The Partition Function of Large Biomolecules, and its Relevance to Infrared and Terahertz Spectroscopy

Description/Abstract

Molecules of biological interest are typically far larger than those studied by infrared or terahertz spectroscopy in non-biological applications. The average size of proteins in one major database is 324 residues, some 5000 atoms. In contrast molecules studied by infrared and terahertz spectroscopy at high resolution in the physical sciences rarely exceed a few tens of atoms. This produces a radical difference in the partition functions of the molecules. The partition function is a normalising function which accounts for the distribution of molecules over thermally excited states. It is conveniently thought of as the reciprocal of the probability of finding a molecule in the ground state. For typical small molecules at 300K this function is slightly exceeds unity, rarely reaching more than a few tens for any molecule whose infrared spectrum has been described in detail. For large biomolecules the mode distribution required to calculate the partition function is not known. However reasonable estimates can be made, and calculations are available in the literature for some cases. We show that for these molecules the partition function at 300K reaches extreme values, exceeding the number of molecules in physical samples by many orders of magnitude. At 77K the values remain very large, and at 4.2K significantly exceeds unity. This leads to the surprising conclusion that at 300K a physical sample contains no ground state molecules. The same is true of any vibrational state, since the number of thermally occupiable states vastly exceeds the number of molecules available to occupy them. The majority of molecules are in vibrational states with quantum numbers of some tens. This implies that comparison of observed spectra to calculated absorption spectra from the ground state must be viewed with caution. We consider the extent of anharmonic broadening of the spectra caused by the thermal distribution of molecules.