Comparing Folding Codes in Simple Heteropolymer Models of Protein Evolutionary Landscape: Robustness of the Superfunnel Paradigm

Abstract

Understanding the evolution of biopolymers is a key element in rationalizing their structures and functions. Simple exact models (SEMs) are well-positioned to address general principles of evolution as they permit the exhaustive enumeration of both sequence and structure (conformational) spaces. The physics-based models of the complete mapping between genotypes and phenotypes afforded by SEMs have proven valuable for gaining insight into how adaptation and selection operate among large collections of sequences and structures. This study compares the properties of evolutionary landscapes of a variety of SEMs to delineate robust predictions and possible model-specific artifacts. Among the models studied, the ruggedness of evolutionary landscape is significantly model-dependent; those derived from more protein-like models appear to be smoother. We found that a common practice of restricting protein structure space to maximally compact lattice conformations results in (i.e., "designs in") many encodable (designable) structures that are not otherwise encodable in the corresponding unrestrained structure space. This discrepancy is especially severe for model potentials that seek to mimic the major role of hydrophobic interactions in protein folding. In general, restricting conformations to be maximally compact leads to larger changes in the model genotype-phenotype mapping than a moderate shifting of reference state energy of the model potential function to allow for more specific encoding via the "designing out" effects of repulsive interactions. Despite these variations, the superfunnel paradigm applies to all SEMs we have tested: For a majority of neutral nets across different models, there exists a funnel-like organization of native stabilities for the sequences in a neutral net encoding for the same structure, and the thermodynamically most stable sequence is also the most robust against mutation.

Biophysics of protein evolution and evolutionary protein biophysics

“…Third, many of the lowest-energy 2D HP structures are highly compact but not maximally compact, as for real globular proteins [158,299], and a significant fraction of compact structures are not encodable by any HP sequence as its unique native structure [158,162]. The latter observation may bear on the question of whether the currently known set of globular protein folds is nearly complete in its coverage of all physically possible compact folds, as discussed in §3.1 [286][287][288][289][290], but one has to also keep in mind that the HP model interaction potential is less heterogeneous, and thus entails fewer encodable structures, than model potentials that contain repulsive interactions or otherwise more heterogeneous interactions [158,322,323]. Fourth, the 2D HP lattice model provides sequences that act like evolutionary bridges [161, 178,204,239,299] (see §3.5), encode for autonomous folding units [292,316,324], and exhibit homology-like behaviours [295], all similar to properties observed in real proteins.…”

“…In several 2D [137,161,293,299,303,304,323] and 3D [305,335] lattice models as well as an off-lattice model [304], protein sequence space was found to be organized as multiple neutral nets. A neutral net is a network of sequences that are connected by single-point substitutions and encoding for the same folded structure.…”

Evolution after gene duplication: models, mechanisms, sequences, systems, and organisms

“…However, such correlations are crucial to understanding neutral mutations and mutational stability. To address these issues, many recent theoretical and computational efforts have focused on constructing models of sequence-structure mapping for proteins motivated by polymer physics theory (Williams et al, 2001;Deeds et al, 2003;Wroe et al, 2005). Because of the immense sizes of the systems, all these models involve significant simplifications (but some of which may be realistic).…”

“…Another modeling approach involves simple exact models, which address general principles of evolution as they permit the exhaustive enumeration of both sequence and structure (conformational) spaces (Wroe et al, 2005). These physical models can then be applied in large scale to make predictions about the mapping between substitutions and molecular phenotypes seen in genomes.…”

Simplification of complexity in protein molecular systems by grouping amino acids: a view from physics

“…During the evolution, the stability of the proteins may tolerate mutations of amino acids, which introduces the redundancy of the sequences even with a constant alphabet [39,51]. Besides, the addition of the new amino acids with innovative function groups could enhance the capability to build functional proteins and improve the fitness in evolution [52].…”