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Paper 4: Biomolecules and Their Interactions

Module 4: Conformational properties of polypeptides and Ramachandran plot

Introduction

Amino acid sequence (primary structure) in the polypeptide chain of protein controls its folding into helix, sheet, turns etc (secondary structure). The intra-molecular interactions amongst different secondary structural subunits give rise to structural motives. Different structural motives pack against each other to give a tertiary structure of the proteins which is the third level of organization in the protein structure. A number of tertiary subunits interact to yield quaternary structure which is the fourth level of protein structure organization. Latter ultimately leads to a ‘Biological assembly’ with active cavities for interaction with specific activators, inhibitors, co-factors and is the business end of the of the ‘protein function’. In order to understand intricacies in protein structure leading to its specific and function, it is necessary first to understand the first level of its organization viz. transition from the primary structure to secondary structure. The process takes place through sequence dependent conformational changes in peptide fragments and formation of hydrogen bonds amongst carboxylic and amino groups of polypeptides. Soon after the discovery for three dimensional structure of myoglobin in the year 1960 by Kendrew et al G. N. Ramachandran in 1963 laid the basic foundation for understanding of polypedide conformarion.

Objectives

The objective of the present module is to:

a) Describe nomenclature and rotational angles in a peptide fragment,

b) Discuss emperical potential energy function for the study of conformational properties of

polypeptides,

c) Discuss generation of a polypeptide chain using standard bond lengths and bond angles,

d) Described methodology for computation of conformational maps for polypeptides, and

e) Elaborate on Ramachandran plot.

4.1 Nomenclature and rotational angles in a peptide fragment

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Figure4.1 Bond lengths, bond angles and dihedran

angles around C, N and C (from Dihedral angles)

We show in figure 4.1 structural details (bond lengths, bond angles and torsional angles of a dipeptide fragment.

The bond lengths and bond angles shown here, usually do not change much. The geometry around the peptide linkage discussed in the previous module is also fairly rigid because of the partial double bonded nature of the H-N= C-O linkage. The structural changes occur because of the rotational freedom around

N-C and C–C bonds (shown in the figure by yellow and green arrow) in the backbone and rotations of the side chains.

Dihedral angle

To define the dihedral angle we consider a four atom sequence A-B-C-D shown in the figure 4.2. The standard IUPAC-IUB definition of a dihedral or torsional angle (both terms used synonymously) rotation takes place around the bond B-C. The view on the right side is along the bond B-C with atom A placed in a 12 O’clock position. The deviation of A-B and C-D bonds from each other measured by the deviation of D from A (rotation around the bond B-C): positive angle corresponds to clockwise rotation.

Backbone torsional angles

We show in the figure 4.3 nomenclatures for atoms forming a peptide linkage. The backbone is a part of a continuous peptide chain. We have selected nth residue for our definitions.

Figure 4.2 Definition of dihedral angle

Figure 4.3 Backbone atom nomenclature for amino acid part of a

polypeptide chain

We have included C=O, and C of the preceeding (n-1) unit and N, HN and C of the following (n+1) units as these are needed for the delimitation of torsional angles. As mentioned earlier the C=O of

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the n-1 th residue and NH of the nth residue remain in a plane. So also are the C=O of nth residue and NH of the n+1 residue. There is flexibility for rotations at C (asymmetric carbon).

• The backbone torsion angles around the N-C, C-C and C-N bonds are denoted , and respectively(table 4.1).

• The torsional angle is defined as rotation around N-C bond (angle Cn-1.-Nn- Cn-Cn),

• Rotational angle is defined as rotation about C- C bond (angle Nn- Cn- Cn-Nn+1),

• Rotational angle about C-N bond (angle Cn-Cn-N n+1-Cn+1) is usually 180° in all trans amino acids. It is 0⁰ in cis amino acids. Peptide bond in proline and some N-substituted amino acids as sarcosine are able to be populated both cis and trans isomers. It is a slow process and impede protein folding (figure 4.4a).

Side chain dihedral angles

• For each side chain dihedral angle, the coordinates of four atoms are required. The table below

contains the list of atoms and rotational angles. The angle 1 is not calculated for Ala and Gly.

