Calculating Potentials of Mean Force

One of the objectives in free energy calculations is to obtain potentials of mean force C(r) along a chosen coordinate r = r(z) defined as 

When calculating the potential of mean force along a fluctuating coordinate r, we can at best observe r (e.g., the instantaneous molecular extension), but we do not set it explicitly. Therefore, r is no longer an externally controlled coupling parameter, and Jarzynski s identity does not immediately apply. However, as was shown in [3], an extension produces the desired result. 

For simplicity, let us consider a molecular system with a Hamiltonian J o(z) that is coupled to a harmonic spring with spring constant k and a time-dependent minimum r(t). The explicitly time-dependent Hamiltonian of the complete system is then 

We can now use the Feynman-Kac relation to obtain the potential of mean force. By multiplying both sides of (5.14) by exp[/3k(r(z) - r(t))2/2]S(r — r(z)) and integrating over z, we obtain an expression for the potential of mean force 

the sums are over different time slices / at which coordinates r and corresponding work values have been saved. 

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Figure 13-11. The snapshots of the S2 (green) and S3 (cyan) binding states predicted from the calculated potential of mean force superimposed with 1ZB6 (grey) and 1ZDW (purple). The electron density contour of 1ZB6 at the level of 0.5cr is shown as a yellow mesh...

Park, S. Schulten, K., Calculating potentials of mean force from steered molecular dynamics simulations, J. Chem. Phys. 2004,120, 5946-5961... 

Fig. 11. (a) Correlation of calculated rate constant at 298 versus average Al-0 bond length on a series of polynuclear clusters and minerals. S1-S5 sites correspond with the notation given in Figure 8. (b) Temperature dependence in calculated potential of mean force and (inset) temperature dependence of the rate computed by integration of the potential of mean force. 

Calculated potentials of mean force for dissociation of the (CHjljC Cl ion pair in... 

Keywords free-energy perturbation QM/MM calculations potentials of mean force polynomial free-energy profiles... 

Figure 2. Calculated potentials of mean force for separating the (CH3)3C Ci ion pair in water at 25°C.

Equations [305]-[310] are strictly valid only for thermodynamic species, which are ordinarily associated with stationary points on the potential energy surface V(R), where R denotes the full set of solute coordinates. However, we also use the SMx solvation models to calculate potentials of mean force, which are called W(R,T). The gradient of W(R,T) gives the force on the solute molecule averaged over a canonical ensemble of solvent molecules and is a generalization of the one-dimensional radial potential of mean force that appears in Debye-Hiickel theory. Thus, we write... 

Such a free energy is called a potential of mean force. Average values of Fs can be computed in dynamics simulations (which sample a Boltzmann distribution), and the integral can be estimated from a series of calculations at several values of s. A third method computes the free energy for perturbing the system by a finite step in s, for example, from si to S2, with... 

M J1990. Calculation of Conformational Ensembles from Potentials of Mean Force. An Approach o the Knowledge-Based Prediction of Local Structures in Globular Proteins. Journal of Molecular Siology 213 859-883. 

PLS (partial least-squares) algorithm used for 3D QSAR calculations PM3 (parameterization method three) a semiempirical method PMF (potential of mean force) a solvation method for molecular dynamics calculations... 

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

MI Sippl. Calculation of conformational ensembles from potentials of mean force. An approach to the knowledge-based prediction of local structures m globular proteins. I Mol Biol 213 859-883, 1990. 

Figure 5 Calculated potential and potential of mean force of acetic acid along the torsional angle of 0(H—O—C—C). The left-hand side shows the total energies and the components, and the right-hand side shows the decomposed A i into the site.

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