Self-consistent mean field (biology)

Like dead-end elimination, the SCMF method explores conformational space by discretizing the dihedral angles of each side chain into a set of rotamers for each position in the protein sequence. The method iteratively develops a probabilistic description of the relative population of each possible rotamer at each position, and the probability of a given structure is defined as a function of the probabilities of its rotamer components.

The basic requirements for an effective SCMF implementation are:

1. A well-defined finite set of discrete independent variables
2. A precomputed numerical value (considered the "energy") associated with each element in the set of variables, and associated with each binary element pair.
3. An initial probability distribution describing the starting population of each rotamer
4. A way of updating rotamer energies and probabilities as a function of the mean-field energy

The process is generally initialized with a uniform probability distribution over the rotamers—that is, if there are ${\displaystyle p}$ rotamers at the ${\displaystyle kth}$ position in the protein, then the probability of any individual rotamer ${\displaystyle r_{k}^{A}}$ is${\displaystyle 1/p}$. The conversion between energies and probabilities is generally accomplished via the Boltzmann distribution, which introduces a temperature factor (thus making the method amenable to simulated annealing). Lower temperatures increase the likelihood of converging to a single solution, rather than to a small subpopulation of solutions.

The energy of an individual rotamer ${\displaystyle r_{k}}$ is dependent on the "mean-field" energy of the other positions—that is, at every other position, each rotamer's energy contribution is proportional to its probability. For a protein of length ${\displaystyle N}$ with ${\displaystyle p}$ rotamers per residue, the energy at the current iteration is described by the following expression. Note that for clarity, the mean-field energy at iteration ${\displaystyle i}$ is denoted by${\displaystyle M_{i}}$, whereas the precomputed energies are denoted by${\displaystyle E}$, and the probability of a given rotamer is denoted by${\displaystyle P_{i}(r_{k}^{A})}$.

${\displaystyle M_{i}(r_{k}^{A})=E_{k}(r_{k}^{A})+\sum _{x=1}^{N}\sum _{y=1}^{p}P_{i-1}(r_{x}^{y})E_{xy}(r_{k}^{A},r_{x}^{y})}$

These mean-field energies are used to update the probabilities through the Boltzmann law:

${\displaystyle P_{i}(r_{k}^{A})=\left(\exp \left(-{\frac {M_{i}(r_{k}^{A})}{kT}}\right)\right)\left(\sum _{y=1}^{p}\exp \left(-{\frac {M_{i}(r_{k}^{y})}{kT}}\right)\right)^{-1}}$

Where ${\displaystyle k}$ is the Boltzmann constant, and ${\displaystyle T}$ is the temperature factor.

Although computing the system energy is not required in carrying out the SCMF method, it is useful to know the overall energies of the converged results. The system energy ${\displaystyle M_{sys}}$ consists of two sums:

${\displaystyle M_{sys}=M_{single}+M_{pair}}$

Where the addends are defined as:

${\displaystyle M_{single}=\sum _{x=1}^{N}\sum _{y=1}^{p}P(r_{x}^{y})E_{x}(r_{x}^{y})}$

${\displaystyle M_{pair}=\sum _{x=1}^{N}\sum _{y=1}^{p}\sum _{a=x+1}^{N}\sum _{b=1}^{p}\left(P(r_{x}^{y})P(r_{a}^{b})E_{xy}(r_{x}^{y},r_{a}^{b})\right)}$