The angle 2 is not calculated for Ala, Cys, Gly, Ser, Thr and Val. The angle 3 is calculated for Arg, Gln, Glu, Lys and Met. The angle 4 is calculated for Arg and Lys and the angle 5 is calculated only for Arg (figure 4.4b).

Table 4.1

Angle Four atom sequence Residues

Cn-1-N-C-C All amino acids

N-C-C-Nn+1 All amino acids

Cn-1-C-N-C All amino acids

N-C-C-C All except Ala and Gly

C-C-C-C All except Ala, Cys, Gly, Ser, Thr and Val

C-C-C-C Lys

C-C-S-C Met

C-C-C-N Arg

C-C-C-O Gln, Glu

C-C-N-C Arg

C-C-C-N Lys

C-N-C-N1 Arg

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Figure 4.4a Cis trans isomerization in proline

Figure 4.4b Side chain torsional angles in Arg. Figure adopted from Dihedral angles

4.3 Emperical potential energy function for study of conformational properties of polypeptides

Conformation of polypeptides can be studied using variety of theoretical methods starting from ab

initio, semi-emperical and empirical quantum chemical methods to purely empirical atom-atom based

additive non bonded potentials. We are quite justified in neglecting all bonded interactions and restrict

ourselves only to non-bonded interactions, since our objective is to understand conformational degrees

of freedom of polypeptides, where energies differ by few kcal/mol. Emperical potential energy functions

have gone through great improvement over the period of time and with the incorporation of statistical

methods as: genetic algorithm, Monte Calrlo method etc these can deliver satisfactory results in a

reasonable period of time.

Hard sphere potential

The origin of the empirical potential energy function can be traced back to the most celebrated work of

G.N. Ramachandran in 1963. He showed that if we rotate the tri-peptide structure around N-C () and

C -C() bonds, and take into account dimensions of the atom, only few values of and are really

allowed, others are forbidden. He plotted these regions on a two dimensional plot of verses

conventionally known as Ramachandran's - map (discussed later), using the van der Waal’s contact

distances given in table 4.2. Contact criterion used here is, in fact equivalent to the hard sphere

potential (figure 4.5). It means that if R > R0, the approach of the two atoms is allowed. If R< R0, the

approach of the atoms is not allowed, as the potential energy becomes very large.

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Figure 4.5Hard sphere potential

Table 4.2 Contact distances (in Å) between atom pairs used for calculating allowed regions for rotations

around N-Cand C-C bonds in Ramachandran plot (Ramachandran et al 1963, 1966).

Pair Ordinary Minimum Pair Ordinary Minimum

C-C 3.0 2.9 O-N 2.7 2.6

C-O 2.8 2.7 O-H 2.4 2.2

C-N 2.9 2.8 N-N 2.7 2.6

C-H 2.4 2,2 N-H 2.4 2.2

O-O 2.7 2.6 H-H 2.0 1.9

Conventional Empirical force field used for conformational analysis

Conventionally we use empirical potential energy function given below.

E= Unon+ Uele+Uhbon+Utor

Unon - Non-bonded interaction

Leach, Nemethy and Scheraga (1966) emphasized the importance of steric interactions on the basis of

the ‘hard sphere potential’. In 'allowed' or 'forbidden' classes of conformations, one cannot find out the

probability of different conformations. De-Santis et al. carried out similar calculations with continuous

potential. Scott and Scheraga (1966) used Lennard-Jones 6-12 potential function given below:

