Conformational statistics of short chains of poly(L-alanine) and poly(glycine) generated by Monte Carlo method and the partition function of chains with constrained ends

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Abstract
We have calculated properties of samples of short poly(L‐alanine) and poly(glycine) chains generated by Monte Carlo method. We analyzed various distributions and averages of these samples. The average square of the end‐to‐end distance <r2> and the average end‐to‐end vector were very close to the values calculated according to the method of Flory. The distribution of <r2> is very different from a Gaussian function for poly(L‐alanine) chains of even 20 units, already close to Gaussian for 10 unit poly(glycine) chains. The average fourth power of the end‐to‐end distance <r4> was compared with that calculated for a Kratky‐Porod type wormlike chain, and the agreement was found to be good for all chain lengths, i.e., including those for which the distribution of r2 is decidedly non‐Gaussian. The distribution of end‐to‐end vectors r was found to be approximately cylindrically symmetrical about the average . This distribution shows a pronounced maximum on the axis parallel to . Poly‐L‐alanine chains with short end‐to‐end distances are quite rare. By analyzing the distribution of samples in which each chain contains at least one (two, three, four) residue(s) in the ``α‐helical'' conformation, it was determined that all chains with very short end‐to‐end distance have several residues in this conformation. This is confirmed by the observation that the calculated average energy of the chains is several kcal/mole higher at short end‐to‐end distance. The results are discussed in terms of the ``stiffness'' of these chains, and in terms of the possibility of loop formation. It is found that probabilities of loop formation of short chains calculated on the basis of a Gaussian distribution may be seriously in error. Finally, a new approach is proposed for using Monte Carlo calculations of this type in order to calculate the partition function for looped chains for any fixed relative position‐orientation of the first and last unit of the chain (including, of course, true cyclic structures).