Predicting RNA folding thermodynamics with a reduced chain representation model

Abstract

Based on the virtual bond representation for the nucleotide backbone, we develop a reduced conformational model for RNA. We use the experimentally measured atomic coordinates to model the helices and use the self-avoiding walks in a diamond lattice to model the loop conformations. The atomic coordinates of the helices and the lattice representation for the loops are matched at the loop-helix junction, where steric viability is accounted for. Unlike the previous simplified lattice-based models, the present virtual bond model can account for the atomic details of realistic three-dimensional RNA structures. Based on the model, we develop a statistical mechanical theory for RNA folding energy landscapes and folding thermodynamics. Tests against experiments show that the theory can give much more improved predictions for the native structures, the thermal denaturation curves, and the equilibrium folding/unfolding pathways than the previous models. The application of the model to the P5abc region of Tetrahymena group I ribozyme reveals the misfolded intermediates as well as the native-like intermediates in the equilibrium folding process. Moreover, based on the free energy landscape analysis for each and every loop mutation, the model predicts five lethal mutations that can completely alter the free energy landscape and the folding stability of the molecule.

FIGURE 1.

(A) The virtual bond scheme for nucleotide backbone (Olson 1980). (B) The bond angles (β_C, β_P) and the torsional angles (η, θ) for the virtual bonds. (C) A vector model used to determine the atomic coordinates from the torsional angles for the virtual bonds in B.

FIGURE 2.

(A) The coordinate system for a base pair (U21-A16) in the helix. s and s′ are the pairing strands. The gray and red colors denote the C₄ and the P atom in the helix, respectively. The magenta color denotes the loop region. (B) Helix–bulge loop junction. The deleted bonds (×) show the fixed configuration of the virtual bonds in the A-form helix without the bulge loop. These bonds become flexible in the bulge loop conformation (see the blue bonds). The blue and magenta colors denote the bonds in the bulge loop. The P and C₄ coordinates in the helix are from the crystal structure of r(CGUAC)dG sequences (Biswas et al. 1998).

FIGURE 3.

The comparison between the experimentally measured loop entropies (+) and the calculated ones (Δ) for (A) hairpin loops, (B) internal loops, and (C) bulge loops. The experimental results for internal and bulge loops are from Serra and Turner (1995) in 1 M NaCl solution. For hairpin loops with loop length >3, we use the parameters in Serra et al. (1997).

FIGURE 4.

(A) The secondary structure of Tar RNA. (B) The configuration for a U-A-U triple base. (C) The entropic difference (S/k_B) for bulge loops with and without the U-A-U triple base.

FIGURE 5.

The density plot for the base-pairing probability and the predicted structure for E. coli 5S rRNA. Circles indicate single-stranded nucleotides as indicated by enzymatic cleavage (Speek and Lind 1982). The row and column indexes in the density plot are the nucleotides along the sequence.

FIGURE 6.

The predicted native structure and the theory–experiment comparisons for the heat capacity melting curves for RNA secondary structures: (A) 72 RNA, (B) 72-C RNA, (C) 72-14 RNA, and (D) B RNA. The experimental melting curves (A–C) are from Gluick and Draper (1994) and (D) from Laing and Draper (1994).

FIGURE 7.

The density plots for the base-pairing probabilities and the stable structures for the wild-type 72 RNA at different temperatures. An intermediate state appears at 45°C. The row and column indexes in the density plots denote the nucleotides along the sequence.

FIGURE 8.

The density plot for the base-pairing probabilities and the stable structures for 72-14 RNA at different temperatures. The row and column indexes in the density plots are the nucleotides along the sequence.

FIGURE 9.

(a) The predicted heat capacity melting curve in 1 M NaCl for the P5abc domain of the Tetrahymena group I ribozyme. (b–d) The density plots for the free energy landscapes F(n, nn) and the stable structures at different temperatures for P5abc. F(n, nn) is the free energy for conformations with n native base pairs and nn non-native base pairs, where the native state is the structure shown in b.

FIGURE 10.

(A) The density plot for the change in the free energy landscape ΔF at different temperatures for different mutations in loops L5a, L5b, and L5c, respectively. The top lines show the color scale. (B) The variations of thermal stabilities at T = 0° C for the wild-type sequence and different mutations in loops L5a, L5b, and L5c, respectively. Loop 5b contains no hotspots of which the mutations can cause drastic changes in the native structure and the stability.

FIGURE 11.

A closed conformation with the closing stack connected to a loop (A ) or to a stack (B ). (C ) The four open conformation types (L , M , R, and LR ). The closed conformation in A is formed from the open conformation in C through the closure of the unstacked loop of length l in A. ( D) The partition function for L-type conformations for a chain froma to b can be computed as the sum of the partition function for a shorter chain from a to b − 1.