Measures of degeneracy and redundancy in biological networks

Degeneracy, the ability of elements that are structurally different to perf orm the same function, is a prominent property of many biological systems r anging from genes to neural networks to evolution itself. Because structura lly different elements mag produce different outputs in different contexts, degeneracy should be distinguished from redundancy, which occurs when the same function is performed by identical elements. However, because of ambig uities in the distinction between structure and function and because of the lack of a theoretical treatment, these two notions often are conflated. By using information theoretical concepts, we develop here functional measure s of the degeneracy and redundancy of a system with respect to a set of out puts. These measures help to distinguish the concept of degeneracy from tha t of redundancy and make it operationally useful. Through computer simulati ons of neural systems differing in connectivity, we show that degeneracy is low both for systems in which each element affects the output independentl y and for redundant systems in which many elements can affect the output in a similar way but do not have independent effects. By contrast, degeneracy is high for systems in which many different elements can affect the output in a similar way and at the same time can have independent effects. We dem onstrate that networks that have been selected for degeneracy have high val ues of complexity, a measure of the average mutual information between the subsets of a system. These measures promise to be useful in characterizing and understanding the functional robustness and adaptability of biological networks.