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).

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)

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

jizzyyxxd ijijijij 222

xi yi

zi

xj

zj

xj

jkji

jkij

jkjijkjijkjiijk

jk

jk

ji

ji

jkji

jkji

ddzzzzyyyyxxxx

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

dddjkklji

ddklji

sinsinsin

sinsin

coscoscos

baba

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

eeeeeeeee

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

rpTRTRTRrrpTRTRr

rpTRrpr

ii

iiiii

ii

i

i

i

d

cossin0sincos0

001

1000cossin0sincos

00 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

rrrrrr

21n d1n

1 n n-1

N. Go and H.A. Scheraga, Macromolecules, 3, 178-187 (1970)

60% 40%

Hybrid of two canonical structures

Peptide bond geometry

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 angles

1=0

1 2 3

Dihedrals with which to describe polypeptide geometry

main chain

side chain

Skan z wykresem energii

Peptide group: cis-trans isomerization

Because of peptide group planarity, main chain conformation is effectively defined by the and angles.

Side chain conformations

The dihedral angles with which to describe the geometry of disulfide bridges

Some and pairs are not allowed due to steric overlap (e.g, ==0o)

The Ramachandran map

Conformations of a terminally-blocked amino-acid residue

C7eq

C7ax

E Zimmerman, Pottle, Nemethy, Scheraga, Macromolecules, 10, 1-9 (1977)

Energy minima of therminally-blocked alanine with the ECEPP/2 force field

- and -turns

-turn (i+1=-79o, i+1=69o) -turns

Types of -turns in proteins

Hutchinson and Thornton, Protein Sci., 3, 2207-2216 (1994)

Older classification