Drenatges, sanejament i canalitzacions

The molecules surrounding the protein behave differently from those in pure solvent. Thus, two groups of solvent can be then defined in a simula- tion: the first solvation shell (FSS) is formed by solvent molecules within 5 Å from the protein; the bulk instead by molecules that lie at a distance from the protein larger than 6Å [11] (#P1 Tab 2, Fig S1A ; #P2 Tab 3). Identification of protein/solvent contacts

To evaluate the contacts that solvent molecules can form with protein residues, I used a criterion based on atomic distance: a contact is formed when at least two heavy atoms belonging to each molecule are closer than 3.5 Å (#P2 Fig 5, 6, S6, #P3 Fig 5,6). Contacts can then be easily grouped according to i) the nature of the residues involved (polar or apolar); ii) the portion that forms the contact (sidechain or backbone) and iii) the posi- tion in the native structure (part of the native protein core or not). Once these contacts are defined other solvent features can be easily extracted, for example the ratio of apolar/polar contacts (#P1 Tab 2, Fig S1,#P2 Tab 3, Fig S5A) or the residence time of a solvent molecule around the protein (#P2 Fig 4B). Among all the contacts, the peculiar nature of the hydrogen bonds allows to identify them with a simple geometrical crite- rion: a cutoff of 3.5 Å for the distance between the two electronegative atoms (the donor and the acceptor) and 120º for the angle between the donor-hydrogen and the acceptor (#P1 Tab 3,#P2 Tab 4, S4).

When two or more co-solvent types are present, the contact prefer- ence with either one co-solvent or the other can be quantified with the Contact Coefficient (CC). Here I specifically employed this metrics to evaluate solvation of proteins by urea and water molecules (CC_UW ) [12], defined as: CCUW = NXU NXW· MW MU    

where N_XU and N_XW are the numbers of atomic contacts of amino acid X with urea and water molecules, respectively. CC_UW is normalized using the total numbers of atoms belonging to urea molecules M_U or to water molecules M_W in the system. A contact coefficient of 1 means that the amino acid has no contact preference for one of the co-solvents; values above 1 indicate preferential interaction with urea while values below 1 with water (#P1 Tab 2, Fig 3B, S7,#P2 Fig 4A).

Figure 3.5. Bond critical points.

Contour map of the electron den- sity. The atomic symbols in denote the positions of the nuclei; yellow spheres the bond critical points and black lines the bond paths.

Energetics of hydrogen bonds (HB): Atoms in Molecule (AIM)

To evaluate the strength of different hydrogen bonds a deeper analysis of the HB structure is needed. The quantum theory of Atoms in Molecule (AIM), pioneered by Richard Bader [13], comes in hand by considering a bond as a 3D entity, the topology of which quantifies its physical and chemical properties. A powerful observable is the spatial topological de- composition of the electron density ρ(r). According to AIM theory the electron density ρ(r) is at maximum at the atomic nuclei, which allows to clearly identifies the atomic position; chemical bonds can then be easily traced to unite the atomic nuclei. The saddle point, the minimum in elec- tron density ρ(r), along the bonding direction identifies the bond critical point. In non covalent interactions, such as the hydrogen bond, the prop-

O C

C C

O H

Critical Point erties of the density field at the crit-

ical point, i.e. the density itself or its Laplacian1 _{, are proportional to} the strength and the energy of the corresponding interaction [14]. In this work I compared the strength of several HB between the protein backbone and the solvent mole- cules, by evaluating the electronic

1 The scalar derivative of the gradient vector field of the electron density. It de- termines where electronic charge is locally concentrated (negative values) and depleted (positive values).

Energetics of solvent interactions

A post processing procedure allows calculating the interactions energies between two subgroups of molecule in a MD trajectory. Similarly to an MD run, the forces can be computed following the same pairwise equa- tion given in Chapter 2. In my projects I calculated the electrostatic and Van der Waals energy contribution between each solvent molecule and the rest of the system, taking into account the relative position to the protein (FSS or bulk) at each frame. The comparison of the interaction energy distribution for all the molecules in the FSS, influenced by the presence of the protein, and in the bulk gives insights on the preferential and more favorable interactions with the protein [11](#P1 Fig S8, #P2 Fig 5; 6, S5B).

Solvent diffusion and MSD – Mean Square Displacement

The collective motion of all particles in a fluid is termed diffusion and its quantified by the diffusion coefficient. This macroscopic property relates to the microscopic thermal motion of individual molecules and relates with their average motility quantified in the mean square displacement MSD: MSD= ∆𝑟𝑟 𝑡𝑡 ! ₌1 N 𝑟𝑟! 𝑡𝑡 − 𝑟𝑟! 0 ! ! !!!  

Where r_i(t) and r_i(0) are the position of particle i at time t and at the reference time 0, respectively. For molecules consisting of more than one atom (i.e. solvent molecules or even proteins), r_i can be taken as the cen- ter of mass positions of the molecules. Albert Einstein, in his PhD thesis, derived a relationship between the macroscopic D diffusion coefficient and the microscopic behavior collected in the MSD [15]:

lim

!→! ∆𝑟𝑟 𝑡𝑡 ! = 6𝐷𝐷𝐷𝐷  

distribution at their critical points. The structure were first extracted by MD simulations runs and geometrically optimized by QM calculation from which electron densities are derived (#P1 Tab 4).

After calculating the MSD in different time windows (n), the diffusion coefficient D can be derived from the slope of the fitting line. Generally only the last half of values is used for the fitting, since the Einstein rela- tion is valid as time approaches infinity (#P2 Fig S5C; #P3 Table 1).