Struktura i Energetyka Białek
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Transcript Struktura i Energetyka Białek
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
ab
j
l
ji kl
cosa ijk cosa jkl
d ij d kl
ab
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 rn1 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