Fundamental relationships

Engelman, D. M.; Moore, P. B.

doi:10.1107/97809553602060000701

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 19.4, pp. 438-439 | 1 | 2 |

Section 19.4.2. Fundamental relationships

D. M. Engelman^a ^* and P. B. Moore^b

^aDepartment of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06520, USA, and ^bDepartments of Chemistry and Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06520, USA
Correspondence e-mail: don@paradigm.csb.yale.edu

19.4.2. Fundamental relationships

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For most purposes, a dilute macromolecular solution can be thought of as a macromolecular gas, and for that reason it is appropriate to apply Debye's theory for gas scatter to macromolecular solutions (Debye, 1915). Debye's master equation can be cast into neutron terms as follows: $[I(Q) \propto I_{0}\textstyle\sum \sum\displaystyle b_{i}b_{j}\sin(Qr_{ij})/(Qr_{ij}), \eqno(19.4.2.1)]$ where I(Q) is the amount of scattered radiation observed at Q, $[I_{0}]$ is the intensity of the incident beam, $[b_{i}]$ and $[b_{j}]$ are the scattering lengths of the ith and jth atoms in the molecule, $[r_{ij}]$ is the distance between atoms i and j, and $[Q = (4\pi / \lambda )\sin \theta]$ , λ being the wavelength of the radiation used and θ being half the scattering angle. When applied to molecules in solution, both summations must include not only all the atoms that are covalent components of the molecule in question, but also all the associated solvent atoms, because when a macromolecule dissolves, the inhomogeneity created includes the counterions associated with it, its solvation layer etc. Equation (19.4.2.1) holds for a single molecule; if the number of molecules contributing to scattering in some sample is N, the scattering profile measured will be N times the profile due to a single molecule. The inhomogeneities responsible for small-angle scatter have linear dimensions of the order of 10 Å or more and, hence, have volumes that contain large numbers of atoms. In addition, interatomic spacings cannot be resolved using small-angle data. Thus, it is appropriate to discuss small-angle scattering in terms of electron densities for X-rays or scattering-length densities for thermal neutrons. The scattering-length density of a volume, ρ, is given by $[\rho \equiv \textstyle\sum\displaystyle b_{i}/V, \eqno(19.4.2.2)]$ where $[b_{i}]$ is the scattering length of the ith atom in volume V and the summation runs over all atoms in the volume.

Recasting the Debye equation in terms of scattering lengths, one obtains $[I(Q) \propto I_{0} \textstyle\int \int\displaystyle \rho ({\bf r}_{i}) \rho ({\bf r}_{j}) [\sin (Q{r}_{ij})/(Q{r}_{ij})] \hbox{ d}V_{i} \hbox{ d}V_{j},]$ where $[\rho({\bf r}_{i})]$ and $[\rho({\bf r}_{j})]$ are the scattering-length densities in volume elements whose positions are described by vectors $[{\bf r}_{i}]$ and $[{\bf r}_{j}]$ , $[r_{ij} = |{\bf r}_{i} - {\bf r}_{j}|]$ , and both integrals run over the volume of the entire macromolecule plus the surrounding perturbed volume. Both this equation and equation (19.4.2.1) implicitly assume that the medium surrounding each molecule is a vacuum, which is not true for the molecules in solution. The effect of solvent on low-angle scattering can be taken care of by subtracting the average scattering length of the solvent, $[\rho_{0}]$ , from the scattering-length densities within the molecule. Thus, $[{I(Q) \propto I_{0} \textstyle\int \int\displaystyle [\rho({\bf r}_{i}) - \rho_{0}][\rho({\bf r}_{j}) - \rho_{0}]_{j} [\sin (Q{r}_{ij})/(Q{r}_{ij})] \hbox{ d} V_{i} \hbox{ d}V_{j}.} \eqno(19.4.2.3)]$ The quantity $[[\rho({\bf r}_{i}) - \rho_{0}]]$ is a contrast, and, as will be shown, contrast manipulation is a major component of SANS experiments.

