# Uniform inter variables in helicoidal DNA

This webpage serves as supplementary material for :

This semester project relies mainly on the following existing work:

• 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 unpublished. 2017.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.

The main goal of this second part of the study is to generate an initial guess for covalently closed (both back bones closed) minicircles configurations for a given sequence of n base pairs and a range of integer linking numbers m (≈ n/10.5). In order to achieve it we divide the study in two steps:

• generate special helicoidal configuration that have specific integer link number m;
• deform the helicoidal equilibrium into a twisted circle.
In this study we focused on the first step, while the second one goes beyond the scope of this semester project. We first construct a helicoidal periodic DNA configuration with uniform inter variables, and periodic intras and phosphates. In particular, we will consider the variable: $$w=(x_1,u,v,x_2,u,v,\dots,x_n,u,v)\in \mathbb{R}^{24n}$$ We transform this variable in a new one z , so that w=Pz, where P is a specific matrix, and: $$z=(x_1,x_2,...,x_n,u,v)\in \mathbb{R}^{18n+6}$$ As a consequence we rewrite the energy function: $$U^*(z)=\frac{1}{2}(Pz-\mu)^{\top} K (Pz-\mu)$$ The aim of the study is to create an helicoidal periodic ground-state that can be closed into a minicircle shape. In order to close it the two end parts must interact appropriately, hence the number of links between the two ends must be an integer number m. But with a generic sequence S there is no reason that φ (the uniform twist angle between two base pairs) will lead to a complete number of turns m. As a consequence we impose the number of links m putting the constraint (22): $$\|u\|=10 \tan \frac{\pi m}{n}=k$$ For this reason we set the Lagrangian function of the new energy function imposing the constraint of the number of links (h(z)=norm(u)-k): $$\mathcal{L}(z;\lambda)=U^*(z)+\lambda h(z)$$ In order the find the optimum value for the constraint optimization problem, we evaluate de gradient and its singular value: $$\nabla \mathcal{L}(z,\lambda)=0 \iff (P^TKP-\lambda E)z=PK\mu$$ We solved this equation on Matlab using mldivide solver which has specific algorithms for sparse matrix, as the one we have. In particular we analized four different sequences:
 - 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.

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.