computer modeling in molecular biology.pdf

6.3.3.1 Application and Techniques6 Theory of Transport in Ion Channels 151The potential of mean force of the ion along the channel axis is calculated using thefree energy simulation technique [80, 811. From Eq. (6-15), the potential of meanforce at a position W(x + Ax) can be expressed in terms of W(x),AW(x+x+ Ax) = W(x+ Ax) - W(x) = -k,T ln(e-A”kB7)U, (6-17)where the bracket with subscript x represents a canonical average with the reactioncoordinate held fixed at x, i. e.,(6-18)and AU is the change in potential energy obtained by displacing the ion from x tox + Ax, i.e., AU is equal to U(R,x + Ax) - U(R,x). The average Eq. (6-18) iscalculated from an ensemble of configurations generated by a computer simulationof the system in thermal equilibrium with the ion fixed at x. Although Eq. (6-17) isformally valid for any Ax, convergence in achievable computer times limits thecalculation to the free energy differences in the neighborhood of x. In practice, theperturbations have to be relatively small to obtain rapid convergence. In practice itis necessary to generate several trajectories of the system with the ion fixed at variousvalues of x. The complete profile is constructed by joining the free energy differencesobtained with Eq. (6-17) at the mid-points between neighboring simulations, that is,W(X,+,) = W(X,) + AW(xn+xn + Ax)-AW(x,+I +xn+l -AX)=n= C [AW(X~+X~ + Ax) - AW(xi+l +xi+l - AX)], (6-19)i=lwhere Ax is the mid-distance between two neighboring simulations,(6-20)W(x) can also be calculated by integrating the reversible work done by the meanforce (F(x)) acting on the ion in the x direction, i. e.,(6-21)