Date: Wed, 21 Nov 2007 15:25:58 -0600
From: Yang Zhang <yzhang@ku.edu>
Subject: Re: I-TASSER potentials again

Hi,

>1) Are you using centroid (geometrical center) or real center
>of mass taking into account the weight of each type of side chain?

The latter, real center of mass.

>2) How do you calculate the radius of gyration? Is it just the
>estimated value of 2.2*N^0.38? If it's a real distance, is it
>measured using just side chain centers of mass, C-alphas or maybe both?

This depends on how you use it. For the calculation of the depth factor,
it is just estimated by 2.2*N^0.38. For the calculation of principal
axes, we use the side-chain center of mass (see below).

>3) How the side chain coordinates from 2-rotamer library are
>chosen? I know that first vector is for extended and second for compact
>local structure but how extend/compact is decided? Are you using here
>the secondary structure prediction (H is compact, else extended) or
>maybe r(i,i+2) distance like in E13 potential?

No, we calculate the angle of C-alpha of (i-1,i,i+1). If the angle>105,
it is extended; otherwise, it is compact. The corresponding parameters
are obtained from PDB statistics.

>4) How the potential is calculated when the residue neighbourhood is
>too small to follow the equations (like at the beginning/end of the
>sequence)? Are the terms impossible to compute just ignored (equal to
>zero) or some defaults are assumed?

We calculate the terms only for the involved residues.

>5) How exactly the local handedness of the three consecutive residues is
>decided? Is it based on angles between CA-CA vectors being within some
>range?

Let a,b,c are the consecutive vectors of neighboring residues, and
d=a(x)b(.)c. d>0 corresponds to right-handed and d<0 to left-handed.

>6) How the length of principal axes of the protein ellipsoid are being
>determined? Is it just half distance between max and min coordinates
>for each dimension?

The principal axes are the eigenvalues of covariance matrix. In the 
following I attach a program for the quick calculation of the 
eigenvalues from the covariance matrix.

Hope this helps.
Yang


**************************************************************
*     Subroutine for calculating eigenvalue and eignevectors
*     A{3,3}-covariance matrix, E{3}-eigenvalue, T{3,3}-eigenvector
**************************************************************
      subroutine eigenvalue
      common/eigen/A(3,3),E(3),T(3,3)

c     #### Eigenvalues ############
      pi=3.1415926
      p1=-1
      p2=A(1,1)+A(2,2)+A(3,3) 
      p3=-A(1,1)*A(2,2)-A(1,1)*A(3,3)-A(2,2)*A(3,3)+
     &     A(2,3)*A(2,3)+A(1,3)*A(1,3)+A(1,2)*A(1,2)
      p4=A(1,1)*A(2,2)*A(3,3)+A(1,2)*A(2,3)*A(3,1)+
     &     A(1,3)*A(2,1)*A(3,2)-A(1,3)*A(2,2)*A(3,1)-
     &     A(1,1)*A(2,3)*A(3,2)-A(1,2)*A(2,1)*A(3,3)
      p5=(-(1.0/3)*(p2/p1)**2+p3/p1)/3 
      ap5=sqrt(-p5)
      p6=((2.0/27)*(p2/p1)**3-(1.0/3)*(p2*p3/p1**2)+p4/p1)/2
      p7=acos(-p6/sqrt(-p5**3))
      p8=2*ap5*cos(p7/3.0)
      p9=-2*ap5*cos((p7+pi)/3.0)
      p10=-2*ap5*cos((p7-pi)/3.0) 
      p11=p2/(3*p1)
      E(1)=p8-p11               !eigenvalue
      E(2)=p9-p11               !eigenvalue
      E(3)=p10-p11              !eigenvalue
      
c     ##### normalized eigenvectors #########
      do i=1,3
         fnorm1=A(2,1)*A(1,2)-(A(1,1)-E(i))*(A(2,2)-E(i))
         x=((A(2,2)-E(i))*A(1,3)-A(1,2)*A(2,3))/fnorm1
         y=((A(1,1)-E(i))*A(2,3)-A(2,1)*A(1,3))/fnorm1
         T(i,3)=1/sqrt(x*x+y*y+1)
         T(i,1)=x*T(i,3)
         T(i,2)=y*T(i,3)
c     write(*,*)i,E(i),T(i,1),T(i,2),T(i,3)
      enddo

      return
      end