parameter θ generally enhances accessibility in fitness landscapes.
- •The HNK model shows that gene regulation, simulated through a masking mechanism (H), can also increase accessibility, suggesting that regulatory effects can facilitate evolutionary pathways.
- •The study reveals that the relationship between ruggedness (number of local optima) and accessibility is not always inversely proportional; increased ruggedness doesn't necessarily hinder evolutionary pathways, as evidenced by enhanced accessibility with higher θ and H.
- •Different interaction patterns (blocked, adjacent, random) in the NK landscape influence accessibility and optimum structure differently. Blocked neighborhoods exhibit a multiplicative decomposition pattern in accessibility that is broken by the HNK model.
- •The HNK model is limited by its requirement that each H-segment locus interacts with precisely K N-segment loci, causing unintuitive behavior when N=K.
- •In the θNK model, for large values of θ, whether θ is even or odd introduces a bias that dominates the overall correlation of the landscape, leading to odd-even alternations in accessibility and optima density.
- •In HNK landscapes, distinction was drawn between cases where the global optimum was fully expressed and those in which it was not, and the accessibility of non-fully expressed global optima was higher than that of fully expressed optima.
Practical Implications
- •The findings have implications for understanding evolutionary processes in biological systems, particularly in the context of drug resistance and adaptation to changing environments.
- •Researchers in evolutionary biology and computational biology can use these models to simulate and analyze the effects of correlated fitness effects and gene regulation on evolutionary trajectories.
- •Practitioners can use the models to explore the accessibility of specific genotypes in complex fitness landscapes, potentially guiding strategies for directed evolution or drug design.
- •Future research can focus on further refining the HNK model by allowing for flexible interactions and on exploring time-dynamic landscapes to model the effects of environmental changes on evolutionary accessibility.
- •The fitting functions developed for the N=K ridge can be used to predict p1 for arbitrary θNK landscapes, facilitating further analysis and prediction.