We deal with a problem of reconstruction of approximate 3D structure of an RNA molecule from its secondary structure. The limited applicability to oligonucleotides of current experimental techniques for determination of 3D molecular conformations justifies the development of alternative computational methods. We propose a new mathematical model as well as its computer implementation which makes it possible to reconstruct large-scale structures of RNA molecules [1].

An RNA molecule is treated as a set of structural elements modelled by thin elastic rods. With adequate choice of parameters of the rod its shape approximates the large-scale structure of the corresponding element of the RNA molecule. The similar rod model has been widely used for investigation of 3D conformations of a DNA, e.g., [2]. We apply the rod model to different elements, both to double helices of stems and to loops of various type (hairpin, interior and bulge). Both ends of the loop are considered fixed. Their relative position and orientation are computed from the stem geometry. The 3D configuration of the continuous loop is governed by the system of ODEs with respect to moments, forces and unit vectors along the principal axes of bending and torsion. The search for a solution of this system is equivalent to minimization of the integral elastic energy of the rod taking into account both end constraints. The corresponding BVP is solved numerically by the shooting method. Its solution depends on elastic coefficients of the model and on the relaxed state. To obtain solutions for various values of parameters a parameter continuation approach has been applied.

We developed a numerical procedure for computation of 3D conformations of elements of given RNA molecule and for assembling the whole molecule from these elements. In particular, this gives a means for observation of the loop shape variation as the parameters are changed. Corresponding energy values are also obtained for a variety of parameters. The approach developed is used for identification of effective elastic coefficients and average characteristics of the relaxed RNA chain by using known free energy data [3]. Large-scale spatial structure of several real RNA molecules was reconstructed, including arginine tRNA of Caenorhabditis elegans, yeast Phenylalanine tRNA, human 5S ribosomal RNA, and Rev responsive element.

The 3D shape of an RNA folding is complex. Knowing spatial conformations of single-stranded elements as well as shapes of double helix stems one may approximate the whole molecule folding. The continuous thin elastic model is suggested to model 3D structure of all different elements of RNA. This approach seems to be especially promising for reconstruction of long RNA molecules.

This research was supported by Russian Foundation for Basic Research Grants N 96-01-01192 and N 96-15-97229.

1. E.I.Kugushev, E.E.Pirogova, E.L.Starostin. Mathematical model of formation of three-dimensional structure of RNA. Moscow, Keldysh Inst. of Appl. Math., Preprint No. 77, 1997, 24~p. [in Russian].
2. E.L.Starostin, Three-dimensional shapes of looped DNA, Meccanica (Intern. J. of Italian Association of Theoretical and Applied Mechanics) 31, 235--271 (1996).
3. S.M.Freier, R.Kierzek, J.A.Jaeger, N.Sugimoto, M.H.Caruthers, T.Neilson, D.H.Turner, Improved free-energy parameters for predictions of RNA duplex stability, Proc. Natl. Acad. Sci. USA 8, 9373--9377 (1986).