A note on the Debye–Waller theory

Using the fluorescence red edge effect to assess the long-term stability of lyophilized protein formulations

Nanosecond relaxation processes in sugar matrices are causally linked through diffusional processes to protein stability in lyophilized formulations. Long-term protein degradation rates track mean-squared displacement (⟨u(2)⟩) of hydrogen atoms in sugar glasses, a parameter describing dynamics on a time scale of picoseconds to nanoseconds. However, measurements of ⟨u(2)⟩ are usually performed by neutron scattering, which is not conducive to rapid formulation screening in early development. Here, we present a benchtop technique to derive a ⟨u(2)⟩ surrogate based on the fluorescence red edge effect. Glycerol, lyophilized trehalose, and lyophilized sucrose were used as model systems. Samples containing 10(-6) mole fraction of rhodamine 6G, a fluorophore, were excited at either 532 nm (main peak) or 566 nm (red edge), and the ⟨u(2)⟩ surrogate was determined based the corresponding Stokes shifts. Results showed reasonable agreement between ⟨u(2)⟩ from neutron scattering and the surrogate from fluorescence, although deviations were observed at very low temperatures. We discuss the sources of the deviations and suggest technique improvements to ameliorate these. We expect that this method will be a valuable tool to evaluate lyophilized sugar matrices with respect to their ability to protect proteins from diffusion-limited degradation processes during long-term storage. Additionally, the method may have broader applications in amorphous pharmaceutical solids.

A Surrogate for Debye-Waller Factors from Dynamic Stokes Shifts

We show that the short-time behavior of time-resolved fluorescence Stokes shifts (TRSS) are similar to that of the intermediate scattering function obtained from neutron scattering at q near the peak in the static structure factor for glycerol. This allows us to extract a Debye-Waller (DW) factor analog from TRSS data at times as short as 1 ps in a relatively simple way. Using the time-domain relaxation data obtained by this method we show that DW factors evaluated at times ≥ 40 ps can be directly influenced by α relaxation and thus should be used with caution when evaluating relationships between fast and slow dynamics in glassforming systems.

The thermodynamic and vibrational properties of H2O ice and D2O ice

The thermodynamic properties of H2O ice and D2O ice have been analysed in terms of the effective harmonic spectrum of lattice vibrations. In this approximation the vibrations are treated as harmonic, but with frequencies which are both volume and temperature dependent. Analysis of the heat capacity then gives properties of the effective spectrum at 0 °K, and the average temperature dependence of the spectrum at constant volume. It has been shown that the parts of the spectrum associated with translational and librational motion of the molecules are distinct, and their contributions to the thermodynamic properties may be separated. The momentsv-2,v-1,v,v2,v4andv6together with the geometric mean frequencyvghave been derived for the separate parts of the spectrum. The reduced librational moments, (vnL)1/n, are independent ofnwithin the experimental uncertainty. This shows that the spectrum of librational frequencies consists of a fairly sharp and symmetrical band. From the very low temperature data the coefficients of the terms inv2,v4andv6in the low frequency expansion of the translational spectrum have been derived, together with the position and approximate weight of the first peak in the spectrum. The ratios of the moments for H20 and D2O show that the intermolecular forces are stronger for D2O than for H2O. From the thermal expansion data, by analysing the Gruneisen function y(T, V) =βV/XsCp, values for some of the parameters y(n) = — (l/n)dlnvn/dlnV, which describe the volume dependence of the moments, have been obtained for the separate parts of the spectrum. These are useful in estimating the volume dependence of various crystal properties. The properties of the effective harmonic spectrum and its average total temperature dependence (including thermal expansion effects) are in good agreement with the results of a variety of spectroscopic experiments. The moments derived from the thermodynamic data have been used to calculate the mean square amplitudes of vibration of O, H and D atoms and the results are in good agreement with the results of X-ray and neutron scattering experiments.