Vector Space of Feynman Integrals and Multivariate Intersection Numbers.

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.

A novel molecular dynamics approach to evaluate the effect of phosphorylation on multimeric protein interface: the αB-Crystallin case study.

BACKGROUND: Phosphorylation is one of the most important post-translational modifications (PTM) employed by cells to regulate several cellular processes. Studying the effects of phosphorylations on protein structures allows to investigate the modulation mechanisms of several proteins including chaperones, like the small HSPs, which display different multimeric structures according to the phosphorylation of a few serine residues. In this context, the proposed study is aimed at finding a method to correlate different PTM patterns (in particular phosphorylations at the monomers interface of multimeric complexes) with the dynamic behaviour of the complex, using physicochemical parameters derived from molecular dynamics simulations in the timescale of nanoseconds. RESULTS: We have developed a methodology relying on computing nine physicochemical parameters, derived from the analysis of short MD simulations, and combined with N identifiers that characterize the PTMs of the analysed protein. The nine general parameters were validated on three proteins, with known post-translational modified conformation and unmodified conformation. Then, we applied this approach to the case study of αB-Crystallin, a chaperone which multimeric state (up to 40 units) is supposed to be controlled by phosphorylation of Ser45 and Ser59. Phosphorylation of serines at the dimer interface induces the release of hexamers, the active state of αB-Crystallin. 30 ns of MD simulation were obtained for each possible combination of dimer phosphorylation state and average values of structural, dynamic, energetic and functional features were calculated on the equilibrated portion of the trajectories. Principal Component Analysis was applied to the parameters and the first five Principal Components, which summed up to 84 % of the total variance, were finally considered. CONCLUSIONS: The validation of this approach on multimeric proteins, which structures were known both modified and unmodified, allowed us to propose a new approach that can be used to predict the impact of PTM patterns in multi-modified proteins using data collected from short molecular dynamics simulations. Analysis on the αB-Crystallin case study clusters together all-P dimers with all-P hexamers and no-P dimer with no-P hexamer and results suggest a great influence of Ser59 phosphorylation on chain B.