Study on the Application of the Combination of TMD Simulation and Umbrella Sampling in PMF Calculation for Molecular Conformational Transitions

Free energy calculations of the potential of mean force (PMF) based on the combination of targeted molecular dynamics (TMD) simulations and umbrella samplings as a function of physical coordinates have been applied to explore the detailed pathways and the corresponding free energy profiles for the conformational transition processes of the butane molecule and the 35-residue villin headpiece subdomain (HP35). The accurate PMF profiles for describing the dihedral rotation of butane under both coordinates of dihedral rotation and root mean square deviation (RMSD) variation were obtained based on the different umbrella samplings from the same TMD simulations. The initial structures for the umbrella samplings can be conveniently selected from the TMD trajectories. For the application of this computational method in the unfolding process of the HP35 protein, the PMF calculation along with the coordinate of the radius of gyration (Rg) presents the gradual increase of free energies by about 1 kcal/mol with the energy fluctuations. The feature of conformational transition for the unfolding process of the HP35 protein shows that the spherical structure extends and the middle α-helix unfolds firstly, followed by the unfolding of other α-helices. The computational method for the PMF calculations based on the combination of TMD simulations and umbrella samplings provided a valuable strategy in investigating detailed conformational transition pathways for other allosteric processes.


Calculations of Interhelical Angle, Hydrogen Bond and Correlation of Atomic Motions
In the calculations of interhelical angle, the program INTERHLX calculates the sign of the angle between two helices by following this convenient role: the two helices are taken to be positioned by helix I being in front of helix II. Helix I (from N to C) is used to define first vertical vector. A second vertical vector is defined with its tail at the C-terminus of helix II. The angle between helices I and II is the rotation required to align the head of the second vector with the N-terminus of helix II. The vector is rotated in the direction that produces an angle of less than 180 degrees with the clockwise or counterclockwise rotation represented by positive or negative sign. This program can also provide other geometry based parameters such as interhelical distances [88].
To examine the unfolding process of the HP35 protein, the occupancies of all possible intra-helix hydrogen bonds at the three helices of the protein were measured by calculating the percentage of snapshots during the simulation that the hydrogen bonds existed. A "hydrogen bond" was defined as a distance of less than 3.5 Å between a hydrogen atom attached to either an oxygen or a nitrogen atom and an acceptor atom, and as an angle formed by a donor, a hydrogen atom and an acceptor being larger than 120° with the corresponding occupancy of ≥50%. To compare the amount of the total possible hydrogen bonds in the various transition conformations, a relative percentage of total hydrogen bond occupancies for a conformation was calculated by defining the relative percentage equal to one hundred percent when the all possible hydrogen bonds occupy each snapshot in the reactant during the HP35 unfolding simulation.
The correlation coefficients were averaged over the regions of protein, and the resulted cross-correlation coefficients are presented in the form of a two-dimensional graph [86]. These structure analyses in the present work were calculated by using the PTRAJ module of the AMBER 9 program [66]. Table S1. Optimized structural parameters for four C atoms in butane molecule along with experimental data (bond: angstrom and angle: degree).

Parametes
Optimization Experiment C1-C2 (a) (b) Figure S1. The dihedral angle values of C-C-C-C and the selected sampling windows of (a) 4 for the second TMD simulation; and (b) 5 for the third TMD simulation. The selected sampling windows are numbered around the red circles. The corresponding expected values and snapshots with the red circles are marked by the dash lines. Figure S2. The checked histograms of the dihedral angles from 19 umbrella sampling simulation trajectories for the rotational transition of butane from −180° to 180°.
(a) (b) Figure S3. The root mean square deviation (RMSD) values of four C atoms and the selected sampling windows of (a) 5 for the second TMD simulation; and (b) 7 for the third TMD simulation. The selected sampling windows are numbered around the red circles. The corresponding expected values and snapshots with the red circles are marked by the dash lines. Figure S4. Root mean square deviation (RMSD) values of all backbone atoms of HP35 protein from its simulation at 340 K with respect to the corresponding starting structure.