Variational Principles in Conformation Dynamics - FU Berlin, FB MI

IV.1. The example system 33221.81.81.61.61.41.41.21.2V(ψ)1V(ψ)10.80.80.60.60.40.40.20.20−4 −3 −2 −1 0 1 2 3 4ψ0−4 −3 −2 −1 0 1 2 3 4ψFigure IV.2.: The potential energy function V (ψ ijkl )=k ijkl (1 − cos(nψ ijkl )) for the dihedralangle, with k ijkl =1and n =2, n =4,respectively.as displayed in Figure IV.2. Ifn =2,thesetwominimaaresituatedatψ ijkl =0;−π. Lastly,we would also like to include Coulomb potentials V C defined by:V C (r ij )= 14π 0e 1 e 2r ij, (IV.7)which describe the electrostatic repulsion or attraction between two atoms. Here, 0 is theelectric constant, and e 1 , e 2 are the charges of the two atoms. In our simulation, the chargeswill be equal to one elementary charge e =1.602 · 10 −19 C in absolute value, but can haveopposite signs. The total potential energy function V is then the sum of all individualenergies.For our computations, we will consider three small systems, which we call system A, systemB and system C.Thefirstisafouratomsystemwhichonlyincludesthethreeharmonicbondinteractions, two bond angle interactions, and a dihedral potential with n =2.Thesecondone also consists of four atoms, but we set n =4for the dihedral and add an attractiveCoulomb interaction between atoms one and four. In the last case, we have N =5,fourbonds, three bond angles and two dihedrals, one with n =4and the other with n =2.If we now add random perturbations and Smoluchowski dynamics, we obtain a diffusionprocess with invariant distributionµ(x) = 1 Zexp(−βV (x)),(IV.8)where Z is the partition function, as before, and x ∈ R 3N is the full position vector containingall coordinates of the atoms.It is important to point out once more that the stationary distribution and all the othereigenfunctions are defined on the 3N-dimensional Euclidean space. As such, they probablytake a highly complicated shape. At least for our examples however, the state of the system