# Sensitivity of cgDNA+min to computational parameters

This webpage serves as supplementary material for :- (♦) Padovano, F. "Sensitivity of cgDNA+min to computational parameters". Master semester project. EPFL, 2022.

The beamer slides of the final presentation are also provided here.

- Beaud, M. "Using cgDNA+ model to compute sequence-dependent shapes for DNA minicircles". Master thesis. EPFL, 2021.

- Manning, R. "Notes on cgDNAmin, Discrete-birod DNA Cyclization". unpublished. 2017.

- Patelli, A. S. "A sequence-dependent coarse-grain model of B-DNA with explicit description of bases and phosphate groups parametrised from large scale Molecular Dynamics simulations". PhD thesis. EPFL, 2019.

- Glowacki, J. "Computation and Visualization in Multiscale Modelling of DNA Mechanics". PhD thesis. EPFL, 2016.

We want to prove numerically the hypothesis that "given the periodic ground-state of a specific sequence, if we change its inter variables, while intras and phosphate are left as they are, we can find a good initial guess for the energy optimization", and for doing that we take advantage of Beaud's Matlab scripts. In order to see if it holds we decided to consider two different initial guesses for the energy optimization:

- configuration obtained with the bBDNA software (will be called bBDNA);

- configuration in which the intra and phosphate coordinates are the same as the periodic ground-state,
while the inter coordinates are the ones of the configuration obtained with the bBDNA software (will be
called bBDNA with gd-intras).

For the minimization step we used fminunc Matlab function, applying both the Trust region and Quasi-Newton methods, depending on the sequence. These two methods aim to minimize locally a quadratic model: $$ m_k(x_k+t)=f(x_k)+\langle\nabla f(x_k),t \rangle +\frac{1}{2} \langle t ,H(x_k)t \rangle $$

**Quasi-Newton**: uses the quadratic method as a Taylor expansion of the function in the current point. It uses as descent vector the deepest direction for the quadratic model, which is $$t=-H(x_k)^{-1} \nabla f(x_k)$$ Since evaluating the Hessian matrix might be very computationally expensive, this method expresses it with some approximations (BFGS). Quasi-Newton has quadratic local convergence properties, but it often fails. Firstly because a Taylor expansion is a good approximation if we remain in a region nearby the current point; secondly because the Hessian of the function at the current point might not be positive definite, therefore the quadratic model is not convex.**Trust Region**: defines a region around the current iterate within which it trusts the model to be an adequate representation of the objective function (in order to use properly the Taylor expansion), and then choose the step to be the approximate minimizer of the model in this region. It is a more accurate method, in fact it uses Newton’s method with safe-guards so that it gains good global convergence properties, while preserving the quadratic local convergence rate of Newton method. However, since each step is constrained to a region centered in the current point, of which radius may or may not vary from iteration to iteration, this method can be very long.

- Kahn & Crothers: | Crothers, D. M. et al. “DNA bending, flexibility, and helical repeat by cyclization kinetics”. In: Methods in Enzymology 212 (1992), pp. 3–29. |

- Pyne et al. 251 bp: | Pyne et al. "Base-pair resolution analysis of the effect of supercoiling on DNA flexibility and major groove recognition by triplex-forming oligonucleotides". In: Nature communications 12.1 (2021), pp. 1053–1053. |

- Pyne et al. 339 bp, | |

- Widom 601: | Coultier, T. and Widom, J. "Spontaneous Sharp Bending of Double-Stranded DNA". In: Molecular cell 14.3 (2004), pp. 255-362. |

For each sequence we run the minimization for the two initial guesses and evaluate the absolute difference between the two obtained optimal configurations, the energies, the eigenvalues and the Hessian condition numbers. The full sequences and all results can be found through the links on the left.

The MATLAB scripts can be found on the separate sub-page.

Further analysis of the results should be done to verify the stability of the solutions.