Evolution of off-lattice model proteins under ligand binding constraints

We investigate protein evolution using an off-lattice polymer model evolved to imitate the behavior of small enzymes. Model proteins evolve through mutations to nucleotide sequences (including insertions and deletions) and are selected to fold and maintain a specific binding site compatible with a model ligand. We show that this requirement is, in itself, sufficient to maintain an ordered folding domain, and we compare it to the requirement of folding an ordered (but otherwise unrestricted) domain. We measure rates of amino acid change as a function of local environment properties such as solvent exposure, packing density, and distance from the active site, as well as overall rates of sequence and structure change, both along and among model lineages in star phylogenies. The model recapitulates essentially all of the behavior found in protein phylogenetic analyses, and predicts that amino acid substitution rates vary linearly with distance from the binding site.

Figure 1

(a) Folded structure, ${\mathbf{x}}^{☆}$, and (b) sample of the folded ensemble $\mathrm{\Delta }{\mathrm{\Gamma }}^{☆}$ recovered by a sequence evolved under ligand binding conditions (see Sec. 2). Amino acids (monomers) are colored blue, light blue, blue-green, green, yellow, orange, and red, in order of increasing affinity to solvent. The binding site monomers and the target ligand (here, a single monomer) are colored black. Ordered binding sites evolve spontaneously under condition (i), and are selected to resemble the active sites of small enzymes for subsequent evolution under condition (ii) (see Sec. 2).

Figure 2

Amino acid exchange probabilities, $p(\mu ,\nu )=A_{\mu \nu }/{\sum }_{\nu }{A}_{\mu \nu }$, for sequences evolved under condition (i), where $A_{\mu \nu }$ is the number of transitions recorded between amino acids $\mu$ and $\nu$. The initial state of a transition is indicated along the lower axis. Model values are indicated by filled red circles. Empirical values obtained from the data of Dayhoff et al. [46] are indicated by open blue circles. The value of $p(\mu ,\nu )$ is indicated by the radius of the corresponding circle.

Figure 3

Amino acid transition rate, $\omega \left(\delta \mathcal{A}\right)$, versus exposed surface area $\delta \mathcal{A}$ for sequences evolved under condition (i). $\omega \left(\delta \mathcal{A}\right)$ is plotted in units of ${10}^{-4}$ accepted mutations per time step. The dashed line is a fit to the data in the region $\delta \mathcal{A}/\mathcal{A}\le 0.8$.

Figure 4

Amino acid transition rate, $\omega \left(\mathcal{Q}\right)$, versus local packing density $\mathcal{Q}$ for sequences evolved under condition (i). $\omega \left(\mathcal{Q}\right)$ is plotted in units of ${10}^{-4}$ accepted mutations per time step; $\mathcal{Q}$ is measured in units of ${\text{Å}}^{-2}$. The dashed line is a fit to the region $0.4\le \mathcal{Q}\le 1.2$.

Figure 5

Amino acid transition rate, $\omega \left(\mathcal{R}\right)$, versus distance from the binding site, $\mathcal{R}$, for sequences evolved under condition (ii). $\omega \left(\mathcal{R}\right)$ is plotted in units of ${10}^{-4}$ accepted mutations per time step; $\mathcal{R}$ is measured in angstroms. The dashed line is a fit to the region $\mathcal{R}\le 12.5$.

Figure 6

Number of mutations, $n^l\left(\tau \right)$, accepted along typical lineage, $l$, evolved under condition (ii). Thick gray segments denote events in the tail of the waiting time distribution $P(\mathcal{T}\ge \tau )$.

Figure 7

Distribution of waiting times, $P(\mathcal{T}\ge \tau )$, for the lineage described in Fig. 6 (circles). The dashed line is a fit to the Pareto distribution in Eq. (4). The solid line is a fit to a Poisson (exponential) distribution. The tail of $P(\mathcal{T}\ge \tau )$ is indicated by the thick gray dashed line.

Figure 8

Plot of the Lindemann parameter, $\lambda \left(\tau \right)$, for the lineage described in Fig. 6. Thick gray segments denote events in the tail of the waiting time distribution $P(\mathcal{T}\ge \tau )$.

Figure 9

Phase diagram defined by Eqs. (5) and (6). Lineages evolved under condition (i) are represented by circles. Lineages evolved under condition (ii) are represented by squares. The shaded region roughly indicates the phase space available to a two-state Poisson process.

Figure 10

Height autocorrelation function, $\langle \mathrm{\Delta }{\mathcal{Y}}^{\mathsf{1}}\rangle$, for a typical lineage (squares), and averages of $\langle \mathrm{\Delta }{\mathcal{Y}}^{\mathsf{1}}\rangle$ and $\langle \mathrm{\Delta }{\mathcal{Y}}^{\mathsf{2}}\rangle$ over all lineages (circles) evolved under condition (ii). Dashed lines are linear fits to the data in the range $k\le 64$ (the logarithms are base 10). Exponents, $\mathsf{H}\left(\mathsf{q}\right)$, obtained from the fits are provided in the text.

Figure 11

Index of dispersion, $\mathcal{I}\left(\tau \right)$, for a typical pair of phylogenies evolved under (a) condition (i), and (b) condition (ii). The dotted line roughly indicates the point at which the average number of accepted mutations per position, per lineage is ${〈n^l/N〉}_l\sim 1$. The lower curves (red) describe the restriction to homologous positions discussed in the text.

Figure 12

Distance from the ancestral fold, $\mathrm{\Delta }x(0,\tau )$, for a typical lineage evolved under condition (i).

Figure 13

Average distance, ${\langle \mathrm{\Delta }x(n,n+k)\rangle }_n$, between structures separated by $k$ accepted mutations for the lineage in Fig. 12 (squares), and average of ${\langle \mathrm{\Delta }x(n,n+k)\rangle }_n$ over all lineages (circles) evolved under condition (i). Dashed lines are fits to the data according to Eq. (10) yielding exponents $\alpha \simeq 0.9$ and $\alpha \simeq 0.8$, respectively. Linear scaling of the lineage average is obtained by a slight reduction in the number of monomers compared in structural alignments, as discussed in the text.

Figure 14

Potential functions, $U^{{\epsilon }^{\prime }}\left(r\right)$, for cross-chain interactions at unit core strength, $\epsilon =1$, unit attraction, ${\epsilon }^{\prime }=-1$ (dot-dashed line), and unit repulsion, ${\epsilon }^{\prime }=1$ (dashed line).