1. Convex Orbits and Orientations of a Moving Protein Domain Marco Longinetti (Università di Firenze) C0NVEX GEOMETRY-ANALYTIC ASPECTS Cortona 2007-June 4-8

2. References (mathematical): Longinetti M., Parigi G., Sgheri L.,Uniqueness and degeneracy in the localization of rigid structural elements in paramagnetic proteins, J. Phys. A: Math. Gen. (2002) 35, 8153-8169. Gardner R.J, Longinetti M., Sgheri L., Reconstruction of orientations of a moving protein domain from paramagnetic data, Inverse Problems (2005) 21, 879-989 Longinetti M., Luchinat C., Parigi G., Sgheri L., Efficient determination of the most favoured orientations of protein domains, Inverse Problems (2006) 22, 1485-1502 Longinetti M.,Sgheri L, Sottile F., On the facial structure and Caratheodory number of Convex Orbit from RDC data, 2007 preprint

3. Proteins: Organic molecules having multiple functions in every living organism Red: Oxygen Gray: Hydrogen Black: Carbide Blue: Nitrogen

4. Protein Folding Spatial determination of the shape of the protein (hints on the function of the molecule) - Protein fold is “almost” unique - No program is able to reconstruct uniquely the fold (i.e. too many interactions) - Only one tenth of the folds of the human proteins are known up to now - Folds are obtained by: a)Fixing some of the variables involved (i.e. distances between bonded atoms) b)Using measures as constrains

5. Paramagnetic NMR Data Effects of the interaction between a metal atom and the protein; Residual Dipolar Coupling (in the metal system) is the paramagnetic tensor (not known, depending on the metal atom), = constant (known), Bad:Measure error quite big, do not guarantee uniqueness. Good: Relatively cheap and quick to obtain (still 1 set ~ 3 months team work).

6. Paramagnetic Tensor is represented by a symmetric 3x3 matrix. Paramagnetic Data do not depend on the trace of the paramagnetic tensor, null trace is assumed. Metal System: reference system where is diagonal and its eigenvalues are in increasing order:

7. non-rigid structures: The Calmodulin (test case) N - terminal Linker zone C - terminal Metal binding site

8. N - terminal Linker zone C - terminal Metal binding site N - terminal Linker zone C - terminal Metal binding site Linker zone Under certain conditions, the long helix breaks and the C-terminal is left free to move

9. Assumptions: The C-terminal and the N-terminal are rigid. Their rigid structures are known. The paramagnetic tensors and the N-terminal are solidal. Results: The RDC data from atoms of the N-terminal are used to find the paramagnetic tensors (rigid case). The translation do not influence RDC data, hence only the orientation of the C-terminal may be reconstructed. The orientation of the C-terminal may be described by a rotation. The rotation centered in the first atom of the short linker is a good approximation of a given orientation. Question (partially open): What can we say about the motion of the C-terminal ?

10. Let be the special orthogonal group of rotations R in. A mathematic model The motion of the C-terminal may be described by a probability measure. RDC data are obtained by: Let: is a symmetric matrix 3x3 with null trace, i.e determined by 5 real numbers, Called the mean paramagnetic tensor.

11. Can we extract information from ? Extremal situations: a) The C-terminal does not move: then: (Dirac), so that: b) The C-terminal moves freely with uniform probability: thenis the uniform Haar measure, and:

12. Convex orbits

13. We are mainly interested on: dim the facial structure of the Charatheodory number of : the minimal number k, s.t., any element of can be represented in k-atomic way.,i.e.

14. Coaxial Convex Orbits of paramagnetic data The following complete results are obtained for coaxial convex orbits. N C u

15. Convex Orbits

16. Convex Orbits of Multiple Tensors Why we need multiple tensors?

17. The problem of symmetries RDC do not change for any i=1,.,3:

18. Overcoming the problem of symmetries: Substitute the metal ion M1 with M2 (may be in the same or in different binding site.) This gives two different paramagnetic tensor and two sets of RDC data Theorem (M. L. – G. Parigi – L.Sgheri 2002): With enough exact data, there is uniqueness in the reconstruction of the spatial position of rigid portions of the protein and the tensors of the metals if and only if two of the paramagnetic tensors defined by the metal ions are not collinear (i.e. they do not have a coordinate axis in common)

19. x y z z’ y’ x’ M1M1 M2M2 x y z=x ’ y’y’ z’ M 1 =M 2 x y z z’ y’ x’ M 1 =M 2 x y z z’ y’ x’ M1M1 M2M2 x y z z’ y’ x’ M1M1 M2M2 CollinearNot Collinear Collinear and not collinear metal systems Note: For RDC data Not Collinear metal systems means distinct eigenvector for related tensors

20. Main results for multiple tensors The results about the facial structures for multiple tensors are not complete. For m=2 we show a 3-dimensional family of coaxial faces:

21. On the facial structure for couple of tensors Open problem: Which is the minimal number of rotations needed to represent any couple of mean paramagnetic tensors? Partial result: Conjectures: for 2 metals, i.e couple of tensors, for 3-5 metals :

22. Most favoured orientations Theorem (R.G., M.L., L.S. 2005)

23. Most favoured orientations for 1-metal tensor 0..

24. The maximal probability of an orientation in case of 2 metals Given a mean paramagnetic tensor, the maximal probability of an orientation is: May not be calculated only from the eigenvalues May be calculated directly from the eigenvalues

25. Simplex type algorithm for the determination of If the boundary does not contain any simplex of maximal dimension, the algorithm converges. This can be proved in our case. The algorithm gives you with a maximal error of Practical: Works Target point

26. Synthetic results: Single privileged orientation with N terminal C terminal (privileged orientation)

27. Synthetic results: Two privileged orientation with:

28. Overcoming the problem of the “ghost cones” with a third strong not-collinear metal ion Single privileged orientation with M1+M2 and M1+M3 (lighter) Common Cone

29. True experimental data: 5% best orientations 20% best orientations

30. Distribution of with respect to the sample: