On random sequence spacing distributions

The random model for allocation of an amino acid A at locations along a protein chain is controlled by an underlying binomial distribution for occurrence of A, with density given by the relative abundance of A. Here we derive analytically the distribution of lengths of inter-A spaces in sequences of arbitrary length n, with arbitrary relative abundance p of occurrencies of A. This provides a well-defined reference structure for comparison with observations. We derive also the distribution function for inter-AA spaces. It turns out that the standard deviation of inter-A space length is approximately proportional to the mean inter-A space length, independently of sequence length n and relative abundance p. This means that, to the extent that the space length can be approximated by a continuous random variable, the distribution of space lengths is represented by a gamma distribution. The significance of this latter approximation is that the gamma family may provide models for spacing length distributions when the underlying process is not binomial, for example, when clustering or evening out of amino acids occurs. The new results show that actual sequences of amino acids also do have standard deviation of spacing lengths proportional to the mean but all exhibit more self-clustering than expected for finite sequences from a random process.