Predicting secondary structural folding kinetics for nucleic acids

Abstract

We report a new computational approach to the prediction of RNA secondary structure folding kinetics. In this approach, each elementary kinetic step is represented as the transformation between two secondary structures that differ by a helix. Based on the free energy landscape analysis, we identify three types of dominant pathways and the rate constants for the kinetic steps: 1), formation; 2), disruption of a helix stem; and 3), helix formation with concomitant partial melting of a competing (incompatible) helix. The third pathway, termed the tunneling pathway, is the low-barrier dominant pathway for the conversion between two incompatible helices. Comparisons with experimental data indicate that this new method is quite reliable in predicting the kinetics for RNA secondary structural folding and structural rearrangements. The approach presented here may provide a robust first step for further systematic development of a predictive theory for the folding kinetics for large RNAs.

Figure 1

The relationship between two helices A [a₁, a₂, a₃, a₄] and B [b₁, b₂, b₃, b₄] can be classified into three types: (a) compatible, (b) partially compatible, and (c) incompatible. For the compatible helices, the formation of helix B does not require unfolding of helix A. For incompatible and partially compatible (incompatible) helices, the formation of helix B requires complete and partial unfolding of helix A, respectively.

Figure 2

(a) A schematic free energy landscape for a pathway from the open chain to a helix. Transition from state 1 to state 2 is the formation of the loop-closing base stack. The values k_f and k_b are calculated from Eqs. 2 and 3. (b) Multiple pathways for the formation of a helix after the first (nucleation) stack formed.

Figure 3

(a) The free energy landscape for the transition between two incompatible helices. Helices A and B contain nucleotides (basepairs) that are incompatible with each other. X_i is the state where helix A is partially melted with the some of the incompatible basepairs disrupted and helix B is partially formed. U is the open state. (b) A schematic free energy profile for the arm-by-arm exchange process. The values k₁ and k′_n are calculated from Eq. 7; k′₁, k₂, ⋯, k_n are calculated from Eq. 2.

Figure 4

The free energy landscape of the two-arm-by-two-arm exchange pathway. State A contains helices h₁ and h₂, state B has a few stacks of helices h₁ and h₂ melted and one stack of helix h₃ and h₄ each formed, state C represents a state where h₁ and h₂ are partially unfolded, state D is the state in which only one stack of helix h₁ and h₂ each remains, and state E is the state with helices h₃ and h₄.

Figure 5

The populational kinetics for the folding of the 27-nt sequence: 5′AUAGGUUAUAUAUCACGUAUAGCCUAU 3′. The dashed lines are from the exact master equation with the complete conformational ensemble; the solid lines are from our helix-based kinetic model.

Figure 6

The reduced states and the transition network of the 82-nt DNA sequence. State 6 is the native structure. The solid line denotes the deletion-addition pathway, the dashed line denotes the arm-by-arm exchange pathway, and the thick dashed line denotes the two-arm-by- two-arm exchange pathway.

Figure 7

(a) Populational kinetics of the open state, intermediate state, and the native state of the 87-nt sequence. (b) The net populational fluxes among the seven states in Fig. 6. 1 → 3 and 1 → 5, 3 → 7, and 5 → 7, coincide with each other. The net flux between other states are very small and thus not shown in the figure.