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.

FIGURE 1

The six heteropolymer models studied in this work. Here the “shifted” and “perturbed-homopolymer” models (Chan and Dill, 1996) are denoted, respectively, by a prime superscript (′) and a “max” notation. The energy matrices (with matrix elements e_ijs) provide the relative interaction energies of pairwise contact between various types of monomers (i, j). Using the energy matrices shown, ground-state conformations are determined by exhaustive enumeration of all possible self-avoiding walks for the HP, HP′, AB, and AB′ models; but enumeration is limited to the maximally compact conformational states (MCSs) for the HPmax and ABmax models. The conformations in this figure are the top-ranking structures, i.e., these ground-state conformations are encoded by the largest number of sequences in their respective models (cf. Table 1). H, P, A, and B monomers (residues) are represented by solid and open circles and solid and open squares, respectively. The sequences shown in this figure are the prototype sequences of their neutral nets.

FIGURE 2

Topology of the largest neutral net in the (a) HP′, (b) HPmax, (c) AB′, and (d) ABmax models. Sequences encoding for the same structures (provided in Fig. 1) are represented by solid symbols (dots, circles, triangles, etc.); and mutational connectivity by a single-point substitution between two sequences is depicted by a line joining a pair of symbols. For a given neutral net, the prototype sequence and the sequence with maximum native stability (as defined in Fig. 3 below) are marked, respectively, by an open circle and an open square. A neutral net conforms to the superfunnel paradigm if both conditions are satisfied by the same sequence. In a, c, and d, different symbols denote sequences with different Hamming distances from the prototype sequence (Bornberg-Bauer and Chan, 1999).

FIGURE 3

Native stabilities of the (a) HP′, (b) HPmax, (c) AB′, and (d) ABmax sequences in the neutral nets in Fig. 2 are indicated by short horizontal levels. They correspond to the (−ɛ/k_BT) value (vertical scale) at the transition midpoint. Neutral single-point mutations (lines in Fig. 2) are indicated here by lines joining horizontal levels. The horizontal scale provides the Hamming distance from the prototype sequence of the given neutral net. Note that each horizontal level in the ABmax neutral net in d represents a sequence as well as the sequence obtained by performing A ↔ B interchange on it.

FIGURE 4

Distribution of neutral net sizes (insets) and native thermodynamic stability of the prototype sequences, for the (a) HP, (b) AB, (c) HP′, (d) AB′, (e) HPmax, and (f) ABmax models. For every model studied, each inverted triangle shows the average native stability (as measured by the (−ɛ/k_BT) at the transition midpoint) among the prototype sequences of neutral nets of a given size; the highest and lowest native stabilities among the same set of prototype sequences are indicated by a square and a dot, respectively. (Note that a higher (−ɛ/k_BT) value here means that the native state is thermodynamically less stable.) Insets show the number of neutral nets 𝒩(ω) as a function of size ω. Solid or dashed lines connecting data points in this figure serve merely as a guide for the eye. We note that panels a and b, for the HP and AB models presented here, are the corrected version of, and should therefore replace, the upper panels of Figs. 3 and 4 in Bornberg-Bauer and Chan (1999). Owing to a technical oversight, instead of recording the thermodynamic stabilities of prototype sequences as they should, the previously published figures erroneously provided the corresponding statistics for the sequences that are most stable in their neutral nets. Despite this error, the discrepancies between the two sets of results are very minor, since a majority of neutral nets are superfunnels with the prototype sequence also being the most stable. Consequently, the general trends exhibited by the two sets of figures are virtually identical, and the conclusions of the previous study remain unchanged.

FIGURE 5

Correlation of structure ranking: (a) HP versus HP′; (b) HP versus HPmax; (c) AB versus ABmax; and (d) HP versus AB models. For a given model, encodable structures are ranked by the sizes of their neutral (convergence) sets. For a pair of models being compared here, the ranks of every structure encodable in both models are provided by the two horizontal scales (the rank can be different in the two models), whereas the number of structures sharing a given pair of ranks is indicated by the vertical scale.