In the calculations, the distribution of similarities between pairwise sequences is computed from a generated multiple sequence alignment (MSA) of a given protein (left). The degree of conservation at each position (not include the gaps) in the MSA is measured by relative entropy (right), and it is to indicate how much an amino acid at a specific position could be conserved. The degree of conservation can also how an amino acid at an evolutionarily conserved position is relavent to functionally important sites of protein, moreover, mutations in these positions could be harmful to protein function.

The evolutionary process of a protein family can be mathematically modeled by a sequence generator at equilibrium¹. The generator produces a sequence $\tau$ with a probability $P(\tau )$ (Eq. 1) from a distribution over the space of all possible sequences in the family.

$$\begin{array}{}\text{(1)}& P(\tau )=\frac{1}{Z}\exp (E(\tau ))\end{array}$$ where $Z$ is the partition function that normalizes the distribution by summing over the Boltzmann factors of all possible sequences in the family. The total energy $E(\tau )$ of the sequence $\tau$ is $$\begin{array}{}\text{(2)}& E(\tau )=\sum _i{\mathbf{h}}_i({\tau }_i)+\sum _{i<j}{\mathbf{e}}_{ij}({\tau }_i,{\tau }_j)\end{array}$$ where ${\mathbf{h}}_i$ and ${\mathbf{e}}_{ij}$ are, respectively, site-specific bias terms of an amino acid and coupling terms between pairwise amino acids.

The coupled interactions between pairwise amino acids of the protein are analyzed by spectrum analysis and ranked by the eigenvalues². To make the residue communities as much statistically independent as possible, the top three residue communities are defined based on two of the top five eigenvalues and their corresponding eigenvectors (${\mathbf{v}}_{k|k=1,\cdots ,5}$) of the matrix ${\mathbf{e}}_{ij}$ (Eq. 2) as:

• Community I consists of residues at the ith position of ${\mathbf{v}}_{k=2}^i>\text{max}({\mathbf{v}}_{k=4}^i,ϵ)$;
• Community II includes residues at the ith position of ${\mathbf{v}}_{k=2}^i<-\text{max}({\mathbf{v}}_{k=4}^i,ϵ)$; and
• Community III includes residues at the ith position of ${\mathbf{v}}_{k=4}^i>\text{max}({\mathbf{v}}_{k=2}^i,ϵ)$.

We use an $ϵ=0.05$ as a threshold to project the amino acids reduced from the coupling matrix and extract meaningful residue communities. The reduced matrix of interactions are achived (left). In order to look into the networks of residues that play important roles in protein function, the residue communities are defined based on the eigenvalues and preserve the positive interactions among amino acids (right).

Complete single mutagenesis. The matrix that is computed by the SAEC method shows ΔE — the energy difference of each mutant sequence with each mutation τ at the ith site and wild-type sequence, negative values representing favourable while positive representing unfavourable mutations.

Based on the best-so-far sequence (designed), one can conduct a signle or multiple (coupled) mutants on the WT sequence, and the energy of the mutant sequence is computed for evaluating the mutant(s). As an example, each two amino acids are combined to be mutated together and the energy differences are to measure quanlity of the coupled mutants.

The energy trajectories of designing the sequence from the WT sequence, and the WT and mutant sequences are listed below.
WT sequence (energy: -147.990):
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VCSEQAETGPCRAMISRWYFDVTEGKCAPFFYGGCGGNRNNFDTEEYCMAVCG

Mutant sequence (best-so-far, energy: -167.354):
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VCSEPAETGPCRAMISRWYYDPKTGKCEPFLYSGCGGNGNNFETKEECEETCK