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non ij

ij

ij

ij

nonR

B

R

AU

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Here, Rij is the distance between non-bonded atom pair i and j. (Note: while calculating energy between

different rotameric states, we assume that bond lengths and bond angles in a molecule are fixed. This is

known as rigid geometry approximation. This means, that if atom i is connected to atom j, the energy of

bond ij is fixed and will not change when we rotate the structure about any single bond. Similarly if

atoms i-j-k make a bond angle, energy between atom pairs ij, jk and ik will not change between different

rotameric states. Energy between atom- pairs separated by at least three bonds will change between

different rotameric states. These pairs are called non-bonded atom pairs). The summation in the

equation is over all the pairs of non-bonded atoms. (This means that we exclude first, second and third

neighbors of any atom and carry out summation only from the fourth neighbor onwards). For making list

of nonbonded pairs of atoms, it is necessary to create a topology file with connectivity. The first term in

equation is a repulsive term, which increases very fast as Rij becomes less than some critical value. The

second term is attractive term. Scott and Scheraga (1966) calculated the values of Bij and Aij from atomic

polarizabilities using the Slater-Kirkwood relationship (Slater and Kirkwood 1931) given below:

jjii

ji

ijNN

meB

//

)/(2/3

Here e and m are electronic charge and mass of atom, ħ - Plank's constant, i, j -atomic

polarizabilities (usually taken from the old spectroscopic data of atoms i and j, Ni, Nj - their effective

number of valence electrons. The values of i, j, Ni, Nj and van der Waal's radii Vi, Vj are the essential

parameters for computation of the non-bonded term. These parameters are stored for different atom

types.

Aij = Bij *Ro6

where Ro - minimum approachable distance (equilibrium distance) between the atom pair i,j . It can be

calculated from van der Waal’s radii Vi and Vj using the relation

R0= Vi + Vj

The values of van der Waal’s radii given by Bondi (1964) were used by Scheraga’s group. Sunderlingam

(1973) added a factor of .2 and modified the above relation as:

R0 = Vi + Vj +.2

Aij, Bij , i, j, Vi , Vj, Ni, and Nj can also be treated as purely empirical parameters and obtained from

the computer fit of the structural data on small molecules with known structures (Hopefinger, 1973).

The latter has become a more common practice. In the empirical conformation energy program for

peptides (ECEPP) developed by Scheraga's group, they combined crystal data and conformation. It

worked successfully for non-aqueous system. Later they developed an empirical potential based on

nuclear interactions Momany et al (1975) which is used for polypeptide conformation calculations.

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Dipole-Dipole interaction Uel

Brant and Flory (1965) incorporated the dipole- dipole interaction term between the non-bonded pair

of atoms. Calculations were compared with the experimentally determined dimensions. The dipole-

dipole interaction term is calculated from the charge on the pairs of non-bonded atoms.

non ij

ji

elR

qqE

.332

Here qi and qj are the charges on the atoms i and j and -dielectric constant. It is taken equal to 3 for

hydrophobic environment. Usually atomic charges on carbon and hydrogen atoms are one order of

magnitude lower than those on nitrogen or oxygen. Partial charges on peptides and nucleotides atoms

are calculated using quantum chemical methods as: AM1, PCILO, Gaussian, MOPAC, EHMO etc.

and incorporated in popular modelling packages as MOE (Molecular Operating Environment) or

HYPERCHEM.

H-bonding terms Uhbon

It is calculated for the atoms pairs involved in H-bonds. Momany et al. (1974,1975) gave the parametric

form:

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hbonhbon hbon

hbonR

BB

R

AAU

where AA and BB are the arbitrary coefficients dependent on atom types and Rhbon -distance between H-

bonding donors-acceptors.

Barrier to internal rotation Utor

The calculation of the energy of internal rotation is based on the assumption that intra-molecular non-

bonded interactions are of the same nature as the inter-molecular interactions. Kitaygorodski (1961),

developed atom-atom summation method for classical organic chemistry, considering rotation of atomic

groups about a single bond as free. This means that any conformation, that arises as a result of internal

rotations (say for example H3C-CH3), has the same energy, or no energy is required for the angle of

rotation to be changed. Study of thermodynamic properties of ethanes and other compounds with

single bonds and structural investigations by NMR and other spectroscopic methods indicate, that these

rotations are not free but have a barrier. Thus, the ethane molecule has the lowest energy in staggered

or trans conformation, and the highest in eclipsed or cis-conformation (figure 4.6). The torsion energy

term is a four-atom term based on the dihedral angle about an axis defined by the middle pair of atoms.

