Abstract: Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single-stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base-pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality. ; © 2007 Society for Industrial and Applied Mathematics. Received by the editors January 27, 2006; accepted for publication (in revised form) March 30, 2006; published electronically January 30, 2007. The first and second authors contributed equally to this work. This work was supported by grants NSF-CNS-PECASE-0093486, NSF-EIA-0113443, NSF-DMS-0506468 (IMAG), NSF-ACI-0204932, and NSF-CCF-CAREER-0448835, the Charles Lee Powell Foundation, and the Ralph M. Parsons Foundation. We wish to thank Z.-G. Wang, M. Cook, and L.B. Pierce for helpful discussions during the course of the work. ; Published - DIRsiamrev07.pdf
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