We study an off-lattice protein toy model with two species of monomers interacting through modified Lennard-Jones interactions. Low energy configurations are optimized using the prunedenriched-Rosenbluth method (PERM), hitherto employed to native state searches only for off lattice models. For 2 dimensions we found states with lower energy than previously proposed putative ground states, for all chain lengths ≥ 13. This indicates that PERM has the potential to produce native states also for more realistic protein models. For d = 3, where no published ground states exist, we present some putative lowest energy states for future comparison with other methods. Predicting the structure of a protein, given its sequence of amino acids, is one of the central problems in computational biology. Since the problem is too difficult to be approached with fully realistic potentials derived from first principles, many authors have studied it in various degrees of simplifications. This involves in particular neglect of solvent water, simplifying the interactions, lumping together smaller groups of atoms, and putting everything on a discrete lattice. Among the most radically simplified models is the HP model of Dill and coworkers [1] where each amino acid is treated as a point particle on a regular (quadratic or cubic) lattice, and only two types of amino acids -hydrophobic (H) and polar (P) -are considered. Apart from the forces responsible for the connectedness of the chain, the only forces are contact forces between nearest lattice neighbours which are different for HH, HP, and PP pairs. Even in this highly simplified model it is far from trivial to predict the native state for a given amino acid sequence [2, 3, 4, 5, 6, 7] . The most efficient algorithms are either deterministic and cannot be generalized to more realistic models at all [7] , or use sequential importance sampling with re-sampling in the form of the prunedenriched-Rosenbluth method (PERM) [6] . Although it was shown that the latter can be applied also to off-lattice homopolymers at higher temperatures [8], it is not obvious that it will be efficient for off-lattice heteropolymers at the low temperatures needed for protein folding. While there are a large number of benchmark cases for lattice protein models in the literature, there exist very few simple off-lattice models with known lowest energy states that can be used as benchmarks for efficient algorithms. One such model is the so-called AB model by Stillinger et al [9, 10] which also uses only two types of monomers, called now "A" (hydrophobic) and "B"(polar). The distances between consecutive monomers along the chain are held fixed to b = 1, while non-consecutive monomers interact through a modified Lennard-Jones potential. In addition, there is an energy contribution from each angle θ i between successive bonds. More precisely, the total energy for a N monomer chain is expressed as where Here r ij is the distance between monomers i and j (with i < j). Each ζ i is either A or B, and C(ζ i , ζ j ) is +1, + 1 2 , and − 1 2 respectively, for AA, BB, and AB pairs, giving strong attraction between AA pairs, weak attraction between BB pairs, and weak repulsion between A and B. This model has been studied in several papers [9, 10, 11, 12, 13] For its 2-d version, putative ground states for various AB sequences and for various chain lengths are given in [9, 10, 13] . Similar models were also studied in [11, 12, 14, 15] , but putative ground states for these generalizations were not given at all or for very short chains only. The methods used to find low energy states of the AB model include neural networks [9] , conventional Metropolis type Monte Carlo procedures [10] , simulated tempering [11], multicanonical Monte Carlo [12], biologically motivated methods [13, 15] , and molecular dynamics [14] . In all cases the stochastic minimization can only lead to some state in the neighbourhood of a local (and hopefully also global) minimum. A greedy deterministic method such as conjugate gradient descent is subsequently applied to reach the minimum itself. It is the purpose of the present paper to see whether PERM can be efficient for energy minimization in the AB model. In particular we shall use the new variant of PERM presented in [6] . We shall restrict ourselves to the subclass of "Fibonacci sequences" studied also in [10] , defined recursively by S 0 = A, S 1 = B, S i+1 = S i−1 * S i . Here * is the concatenation operator. The first few sequences are S 2 = AB, S 3 = BAB, S 4 = ABBAB, etc. They have lengths given by N i+1 = N i−1 + N i , i.e
In REF the low energy configurations are optimized using the pruned-enriched-Rosenbluth method (PERM).
Structure optimization in an off-lattice protein model
Phys. Rev. E 68, 037703 (2003)
Physics Biology Medicine