Equation (19.4.2.3) can be evaluated a second way, because the $[(\sin x)/x]$ term in the integral depends only on the distances between volume elements, not on their locations in space. Thus, a function [p(r)] can be defined as follows: $[p(r) \equiv \textstyle\int \int\displaystyle [\rho ({\bf r}_{i}) - \rho_{0}] [\rho ({\bf r}_{i} + {\bf r}) - \rho_{0}] \hbox{ d}V_{i} \hbox{ d}V{\bf r},]$ where the integral in r runs over all r such that $[|{\bf r}| = r]$ , and the integral in $[{\bf r}_{i}]$ runs over the entire molecular volume. Written in terms of [p(r)] , equation (19.4.2.3) becomes $[I(Q) \propto I_{0}\textstyle\int\displaystyle p(r) [\sin (Qr)/(Qr)] \hbox{ d{\it r}}, \eqno(19.4.2.4)]$ where the integral runs from [r = 0] to $[r_{\max}]$ , the maximum atom-to-atom length within the molecule.

Note that if contrast was constant within a macromolecule, [p(r)] would be proportional to the distribution of interatomic distances in the molecule, and for that reason [p(r)] is often called the length distribution . Note also that [p(r)] is simply the molecule's Patterson function, rotationally averaged about its origin. Note, finally, that [p(r)] is the summation of a large, but finite, number of sharp, discrete interatomic distance peaks, each with its own weight. If the individual interatomic peaks in this `length spectrum' could be assigned, i.e., if the atoms responsible for each one could be identified, it would be possible to determine the three-dimensional structure of the molecule in question, save for uncertainty about its hand.

Since solution-scattering profiles can be computed by sine transformations of length distributions , it is reasonable to hope that a transformation might exist that enables one to compute length distributions once solution-scattering profiles have been measured. There is (Debye & Bueche, 1949; Debye & Pirenne, 1938): $[p(r) \propto r\textstyle\int\displaystyle QI(Q)\sin (Qr) \hbox{ d}r. \eqno(19.4.2.5)]$ Two practical issues must be addressed when carrying out the operation implied by equation (19.4.2.5) because the integral it contains runs from [Q = 0] to ∞. Firstly, scattering is never measured at [Q = 0] due to interference with the direct beam. Secondly, the largest value of Q for which [I(Q)] is measured is always less than ∞. The absence of data at very small values of Q is easily addressed, because a soundly based method exists for extrapolating the low-angle data to [Q = 0] (see below). The lack of data at high Q is harder to cope with, but it can be dealt with approximately using Porod's Law (Porod, 1951, 1952) and the impact of its absence on molecular parameters deduced from small-angle data is easy to estimate. In any case, it is important to realize that length distributions represent the sum total of the information that can be extracted from solution-scattering experiments.

The problem of extrapolating small-angle data to [Q = 0] was solved by Guinier (1939). He demonstrated that, at very small angles, $[I(Q) \propto I(0)\exp [-(QR_{g})^{2}/3], \eqno(19.4.2.6)]$ where $[R_{g}]$ is the radius of gyration, and $[{R_{g} \equiv (\{\textstyle\int\displaystyle [\rho ({\bf r}) - \rho_{0}] |{\bf r}|^{2} \hbox{ d}V\} / \{\textstyle\int\displaystyle [\rho ({\bf r}) - \rho_{0}] \hbox{ d}V\})^{1/2}.}\hfill\!\! \eqno(19.4.2.7)]$ The origin of the vector r in this equation is the centre of gravity of the macromolecule's scattering-length density distribution, i.e., it is the point where $[0 = \{\textstyle\int\displaystyle [\rho ({\bf r}) - \rho_{0}]{\bf r} \hbox{ d}V\} / \{\textstyle\int\displaystyle [\rho ({\bf r}) - \rho_{0}] \hbox{ d}V\}.]$ It follows from equation (19.4.2.6) that if the lowest-angle data collected are plotted in the form ln[ [I(Q)] ] versus $[Q^{2}]$ , a straight line should result, the slope of which is $[(R_{g}^{2}/3)]$ and the intercept of which at [Q = 0] is [I(0)] . Note that data have to be obtained at scattering angles well inside the region where $[I(Q) \sim I(0)/2]$ in order for this formula to hold; if the data are thus obtained, a radius of gyration estimate will emerge. The radius of gyration of an object is the root-mean-squared distance between its centre of gravity and the elements of which it is composed.

