Ewald summation and Particule Mesh Ewald (PME)

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UV-visible absorption

As described in Figure 1.1 (black arrows), when a sample containing light sensitive molecules is hit by electromagnetic waves, especially in the ultra-violet and visible wavelength range, absorption of energy will occur resulting on electronic excitations (namely electronic transitions from lower occupied energy levels to higher’s). The absorption efficiency of the irradiating beam by the sample will depend on the vicinity between its energy and the energetic excitation gap of the excited state energy levels.
The absorption intensity (I) of a light beam of wavelength λ traveling through a given sample of length (l) will depend on the absorption coefficient ( ) and molar concentration (C) of the molecules it contains, as well as the initial intensity (I0): I = I0 e− 0 C = I0 10− C (1.2).

Fundamental building blocks of DNA

DNA is a biopolymer made of monomers called nucleotides which are composed of three organic chemical entities:
• Pyrimidines (thymine, cytosine) and purine (adenine, guanine) nitrogen bases.
• pentose sugars (deoxyribose in DNA).
• phosphate (PO−4) groups.
As presented in 2.2, the pentose sugar works as an anchors to fix the base to the backbone of the polymer. Then the nucleotides are conventionnaly arranged in a 3’ – 5’ sequence where the phosphate group of each monomer (attached to the C5’ of the sugar) is linked to the C3’ of the following sugar moiety.

DNA structural parameters

Even though DNA in its double helical form is highly flexible, an ensemble of well de-fined parameters can be employed to efficiently and completely describe its structure and behavior. Indeed, these parameters can either characterize nucleotides (intra base pairs parameters), stacked base pairs (inter base pair parameters) and grooves (width and depth of the major and minor grooves). All schemes of parameters are given in Appendix A.
These parameters can be efficiently obtained by accessible tools such as Curves+ [79] and 3DNA [80]. The information that can be extracted from such data are of high interest in the study of compounds interacting with DNA, global and local deformations of the strand, interaction with proteins and even more specific phe-nomena such as base flipping for example.

DNA photosensitivity

Since it is present in every cells of every leaving animal, DNA is facing the influence of sunlight [81]. Indeed, the electronic aromaticity of the bases causes there highly efficient light absorption in the UV, with a maximum at approximately 260 nm [82]. Their absorption spans the UVC region as well as all the UVB wavelengths, and also have a small absorption probability in the UVA due too excitonic couplings. In order to prevent the photoreaction and degradation of the nucleobases, thus inducing mutations or cell death, the chemical structure of the absorbing nitrogen bases evolved toward a highly photostable ensemble. Photostability in DNA is assured through several mechanisms:
Fluorescence The nucleobases excited states decay at the subfemtosecond time scale [83]. The direct fluorescent deexcitation has, both in isolated nucleobases or in DNA, a low quantum yield, respectively 10−4 and 3.10−4. This means that the excess of energy is mainly released in a non radiative way through IC and ISC.
Non-radiative deexcitation It has been demonstrated that the non-radiative re-laxation of nucleic acids excited states differs between monomers and mono-or double-strands. In the case of monomers, the S0 → S1 transition of π − π∗ nature will either decay directly through a conical intersection between the two potential energy surfaces (60%) or evolve toward a 1n − π ∗ state followed by a ISC to the triplet state (picosecond timescale) [84–86]. In the case of base stacks and double strands, charge transfer states and charge recombination is considered to play the highest role in the decay toward S0;
Charge recombination When the excited nucleobase is embedded in a single or double strand, the π-stacking interactions and hydrogen bonds of the Watson-Crick arrangement play a modulating role in the excited-states lifetimes. In-deed, for single strands an electron can be transferred to the adjacent base and induce the formation of radical ions. Then charge recombination prevents any further reactions and allows both bases to relax to their canonical states [23]. In double strands, it is the hydrogen pairing that plays the highest role. After excitation a charge separation may occur between the two nucleobases of the base pair followed by a proton transfer. Then, through conical intersections between the charge transfer state and the ground state, the system can decay non-radiatively to its ground state.

