W0087

On Static and Dynamic Elastic Eonstants of Protein Crystals. A.A. Chernov, Universities Space Research Association at NASA, Marshall Space Flight Center, Mail Code SD46, MSFC, AL 35812, alex.chernov@msfc.nasa.gov

Reciprocal compressibilities of protein molecules in solution, i.e. elastic bulk modulus, K_p ≅ 8 – 26 GPa. Bending or low frequency (2 – 60 Hz) vibration measurements of Young moduli, E, for crystals result in E = 0.1 – 1 Gpa. On the other hand, from resonance light scattering on thermal hypersound at frequency ~10¹⁰ Hz, waves in a lysozyme crystal elastic constants are 6 – 12 GPa. Contradiction disappears if we realize that intracrystalline water has or does not have enough time to flow between compressed and expanded regions under static and dynamic conditions, respectively. This flow is due different compressibilities of molecules (K_p = 26 GPa for lysozyme) and water (K_w = 7 GPa). The water flow redistribution time, τ, through intermolecular channels of assumed triangular cross section with perimeter 3l was estimated to be

τ ≅ 200 (η λ Ω/l⁴) dθ/dp ≅ 10^-8 s

at water viscosity η = 10^-2 g/cm s, hypersound wavelength λ ≅ 2.3·10⁵ cm, intracrystalline water volume per protein molecule Ω ≅ 10^-20 cm³, l ≅ 10^-7 cm, and dθ/dp = θ(1 - θ) (1/K_w – 1/K_p), θ is water fraction in crystal, p is pressure. Thus, at high frequency (≥ 10¹⁰ s – 1), protein lattice and intermolecular water vibrate as a whole with the average bulk modulus K = (θ/K_w + (1 - θ)/K_p) ≅ 12 GPa for lysozyme.