• Proteins
• Coarse Grained models

Proteins are biopolymers highly specialized for specific biochemical purposes, with a complex hierarchical structure. Particularly relevant to this Thesis work is the distinction between the primary structure, indicating the sequence of the amino-acid linked to form the polypeptide, and the secondary structure, namely the local fold in which the polypeptide chain organizes (basically in helical or flat structures).

Computer modeling and simulations have revealed invaluable tools to investigate proteins dynamical behavior, interpret experimental measurements, and give fundamental support in many application fields such as drug design
or molecular medicine. However, their relatively large size (nanometers) limits the length of simulations, when these are performed at the atomistic level. Even considering a purely classical dynamics approach, with chemical bonds
implicitly represented by empirical potentials, a protein includes hundreds of thousands of degrees of freedom, which currently allows to reach the nanosecond (microsecond on heavily parallel systems) on simulations of a single protein, but prevent reaching biologically interesting scales or the simulations of proteins interactions within the heavily crowded cell environment.

For this reason, degrees of freedom are often reduced by treating a protein at lower level resolution (namely, ”coarse graining”). This can be done
exploiting their hierarchical organization: a typical coarse graining level is one interacting center (bead) per amino acid. More specifically, among the one-bead models, a particularly interesting subclass is that of the ”minimalist” ones, in which the interacting bead representing the amino acid is placed on the Cα carbon. In previous work, this choice was mainly due to some
practical facts: the Cα carbon is located at a hinge point of the protein backbone and directly ligates the side chain and the two peptide bonds connecting the preceding and following amino acid. This choice also implies that the polypeptide is represented as a chain of interconnected rigid rods. This extremely simplified representation raises some fundamental questions:(i) are all the possible 3D folding of the protein still representable with this
model, (ii) are them still distinguishable (namely it is possible to back map from
the minimalist representation to the atomistic one) and (iii) which are the minimal properties that a minimalist model must have,
e.g. in terms of interacting Hamiltonian?
In spite of the importance of these questions, they were addressed in previous studies with a mostly empirical approach, basically by verifying that (i) and (ii) are satisfied in the most relevant real cases, while very little attention
was payed to the question (iii).

The aim of this Thesis work is to address these problems at a formally rigorous level. This is done by first deriving the transformation operator between the internal variables in the atomistic and in the minimalist representation (mapping operator). This has a very complex analytical form, which is studied numerically in general, and analytically in some simplifying conditions. The symmetries of the transformation are analyzed and interpreted on the basis of fundamental symmetries of the proteins (related to the chirality of the amino acids). The study of the Jacobian of the transformation gives
rigorous answers to (i) and (ii), allowing to interpret the ”empirical” rules and reveling some exceptions, which should be taken into account when treating proteins in the minimalist representation. In addition, this study allows to
address question (iii), and directly yields an effective interaction potential for the minimalist model merely due to the transformation. This is studied in detail and interpreted on the basis of the chemical atomistic structure of the polypeptide, which is hidden once the coarse graining to the minimalist
representation is operated. The consequences of these results on the coarse graining modeling of proteins are finally drawn.