How global DNA unwinding causes non-uniform stress distribution and melting of DNA.

DNA unwinding is an important process that controls binding of proteins, gene expression and melting of double-stranded DNA. In a series of all-atom MD simulations on two DNA molecules containing a transcription start TATA-box sequence we demonstrate that application of a global restraint on the DNA twisting dramatically changes the coupling between helical parameters and the distribution of deformation energy along the sequence. Whereas only short range nearest-neighbor coupling is observed in the relaxed case, long-range coupling is induced in the globally restrained case. With increased overall unwinding the elastic deformation energy is strongly non-uniformly distributed resulting ultimately in a local melting transition of only the TATA box segment during the simulations. The deformation energy tends to be stored more in cytidine/guanine rich regions associated with a change in conformational substate distribution. Upon TATA box melting the deformation energy is largely absorbed by the melting bubble with the rest of the sequences relaxing back to near B-form. The simulations allow us to characterize the structural changes and the propagation of the elastic energy but also to calculate the associated free energy change upon DNA unwinding up to DNA melting. Finally, we design an Ising model for predicting the local melting transition based on empirical parameters. The direct comparison with the atomistic MD simulations indicates a remarkably good agreement for the predicted necessary torsional stress to induce a melting transition, for the position and length of the melted region and for the calculated associated free energy change between both approaches.


List of Figures
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S1: Free Energy profiles obtained from subintervals of the Umbrella Sampling simulations (for the at-sequence).

Convergence of Umbrella Sampling free energy simulations
The convergence of our Umbrella Sampling Simulations was checked through splitting the trajectories in subintervals (corresponding to the first 25 %, 50 %, 75 % or 100 % of the Umbrella sampling trajectories). The resulting PMFs indicate only little changes of less than 0.5 kcal/mol especially between using 75 % or 100 % of the Umbrella sampling data ( Figure S1, S2).

Calculation of Stiffness and Covariance Matrices
Computed stiffness matrices are shown in Figure S3 and S4. Covariance matrices under relaxed and restrained cases are depicted in Figure S5-S8.

Elastic Energy Profiles
For the unrestrained simulation as wells as every Umbrella window we calculated the average elastic energy of all central 43 base-pair steps by using the harmonic model as explained in the previous section. The energy was projected on every step by the sum of its diagonal and half of the coupling terms. As shown in Figure S9 and S10, energy equipartitioning is accurately obtained in the unrestrained simulation. The profiles of the Umbrella Sampling windows are shown in Figurere S20-S31.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S2: Free Energy profiles obtained from subintervals of the Umbrella Sampling simulations (for the gc-sequence). Figure S3: Computed stiffness-matrix for at-sequence, in units of kcal mol·deg 2 . Note, the matrix is presented with the twist parameter of each base pair step along the sequence, followed by roll and tilt for each base pair step on the x-and y-axis. Each colored dot represents a stiffness force constant (color-coded and in units of kcal mol·deg 2 ). Only diagonal stiffness parameters and nearestneighbor parameters deviate significantly from zero whereas the white areas represent near-zero stiffness entries (coupling beyond nearest neighbors is weak).
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S4: Computed stiffness-matrix for gc-sequence, in units of kcal mol·deg 2 . The order of matrix entries is the same as explained in legend of Figure S3. How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S12: Average elastic energy (black line) for all central 43 base-pair steps for the at-sequence simulated in the second Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S13: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the third Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S14: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the fourth Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S15: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 5th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S16: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 6th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S17: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 7th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S18: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 8th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S19: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 9th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S20: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 10th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S21: Average elastic energy for all central 43 base-pair steps for the at-sequence simulated in the 11th Umbrella window. The plot was generated in the same way as Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S22: Average elastic energy for all central 43 base-pair steps for the gcsequence simulated in the first Umbrella window (minimal torsional stress). Changes in the backbone population are given as relative changes.
Cyan bars indicate increase in BI, orange bars increase in BII population. Bottom colorbar indicates local twist-stiffnes, see Figure S11.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S23: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the second Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S24: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the third Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S25: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the fourth Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S26: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the 5th Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S27: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the 6th Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S28: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the 7th Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S29: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the 8th Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S30: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the 9th Umbrella window. The plot was generated in the same way as Figure S22.
How global DNA unwinding causes non-uniform stress distribution and melting of DNA Figure S31: Average elastic energy for all central 43 base-pair steps for the gc-sequence simulated in the 10th Umbrella window. The plot was generated in the same way as Figure S22. How global DNA unwinding causes non-uniform stress distribution and melting of DNA

Ising model
In our implementation of the Ising model, we considered only macrostates with at most one denaturation bubble. Thereby, we iterate through all steps of the sequence as bubble-initiation steps and consider various bubble-lengths. We considered bubble sizes smaller than 6 molten base-pairs. Expected bubble-sizes for our system are quite short, ∼ 1 − 2bp ( Figure S32).

Calculation of Geometric parameters through Rigid-Body Transformation
In order to quantify TATA-Box geometries, we set-up a new protocol, which allows us to calculate all base-pair and base-pair-step parameters. The major differences to other protocols (such as Curves+) are: • All rotational parameters are Euler angles in our method (including also opening and buckle).
• We can easily obtain a reference-axis system for every base and base-pair. This also enables us to measure bending as the angle of the axis-vectors, which are orthogonal to the respective base-pairs.
• No specifications are required. Our script requires only topology-and trajectory files as input.
In the following, we outline our protocol: First, we span a local reference system on every base. The local x-vector is the glycosidic bond. We then define a preliminar vector y p as the vector-difference between the two base-carbon atoms next to the glycosidic nitrogen (e.g. C2, C6 in case of thymine). In the Crick-strand we define this vector in opposite direction. The local z-vector is obtained by: z = x ×y p . We then calculate the actual y-vector through y=x × z. We place this axis-system, A, on the corresponding C1' atom, which we denote as anchor-points. Second, we calculate the six intra base-pair parameters. The three translational parameters are given by the vectors connecting the anchor-points of paired bases. In a next step, we calculate the rotation matrix R, which transforms the Watson-Axis system into the Crick-Axis system: R = A Crick A T W atson . From this matrix, the three intra base-pair parameters are obtained as Euler-angles. Third, we calculate the six base-pair-step parameters. For this, we first determine a mid-axis system for every base-pair by performing half-translation and half-rotation between the axis-systems of paired bases. The base-pair-step parameters follow from rigid-body-transformations between successive mid-axis systems analog to the calculation of intra base-pair parameters. Bending angles are calculated as the angle between the z-vectors of the mid-axis system of the chosen base-pairs. How global DNA unwinding causes non-uniform stress distribution and melting of DNA