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Local geometry of polypeptide chains Elements of secondary structure ( turns )

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Enantiomers

Phenomenological manifestation of chiraliy: optical dichroism (rotation of the plane of polarized light).

- Cartesiancoordinates
- describeabsolute geometry of a system,
- versatilewith MD/minimizing energy,
- need a moleculargraphics program to visualize.

- describelocal geometry of an atom wrt a selectedreferenceframe,
- withsomeexperience, local geometry can be imaginedwithout a moleculargraphics software,
- mightcauseproblemswhendoing MD/minimizing energy (curvilinearspace).

Cartesian coordinate system

z

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)

H(6)

O(2)

H(4)

C(1)

yH(6)

xH(6)

x

H(5)

y

H(3)

Internal coordinate system

idijaijkbijkl j k l

C(1)

O(2)1.40000 * 1

H(3)1.08900 * 109.47100 * 1 2

H(4)1.08900 * 109.47100 * 120.00000 * 1 2 3

H(5)1.08900 * 109.47100 * -120.00000 * 1 2 3

H(6)0.95000 * 109.47100 * 180.00000 * 2 1 5

H(6)

O(2)

H(4)

C(1)

H(5)

H(3)

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

zj

zi

xi

yi

xj

xj

Bond angle calculation

j

aijk

i

k

Dihedral angle calculation

i

bijkl

k

j

l

Calculation of Cartesian coordinates in a local reference frame from internal coordinates

H(5)

z

H(6)

d26

C(1)

a426

H(3)

b3426

O(2)

y

x

H(4)

Need to bring the coordinates to the global coordinate system

Polymer chains

qi+2

qi+2

wi+1

wi+1

qi+1

i+1

i+1

di+1

di+1

i

i

wi

pi-1

di

ai

wi-1

wi-1

qi-1

qi-1

i-1

i-1

di-1

di-1

qi

i-2

i-2

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

3

4

q3

w4

2

d2

n-3

1

a21n

d1n

a1 n n-1

wn

n

n-2

dn

qn

n-1

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

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

c1

c2

c3

c1=0

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.

E

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

C7eq

C7ax

g-turn (fi+1=-79o, yi+1=69o)

b-turns

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