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Free energy calculation General theory. There is a variety of methods for
computing the free energy difference between given equilibrium states of a
molecule or complexes of molecules (see e.g. "Computer Simulations of
Biomolecular Systems", Vol 1 and 2, W. F. van Gunsteren et al. , Escom,
1989, 1993). These methods refer to transitions that are induced by changing
the Hamiltonian. When
we started to simulate conformational changes along constrained reaction
coordinates of the distance type it turned out that no practicable method was
available to compute the corresponding free energy changes. Reaction coordinates are a useful means for
inducing chemical reaction and conformational changes. Note that a constraint
imposed to such a coordinate leaves most degrees of freedom really free (for
a protein in solution 10 – 100 thousands). For TMD and
XTMD we use reaction coordinates of
the distance type like the rms distance between conformers or the radius of gyration.
Our 2001 paper is a survey of the particular properties of such coordinates
which make them a useful tool for the simulation and thermodynamic evaluation
of conformational changes. Free energy of a reaction
coordinate. In the early nineties
there was no practicable method for
computing free energy profiles along an arbitrary reaction coordinate (rc).
The formulation along the lines of thermodynamic integration would require
definition of a complete new set of coordinates including the rc, and
computation of all corresponding partial
derivatives as stated most clearly by den Otter and Briels, 1998. This is
possible only in very simple cases, but not in applications to
macromolecules. We solved the problem in two steps. ·
Constraint
force. In 1996 we considered systems for which the value of
the constrained rc is continuously changed (without releasing the constraint). For this case we established
the connection between the mean constraint force, which is
always numerically available, and the "mean force" of
thermodynamic integration that determines the free energy difference:
they are identical up to the sign (-1). This finding allows the easy
evaluation of simulations. It turns out that the very problem can be slow
relaxation and slow convergence of the mean force. ·
Correction.
It was clear that a correction had to be found in order to adapt the free
energy profile to the case without
constraint. The constraint should be used merely to enable
computation. Corrections had been proposed that not only were complicated,
but also again entailed the old problem of partial derivatives with respect
to a new complete set. In 2003 we could prove that the correction can be
given a concise shape lacking of those problems:
A = free energy, r = reaction coordinate, l = constraint force, z
= Fixman determinant The correction is the determinant
of the well-known Fixman matrix. For interesting classes
of rc’s it can be shown to vanish. When other coordinates are constrained in
addition to the rc the multiple constraints have to be regarded in the matrix.
Even so it is a constant in many cases of practicable interest and can be
omitted. References: Schlitter, J. and Klähn, J. The free energy of a reaction coordinate
at multiple constraints: A concise formulation. J. Mol. Phys. 101, 2003,
3439-44, 2003 Schlitter, J. and Klähn, M. A new concise expression for the free
energy of a reaction coordinate J. Chem. Phys. 118 (5), 2003, 2057-60, 2003 Schlitter, J., Swegat, W. and
Mülders, T. Distance- type reaction coordinates for modeling activated
processes. J. Mol. Mod. 7: 171-177, 2001 Mülders, T., Krüger, P., Swegat,
W., and Schlitter, J. Free energy as the potential of mean constraint force. J. Chem.
Phys. 104:4869-70, 1996.
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