For this term, the energy constant may be negative (indicating a maximum at the cis conformation), and

there may be several contributions with different periodicities for a given set of four atoms. A typical

three- fold potential is represented in figure 4.7 as:

3cos12/1 0 UU tor

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Here Uo is the height of the barrier Utor increases when hydrogen atoms are replaced by bulkier atoms.

Summation is over all the flexible angles. Rigorous calculation of Utor is a formidable task since Utor is

much smaller than the chemical bond energies.

Figure 4.6 Staggered and eclipsed

conformation

Figure 4.7 A three fold potential barrier

4.3 Generation of a poly peptide structure using standard bond length and bond angles

First step for computation of potential energy function of a polypeptide chain is to obtain 3D coordinates of all the atoms in the chain. It can be generated using sketchers in a modeling package. One can start either from a known geometry or generate some starting geometry based on the internal coordinates: bond lengths, bond angles and torsional angle. In first two methods one can generate 3D coordinates for different combinations of torsional angles (generally called as conformations) by applying a rotational transformation by Jeffery and Jeffery 1950 given below. The transformation

giving rotation of a point P with coordinates Xi, Yi and Zi about an axis passing through the origin

with direction cosines , , by an angle is given by:

ABCACA

CAABCA

CACAAB

M

cos1A , cosB , sinC .

The new coordinates are obtained by matrix multiplication between M and Cartesian coordinates

Xi, Yi, Zi . If we want to give rotation about an axis not passing through the origin, first we shift the origin

at the foot of the coordinate axes by subtracting the coordinates of the new origin and then apply the

transformation. After the transformation we shift back the origin to its starting position.

In the second method we use internal coordinates R j , j and j and generate a 4X4 matrix Bj

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sinsincossincossinsin

cossinsincoscoscossin

cos0sincos

jjjjj

jjjjjjjj

jjjj

j

R

R

R

B

This is used to generate a 4X4 matrix Aj using relation

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Aij= BijXAi’J where Ai’j is the A matrix of the connected atom (Thompson 1967). The terms A1,4, A2,4 and

A3,4 represent the X, Y and Z coordinates. For the first atom Aj =Bj =U(unitary matrix). Thus for example,

in case of molecule shown in Figure 4.8, atom 1 is connected to none, atom 2 to atom 1, atom 3 to atom

2, atom 4 to atom 2, atom 5 to atom 2, atom 6 to atom 3 and so on. Atom 9 is connected to atom 6,

atom 6 to atom 3, atom 3 to atom 2 and atom 2 to atom 1. The A matrix for atoms 2, 3, 4, 5, 6 and 9 are

obtained by multiplying B matrices of these atoms by A matrix of the connected atom. This leads to

continuous product of B matrices till the first atom.

Figure 4.8 Atom numbering of a typical molecule

Giving rotation about a particular bond is no problem in this method as one can change any torsional angle from a minimum value to a maximum value with desired increament. The starting geometry for

standard secondary structures can be generated on the basis of values for the structure from the table given in table 4.3. Generally all sketchers (Gasteiger et al 1967) build molecules using internal coordinates from their in-built databases.

4.4 computation of the conformational maps for polypeptides

For computation of conformation energy maps of polypeptides we use rigid geometry approach, meaning thereby the bond lengths and bond angles are considered constant. Out of the three

backbone torsional angles, rotational angle about C - N bonds between two adjacent amino acids is always fixed (180°) because of its partial double bond nature. The other two torsional angles

(rotations about the N-C ( and C-C ( bonds depend on the secondary structure assignment. The side chains of any peptide are flexible. There are several statistical correlations available for side chain assignment. It is possible to feed the torsional angles data, if it is available. Different software

packages adopt different strategies.

Torsional angles for helices, sheets and two central residues for type- I, type-II and type-III bends are

given in table 4.3. Sibanda et al (1988) defined a large variety of bends which can be used if needed.

