Steric Constraint Networks and Protein Folding The Levinthal's paradox of protein folding basically contrasts the time that a uniform random sampling of the configuration space for an N-monomer protein would take (which is on the order of 10^N and would take astronomical times even for peptides) with the actual observed times for folding from a denaturated state to the native state (which ranges from nano-seconds to minutes). Recent studies have taken a networks approach to protein folding by identifying secondary structures with nodes, and folding through small-energy barriers between two such configurations as links (Rao and Caflisch, 2004). Molecular Dynamics simulations show that the protein folding network is a scale-free graph (Albert and Barabasi, 1999) with an exponent of -2, that seems to be independent on the ordering of the monomers in the chain. Here we introduce the notion of gradient flow networks as directed substructures on graphs generated by following the gradients of a scalar field distributed on the nodes of this graph. We then show that in general these gradient networks have a scale-free degree distribution, with an exponent that depends on the correlations between the scalars (energies) at nodes and local graph properties such as degree and clustering. This formalism then allows us to give a simple resolution to the Levinthal paradox and recover the measurements obtained via MD simulations.
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