LOCAL GEOMETRY OF POLYPEPTIDE
CHAINS
ELEMENTS OF SECONDARY
STRUCTURE (TURNS)
Levels of protein structure organization
Atom symbols and numbering in amino acids
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).
Cartesian coordinate system
z
zH(6)
H(6)
O(2)
H(4)
Atom
C(1)
O(2)
H(3)
H(4)
H(5)
H(6)
x (Å)
0.000000
0.000000
1.026719
-0.513360
-0.513360
0.447834
y (Å)
0.000000
0.000000
0.000000
-0.889165
0.889165
0.775672
C(1)
yH(6)
x
xH(6)
H(5)
H(3)
y
z (Å)
0.000000
1.400000
-0.363000
-0.363000
-0.363000
1.716667
Internal coordinate system
H(6)
H(4)
i
C(1)
O(2)
H(3)
O(2) H(4)
H(5)
H(6)
1.40000
1.08900
1.08900
1.08900
0.95000
C(1)
H(5)
H(3)
aijk
dij
*
*
*
*
*
109.47100
109.47100
109.47100
109.47100
bijkl
j k l
1
*
1 2
* 120.00000 * 1 2 3
* -120.00000 * 1 2 3
* 180.00000 * 2 1 5
Bond length
Bond (valence) angle
Dihedral (torsional) angle
The C-O-H plane is rotated counterclockwise about the C-O bond from
the H-C-O plane.
Improper dihedral (torsional) angle
Bond length calculation
dij 
x
 xi   y j  yi   z j  zi   i j
2
j
2
2
zj
zi
xi
xj
yi
xj
Bond angle calculation
cosa ijk


x  x x

ji  jk
ji jk
i

j
ji
ji

jk
jk
k
 x j    yi  y j  yk  y j   zi  z j z k  z j 
d ij d jk
 uˆ ji  uˆ jk
j
aijk
i
k
Dihedral angle calculation

a
k
i
bijkl

b
 
ab
j
l
ji  kl
 cosa ijk cosa jkl


d ij d kl
ab
cos b ijkl    
sin a ijk sin a jkl
ab
sin b ijkl 
 ji  kl  jk
d ij d jk d kl sin a ijk sin a jkl
Calculation of Cartesian coordinates in a local reference
frame from internal coordinates
H(5)
z
H(6)
d26
a426
C(1)
b3426
O(2)
y
x
H(4)
xH(6)   d 26 cosa 426
yH(6)  d 26 sin a 426 cos b 3426
z H(6)  d 26 sin a 426 sin b 3426
H(3)
Need to bring the coordinates to the
global coordinate system
i
 xiglobal   e11i e21

 
global
i
i
 yi
   e12 e22
 global   i
i
 zi
  e13 e23


global
T
R
 E R local
local

e  xi 
 local
e  yi 
 local 

e  zi 


i
31
i
32
i
33
Polymer chains
qi+2
qi+2
wi+1
wi+1qi+1
i+1
i+1
di+1
pi-1
di+1
i
di
i
wi
ai
wi-1 q
i-1
i-1
qi di-1
i-2
qi  1800  ai
wi-1
i-1
qi-1
di-1
i-2
r1  p1
r2  R 2 T2p 2  r1
r3  R 2 T2 R 3T3p 3  r2
r4  R 2 T2 R 3T3 R 4 T4p 4  r3

ri  R 2 T2 R 3T3  R i 1Ti 1R i Ti p i  ri 1

rn  R 2 T2 R 3T3  R n 1Tn 1R n Tnp n  rn 1
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).
 di 
 
pi   0 
0 
 
 cosq i

Ti   sin q i
 0

 sin q i
cosq i
0
0

0
1 
0
1

R i   0 coswi
 0 sin w
i



 sin wi 
coswi 
0
Ring closure
rn  r1  d1n
3
q3
r2  r1   rn  r1   cosa
4
d 2 d1n
w4
rn  r1   rn1  rn   cosa
d n d1n
2
d2
n-3
1
d1n
a21n
a1 n n-1
n
12 n
wn
n-2
dn
n-1
qn
N. Go and H.A. Scheraga, Macromolecules, 3, 178-187 (1970)
1n n 1
Peptide bond geometry
Hybrid of two canonical structures
60%
40%
Electronic structure of peptide bond
Peptide bond: planarity
The partially double
character of the peptide
bond results in
•planarity of peptide
groups
•their relatively large
dipole moment
Side chain conformations: the c angles
c1 c 2 c 3
c1=0
Dihedrals with which to describe polypeptide geometry
side chain
main chain
Peptide group: cis-trans isomerization
Skan z wykresem energii
Because of peptide group planarity, main chain conformation is
effectively defined by the f and y angles.
Side chain conformations
The dihedral angles with which to describe
the geometry of disulfide bridges
Some f and y pairs are not allowed due to steric
overlap (e.g, f=y=0o)
The Ramachandran map
Conformations of a terminally-blocked amino-acid residue
E
Zimmerman, Pottle, Nemethy, Scheraga,
Macromolecules, 10, 1-9 (1977)
C7eq
C7ax
Energy minima of therminally-blocked alanine with
the ECEPP/2 force field
g- and b-turns
g-turn (fi+1=-79o, yi+1=69o)
b-turns
Types of b-turns in proteins
Hutchinson and Thornton, Protein Sci., 3, 2207-2216 (1994)
Older classification