# Statistical Coupling Analysis - Definition of Statistical Coupling Energy

Definition of Statistical Coupling Energy

Statistical coupling energy measures how a perturbation of amino acid distribution at one site in an MSA affects the amino acid distribution at another site. For example, consider a multiple sequence alignment with sites (or columns) a through z, where each site has some distribution of amino acids. At position i, 60% of the sequences have a valine and the remaining 40% of sequences have a leucine, at position j the distribution is 40% isoleucine, 40% histidine and 20% methionine, k has an average distribution (the 20 amino acids are present at roughly the same frequencies seen in all proteins), and l has 80% histidine, 20% valine. Since positions i, j and l have an amino acid distribution different from the mean distribution observed in all proteins, they are said to have some degree of conservation.

In statistical coupling analysis, the conservation (ΔGstat) at each site (i) is defined as: .

Here, Pix describes the probability of finding amino acid x at position i, and is defined by a function in binomial form as follows:

,

where N is 100, nx is the percentage of sequences with residue x (e.g. methionine) at position i, and px corresponds to the approximate distribution of amino acid x in all positions among all sequenced proteins. The summation runs over all 20 amino acids. After ΔGistat is computed, the conservation for position i in a subalignment produced after a perturbation of amino acid distribution at j (ΔGi | δjstat) is taken. Statistical coupling energy, denoted ΔΔGi, jstat, is simply the difference between these two values. That is:

, or, more commonly,

Statistical coupling energy is often systematically calculated between a fixed, perturbated position, and all other positions in an MSA. Continuing with the example MSA from the beginning of the section, consider a perturbation at position j where the amino distribution changes from 40% I, 40% H, 20% M to 100% I. If, in a subsequent subalignment, this changes the distribution at i from 60% V, 40% L to 90% V, 10% L, but does not change the distribution at position l, then there would be some amount of statistical coupling energy between i and j but none between l and j.

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