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Chirality
Enantiomers
Phenomenological manifestation of chiraliy: optical dichroism (rotation of the plane of polarized light).
Representation of geometry of molecular systems
• Cartesian coordinates• describe absolute geometry of a system,
• versatile with MD/minimizing energy,
• need a molecular graphics program to visualize.
• Internal coordinates• describe local geometry of an atom wrt a selected reference
frame,
• with some experience, local geometry can be imagined without a molecular graphics software,
• might cause problems when doing MD/minimizing energy (curvilinear space).
z
x yxH(6)
yH(6)
Cartesian coordinate system
Atom x (Å) y (Å) z (Å) C(1) 0.000000 0.000000 0.000000 O(2) 0.000000 0.000000 1.400000 H(3) 1.026719 0.000000 -0.363000 H(4) -0.513360 -0.889165 -0.363000 H(5) -0.513360 0.889165 -0.363000 H(6) 0.447834 0.775672 1.716667
zH(6)
C(1)
O(2)
H(3)
H(4)
H(5)
H(6)
Internal coordinate system
i dij ijk ijkl j k lC(1) O(2) 1.40000 * 1H(3) 1.08900 * 109.47100 * 1 2H(4) 1.08900 * 109.47100 * 120.00000 * 1 2 3H(5) 1.08900 * 109.47100 * -120.00000 * 1 2 3H(6) 0.95000 * 109.47100 * 180.00000 * 2 1 5
C(1)
O(2)
H(3)
H(4)
H(5)
H(6)
Dihedral (torsional) angle
The C-O-H plane is rotated counterclockwise about the C-O bond from the H-C-O plane.
jkji
jkij
jkjijkjijkjiijk
jk
jk
ji
ji
jkji
jkji
dd
zzzzyyyyxxxx
uu ˆˆ
cos
ijk
i
j
k
Bond angle calculation
i
j
k
l
ijkl
a
b
ba
Dihedral angle calculation
jklijkkljkij
ijkl
jklijk
jklijkklij
ijkl
ddd
jkklji
ddklji
sinsinsin
sinsin
coscos
cos
ba
ba
yx
z
342642626H(6)
342642626H(6)
42626H(6)
sinsin
cossin
cos
dz
dy
dx
3426
426
d26
C(1)
H(3)
O(2)
H(4)
H(5)
H(6)
Calculation of Cartesian coordinates in a local reference frame from internal coordinates
Need to bring the coordinates to the global coordinate system
localTglobal
locali
locali
locali
iii
iii
iii
globali
globali
globali
z
y
x
eee
eee
eee
z
y
x
RER
332313
322212
312111
i-2
i-1
i
i+1
di-1
di
di+1
i-1
i
i+1
i+2
i
Polymer chains
i-2
i-1
i
i+1
di-1
di+1
i-1
i+2
i-1
i+1
i-1
i+1
pi-1
ii 0180
i
1113322
1113322
344433224
2333223
12222
11
nnnnnnn
iiiiiii
rpTRTRTRTRr
rpTRTRTRTRr
rpTRTRTRr
rpTRTRr
rpTRr
pr
ii
iiiii
ii
i
i
i
d
cossin0
sincos0
001
100
0cossin
0sincos
0
0 RTp
For regular polymers (when there are „blocks” inside such as in the right picture, pi is a full translation vector and TiRi is a full transformation matrix).
Ring closure
n-1
n
1
2
dn
d2
n
3
n
4
n-2
3 4
n-3
11
1
11
1212
112
11
cos
cos
nnnn
nnn
nn
n
nn
dd
dd
d
rrrr
rrrr
rr
21n d1n
1 n n-1
N. Go and H.A. Scheraga, Macromolecules, 3, 178-187 (1970)
Peptide bond: planarity
The partially double character of the peptide bond results in
•planarity of peptide groups
•their relatively large dipole moment
Because of peptide group planarity, main chain conformation is effectively defined by the and angles.
Conformations of a terminally-blocked amino-acid residue
C7eq
C7ax
E Zimmerman, Pottle, Nemethy, Scheraga, Macromolecules, 10, 1-9 (1977)