FIG. 3: (a) The bubble size distribution P$_{b}$ ( n ) at 310 K for contour length L = 55, 100, 150, and 300 bp. (b) The fork size distribution P$_{f}$ ( n ) at 310 K for the same lengths. where α is the statistical factor for a loop formation and Δ is the average energy in k$_{B}$T to unbind a bp. These factors are independent of the length and given by those for a long length, α = 1 . 7, Δ = 0 . 33, at 310 K [8]. The average size of the bubbles for various contour lengths is 1.7 bp at 310 K. This is in agreement with the Peyrard-Bishop-Dauxois model simulation [15]. The length-independence of their distribution and small average size implies that the thermal bubbles are transiently excited due to short-range elastic interaction in unconstrained DNA. This is to be contrasted with the dsDNA under mechanical constraints, where the average bubble size is found to be much larger depending on the (negative) twist as well as the contour length [16]. The distribution of n bp open at a free end is given by where β = 1 . 1, Δ = 0 . 33 are also independent of the contour length within the errors. The average size of the forks is 1.9 bp. Since L$_{ss}$ is 4 nm in these simulations, they are semiflexible forks with β much larger than that of the flexible forks [17] Last, we obtain the N$_{b}$ ( m ), the relative probability of finding m bubbles simultaneously in a DNA of given length. The result obtained from simulation is presented in Fig. 4, showing that the single bubble occurrence becomes predominant as the DNA length gets shorter. This is mainly due to the large energy cost of bubble initiation; namely, once a bubble is formed, increasing its size is energetically more favorable than opening a new bubble P$_{f}$ ( n ) ∼ e − n Δ n β , (7) P$_{b}$ ( n ) ∼ e − n Δ n α , (6) 6