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Of special interest in this regard are biological systems that display complex emergent behavior, for example, through long-range chemical signaling among a collection of cells in social amoebas (Goldbeter, 1996; Brännström and Dieckmann, 2005; Schaap et al., 2006), cooperation among consortia of host organisms and symbionts (Moran, 2007, Chapter 9, this volume), or colonies of social insects (Solé and Goodwin, 2000; Camazine et al., 2001; Strassmann and Queller, 2007, Chapter 8, this volume). We propose that functional information can be applied, at least in principle, to any such emergent system that has the ability to perform a function.

Many emergent systems can be analyzed in terms of their ability to dissipate energy or maximize entropy production (Nicolis and Progogine, 1977; Lorenz, 2003; Whitfield, 2005; Emanuel, 2006). For example, consider the functional information of an assemblage of sand grains subjected to a steady flow of wind or water (e.g., Bagnold, 1988; Hansen et al., 2001). The formation of periodic sand dunes or ripples serves to initiate turbulent flow and thus increase energy dissipation. Functional information of the system can thus be measured as the fraction of all possible sand configurations, F(Ex), that achieve at least the corresponding energy dissipation, Ex. Such a problem might be analyzed with Monte Carlo simulations of numerous gravitationally stable sand configurations. The analytical challenge remains to determine the degree of function of a statistically significant random fraction of all possible configurations of the system so that the relationship between I(Ex) and Ex can be deduced.

CONCLUSIONS

A complexity metric is of little utility unless its conceptual framework and predictive power result in a deeper understanding of the behavior of complex systems. Analysis of complex systems in terms of functional information reveals several characteristics that are important in understanding the behavior of systems composed of many interacting agents. Letter sequences, Avida genomes and biopolymers all display degrees of functions that are not attainable with individual agents (a single letter, machine instruction, or RNA nucleotide, respectively). In all three cases, highly functional configurations comprise only a small fraction of all possible sequences. Furthermore, these three examples reveal that several discrete classes of functional configurations exist, a situation that can lead to distinctive step features in plots of information versus function.

The functional information formalism may also point to key factors in the origin and emergence of biocomplexity. In particular, functional information quantifies the probability that, for a particular system, a configuration with a specified degree of function will emerge. Furthermore, analysis of the relationship between information and function may reveal



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