Conformational energy
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص675-677
2025-12-18
65
Conformational energy
A polypeptide chain adopts a conformation corresponding to a minimum Gibbs energy, which depends on the conformational energy, the energy of interaction between different parts of the chain, and the energy of interaction between the chain and sur rounding solvent molecules. In the aqueous environment of biological cells, the outer surface of a protein molecule is covered by a mobile sheath of water molecules, and its interior contains pockets of water molecules. These water molecules play an import ant role in determining the conformation that the chain adopts through hydrophobic interactions and hydrogen bonding to amino acids in the chain. The simplest calculations of the conformational energy of a polypeptide chain ignore entropy and solvent effects and concentrate on the total potential energy of all the interactions between nonbonded atoms. For example, these calculations predict that a right-handed α helix of L-amino acids is marginally more stable than a left-handed helix of the same amino acids. To calculate the energy of a conformation, we need to make use of many of the molecular interactions described in Chapter 18, and also of some additional interactions: 1 Bond stretching. Bonds are not rigid, and it may be advantageous for some bonds to stretch and others to be compressed slightly as parts of the chain press against one another. If we liken the bond to a spring, then the potential energy takes the form of Hooke’s law
Vstretch = 
kstretch (R − Re)2
where Reis the equilibrium bond length and kstretch is the force constant, a measure of the stiffness of the bond in question. 2 Bond bending. An O-C-H bond angle (or some other angle) may open out or close in slightly to enable the molecule as a whole to fit together better. If the equilibrium bond angle is θe, we write
Vbend = 
kbend (θ − θe)2
where kbend is the force constant, a measure of how difficult it is to change the bond angle.
3 Bond torsion. There is a barrier to internal rotation of one bond relative to another (just like the barrier to internal rotation in ethane). Because the planar peptide link is relatively rigid, the geometry of a polypeptide chain can be specified by the two angles that two neighbouring planar peptide links make to each other. Figure 19.26 shows the two angles φ and ψ commonly used to specify this relative orientation. The sign convention is that a positive angle means that the front atom must be rotated clockwise to bring it into an eclipsed position relative to the rear atom. For an all-trans form of the chain, all φ and ψ are 180°. A helix is obtained when all the φ are equal and when all the ψ are equal. For a right-handed helix, all φ =−57° and all ψ =−47°. For a left-handed helix, both angles are positive. The torsional contribution to the total potential energy is
Vtorsion = A (1 + cos 3φ) + B (1 + cos 3ψ)
in which A and B are constants of the order of 1 kJ mol−1. Because only two angles are needed to specify the conformation of a helix, and they range from −180° to +180°, the torsional potential energy of the entire molecule can be represented on a Ramachandran plot, a contour diagram in which one axis represents φ and the other represents ψ. 4 Interaction between partial charges. If the partial charges qi and qj on the atoms i and j are known, a Coulombic contribution of the form 1/r can be included (Section 18.3):
V Coulomb = 
Where ε is the permittivity of the medium in which the charges are embedded. Charges of −0.28e and +0.28e are assigned to N and H, respectively, and −0.39e and +0.39e to O and C, respectively. The interaction between partial charges does away with the need to take dipole–dipole interactions into account, for they are taken care of by dealing with each partial charge explicitly. 5 Dispersive and repulsive interactions. The interaction energy of two atoms separated by a distance r (which we know once φ and ψ are specified) can be given by the Lennard-Jones (12,6) form (Section 18.5):
VLJ = 
6 Hydrogen bonding. In some models of structure, the interaction between partial charges is judged to take into account the effect of hydrogen bonding. In other models, hydrogen bonding is added as another interaction of the form
VH bonding = 
The total potential energy of a given conformation (φ,ψ) can be calculated by summing the contributions given by eqns 19.38–19.43 for all bond angles (includ ing torsional angles) and pairs of atoms in the molecule. The procedure is known as a molecular mechanics simulation and is automated in commercially available molecular modelling software. For large molecules, plots of potential energy against bond distance or bond angle often show several local minima and a global minimum (Fig. 19.27). The software packages include schemes for modifying the locations of the atoms and searching for these minima systematically. The structure corresponding to the global minimum of a molecular mechanics simulation is a snapshot of the molecule at T = 0 because only the potential energy is included in the calculation; contributions to the total energy from kinetic energy are excluded. In a molecular dynamics simulation, the molecule is set in motion by heating it to a specified temperature, as described in Section 17.6b. The possible trajectories of all atoms under the influence of the intermolecular potentials correspond to the conformations that the molecule can sample at the temperature of the simulation. At very low temperatures, the molecule cannot overcome some of the potential energy barriers given by eqns 19.38–19.43, atomic motion is restricted, and only a few conformations are possible. At high temperatures, more potential energy barriers can be overcome and more conformations are possible. Therefore, molecular dynamics calculations are useful tools for the visualization of the flexibility of polymers.
Fig. 19.26 The definition of the torsional angles ψ and φ between two peptide units. In this case (an α-l-polypeptide) the chain has been drawn in its all-trans form, with ψ=φ=180°.
Fig. 19.27 For large molecules, a plot of potential energy against the molecular geometry often shows several local minima and a global minimum.
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