A combinatorial algorithm for effective generation of long maximally compact lattice chains

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Abstract
We investigate the problem of generation of maximally compact lattice chains which are useful in understanding folding of model proteins. The term, maximally compact chain, refers to a lattice self-avoiding walk that visits every lattice site. Generation of a representative sample of compact conformations is extremely difficult by conventional simulation methods such as static growth methods or dynamic Monte Carlo techniques. Growing a random walk is ineffective for generating long walks in a compact shape because a large number of walks are rejected due to overlap (attrition). In the interest of an unbiased sample, one needs to enumerate all possible compact conformations that are realizable or produce a representative sample, the former of which is intractable for long chains. In this paper a method is proposed for generation of compact chains on a lattice based on a mathematical programming approach. The method, which we refer to as the Hamiltonian path generation method, generates a random sample of lattice filling self-avoiding walks. A detailed description of a randomized generation algorithm is presented, which is effective for producing a static sample of compact lattice chains. There is a statistical evidence of fair generation of conformations from the conformational space using this scheme. This method generates a compact conformation on a 60×60×60 cubic lattice in forty minutes on a Sparc-2 workstation.