As might be expected, [I(0)] and $[R_{g}]$ can also be computed from [p(r)] . Consider the magnitude of [I(Q)] at [Q = 0] . Since the $[\sin x/x]$ term in equation (19.4.2.4) is 1 at [Q = 0] , $[{I(0) \propto I_{0}\textstyle\int \int\displaystyle [\rho ({\bf r}_{i}) - \rho_{0}] [\rho ({\bf r}_{j}) - \rho_{0}] \hbox{ d}V_{i} \hbox{ d}V_{j} = \textstyle\int\displaystyle p(r) \hbox{ d}r.} \eqno(19.4.2.8)]$ Thus, [I(0)] , the forward scatter, is proportional to the integral of the length distribution. It is easy to show that $[R_{g}]$ equals $[(M/2)^{1/2}]$ , where M is the second moment of [p(r)] given by $[M = [\textstyle\int\displaystyle r^{2}p(r) \hbox{ d}r]/[\textstyle\int\displaystyle p(r) \hbox{ d}r]. \eqno(19.4.2.9)]$ The average atom-to-atom distance in a molecule, $[r_{\rm ave}]$ , is easy to compute if [p(r)] is known from $[r_{\rm ave} = [\textstyle\int\displaystyle rp(r) \hbox{ d}r]/[\textstyle\int\displaystyle p(r) \hbox{ d}r]. \eqno(19.4.2.10)]$

The reason forward scatter, [I(0)] , is interesting is its dependence on molecular weight. As equation (19.4.2.8) suggests, the forward scatter measured for a sample is proportional to N times the square of the product of the average contrast between a molecule and its solvent and the molecular volume, where N is again the number of molecules contributing to the signal observed. Since average contrasts can be estimated from chemical compositions and partial specific volumes, [I(0)] measurements can be used to estimate molecular weights. If the [I(0)] values of solutions of a set of molecules of similar chemical composition are compared, it will be found that [I(0)] divided by the weight concentration of each sample is proportional to molecular weight.

This procedure for estimating molecular weights can fail. Suppose $[\int[\rho({\bf r}_{i}) - \rho_{0})\hbox{ d}V = 0]$ , i.e., the scattering-length density of the solvent is the same as the average scattering-length density of the macromolecule. Then [I(0)] will be zero, the solution-scattering profile will lack a peak at small angles and no molecular-weight estimate will result. Under these conditions, the macromolecule is said to be `contrast matched'. It is easy to contrast-match biological macromolecules in the context of SANS experiments, because all biological macromolecules that have not been labelled with ²H have average scattering-length densities between those of H₂O and D₂O (see below).

References

Debye, P. (1915). Zerstreuung von Röntgenstrahlen. Ann. Phys. (Leipzig), 46, 809–823.Google Scholar

Debye, P. & Bueche, A. M. (1949). Scattering by an inhomogeneous solid. J. Appl. Phys. 20, 518–525.Google Scholar

Debye, P. & Pirenne, M. H. (1938). Über die Fourieranalyse von interferometrischen Messungen an freien Molekülen. Ann. Phys. (Leipzig), 33, 617–629.Google Scholar

Guinier, A. (1939). Diffraction of X-rays of very small angles – application to the study of ultramicroscopic phenomena. Ann. Phys. (Paris), 12, 161–237.Google Scholar

Porod, G. (1951). The X-ray small-angle scattering of close-packed colloid systems. I. Kolloid-Z. 124, 83–144.Google Scholar

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International Tables for Crystallography (2006). Vol. F. ch. 19.4, pp. 438-439