Functional information and the emergence of biocomplexity

Abstract

Complex emergent systems of many interacting components, including complex biological systems, have the potential to perform quantifiable functions. Accordingly, we define "functional information," I(E(x)), as a measure of system complexity. For a given system and function, x (e.g., a folded RNA sequence that binds to GTP), and degree of function, E(x) (e.g., the RNA-GTP binding energy), I(E(x)) = -log(2)[F(E(x))], where F(E(x)) is the fraction of all possible configurations of the system that possess a degree of function > or = E(x). Functional information, which we illustrate with letter sequences, artificial life, and biopolymers, thus represents the probability that an arbitrary configuration of a system will achieve a specific function to a specified degree. In each case we observe evidence for several distinct solutions with different maximum degrees of function, features that lead to steps in plots of information versus degree of function.

Fig. 1.

Distribution of the not/and (NAND) function in 300-line Avida genomes in a randomly generated sample of 10⁷ genomes. The degree of function, E, is the number of times NAND is executed by the genome, whereas functional information, I (in bits), is −log₂ of the fraction of all sequences that achieves at least that degree of function, F(E). Note the discontinuities, which are a recurrent feature in these experiments.

Fig. 2.

The frequency of the ADD function in 100-, 200-, 300-, and 500-line linear Avida genomes in randomly generated samples of 10⁶ genomes. Degree of function, E, is the number of times the ADD function is executed by the genome, whereas functional information, I (in bits), is −log₂ of the fraction of all sequences that achieves at least that degree of function, F(E). Note that maximum E increases with genome length.

Fig. 3.

Schematic representation of four discrete functional classes, or “islands,” of solutions that display function. The vertical axis is degree of function, E, whereas the horizontal plane represents a two-dimensional projection in sequence space. The number of sequences with degree of function ≥E corresponds to the area intersected by the horizontal plane at that height along the E axis. Increasing E above the heights of the flat-topped islands A and B will result in discontinuities in the function E versus I, as illustrated in Figs. 1 and 2. Island C is a cone-shaped distribution, and island D represents a discrete solution of the type that might not be discovered in random sampling experiments.

Fig. 4.

I(E) versus E for the statistically random system, where E is the number of times the digit 1 appears at least that many times in a sequence of 100 digits. This statistically random case is not stepped, in contrast to the topology of Avida genomes.