Copyright © 2020, Molsoft LLC Jun 20 2021

[ Main concepts  Crystal symmetry transformation  Define axis  Combine transformations  Build helix ] This section describes geometrical transformations of molecular objects and manipulations with crystallographic symmetry.
Molecular objects and 3D density maps may contain symmetry information. This information allows one to generate symmetry related parts of the density or molecular objects. An elementary space transformation is defined by a rarray where values {a1,a2,...,a12} define 3x3 rotation matrix and translation vector {a4,a8,a12}. The complete augmented affine 4x4 transformation matrix in direct space can be presented as: a1 a2 a3  a4 a5 a6 a7  a8 a9 a10 a11  a12 + 0. 0. 0.  1.The related commands and functions (transformation vector will be referred to as R_tv):
There are three steps:
Example: see example in the transform command or paste the following lines into your ICM session: read csd "qfuran" for i=2,Nof(Symgroup(Symgroup(a_)))/12 * 2 # 2 elem.cells will be filled copy a_1. "a"+i transform a_a$i. i1 display center endfor color ml a_*. gcell=Grob("cell",Cell( )) obl=Augment(Cell( )) # this will work also for any oblique matrix g1 = gcell + (1)*obl[1:3,2] # shift cell by avector g2 = g1 + obl[1:3,3] display gcell g1 g2 print " cell=" Cell( ) "\n Symgroup=" Symgroup(a_) \ "Nof.sym.=" Nof(Symgroup(Symgroup(a_)))/12
Steps:
Examples: read obj "crn" # let us display an axis of the alpha helix 7:17 ds a_/7:17 R1=Mean(Xyz(a_/7:17/ca)) # this point will be projected onto the axis show Rmsd(a_/7:16/ca a_/8:17/ca exact ) # find the transformation vv=R_out aa=Axis(vv) # now R_out contains xyz of a point at the axis print "The rotation angle is ",r_out # grob + V3 translates grob by V3 gg=Grob("arrow",aa )*10.+ Sum((R1R_out)*aa)*aa+R_out # projection of R1 on aa display gg center pause #Symmetry of an extended fragment delete object build string "se ala ala ala ala ala ala ala ala ala ala ala ala" R1=Mean(Xyz(a_/*/ca)) # this point will be projected onto the axis show Rmsd(a_/2:12/ca a_/1:11/ca exact ) vv=R_out aa=Axis(vv) print "The rotation angle is ",r_out gg=Grob("arrow",aa )*10.+ Sum((R1R_out)*aa)*aa+R_out # projection of R1 on aa display gg center Now you can rotate your selection of molecules around the axis aa with the rotate command and the Rot( R_center R_axis r_angle ) function, e.g. for i=0,360,30 rotate a_1 Rot( R1 aa Real(i) ) endfor
To combine several transformations simply multiply their augmented matrices. Examples: tv = Symgroup(19) # 12*4 vector of the P212121 symmetry group tv1 = tv[13:24] # grab the second transformation of the symmetry group # the first is the identity transformation tv2 = tv[37:48] # grab the last transformation of the symmetry group tvComb = Vector(Augment(tv1)*Augment(tv2)) # selfexplanatory show tvComb # you can apply it now with the transform command
Two steps:
Example: read pdb s_icmhome + "chq" # download two molecules related by a rot. symmetry display a_*. R1=Mean(Xyz(a_/*/ca)) # this point will be projected onto the axis show Rmsd( a_2 a_1 exact) # we assume that molecules are the same tv = R_out # this is the transformation 12vector ha = r_out # rotation angle (just FYI) delete a_2 # we will regenerate the 2nd molecule anyway rename a_1. "s1" for i=1,4 # build a helix with 4 new elements j=i+1 copy a_s$i. "s"+j # names for a new subunits: s2,s3,s4,s5,s6 transform a_s$j. tv # apply tv to the last copied object endfor aa=Axis(tv) print "The rotation angle is ",r_out g_ax=Grob("arrow",aa )*10.+ Sum((R1R_out)*aa)*aa+R_out # projection of R1 on aa display a_*. g_ax color a_*. Count(Nof(a_*.))

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