DNA photolesions

Even if the photostability of DNA is high and prevents the formation of lesions and damages, since it is composed of approximately 3 billion nucleobases, the occurrence of this phenomenon cannot be ruled out. For example, the increasing number of skin cancers is due to the direct exposure of the skin to UV light, and especially the pho-tosensitization of DNA and the photochemical reactions of the nucleobases inducing photolesions. The main DNA lesions due to irradiation are the cyclobutane pyrim-idine dimers (CPD), the 6-4 photoproduct (6-4PP) and the Dewar photoproduct (DewPP) [87, 88], represented in 2.5.

Integration of equations of motion

Molecular dynamics is based on classical Newton’s laws of motion, especially the second one: 2 ∂ xi = Fxi (4.1).
∂t2 mi 2 mi ∂ x2i = −∂Vxi (4.2).
where xi is the position of the atom i of mass mi submitted to a force Fxi for a particle moving in one dimension. Moreover, this force can be determined by the calculation of the first differential of the potential energy equation given by the force field formula 3.1.
In most molecular dynamics softwares, the molecular dynamic simulations are conducted by numerically solving the differential equations of motions with finite difference methods. The concept of these methods is to split the integration into small intervals of a fixed time length ∂t. Then, the total force applied on each atom for a given configuration at a time t is computed as the vector sum of its interactions with every other atom. Now that the forces are generated, the accelerations can be easily obtained. And finally, with positions, velocities and accelerations at time t the system can evolve toward the next step at time t + ∂t and so on.
In order to achieve a better stability, the Amber software employs a common algorithm for simulations called leapfrog and is described as follows:
r(t + ∂t) = r + ∂tv t + 1 ∂t (4.3).
v t + 1 ∂t = v t − 1 ∂t + ∂ta(t) (4.4).
In this algorithm the velocities at a time t + 12 ∂t are calculated from the ones at time t − 12 ∂t and the accelerations at time t. And from equation 4.3, the positions at time t + ∂t are generated from the previous velocities and the positions at time t. And finally to synchronize the positions and velocities (respectively calculated at times t + ∂t and t + 12 ∂t), the velocity at time t is determined as follows:
v(t) = 1 v t + 1 ∂t − v t − 1 ∂t (4.5).


Periodic Boundary Conditions (PBC)

The computational cost of MD simulations is dependent on the numbers of particles. If one wants to consider a protein in solution, the number of particles that should be taken into account in the calculation compared to the size of the protein would be basically infinite, thus leading to impossible simulations. In order to bypass this issue, the system can be placed in a finite sphere of explicit solvent molecules, but then important border effects will appear if the solute reaches the edges of the sphere during the MD run leading to non-meaningful results. So to avoid the border effect problem and still reduce the computational cost of the simulations using reasonable solvation box sizes, one can rely on the use of the so called Periodic Boundary Conditions (PBC), Figure 4.1. The aim of this tool is to surround a box of explicit particles, with replicated images of itself in all directions to give a periodic array. The principle of this method, described in 2D on the Figure 4.1, is to replace a particle, leaving the box in one side, by an identical particle in the opposite side, thus, the number of particles remains the same.
The simplest periodic system is the cubic box, surrounded by 6 similar replica, but it is possible to reduce the cost of the simulation by using a more complex shape. In the studies related to this manuscript, the structure of the periodic box is a truncated octahedron. It is surrounded by 8 neighbors and have half of the volume of a cubic box which reduces much more the costs of simulations. The only inconvenience of this method is that the system has to be strictly smaller than the length of the box, otherwise an overlap between the solute molecules in the periodic cells will occur resulting in non-physical results.

