2.1 Overview

2.2 Mean field theory

• The effective system: multicopy of the molecule

 

We define the coordinate vector of all atoms in the molecular system as X. The probability of finding the molecular coordinates between X and X+dX is denoted by µ(X)dX, where µ(X) is the probability density of the coordinates, normalized to 1. The total energy of the system is given by:
 



where U is the potential energy function.
If the system contains a single molecule with a unique conformation whose coordinates are X⁰, Equation (1) becomes:
 



The native conformation X_nat of the molecular system corresponds to the global minimum of E_eff. The search for this global minimum is hindered by the presence of many local minima. One way to alleviate this problem is to consider and effective, larger system as a computational tool to enhance the sampling protocol of the minimization procedure. This larger system is obtained by considering multiple copies of the molecule, or part of the molecule.

Assumption 1: Hartree approximation

The probability density µ is replaced by a product of independent probability densities of different systems, using a Hartree product [3]:
 



As an example of such a partition into subsystems, a ligand and a protein can be considered separately [4,5]. Another example is the partition of a protein into backbone and side-chains [6]. The effective system is then built by considering multiple copies of each subsystem j.
 

 
Multicopy sampling of the conformation of a protein.
The protein is divided into its backbone (subsytem 0), and all its side-chain.
In the simple case presented here, the Tyrosine at position j in the sequence
has 3 possible conformations, corresponding to the 3 "copies" of subsystem j.
Each copy k is given a weight or probability P(j,k), which are normalised to 1 over all copies.


 

Assumption 2: Discrete sets

The probability density µ_j of each subsystem is expanded into a finite number of delta functions:



where k_j runs over all K_j copies of the subsystem j. P(j,k_j) are normalization factors or probabilities veryfying:



For practical reasons, K_jis always finite.
 
 

• The energy of the effective system

 
 

Substituting Equations (3) and (4) into Equation (1) and integrating over the spatial variables leads to the following functional form for the energy of the effective system:



Assumption 3: Pair-wise potential

The potential energy function U is assumed to be a one and two-body potential:



in which case Equation (6) reduces to:



Equation (8) represents the effective energy function used in most MFT bio-applications. In essence, MFT replaces the problem of finding the global minimum energy given by equation (2) by the problem of finding the minimum of the "effective" potential energy described by equation (8). Its major advantage is a significant reduction of the variable space. For example, there are K_ix K_j alternative configurations of subsystems i and j that can be examined using a single configuration of the effective system.
 
 

• Minimization of the effective energy

 
 

The minimization of E_eff can be obtained by three different routes:
 
 

• Fixed probabilities, variable conformations: the LES protocol

 

The normalisation factors P in Equation (8) are kept constant (usually taken as P(i,k_i)=1/K_j, where K_j is the number of copies for subsystem j). The positions of each copy for each subsystem are then optimized by solving Newton-like equations of motion. This procedure was initially proposed as the LES (Locally Enhanced Sampling) method [3,7].
 
 

• Fixed conformations, variable probabilities: the SCMF procedure

 

The position of the various copies of the subsystems are supposed to be known and fixed in space. The effective system is then described by its free energy F:



where T is the temperature, and S the entropy, defined as:



The free energy F is a function of all factors P which are stored in a matrix P, whose current element P(j,k) is the probability that subsystem j is described by its possible state k.

Minimization of F with respect to P leads to:



where W(j,k) corresponds to the molecular field potential "felt" by the copy k in subsytem j:



The system of equation (8) and (11) is then iterated until convergence, that is, until self-consistency is achieved. This procedure is referred to as SCMF (Self-Consistent Mean Field) method [2,6]
 
 

• Simultaneous variation of conformations and probabilities

 

With simultaneous variation of both coordinates and probabilities, it is theoretically possible to combine the advantages and avoid the disadvantages of the previous methods. A correct weighting (instead of weighting) between the different copies of each subsystem can be obtained because the probabilities are allowed to vary. Furthermore, the choice of the basis set of conformations is no more critical. For a detailed description of this approach, see [8].