Structural evolution of proteinlike heteropolymers

The biological function of a protein often depends on the formation of an ordered structure in order to support a smaller, chemically active configuration of amino acids against thermal fluctuations. Here we explore the development of proteins evolving to satisfy this requirement using an off-lattice polymer model in which monomers interact as low resolution amino acids. To evolve the model, we construct a Markov process in which sequences are subjected to random replacements, insertions, and deletions and are selected to recover a predefined minimum number of solid-ordered monomers using the Lindemann melting criterion. We show that polymers generated by this process consistently fold into soluble, ordered globules of similar length and complexity to small protein motifs. To compare the evolution of the globules with proteins, we analyze the statistics of amino acid replacements, the dependence of site mutation rates on solvent exposure, and the dependence of structural distance on sequence distance for homologous alignments. Despite the simplicity of the model, the results display a surprisingly close correspondence with protein data.

Figure 1

Sample of the ensemble, $\Delta {\Gamma }^{☆}$, corresponding to the dominant energy basin recovered by a typical viable sequence. The structures are obtained by folding many replicas of the sequence in parallel using methods described in the text. To indicate amino acid type, monomers are colored blue, light blue, blue-green, green, yellow, orange, and red, in order of increasing affinity to solvent.

Figure 2

Comparison of model and empirical amino acid frequencies, $p\left(\nu \right)$. Model results are indicated by circles. Differences between model and empirical results are indicated by one-sided error bars. Amino acid labels on the lower axis are arranged in order of increasing affinity to solvent (see Appendix).

Figure 3

Comparison of model and empirical amino acid mutabilities, $m\left(\nu \right)$. Model results are indicated by circles. Differences between model and empirical results are indicated by one-sided error bars. Model results are scaled to the empirical mutability for alanine.

Figure 4

Comparison of model and empirical amino acid replacement probabilities, $p(\mu ,\nu )$. The plot describes the replacement of amino acids of type $\mu$ by amino acids of type $\nu$ with probabilities indicated by circle radii. Solid red circles indicate model values; open black circles indicate empirical values computed by Dayhoff et al. (correlation coefficient $\sim 0.55$).

Figure 5

Mutability, $m\left(\lambda \right)$, of monomers with Lindemann parameter ${\lambda }_j=\lambda$. Dotted lines roughly indicate the transition (1) from solid to liquid and (2) from liquid to disordered phases within folded globules.

Figure 6

Mutability, $m\left(\delta \mathcal{A}\right)$, of monomers with exposed surface area $\delta {\mathcal{A}}_j=\delta \mathcal{A}$. The exposed area of an isolated monomer is $\mathcal{A}=4\pi {\varrho }^2$, where $\varrho =0.65l$. The dashed line is a linear fit to the data in the region $\delta \mathcal{A}/\mathcal{A}<0.7$ corresponding to one or more cross-chain (sphere) contacts.

Figure 7

Aligned distance between structures, $\mathcal{D}\left(n\right)$, as a function of mutational distance, $n$. The dotted line is a linear fit to the data (points are plotted every fourth mutation for clarity in the figure).

Figure 8

Fraction of sites, $\mathcal{Q}\left(n\right)$, in the alignment of sequential structures, ${\mathbf{x}}^{☆}\left(\tau \right)$ and ${\mathbf{x}}^{☆}\left({\tau }^{\prime }\right)$, preserved in later alignments of ${\mathbf{x}}^{☆}\left(\tau \right)$ with structures ${\mathbf{x}}^{☆}\left({\tau }^{\prime \prime }\right)$. Data points represent averages over paths with $n(\tau ,{\tau }^{\prime \prime })=n$. The dotted line is a power-law fit to the data.

Figure 9

Aligned distance between structures, $\mathcal{D}\left(q\right)$, as a function of the percentage of nonidentical amino acids, $q$. The data used to compute $\mathcal{D}\left(q\right)$ are the same as in Fig. 7. The solid line is an exponential fit to the data; the dotted line is a power-law fit.

Figure 10

Evolution of polymer length, $N\left(\tau \right)$, for sequences evolved under the condition $\delta N\ge 15$. The lower axis, $\tau$, in each panel denotes elapsed time along a trajectory measured in mutation attempts. (a) The system evolves from a single sequence which stems from a designable structure. (b) The system evolves from multiple sequences which stem from less designable structures. Paths are numbered for later reference in Fig. 11.

Figure 11

Evolution of complexity (average loop length), $\langle \ell \left(\tau \right)\rangle$, for sequences evolved under the condition $\delta N\ge 15$. Each point represents an average over a small group of structures in $\Delta {\Gamma }^{☆}\left(\tau \right)$ closest to ${\mathbf{x}}^{☆}\left(\tau \right)$. Path colors and numbers correspond with Fig. 10.

Figure 12

Replica average specific heat function, ${\langle C\left(T\right)\rangle }_{\Gamma }$, for an evolved sequence of length $N=35$ monomers. Numbers along the top edge of the figure indicate equilibration temperatures in Eq. (1).