= Multiple isomorphous replacement
\b[Multiple isomorphous replacement] combines a native <X-ray diffraction> data set with data from two or more <heavy-atom derivatives> to estimate the native <structure factor> phases. In the ideal <crystal isomorphism> model, each derivative adds a constraint $|F_P+F_{H,j}|=|F_{PH,j}|$. Consistently referenced heavy-atom models can distinguish the two alternatives left by <single isomorphous replacement>.<Phasing diagrams in CCP4 Basic Maths for Protein Crystallographers>[MIR, third circle and matching origin/hand]{ref}
Continue the original example in <single isomorphous replacement>, where $F_P=4\pm3i$ both fit the native amplitude5 and first derivative with $F_{H,1}=3$. Choose a second reference $F_{H,2}=2i$. Its predicted squared amplitudes differ:
$$
|(4+3i)+2i|^2=|4+5i|^2=41,\qquad
|(4-3i)+2i|^2=|4-i|^2=17.
$$
A measured second-derivative amplitude $\sqrt{41}$ therefore selects $4+3i$ in this exact model. If the second reference had instead been real, both candidates would still have equal magnitudes after adding it. An additional derivative must contribute useful geometry, rather than merely repeat a constraint.
The heavy-atom models must share a consistent coordinate origin and handedness. Resolving a circle-crossing ambiguity does not by itself determine absolute handedness: conjugating all the amplitudes preserves their magnitudes. CCP4's diagrams explicitly retain this distinction.<Phasing diagrams in CCP4 Basic Maths for Protein Crystallographers>[MIR, conjugate phases and enantiomorphic maps]{ref}
The method has a concrete experimental history. The opening of the1960 <myoglobin> structure report describes several mercury- or gold-containing derivatives whose <X-ray diffraction> patterns were compared with the unsubstituted protein. These measurements had supplied phases for the earlier map at spacings greater than6Å; the new report advanced the analysis to2Å.<Structure of Myoglobin: A Three-Dimensional Fourier Synthesis at 2 Å. Resolution by J. C. Kendrew and others>[p422, opening paragraph]{ref} Different labeled crystals supplied information unavailable from native intensities alone.
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