## Dynamics and dynamical transitions in proteins

##### Date

2014

##### Authors

Vurnal, Derya

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##### Publisher

University of Delaware

##### Abstract

Neutron scattering experiments and molecular dynamics simulations are the most effective tools to explore the dynamics of hydrogen in proteins. The mean square displacement (MSD) of hydrogen (H ) in proteins has been extensively measured using neutron scattering and calculated using molecular dynamics simulations. A small MSD is observed at low temperatures and the slope of the MSD significantly increases at a specific temperature TD . This increase in the slope of the MSD is identified as a dynamical transition, and the temperature it takes place at a specific temperature which is denoted a dynamical transition temperature T D . The observed MSD in neutron scattering experiments depend on the energy resolution of the instrument. In this thesis, we first focus on the resolution dependent of the observed MSD [left angle bracket]r2 [right angle bracket]exp in neutron scattering experiments. We propose a method for obtaining the intrinsic MSD [left angle bracket]r2 [right angle bracket]exp of H, which is independent of the resolution of the instrument employed, in proteins. The intrinsic MSD is defined as the infinite time value of MSD which appears in the well-known Debye-Waller factor. In this method, a model of the resolution broadened elastic incoherent dynamic structure factor SR (Q, ω = 0) is developed to extract the intrinsic MSD from the resolution dependent data. The model contains the intrinsic MSD, the instrument resolution width and a relaxation frequency characterizing the motions of H in proteins. The model of SR (Q, ω = 0) is fitted to the resolution broadened DSF data already published in the recent literature and the intrinsic MSD in three proteins was successfully obtained. Later, we constructed a model for the incoherent intermediate scattering function I (Q, t ) to obtain the intrinsic, long-time MSD of H in proteins from finite time molecular dynamics simulations. In the literature, the simulated MSD increases with increasing time and does not reach a certain limiting value at even 10 ns. The infinite time MSD, [left angle bracket]r2 [right angle bracket], is the long time value of the simulated MSD. The model I ( Q, t ) fits to the simulated Iinc ( Q, t ) to obtain the intrinsic long-time MSD [left angle bracket] r2 [right angle bracket] which is as a parameter in the model I (Q, t ). The intrinsic MSD [left angle bracket] r2 [right angle bracket] of hydrated lysozyme powder (h = 0.4 g water/g protein) over a temperature range between 100 K and 300 K obtained from data out to 1 ns and to 10 ns is found to be the same. The intrinsic [left angle bracket]r2 [right angle bracket] is approximately twice the value of the MSD that is reached in simulations after times of 1 ns. The simulated MSD in 1 ns corresponds to the observed MSD measured by neutron scattering instruments having 1 μe V energy resolution width. The observed MSD [left angle bracket]r2 [right angle bracket]exp extracted from an elastic dynamical structure factor measured in neutron scattering experiments is also found to be Q-dependent. In the second part of this thesis, we analyse the possible origins of the Q-dependence of the observed MSD [left angle bracket]r 2 [right angle bracket]exp . We show that this dependence does not arise from the Gaussian approximation, commonly used in the analysis of the neutron scattering data. The Q-dependence of the MSD is the artificial consequence of neglecting the dynamical diversity in the model that are used to analyse the data. Finally, we illustrate the dynamical transition in proteins by reproducing the change in the slope of MSD versus temperature at TD within a simple model of vibration, a single particle in an anharmonic potential. Using Self-Consistent-Harmonic theory, we investigate dynamics of a particle in different potential models. A simple Gaussian potential or a potential containing a hard wall and a soft wall is particularly effective to reproduce the change in the slope of MSD versus temperature. The MSD data of myoglobin and purple membrane in the literature is reproduced well using a potential containing hard wall and a Gaussian potential.