A native protein crystal gives a reflection with structure-factor magnitude…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Biophysics & Medical Physics · Crystal structure, bonding and diffraction

Forces, Machines and Signals, Ch Seeing molecules: diffraction, cryo-electron microscopy and fluorescence

Problem

A native protein crystal gives a reflection with structure-factor magnitude ∣FP∣=50|F_P| = 50 (arbitrary units). A single mercury derivative, isomorphous with the native crystal, gives ∣FPH∣=68|F_{PH}| = 68 for the same reflection. From the known position of the mercury atom, its own contribution has magnitude ∣FH∣=30|F_H| = 30 and phase φH=40∘\varphi_H = 40^\circ. ((a)) Using the vector relation FPH=FP+FH\mathbf{F}_{PH} = \mathbf{F}_P + \mathbf{F}_H and the law of cosines, find the two candidate values of the native phase φP\varphi_P that the Harker construction (one circle of radius ∣FP∣|F_P| about the origin, one of radius ∣FPH∣|F_{PH}| about the tip of −FH-\mathbf{F}_H) allows. ((b)) State, in one sentence, why a single derivative leaves this two-fold ambiguity and what a second, independent heavy-atom derivative (or an anomalous-scattering measurement) does to resolve it.

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