Ewald summation and Particule Mesh Ewald (PME)

Based on the Ewald summation that allows the computation of charge and dipole interactions in periodic systems, the so-called Particle Mesh Ewald (PME) algorithm has been developed to improve the efficiency of MD simulations and remove the bias created when a simple cut-off is used for Coulombic interactions.
The Coulombic potential VC for N particles in a periodic box of size L is given by: 1 N N qiqj VC = X X X (4.6).
where qi and qj are the charges of the particules i and j that are separated by distance rij and n is the translation vector of the periodic cell: n = (nxL, nyL, nzL) (4.7)

Table of contents :

I Acknowledgments/Remerciements 
II Outreach activities 
0.1 Introduction
0.2 Pretexte
0.3 Esperluette
0.4 MT180
III List of publications
IV Introduction générale 
V General introduction 
VI Main concepts and systems 
1 Photochemical and photophysical processes 
1.1 Photochemistry
1.2 Spectroscopies
1.2.1 UV-visible absorption
1.2.2 Circular dichroism
1.2.3 Fluorescence
1.3 Photosensitization
1.4 Photodynamic therapy
2 DNA 
2.1 Introduction
2.2 Fundamental building blocks of DNA
2.3 Double helix
2.4 DNA structural parameters
2.5 DNA photosensitivity
2.6 DNA photolesions
2.7 Repair processes
VII Methods 
3 Molecular Mechanics 
4 Molecular Dynamics 
4.1 Integration of equations of motion
4.2 Periodic Boundary Conditions (PBC)
4.3 Ewald summation and Particule Mesh Ewald (PME)
4.4 Biased Molecular Dynamics
4.5 Alchemical simulations
5 Quantum chemistry 
5.1 The six postulates of quantum mechanics
5.1.1 Postulate 1: the wave function
5.1.2 Postulate 2: Hermitian operators
5.1.3 Postulate 3: eigenstates and eigenvalues
5.1.4 Postulate 4: expectation of the wave function
5.1.5 Postulate 5: Time dependent Schrödinger equation
5.1.6 Postulate 6: wave function collapse
5.2 The Schrödinger equation in quantum chemistry
5.3 Born-Oppenheimer approximation
5.3.1 Variational Principle
5.4 Ground state
5.4.1 Hartree-Fock approximation
5.4.2 Density Functional theory (DFT) Principle The electronic density The Hohenberg-Kohn theorems The Kohn-Sham equations LDA GGA meta-GGA Hybrid GGA and hybrid meta-GGA Double hybrids
5.5 Excited states
5.5.1 Time Dependent-Hartree Fock (TDHF)
5.5.2 Time Dependent-Density Functional Theory (TDDFT) The Runge-Gross theorem Time-Dependent Kohn-Sham equation
5.5.3 Frenkel excitons theory
6 QM/MM 
6.1 QM/MM energy
6.1.1 QM/MM embeddings Mechanical Embedding (ME) Electrostatic Embedding (EE) Polarisable Embedding (PE)
6.2 QM/MM boundary
6.2.1 Link Atom (LA)
VIII Results and discussions 
7 Introduction 
8 Electronic Circular Dichroism (ECD) modeling 
8.1 ECD of B-DNA
8.2 ECD of G-quadruplexes
9 Binding free energy of Benzophenone 
10 Photosensitization of DNA 
10.1 Benzophenone: Hydrogen abstraction
10.2 Nile Blue, Nile Red: Electron transfer
10.3 BMEMC: Radical species
10.4 Pyo: Triplet-Triplet Energy Transfer (TTET)
11 Damaged DNA: structure and recognition 134
11.1 Cluster abasic sites
11.1.1 Recognition by APE1
11.2 64-PP and CPD
XI Appendices 
A DNA structural parameters 
C ECD of G-quadruplexes 
D Binding free energy of Benzophenone 
E Benzophenone: Hydrogen abstraction 
F Nile Blue, Nile Red: Electron transfer 
G BMEMC: Radical reactive species 
H Pyo: Triplet-Triplet Energy Transfer (TTET) 
I Cluster abasic sites 
J 6-4PP and CPD


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