Girse nd part: examples Relaxation processes, molecular motions and electron spin resonance Antonino Polimeno Università degli Studi di Padova Applications

Girse nd part: examples T4 lysozyme  Biological Macromolecules 3. Liang, Z.; Lou, Y.; Freed, J.H.; Columbus, L.; Hubbel, W. L. J. Phys. Chem. B 2004, 108,  Labelled Peptides 1. Carlotto, S.; Cimino, P.; Zerbetto, M.; Franco, L.; Corvaja, C.; Crisma, M.; Formaggio, F.; Toniolo, C.; Polimeno, A.; Barone, V. J. Am. Chem. Soc. 2007, 129,  Materials 2. Barone, V.; Brustolon, M.; Cimino, P.; Polimeno, A.; Zerbetto, M.; Zoleo, A. J. Am. Chem. Soc. 2006, 128, 1586 Applications

Girse nd part: examples Interpretation based on Stochastic Liouville Equation (SLE) defined by the direct inclusion of motional dynamics via stochastic operators plus super Hamiltonian H DIPOLE- DIPOLE TENSOR g, A TENSORS DIFFUSION TENSOR ELECTRONIC STRUCTURE Dissipative parameters, e.g. rotational diffusion tensors, can be determined via hydrodynamic modelling Principal values and orientation of electron Zeeman tensor and hyperfine coupling tensors Additional interactions; e.g. in double labeled systems, dipolar interaction based on the molecular structures beyond the point approximation MOLECULAR GEOMETRY Quantum Mechanical calculation pursued by Density Functional Theory (DFT) via adoption of mixed quantum-mechanical / molecular mechanical (QM/MM) methods Barone, V.; Polimeno, A. Phys. Chem. Chem. Phys. 2006, 8, 4609 Integrated approach

Girse nd part: examples The SLE describes the variation in time of the system density matrix. The density matrix / probability depends upon quantum pseudo- coordinates and classical (stochastic) variables 1. The cw-ESR spectrum is obtained from the spectral density 1. Schneider, D. J.; Freed, J. H. Adv. Chem. Phys. 1989, 73, 387 SLE

Girse nd part: examples Numerical implementation (1) The cw-ESR spectrum given is now identified as the real part of the spectral density for the auto-correlation function for the observable, usually named ‘starting’ vector corresponding to the x-component of the magnetization The spectrum is obtained by numerically evaluating the spectral density and this is usually attained via iterative algorithms, like Lanczos or conjugate gradients

Girse nd part: examples Numerical implementation (2) The spectrum can be written in the form a continued fraction

Girse nd part: examples Numerical Implementation (3) The computational task is carried on in finite arithmetic, by projecting the symmetrized time evolution operator and the starting vector on a basis set Symmetry arguments can be employed to significantly reduce the number of basis function sets required to achieve convergence, A numerical selection of a reduced basis set of functions based on the ‘pruning’ of basis elements with negligible contribution to the spectrum can be also used The matrix-vector expression for the c.f. coefficients is

Girse nd part: examples Numerical implementation (4) To evaluate matrix elements, one needs to make explicit the dependence of the Liouvillean from magnetic and orientational parameters. We adopt a spherical irreducible tensorial representation LF GF AnFAnF  MF  GF  MF  AnF B0B0 MF  LF  MF

Girse nd part: examples Liouvillean symmetrization The Liouville operator is spanned in the set of basis functions This basis set generally leads to a Hermitean matrix representation of the Liouvillean, but Lanczos algorithm is written for symmetric matrices Symmetrization of the Liouvillean is achieved by the basis transformation The matrix elements in the new basis set are expressed in function of the matrix elements in the old basis

Girse nd part: examples Pruning Basis functions are kept or discharged in function of their relative importance on determining the line shape Weights are calculated with the criterion [7] [7] D. J. Schneider, J. H. Freed, Adv. Chem. Phys. 73, 387 – 527 (1989)

Girse nd part: examples int main (void) { int i; long double dw, Iw, omega0; FILE *ouf, *ouf2; phisics(); basi(); matrix(); stvec(); omega0=(g[0][0]+g[1][0]+g[2][0])/3.0; […] printf("\n\nnstep = "); scanf("%d",&nstep); lanczos(); […] for (i=-10000;i<=10000;i++) { dw=(long double)i/ ; Iw=fracinf(dw-omega0); fprintf(ouf,"%Lf %Le\n",dw,Iw); } Main Input physical parameters 1 Build Liouville operator matrix 3 Project basis functions on v 4 Lanczos tridiagonalization 5 Generate basis functions indexes 2 ESR line calculation 6

