LAC--DNA project.

Elastic rod modelling of DNA connecting loop in the LAC repressor structure

Brief Description :

Primary goal: Things to take into account: The project extensions:
  1. To complete the modelling replacing the DNA-rod with the all-atom model of DNA and performing minimization/molecular dynamics to stabilize the structure (simulated annealing too, if it proves to be necessary).
  2. Solve the problem taking into account the DNA internal curvature and sequence-dependency of the elastic moduli in the connecting loop.
  3. To re-model the structure of the loop replacing part of it with the DNA-CAP protein complex.
  4. To extend the elastic rod representation on the protein-bound segments; estimate the force imposed on the DNA by the protein and the interaction energy; compare to the experimental data and the all-atom model results.
  5. Consider the third operator sequence of LAC repressor (downstream from the former two ones); reproduce the DNA sequence when the protein is bound to that site.
  6. To correct the structure of the protein--DNA complex taking into account the force which the DNA exerts on the protein (use NAMD).

Project Personnel:

Project Log:

Foreword: The shape of the loop is to be obtained by solving the 13-th order system of differential equations describing the elastic rod with internal twist. The rod is characterized by its elastic moduli alpha and beta (for bending in the directions of the two principal axes of the rod's cross-section). We have a boundary value problem very sencitive to the boundary conditions. Hence, the solution is going to be obtained by a smooth changing from a primitive shape of the rod (flat closed loop) towards the desired conditions. The starting loop is located in the XY-plane.

Week Goals Progress
Feb. 24 - Mar. 2
  1. To move the ends of the loop by 0.3 units along the Z-axis from each other (the contour length of the rod being equal to 1). This is approximately the distance between the ends of the DNA loop in the complex with LAC repressor.
  2. To bend the ends of the DNA, rotating the normal vectors to the lateral cross-sections by +45° (s=0) and -45° (s=1), approximately being the real values too. Before using the real values, we'd try some straightforward but close test values first.

Step one was quickly and successfully completed. The ends of the "DNA" were moved from each other. After some debugging, ends were successfully bent too.
Mar. 3 - Mar. 9
  1. Transform the solution obtained to the solution for alpha not equal to beta. In classical interpretation, the non-uniform bending accounts for the non-circularity of the cross-section.
  2. Finally, the ends are not only bent, but also twisted relative to each other. The twist of ½pi is to be put in.
  3. Do these two steps in different order (see "Progress"). First twist, then change alpha-to-beta ratio.

The moduli were succesfully changed to their values in the DNA (alpha of 0.55, beta of 1.3). But then we ran into problems. The "non-circular" solution proved to be very sensitive to the changes of the twist, and did not converge at some point. Increasing the number of the step did not help. An extensive debugging showed that everything seemed to be correct. Then a decision was made to move one step back and try to introduce the twist first, since the the rod with an intrinsic twist is very sencitive to this parameter. This was done, and the "circular" solution accepted the additional twist. However, even the circular solution did not converge until the required precision was dropped to 1e-3 from the initial value of 1e-5.

Mar. 10 - Mar. 16
  1. Improve the precision of the solution obtained on the current step.
  2. Make the last step, driving alpha and beta elastic moduli to their values for real DNA (alpha about 0.55, beta about 1.3).

The precision of the final solution for the twist of ½pi and alpha = beta = 0.5 was succesfully raised back to the value of 1e-5. Then the attempt to change the moduli was undertaken. Alas, it did not work. When about 1/5 of the required path was passed, the solution stopped converging. Nothing helped. The only thing remained was to ascribe this unstable behaviour to some inner symmetry of the solution for the chosen values of twist and bend. The change of the alpha-to-beta ratio breaks the symmetry and yields to the branching of the solution (like, for example, bending to the right or to the left). If this is true, at the point of bifurcation of the two branches, the non-convergence may happen. Hence, we decided to abort the "test" runs, to obtain the bend/twist values for the real DNA, that probably lack such symmetry, and to try to obtain solution for these values.

Mar. 17 - Mar. 23
  1. Obtain the boundary conditions from the real DNA.
  2. Try to obtain the solution with them.

The parameters of the real DNA are:
  • internal twist of 14.61538461538461 pi,
  • end-to-end distance of about 0.17,
  • polar angles of the normal vector at s=1 equal to psi=0.40 rad, theta=1.75 rad (normal vector at s=0 being defined as the z-axis),
  • imposed (apparent) twist of pi-0.034 rad,
  • alpha = 0.55, beta = 1.3.
All the solution-obtaining cycle (move ends -- bend -- twist -- change alpha/beta) was repeated for these values. And this was a real success! At all the steps the solution was very well converging, it was even possible to change the bend and twist simultaneously. One thing to note was that the rotation of the cross-section mattered: to obtain the solution with one superhelical loop one must impose the apparent twist shown above, i.e. rotate the last cross-section counterclockwise; the attepmt to rotate it clockwise (i.e., to the apparent twist different by -2pi, -0.034-pi rad) yields the solution with 3 superhelical loops (with greater writhe).

One mistake was made though: with our assignment of the cross-section principal axes (Y axis points to the major groove) the moduli should be alpha=1.3 and beta=0.55 rather than vice versa (it's easier to bend towards a groove). Thus, the last step of the process is to be re-run.

Mar. 24 - Mar. 30
  1. Repeat the calculations with two elastic moduli correctly assingned.
  2. Improve the precision of the solution obtained.

The calculations for the re-assigned moduli succesfully finished, and thus, the problem was succesfully solved. However, the precision of the solution was low (1e-3) and still needed to be improved. This was done too: the precision was raised to 1e-5 without any problems (in as few as 20 steps!).

Mar 31 - Apr. 6
  1. Replace the "elastic rod" of DNA with a real DNA structure.
  2. Minimize it.
  3. Compare the resulting shape of the loop to the modelling results .

Apr. 7 - Apr. 13
  1. Replace part of the loop with DNA from CAP-DNA compex (as it was done in the paper by Lewis et al.). Start building the elastic model for the rest of the loop.

Apr. 14 - Apr. 15



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