Ch 23 Pages 580-592

Ch 23Pages 580-592

Lecture 17 – Intramolecular interactions

We have partitioned the overall interaction energy between molecules or within molecules into terms that describe individual interactions as follows

Summary of lecture 16

...612int ij

ij

ij

ij

ij

jiramolecula r

ArB

rqq

CUU

The first term reflects interactions between atoms within a molecule, the second term is the Coulomb interaction between charged particles; the third reflects the strong repulsion between atoms and molecules at short distance and the fourth weak attractive forces between molecules (London dispersion forces)

A very important interaction between biological molecules is the electrostatic potential expressed by Coulomb’s law, for two charges separated by a distance r:

Summary of lecture 16

The Lennard-Jones potential describes weak attractive and strong repulsive forces between molecules.

rqq

rU4

)( 21

612

4)(ij

ij

ij

ij

rA

rB

rU

For a given atom or molecular pair, it can be written as:

612

4)(rR

rRrU

Intramolecular Interactions

Thus far we have considered intermolecular interactions in systems of molecules that have no internal motions. However, except for monatomic systems, all molecules display internal motions

For example, diatomic molecules undergo whole-molecular rotation (i.e. rigid body rotation) and changes in the length of the bond (i.e. bond vibration).


When biological molecules interact with each other or as we change temperature or denature them, their chemical structure (configuration) does not change, however, their three-dimensional structure can and often does change

Their structure is determined by the rotation about chemical bonds whose length and angles are fixed by covalent properties of the chemical bonds that hold these molecules together

The conformation of a molecule is dictated by the value of each of the angles that are more or less free to rotate (C-C double bonds, for example, are not free to rotate).


For a large molecules like protein or DNA, there are very many possible conformations, but many are not populated because energetically unfavorable

For example, for proteins, only certain regions of the conformational space in the peptidic backbone are populated due to steric clashes between amino acid side chains (Ramachandran’s plot) and similar considerations apply to nucleic acids as well


Bond lengths and bond angles are determined by interactions of electron and nuclei that define the chemical bonds, and are eminently of quantum mechanical nature. However, the forces that change them can be treated by simple classical physical models

The energy of moving atoms so as to stretch or compress a bond, or to change a bond angle, depends (close to the value which is most favored, or equilibrium value) on the square of the change in bond length or the square of the change in bond angles:

21 ( )2vib vib eqU k r r

21 ( )2rot rot eqU k


Here the differences represent deviations from the equilibrium value that is the bond length and angle that are quantum mechanically most favored

The energy increases when the bonds are perturbed from their equilibrium positions and, as it happens, it takes far more energy to change a bond length than a bond angle.

21 ( )2vib vib eqU k r r 21 ( )

2rot rot eqU k


Rotations about single bonds cause large changes in molecular conformations and dictate the three-dimensional structure of biological molecules. These rotations do not require much energy (unless there are double bonds involved, in which case the energy are substantial)

For example, the phenyl ring in phenylalanine in proteins rotate rapidly around the bond connecting it to the polypeptide chain at room temperature, the barrier is only about 2.5 kJ/mole.


Torsion energies will have various minima corresponding to most favored states, and can be represented by a potential that has the following form:

))3cos(1(2

torVU

Where is the torsion angle. In order to rotate a double bond, it takes much more energy because the p bond has to be broken, while for a bond with partial double character, like the C-N bond in formamide or in peptides, the energy is intermediate.

Rigid Rotors

Let us consider rotations and vibrations in a more formal context. A homonuclear diatomic molecule (e.g. H2, N2, etc.) may be classically represented as a dumbbell. Consider the dumbbell as composed of two spheres with masses m, which are intended to represent atoms, connected by a mass-less bar, which is intended to represent a chemical bond. The bond length is r.

Rigid Rotors

If the dumbbell does not interact with other dumbbells, the total energy is purely kinetic, as usual, i.e

E E Etrans rotate

where Etrans is the kinetic energy of translational motions and Erotate is the kinetic energy associate with rotational motions

Rigid Rotors

The translational energy is, in Cartesian coordinates

The rotational motions are best represented in a spherical coordinate system, in which angular motions are described in terms of the angles and , see diagram below. In this coordinate system the angular kinetic energy is

Ep p p

mtransx y z 2 2 2

2

Ep p

Irotate

2 2

2

Where I is the moment of inertia

I Rm mm m

R mR

2 1 2

1 2

22

2

Vibrations and harmonic oscillators

The vibration of the bond of a dumbbell can be represented as the motion of a harmonic oscillator. The energy of a diatomic vibrator, modeled as a classical harmonic oscillator is:

Where is again the reduced mass

If m1=m2=m, and is the spring constant of the bond

E K Up R

vib vib vibR 2 2

2 2

m mm m

m1 2

1 2 2

The Trouble with Oscillators: Black Body Radiators & Ultraviolet Catastrophe

Fundamental properties of nature like the structure of the atom, atomic spectra, and the distribution of radiant energy emitted by heated materials cannot be properly explained by classical physics

A famous example, which was very important in the early days of questioning classical mechanics as an appropriate description of matter and directly led to quantum mechanics is provided by black body radiation.

The Trouble with Oscillators: Black Body Radiators

When a material, like a metal, is heated, it first glows dull red (around T=1100K), then as it gets hotter its color shifts to the blue end of the visible spectrum. This is the effect responsible for the luminosity of light bulbs. The graph of the intensity of emitted radiation as a function of wavelength for several temperatures is shown here


A blackbody is an ideal version of a heated object. To assure that the emitted radiation is not simply being reflected from the body, the body is painted black (i.e. so it does not reflect radiation) and light is emitted only through a pinhole. The radiation profile from such object (also called a spectral distribution) is called “black body radiation”.


As the metal is heated, atoms in the metals vibrate (they behave as harmonic oscillators) and emit radiation into the hollow cavity (because they consist of charged particles, which emit radiation at the frequency of their vibrational motion)

When the density of energy in the cavity equals the density in the metal, the black body is at thermal equilibrium and emission into the cavity stops. Classical physics predicted that the intensity of radiation in the cavity as a function of temperature and wavelength is given by an equation called Rayleigh-Jean Law (<E> is the average energy per oscillator)

ETI 4

8),(


The average oscillator energy can be calculated from the vibrational partition function:

ETI 4

8),(

E EN

kTqTvibrate

2 ln

The partition function for a classical harmonic oscillator can be easily evaluated:

kTmkTkTmkTdxkTxdp

mkTpqvibrate

2/222

exp2

exp4 2/12/1

0

2

0

2

Where is the frequency of the vibrational motion and

m 2


Thus:

ETI 4

8),(

The evaluation of the partition function has just yielded the result expected from the equipartition principle. This result substituted into the Rahleigh-Jean Law gives:

kTmkTkTmkTdxkTxdp

mkTpqvibrate

2/222

exp2

exp4 2/12/1

0

2

0

2

kTTkTkTE

/ln2

kTETI 44

88),(

The Trouble with Oscillators: UV catastrophy

ETI 4

8),(

This result means that a common blackened material, when heated, should emit intense levels of radiation at short wavelengths, i.e. at very high frequencies. This is contrary to common experience. One cannot expect a heated material to emit large quantities of ultraviolet radiation, which is what classical physics claims. The prediction of the Raleigh-Jean theory is in such serious disagreement with experiment, that it was called the “ultraviolet catastrophe”.

kTETI 44

88),(

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Ch 23 Pages 580-592