Girse nd part: examples Calculation of the spectrum is based on stochastic methods and in particular on the solution of the stochastic Liouville equation. In this equation dynamics is added as stochastic operators in the Liouville operator that describes the time evolution of the density matrix of the system. Full diffusion tensor is calculated via a hydrodynamic model that describes the molecule as an ensemble of interacting spherical beads surrounded by a locally isotropic continuous fluid. Diffusion tensor depends on the geometry of the molecule and on the viscosity of the solvent. Non-rigid molecules can be described with internal degrees of freedom represented by torsional angles. Quantum Mechanical calculations are employed to calculate: (i) structural properties like geometry and torsional potentials; (ii) magnetic properties, i.e. the magnetic tensors. E-SpiReS automatically generates an input file for Gaussian that can be edited by the user and submitted directly from the GUI. A number of parameters can be chosen to refine via a non linear least squares minimization routine based on the Levenberg - Marquardt method. Also an experimental spectrum can be loaded as reference. Quantum Mechanics Hydro Dynamics Stochastic Methods CW-ESR spectra prediction Molecular geometry is given in Z-matrix or PDB format Graphical definition of the form of the stochastic Liouville operator All physical and calculation parameters are set here Electron Spin Resonance Simulation is a multiscale software for the ab-initio calculation of cw-ESR spectra. The core of E-SpiReS is written in C and parallelized under the MPI paradigm. The graphic user interface (GUI) is written in JAVA to ensure good portability in every operating system. Polimeno, A.; Zerbetto, M.; Barone V. Comp. Phys. Comm. – submitted

Girse nd part: examples Electron Spin Resonance Simulation is a multiscale software for the ab-initio calculation of cw-ESR spectra. The core of E-SpiReS is written in C and parallelized under the MPI paradigm. The graphic user interface (GUI) is written in JAVA to ensure good portability in every operating system. The web interface, accessible from any browser, communicates with the cluster sending to it the requests of calculation and giving back to the user the spectrum hhhh tttt tttt pppp :::: //// //// wwww wwww wwww.... cccc hhhh iiii mmmm iiii cccc aaaa.... uuuu nnnn iiii pppp dddd.... iiii tttt //// llll iiii cccc cccc

Girse nd part: examples Polimeno, A.; Zerbetto, M.; Franco, L.; Maggini, M.; Corvaja, C. J. Am. Chem. Soc. 2006, 128, 4737 Isomer 2 Isomer 1 Stochastic model: free Brownian rotator Geometry & shape: MM level calculation Diffusion tensor: hydrodynamic model Interaction energy J obtained via fitting Diffusive operator Spin Hamiltonian

Girse nd part: examples Barone, V.; Brustolon, M.; Cimino, P.; Polimeno, A.; Zerbetto, M.; Zoleo, A. J. Am. Chem. Soc. 2006, 128, 1586 Stochastic model: Brownian rotator + conformational dynamics Geometry & shape: QM level calculation (Gaussian, DFT-B3LYP) Diffusion tensor: hydrodynamic model Diffusive operator Spin Hamiltonian

Girse nd part: examples The oligopeptide is labeled with two nitroxide radicals, in the form of  -amino acid TOAC (2,2,6,6-tetrametyl- 1oxyl-4amino-4-carboxylic acid) T2FT2F MF LF B0B0 Zerbetto, M.; Carlotto, S.; Polimeno, A.; Corvaja, C.; Franco, L.; Toniolo, C.; Formaggio, F.; Barone, V.; Cimino, P. J. Phys. Chem. B 2007, 111, 2668 Carlotto, S.; Cimino, P.; Zerbetto, M.; Franco, L.; Corvaja, C.; Crisma, M.; Formaggio, F.; Toniolo, C.; Polimeno, A.; Barone, V. J. Am. Chem. Soc. 2007, 129, 11248

Girse nd part: examples At short distances between the two spin probes the calculation of the dipolar interaction tensor must consider the distribution of the unpaired electrons over the anti-bonding orbitals. R versus distance calculated via approximated expression (dashed line) and exact treatment (solid line).

Girse nd part: examples Simulations of cw-ESR spectra of the peptide in four different solvents. Red solid line are experimental spectra, black dashed line are theoretical spectra AcetonytrileChloroformMethanolToluene

Girse nd part: examples Polimeno, A.; Carlotto, S.; Zerbetto, M.; Huber, M.; Toniolo, C. – in preparation This preliminary study is based on Molecular structure from molecular mechanics calculations Magnetic tensors from literature Diffusion tensor calculated via a hydrodynamic method X-band cw-EPR spectrum in acetonitrile at 293 K W-band cw-EPR spectrum in acetonitrile at 293 K Structure of the peptide: Fmoc-Aib-TOAC-(Aib) 5 -TOAC-Aib-OMe

Girse nd part: examples Zerbetto, M.; Polimeno, A.; Cimino, P.; Barone, V. J. Chem. Phys. 2008, 128, Diffusive operator Spin Hamiltonian Isotropic phase Nematic phase

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Girse nd part: examples We investigate the temperature dependence of cw-EPR line shape measured at the early stages of the methyl methacrilic polymerization. Hermosilla, L.; Sieiro, C.; Calle, P.; Zerbetto, M.; Polimeno, A. J. Phys. Chem. B 2008, 112, We assume the following model for the dynamics: with a diffusive part describing rotation about the C  – C  bond: and a random-walk component that we interpret as the effect of the propagation reaction that imposes random changes to the internal angle 

Girse nd part: examples Quantum mechanical calculations conducted ad DFT level, B3LYP functional, 6-31G* basis set Dependence of internal potential on  Internal torsional potential: Dependence of  and  ’ hyperfine constants on  Spin Hamiltonian:

Girse nd part: examples