Shannon Mutual Information Applied to Genetic Systems

Abstract

Shannon information has, in the past, been applied to quantify the genetic diversity of many natural populations. Here, we apply the Shannon concept to consecutive generations of alleles as they evolve over time. We suppose a genetic system analogous to the discrete noisy channel of Shannon, where the signal emitted by the input (mother population) is a number of alleles that will form the next generation (offspring). The alleles received at a given generation are conditional upon the previous generation. Knowledge of this conditional probability law allows us to track the evolution of the allele entropies and mutual information values from one generation to the next. We apply these laws to numerical computer simulations and to real data (Stryphnodendron adstringens). We find that, due to the genetic sampling process, in the absence of new mutations the mutual information increases between generations toward a maximum value. Lastly, after sufficient generations the system has a level of mutual information equal to the entropy (diversity) that it had at the beginning of the process (mother population). This implies no increase in genetic diversity in the absence of new mutations. Now, obviously, mutations are essential to the evolution of species. In addition, we observe that when a population shows at least a low level of genetic diversity, the highest values of mutual information between the generations occurs when the system is neither too orderly nor too disorderly. We also find that mutual information is a valid measure of allele fixation.