Crick’s excel­lent paper on X-ray dif­frac­tion

An inter­est­ing paper by Fran­cis Crick almost suc­cess­fully explain­ing X-ray dif­frac­tion to biol­o­gists.

In the process of re-for­mu­lat­ing the cur­ricu­lum for spe­cial­i­sa­tion in con­densed matter physics at our uni­ver­sity, I chanced upon a review paper by Fran­cis Crick titled X-ray dif­frac­tion of pro­tein crys­tals’. One of the papers I am teach­ing this year has a lot to do with the the­o­ret­i­cal aspects of this: the phe­nom­ena of scat­ter­ing, the struc­ture factor of an atom, exper­i­men­tally observed inten­si­ties, deter­mi­na­tion of crys­tal sym­me­tries and their rela­tion­ships, and so on.

Inter­est­ingly, Crick’s paper begins with this con­cise note on why biol­o­gists cannot care less about X-ray dif­frac­tion and seems to me to be in agree­ment with an argu­ment I have made for years now that biol­o­gists and chemists seem unusu­ally sat­is­fied with a sur­face-level under­stand­ing of phe­nom­ena, because if they ever decide to dig any deeper they would all end up at quarks and become physi­cists. From Crick:

The X-ray study of pro­tein crys­tals is a dif­fi­cult and highly spe­cial­ized field. Even­tu­ally, when the struc­ture of a few pro­teins has been unrav­eled, the results will be of vital inter­est to ensy­ti­ol­o­gists, since they should give infor­ma­tion about the spa­tial arrange­ment of the active site” of the enzyme. But mean­while X-ray meth­ods will be only of sec­ondary inter­est to enzy­mol­o­gists.

The paper is rather vague in its descrip­tions, which makes it an excel­lent read for anyone start­ing out with it. Now that is not to say Crick did not know any better than what he wrote; in fact, he prob­a­bly know a lot more than he gave away in this paper. But it is mildly sur­pris­ing, even if not pleas­antly so, that only two equa­tions exist in a paper explain­ing how the mol­e­c­u­lar weight, den­sity, com­po­si­tion and shape of crys­tals are deter­mined.

Both these equa­tions are simple: one of them, used to mea­sure the mol­e­c­u­lar weight $A$ of a crys­tal in terms of an empir­i­cal con­stant denoted by $k$, is given by $$A = \frac{1}{k}\,\frac{V}{n}$$for a unit cell of volume $V$ with $n$ asym­met­ric units within it. How­ever, the pro­ce­dure is not as straight­for­ward as this equa­tion might make it seem.

For starters, the most effec­tive way to deter­mine the mol­e­c­u­lar weight of a crys­tal is to use it in a solu­tion and resort to SAXS. This is nowhere near as accu­rate as straight up mass spec­troscopy, but it works more prac­ti­cally since crys­tals are, quite often, found/​grown in a solu­tion. In this case, the for­mula really becomes $$A_x = \frac{I(0)_x}{C_x}\,\frac{A_c C_c}{I(0)_c}$$as out­lined in the paper,Accu­racy of mol­e­c­u­lar mass deter­mi­na­tion of pro­teins in solu­tion by small-angle X-ray scat­ter­ing’, by E. Mylonas and D. Sver­gun. The sub­script $x$ refers to the unknown crys­tal and $c$ to the known one. To mea­sure the mol­e­c­u­lar weight then we use such a com­par­i­son tech­nique between two crys­tals involv­ing their zero-angle scat­ter­ing inten­si­ties $I(0)$ and the con­cen­tra­tions $C$ of their respec­tive solu­tions. In other words, this is some­thing like Crick’s for­mula, but not quite.

Of course Crick’s own paper goes on to qual­i­ta­tively describe that crys­tals can be used in three states while mea­sur­ing their mol­e­c­u­lar weight: wet (in a solu­tion), air-dried (with sol­vent mol­e­cules present between the lat­tice), and vacuum-dried (with no left­over sol­vent mol­e­cules what­so­ever), point­ing to the fact that the former is a rather con­ve­nient method and that, in the wet crys­tal method, the Adair – Adair for­mula helps deter­mine the ratio of water to dry crys­tal, usu­ally in grams: $$w = \left( \frac{D_p — D_b}{D_b — D_w}\right) \frac{D_w}{D_p}$$where $D_b$ is the den­sity of the crys­tal, $D_w$ the den­sity of water, and $D_p$ the rec­i­p­ro­cal of the par­tial spe­cific volume of the crys­tal.

How­ever, $D_p$ itself demands an exper­i­ment for its value to be deter­mined, and Crick care­fully avoids some basic cal­cu­lus here: the par­tial spe­cific volume is the par­tial deriv­a­tive of the volume of a com­pound with respect to its mass. A den­sit­o­me­ter, an instru­ment that mea­sures trans­mit­tance through a solid, is gen­er­ally used to find this out with a buffer of (read, a cham­ber filled with) suit­able mate­r­ial and another of the crys­tal in ques­tion and then this: $$D_p = \frac{1}{D^\prime}\left( 1 — \frac{D_c — D^\prime}{C_c} \right)$$with $D^\prime$ being the den­sity of the buffer and the rest of the terms as explained ear­lier.

This is not a cri­tique of Crick’s paper. In fact, I thought it would be useful enough and have added it in as quick read­ing mate­r­ial for stu­dents of the intro­duc­tory course since there are sev­eral good sum­maries Crick makes (e.g. pp. 142 and 143 in the linked paper). What I wanted to pro­vide were some clar­i­fi­ca­tions that the meth­ods of XRD are gen­er­ally more com­pli­cated than descrip­tion makes them seem. But what really makes this useful read­ing is that Crick’s sum­maries are valid even for today’s SAXS exper­i­ments.

This is one of those basic overviews I wish some­one had given me when I was study­ing crys­tal­log­ra­phy: it is a great read for some­one who is start­ing out in the field but the glar­ing omis­sion of math­e­mat­ics alone should be enough to sug­gest to any reader that it should only be used as a primer and not as spe­cialised read­ing. Over­all, a worth­while read.

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