Chapter 3. A simple model of coherent energy transfer
3.3 The simple model
We have seen in Table 3.1 that trans secondary amides, polypeptides and proteins - that is compounds containing a trans peptide group - are likely to contain Fermi resonance coupling between the NH stretch and the Amide-I and perhaps the Amide-II modes. Before looking further into prospects for energy transfer we must first understand the character of the relevant amide group modes2,25 of the peptide moiety.
The Amide-I mode, Fig 3.1a, is characterised largely by C = 0 stretching (=80% in potential energy distributions) and to a lesser extent NH in-plane wagging (=10%) and CN stretching (=10%). In some descriptions of the Amide-I mode the NH wagging component is considered too small to include in potential energy distributions. However the relatively light mass of the hydrogen means the wagging amplitude is still significant.
The Amide-II mode, Fig 3.1b, involves NH in-plane wagging (=60% ), CN stretching (=40%) and some smaller contributions from NC stretching and CO in-plane wagging. The Am ide-M mode, Fig. 3.1c, again contains mainly NH wagging motions (=30%) and CN stretching (=30%) with the CC and NC stretch and CO wagging becoming further involved. The frequency of the Amide-III mode in the condensed state is around 1300 cm '1! Upon deuteration, the ND wagging disappears from the Amide-I and Amide-II modes ( leaving essentially CO stretch and CN stretch modes, respectively) and becomes confined to the Amide-En modes which drops greatly in frequency to below +See the note in the addendum.
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Figure 3.1 Group modes of the amide/peptide moiety.
1000 c m '1. The NH stretching mode, which is sometimes called the Amide-A mode, contains, not surprisingly, wholly NH stretching motions.
Hydrogen bonding occurs between NH and CO groups on adjacent amide groups in the condensed state; for example in crystalline N-methylacetamide (NMA) a hydrogen bonded network is set up along the
c
axis of the crystal26 with the individual molecules hydrogen bonding in a head to tail fashion. Alternatively, in the a-helix, three linear hydrogen bonding networks or spines are also set up as hydrogen bonding occurs between peptide groups on adjacent turns. The three hydrogen bonded spines are illustrated in Fig 3.2.This hydrogen bonding directly couples modes involving the NH and CO stretching via both harmonic and anharmonic terms in the potential energy. As we shall see directly, anharmonic coupling is the most significant. A strong feature o f the vibrational spectroscopy of amides2,3,25 is the large frequency shift of a number of modes from their values in the gas phase to their value in condensed phases where hydrogen bonding occurs. For example, the NH stretching modes shift from about 3500 c m '1 in the gas phase to a broad band peaking around 3280-3310 c m '1 in peptides and condensed secondary amides. The Amide-I mode in N-methylacetamide shifts from near 1730 c m '1 (gas phase) to 1650-1670 c m '1 (condensed phases). This appears to be due partly to a weakening o f the C = 0 diagonal force constant on hydrogen bonding. Thus we might expect non-negligible terms in the potential energy involving the C = 0 bond length squared times either the 0 ---H or NH bond lengths. Given the near linear configuration of the NH and CO bonds at equilibrium, and the mass ratios, either O • • H contraction or NH stretching could lead to a further weakening of the effective C = 0 force constant. Thus we would expect significant cubic coupling terms in the potential energy, which are linear in the NH stretching coordinate and quadratic in the lower frequency Amide-I and Amide-II modes on the adjacent amide group.
The particular interest in peptide groups, therefore, comes from the combination of three features:
1 . The very good likelihood of the 2:1 Fermi resonance occurring between the NH stretch and the Amide-I mode.
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alpha carbon
peptide carbon
( 3
nitrogen
^
hydrogen bonds
Figure 3.2.
Hydrogen bonded spines of an a-helix.
Only the backbone atoms are shown and hydrogen
bonds, N-H---0=C, are drawn as N---C.
2 . The involvement of the CO stretch in the Amide-I mode, which, owing to the head to tail arrangement of hydrogen bonding in secondary amides and proteins, is directly bonded to the next NH stretch.
3 . The reasonable possibility o f suitable cubic coupling terms existing across the hydrogen bond.
This means that the following scenario becomes possible: upon excitation o f the NH stretch, energy will be transferred into the Amide-I mode (possibly in conjunction with the Am ide-II mode) by the intram olecular (stretch-wag) Fermi resonance and therefore into the CO stretch. This bond is not only hydrogen bonded to the NH on the next peptide group, but by virtue o f the success of the intramolecular energy transfer is already in 2:1 frequency correspondence with the next NH stretch. If there exists some suitable anharmonic coupling between the CO (Amide-I) and the NH stretch across the hydrogen bond, then energy will be transferred by an intermolecular Fermi resonance into the next molecule where it can continue further by another intramolecular transfer and so on. A concerted flow of energy by nonlinear resonances becomes possible either along a hydrogen bonded chain of secondary amides or along the hydrogen bonded spine o f an a-helix.
The simple model is an idealised representation of a chain of secondary amides which have the stable trans configuration and which are hydrogen bonded in head-to-tail fashion. To simplify the vibrational dynamics, each trans secondary amide group in a hydrogen bonded chain is allowed just two relevant internal vibrational modes. One is a high frequency mode representing the NH stretch. The second is a mode whose harmonic frequency is exactly half that of the high frequency mode and represents both the Amide-I and Amide-II modes.