We are studying the stability of RNA folding intermediates using a motion planning approach. We have invented a new geometric sampler for coarse-grained RNA models that can generate low-energy RNA conformations/folding-intermediates which are free from steric clashes and maintain their helical rigidity and tertiary contacts (Figures 2 & 3). This new geometric sampler is used to construct a simplified representation of the RNA conformation space using Probabilistic Roadmaps (PRMs).

Comprised of only four bases, RNA biopolymers adopt surprisingly complex three-dimensional structures and are involved in a wide variety of cellular functions. The observed structural diversity in RNA is a consequence of the diverse hydrogen bonding potential of the four bases and a flexible sugar-phosphate backbone. Most large RNAs are comprised of multiple helices connected by semi-flexible junctions and stabilized by specific tertiary contacts.

A tertiary contact is defined as a long-range interaction between one or several nucleotides in the RNA. These contacts are mediated by specific three-dimensional arrangements of nucleotides that often occur multiple times in RNA structures. Tertiary contacts are usually identified in RNA structures by co-variation sequence analysis and confirmed by x-ray crystallography. For the L-21 T. thermophila group I intron five tertiary contacts have been identified, \alpha, \beta, \gamma, \delta, and \varepsilon in Figure 1-A. Three of these contacts (\gamma, \delta, and \varepsilon) have been confirmed by X-ray crystallography as illustrated in Figure 1-B.

We define connectivity as the extent to which the formation of one contact affects the stability of another. This term is not to be confounded with macroscopic cooperativity, a thermodynamic quantity generally associated with protein kinetics . In protein biopolymers highly connected networks of tertiary contacts between amino acids, along with a hydrophobic collapse, stabilize the native state. The absence of hydrophobic residues in RNA, and the comparatively small diversity of biopolymer residues (four nucleotides compared to 20 Amino Acids) result in a much sparser tertiary contact map for RNA compared to proteins.

Tertiary contacts stabilize the fold of and RNA through the formation of specific atomic contacts (Hydrogen bonds, inter-helical base stacking). Site-specific mutations can easily be introduced that eliminate a tertiary contact in an RNA experimentally. In collaboration with the Herschlag lab, we have undertaken a systematic analysis of tertiary contact formation in the L-21 T. thermophila group I intron. We have an undertaken a series of experiments in which we systematically mutate one to four tertiary contacts in the Tetrahymena ribozyme (Figure 1). We then monitor the effect of the mutation on the relative stability of the intact tertiary contacts by hydroxyl radical footprinting. This analysis has revealed a rather complex network of connectivity between the tertiary contacts that is not easily explained by a simple structural analysis of the RNA.

One outstanding question in the RNA folding literature is the role of the native state topology of the molecule in determining stability and folding pathway. Specifically, how the molecular architecture of the RNA (rigid A-form helices connected by junctions and tertiary) affects its folding dynamics. The relationship between the RNA tertiary connectivity and its dynamics is complex, and given the large size of RNA molecules, all atom molecular dynamics may never be able to address this question. We therefore set out to develop a very simple model of RNA structure to begin to address the issues of topology in RNA folding.

We chose as our initial system the P4P6 subdomain of the L-21 T. thermophila group I intron. In Figure 1, this corresponds to nucleotides 104-260. This domain is comprised of 13 helices which pack co-axially, held together by a flexible junction and two tertiary contacts (\delta and \varepsilon). Recent single molecule experiments on this domain have revealed that the cooperativity of the two tertiary contacts is approximately 3 kcal/mol (B. Sattin, personal communication). Therefore, mutating one contact destabilizes the other by approximately 3 kcal/mol. The fundamental question these experiments cannot answer is the origin of these 3 kcal/mol. It is likely that most of the destabilization is due to entropic effects related to the number of conformations available to the system when one tertiary contact is mutated.

We therefore decided to attempt a simulation of the system making the following assumptions.

• Each helix is assumed to be a rigid body attached to its neighboring helices by flexible joints which use the NasT angle and dihedral constraints.
• Junctions are treated as a string of beads using the same repulsive term as that used in NasT.
• Tertiary contacts are treated as fixed length constraints on the rigid body system.
• A motion planning algorithm (Probabilistic Roadmaps) is used to generate alternative, collision free conformations of the RNA given the set of constraints on the system. A new geometric sampler, based on the one that we used to generate protein loop decoys (http://ai.stanford.edu/~mitul/loop/loop.html), is used by the PRM planner to sample low-energy RNA conformations, MAINTAINING helical rigidity and desired tertiary contacts.
• The conformational entropy of the system is related to the relative volume of motion of the individual bodies in the system.

Our approach is to use motion planning to generate alternative conformations of the P4P6 subdomain that are sterically correct given one or both tertiary constraints. We then measure for each helix the total spatial volume given all the conformations generated (in this case, 15,000). Four models were built: In the reference model (Model R, Figure 4), no tertiary constraints were applied and the Volumes of the helices were measured. Two models were built with either the \delta or the \varepsilon constraint (E and D models), and the native model had both constraints (N). We then measured the relative accessible volumes of the helices and report these values in Figure 4.

In Figure 4, the fractional accessible volumes of each helix are shown for the R, E, D, and N models respectively. The values are normalized such that the R (both tertiary contacts missing) model's volumes are always 1.0. As expected, the addition of the tertiary contacts constrains the system and decreases the volumes. The most constrained molecule is in fact the N model. These results therefore demonstrate that we can create a coarse-grained model of RNA and measure the conformational entropy changes due the presence and absence of tertiary contacts. These differences are also spatially specific, as the tertiary contacts affect the relative accessible volumes of the helices differently.

These very preliminary results are promising in terms of using robotics algorithms to answer fundamental questions about RNA topology. Possible future directions will include:

• Establishing the correct relationship between accessible volume and conformational entropy.
• Evaluating the effects of tertiary contact mutation on the entire L-21 T. thermophila group I intron (5 contacts) and relating the changes to the experimentally observed changes in stability.
• Using these robotics algorithms instead of MD for RNA simulations in general.