[MMTK] getting hessian matrix (and equation) and force vector of a protein conformaion
Hyuntae Na
htna at hotmail.com
Tue Jan 10 20:37:15 UTC 2012
Hello all,
I just begins to use MMTK for the protein deformation simulation with minimizing its conformational potential energy including hydrogen.
I am interested in how to get a hessian matrix of a protein conformation and its equations. As far as I understand, the hessian matrix is the second derivative of a potential energy. In the mean time of implementing the hessian matrix by my self, because I have a little confusion of implementing the equation of the second derivative of the potential energies, I decided to use MMTK and to send this email in order to understand it clearly.
My first thought about the second derivative of the (likely say) van der waals (vdW) potential energy was like the follows:
$$
d^2E / d dx^2 = eps_ij * (12*13*r_ij^12/ r^12 - 2*6*7*r_ij^6/r^6) * (dx/r)^2,
d^2E / d dx dy = eps_ij * (12*13*r_ij^12/ r^12 - 2*6*7*r_ij^6/r^6) * (dx/r)*(dy/r),
$$
where eps_ij = sqrt(eps_i * eps_j), r_ij = vdW_rad_i + vdW_rad_j, dx (and dy) is the difference of x(and y)-coordinate of atoms i and j, and r=sqrt(dx^2+dy^2+dz^2).
However, after running a symbolic operation tool (Mathematica) to check its validity, the program returns a more complex equations:
$$
d^2E / d dx^2 = eps_ij * (168*r_ij^12/ r^12 - 96*r_ij^6/r^6) * (dx/r)^2 + (-12*r_ij^12/r^12 + 12*r_ij^6/r^6)*(1/r)^2,
d^2E / d dx dy = eps_ij * (168*r_ij^12/ r^12 - 96*r_ij^6/r^6) * (dx/r)*(dy/r).
$$
There exist the equation inconsistencies of the second derivation for the electrostatic potential also.
Do you know if my derivation (first equations) are correct and the running mathematica was wrong? or my derivation was incorrect and the mathematica is right?
Additionally, would you tell me how to get the hessian matrix and forces of each atoms of a protein using MMTK?
Thank you very much.
Have a nice day~
Best regards,
-- Hyuntae
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