Taming combinatorial explosion

Assembling objects from building blocks by means of predefined combination rules leads to combinatorial explosions. Indeed, it does not matter how many classes of building blocks or alternatives of combination rules are given, provided one of both is two or larger, because then the number of possible objects commonly increases exponentially with the number of elements and soon exceeds the realizations that can be sustained by taking together all available resources (for a few simple examples, see Fig. 1). Problems of this kind are encountered in biophysical chemistry when biopolymer molecules are built from several classes of monomers, in combinatorial chemistry when new molecules are formed through combination of several reactions, or in molecular genetics when regulation and control networks are considered. Thus, combinatorial explosion is a universal threat to biopolymer sequences and structures as well as reaction and controlling networks. Examples are known from biology, in particular metabolic, genetic, developmental, signaling, and neural networks. If this is true, how, then, can organized objects originate? Must not all processes that are not regulated externally end up in a highly diverse mess of molecular species, each one at best realized only in a few molecules? The frequently given answers invoke self-organization as a (universal) principle introducing order into diverse manifolds. In general, self-organization requires non-equilibrium conditions and some kind of self-enhancement. Both criteria are often fulfilled and commonly met in biology under realistic conditions. The main questions, nevertheless, remain: Which are the chemical driving forces or reaction mechanisms that shape organized networks: for example, those we see in nature? What limits the size, the diversity, and the complexity of chemical reaction networks or, in particular, what determines the properties of the ones that operate in living organisms? Figure 1 Examples of combinatorial explosion and a catalytic cycle reducing the numbers of possible objects. …