Kinetic Monte Carlo method applied to nucleic acid hairpin folding

Kinetic Monte Carlo on coarse-grained systems, such as nucleic acid secondary structure, is advantageous for being able to access behavior at long time scales, even minutes or hours. Transition rates between coarse-grained states depend upon intermediate barriers, which are not directly simulated. We propose an Arrhenius rate model and an intermediate energy model that incorporates the effects of the barrier between simulated states without enlarging the state space itself. Applying our Arrhenius rate model to DNA hairpin folding, we demonstrate improved agreement with experiment compared to the usual kinetic Monte Carlo model. Further improvement results from including rigidity of single-stranded stacking.

Figure 1

Free energies and secondary structures of the T${}_{12}$ sequence according to the SantaLucia free-energy model at $0.1M$ NaCl (see below). An almost closed unbound intermediate state forms a true barrier between open and closed states.

Figure 2

Diagram of the true barrier state between the open chain and a nucleic acid chain with a single base pair. Loop and stack energy contributions to the intermediate state are diagramed at the bottom. Note that the intersection of base pairs here is the empty set, so there is no double-stranded or dangling-end stacking contribution to the free energy of this intermediate, which was selected for being the overall rate-limiting step in the systems studied here.

Figure 3

Three types of move allowed by kinfold [16].

Figure 4

Comparison of experiment [19] and calculated rates. All lines shown are a linear regression of the correspondingly colored data. The left-hand column shows sequences T${}_n$ for various $n$. The right-hand column shows sequence A${}_{21}$ at various salt concentrations. (Top row) Experimental opening ($k_{\mathrm{open}}$) and closing ($k_{\mathrm{close}}$) rates. (Middle row) Simulations utilizing the Kawasaki rate model. (Bottom row) Equilibrium constants $K=k_{\mathrm{open}}/k_{\mathrm{close}}$ compared between experiment and calculation.

Figure 5

Simulated opening and closing rates of T${}_n$ sequences. (Left) Kawasaki rate model as previously shown in Fig. 4. (Right) Arrhenius rate model with a Kawasaki cutoff. Lines shown are linear regressions of the corresponding colored data.

Figure 6

Simulated opening and closing rates of A${}_{21}$ sequences. Lines shown are linear regression of corresponding colored data points. (Top left) Conventional Kawasaki as previously shown in Fig. 4. (Top right) Arrhenius rate model with a Kawasaki cutoff. (Bottom left) Kawasaki rate model including stacking rigidity of 0.5 kcal/mol/nt. (Bottom right) Arrhenius rate model including stacking rigidity.