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Proteins are complex molecular systems whose internal dynamics is

characterized by a vast spectrum of time scales, ranging from

sub-picoseconds for vibrations of chemical bonds to seconds and beyond

for large conformational rearrangements. Using a "minimalistic"

multi-time scale model for the relaxation dynamics of

proteins~[1,2], we show here that even small changes due to external stress, such as temperature, solvent modification or ligand binding,

can be elucidated by quasi-elastic neutron scattering (QENS).

The neutron intermediate scattering function is here written in the form $F(q,t)=EISF(q)+(1-EISF(q))\phi(q,t)$, where $EISF(q)$ is the elastic incoherent structure factor which gives information about the motional amplitudes of the hydrogen-atoms in hydrogen-rich systems,

and $\phi(q,t)$ is a relaxation function which is chosen to be

the "stretched" Mittag-Leffler function, $\phi(q,t)=E_\alpha(-(t/\tau)^\alpha)$ in order to account

for the asymptotically self-similar relaxation dynamics of proteins.

An important technical point is the estimation of the EISF

on the basis of its measured counterpart and the model

parameters

of the relaxation function, which are the $q$-dependent

form parameter $\alpha $ and the time scale parameter $\tau$ [3].

Our first example concerns the intrinsically disordered protein

Myelin Basic Protein (MBP) in solution, which is studied

in pure D$_2$O-buffer and in a mixture of D$_2$O-buffer with

30\%\, deuterated Trifluoroethanol at different temperatures,

in order to evaluate the impact of formation of secondary structure

elements on the internal dynamics [4]. The second example concerns

the change of the internal dynamics of myoglobin in solution

in presence of denaturing agents, and the third example

is devoted to understanding the functional dynamics

of the enzyme Phosphoglycerate kinase. Here the model allows

for determining unambiguously the amplitude of the

inter-domain fluctuations which are important for its catalytic function.

References:

[1] G. R. Kneller. PNAS USA, vol. 115, no. 38, pp. 9450-9455, 2018

[2] M. Saouessi, J. Peters, and G. R. Kneller. J. Chem. Phys.,

vol. 150, p. 161104, 2019.

[3] A. N. Hassani, A. M. Stadler, and G. R. Kneller. J. Chem. Phys.

vol. 157, p. 134103, 2022.

[4] A. N. Hassani, L. Haris, M. Appel, T. Seydel, A. M. Stadler,

and G. R. Kneller. J. Chem. Phys. vol. 156, p. 025102, 2022.