Marco Longinetti

Efficient determination of the most favoured orientations of protein domains from paramagnetic NMR data

We study the inverse problem of the determination of the most favoured relative orientations of moving protein domains from residual dipolar coupling measurements. We present a numerical procedure based on the simplex method for the efficient determination of the maximum probability of a given relative orientation. We prove the convergence of the algorithm and present the results obtained both on synthetic and on experimental data.

On minimal surfaces bounded by two convex curves in parallel planes

Abstract The curvature of the intersection of a minimal surface S with parallel planes {z = t}, between plane parallel convex curves Γ0 and Γ1 on S, takes its minimum on Γ0 ∪ Γ1. A sharp lower bound for the curvature of S ∩ {z = t} is derived. Similarly upper and lower bounds for the gradients of these curves and for their distances from a perpendicular axis are derived too, together with a differential inequalities for their lengths which implies an explicit necessary condition for the surface S to exist.

Efficiency of paramagnetism-based constraints to determine the spatial arrangement of alpha-helical secondary structure elements.

A computational approach has been developed to assess the power of paramagnetism-based backbone constraints with respect to the determination of the tertiary structure, once the secondary structure elements are known. This is part of the general assessment of paramagnetism-based constraints which are known to be relevant when used in conjunction with all classical constraints. The paramagnetism-based constraints here investigated are the pseudocontact shifts, the residual dipolar couplings due to self-orientation of the metalloprotein in high magnetic fields, and the cross correlation between dipolar relaxation and Curie relaxation. The relative constraints are generated by back-calculation from a known structure. The elements of secondary structure are supposed to be obtained from chemical shift index. The problem of the reciprocal orientation of the helices is addressed. It is shown that the correct fold can be obtained depending on the length of the α-helical stretches with respect to the length of the non helical segments connecting the α-helices. For example, the correct fold is straightforwardly obtained for the four-helix bundle protein cytochrome b562, while the double EF-hand motif of calbindin D9k is hardly obtained without ambiguity. In cases like calbindin D9k, the availability of datasets from different metal ions is helpful, whereas less important is the location of the metal ion with respect to the secondary structure elements.

Reconstruction of orientations of a moving protein domain from paramagnetic data

We study the inverse problem of determining the position of the moving C-terminal domain in a metalloprotein from measurements of its mean paramagnetic tensor . The latter can be represented as a finite sum involving the corresponding magnetic susceptibility tensor χ and a finite number of rotations. We obtain an optimal estimate for the maximum probability that the C-terminal domain can assume a given orientation, and we show that only three rotations are required in the representation of , and that in general two are not enough. We also investigate the situation in which a compatible pair of mean paramagnetic tensors is obtained. Under a mild assumption on the corresponding magnetic susceptibility tensors, justified on physical grounds, we again obtain an optimal estimate for the maximum probability that the C-terminal domain can assume a given orientation. Moreover, we prove that only ten rotations are required in the representation of the compatible pair of mean paramagnetic tensors, and that in general three are not enough. The theoretical investigation is concluded by a study of the coaxial case, when all rotations are assumed to have a common axis. Results are obtained via an interesting connection with another inverse problem, the quadratic complex moment problem. Finally, we describe an application to experimental NMR data.

Reconstructing plane sets from projections

We give some uniqueness results for the problem of determining a finite set in the plane knowing its projections alongm directions. We apply the results to the problem of the reconstruction of a homogeneous convex body with a finite set of spherical disjoint holes. Ifm X-ray pictures with directions in some plane are given, then the problem is well posed provided the number of the holes is less than or equal tom and the set of the directions satisfies a suitable condition.

Some isoperimetric inequalities for the level curves of capacity and Green's functions on convex plane domains

The perimeter and the area of the convex level sets of capacity and Green’s functions in convex plane domains are shown to satisfy sharp differential inequalities. Isoperimetric inequalities for capacity problems for optimal conductors are derived.

Uniqueness and degeneracy in the localization of rigid structural elements in paramagnetic proteins

The uniqueness problem in the localization of some rigid structural elements is studied using constraints available for proteins containing paramagnetic metal ions. The degeneracy arising with a single set of data is investigated, and uniqueness is restored using multiple magnetic tensors. An efficient numerical strategy to deal with multiple datasets is presented.

Convex Hulls of Orbits and Orientations of a Moving Protein Domain

We study the facial structure and Carathéodory number of the convex hull of an orbit of the group of rotations in ℝ3 acting on the space of pairs of anisotropic symmetric 3×3 tensors. This is motivated by the problem of determining the structure of some proteins in an aqueous solution.

Differential inequalities for Minkowski functionals of level sets

The Minkowski functionals of the level surfaces of a function u are differentiated with respect to the level parameter. When u is a solution to some classical boundary value problems the Minkowski functionals satisfy sharp differential inequalities.

Affinely Regular Polygons as Extremals of Area Functionals

Abstract For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.