Table 4.3 Dihedral angles for different secondary structures of proteins

Conformation Residue

helix 1 -57° -47°

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310helix 1 -60° -30°

sheet anti-parallel 1 -139° 135°

sheets parallel 1 -119° 113°

Type- I turn 2 -60° -30°

3 90° -0°

Type- II turn 2 -60° 120°

3 80° 0°

Type- III Turn 2 -60° -30°

3 -60° -30°

The main steps in conformation energy calculation of a polypeptide chain are:

Get the starting geometry of a polypeptide (either from the literature) or generate the same using sketchers or using torsional angle information on the basis of secondary structure using a program available in the package,

Select rotational angle pairs,

Finalize initial, incremental and final value for each rotational angle (normally one uses 0-360⁰

with increment of 20⁰. However in case of polypeptide backbone rotations and this is taken from -180⁰ to to +180⁰ consistent with original definitions of these angles by Ramachandran et al 1963. According to which the trans orientation was given the dihedral angle value of 0⁰. Now in accordance with IUPAC-IUB nomenclature the trans orientation is given value 180⁰ but the

maps are plotted from -180⁰ to +180⁰.

Generate Cartesian coordinates for the selected values of the two dihedral angles. Compute potential energy (non-bonded, electrostatic, H-bond and torsional etc) for each combinations of rotational angles (conformation) based on the geometry and create a 2D database,

Find out the conformation (combination of two dihedral angles) with the lowest energy,

Find out points within 1 kcal/mole from the lowest energy, and join them by a contour,

Repeat the procedure for energy interval between 2-3, 3-4, 4-5 kcal/ mole. Join the boundaries of each region as in contour maps.

4.5 Ramachandran plot

A Ramachandran plot (alternately known as ,map) was developed by G. N. Ramachandran C.

Ramakrishnan and V. Sasisekharan in 1963. This is a way to visualize backbone torsional angles ,of

an amino acid residues in protein.

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Figure4.9 G. N. Ramachandran 1922-2001. Photo from Wiki

Figure 1Figure 4.10 Ramachandran plot figure from wiki

We show in figure 4.9 file photo of G.N. ramachandran from a wiki. As discussed earlier authors used

distance criterion (hard sphere potential) to compute sterically tolerable angle combinations of the

two angles and . They defined two distance criterion given in table 4.2. Figure 4.10 shows

allowed backbone conformational angles. The continuous line shows normal ‘allowed region’ or

‘full radius’ in conformational space. The dotted line outside shows ‘extra allowed region’ if some

adjustments for van der Waal’s allowed distances is done and reduced radius given in table 4.2 is used.

Also shown in this map are the torsional angle for different secondary conformations as -

helix and -sheets.

The importance of Ramachandran plot is that the first Ramachandran plot was plotted in 1963 soon

after the structure of Myoglobin was given by Kendrew et al (1960). The computation technique and

force field parameters improved substantially later. Scott and Scheraga (1966) gave the conformational

map for Ala-Ala-Ala (figure 4.11). One can notice that the general features in figure 4.10 and 4.11 are

similar. The second most important point to note is that with the exception of Gly, Pro and few residues

in the turn region the general features of the Ramachandran plot remain valid over the last five decades.

Third most important outcome of the Ramachandran plot is its application for understanding quality of

protein structure. Number of high resolution protein structures are now available. Several computer

programs as ProCheck, MolMol etc are now available which can immediately plot values of the

protein backbone from PDB coordinates on Ramachandran map. These can be used for the assessment

of protein structure. A typical example of PCNA trimeric protein 1AXC is depicted in figure 4.12.

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Figure 4.11Ramachandran plot fo Ala-Ala-Ala by Scott & Scheraga 1966

Figure 4.12 Rachandran map for PCNA trimeric DNA clamp protein with both helix and -sheet. PDB ID 1AXC. Red,

brown and yellow regions depict favored, allowed and generously allowed. Figure by PROCHECK from wiki

Summary

We have discussed in the present module conformational flexibility and rotational angles of a poly-peptide chain. We have described origin of empirical potential energy function and conventional potential energy functions used for the study of polypeptide chains. We have enumerated different ways of generating polypeptide conformations. Mathematical foundation for computation of conformation energy maps is given. We have elaborated on Ramachandran plot, its historical importance and use in protein structure.