Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 376-377

Section Structure determination in single-particle reconstruction

P. A. Penczekg Structure determination in single-particle reconstruction

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The goal of single-particle reconstruction is to determine the 3D electron-density map f of a biological macromolecule such that its projections agree in a least-squares sense with a large number of collected 2D electron-microscopy projection images, [d_n \in R^{n^2 } ](n = 1, 2, …, N), of isolated (single) particles with random and unknown orientations. Thus, we seek a least-squares solution to the problem stated by ([link] [or, equivalently, in Fourier space, to ([link]]. This is formally written as a nonlinear optimization problem (Yang et al., 2005[link]),[\eqalignno{\min \limits_{\psi _n ,\theta _n ,\varphi _n ,t_{x_n } ,t_{y_n } ,f,\Delta f_n ,q, \ldots } L(\psi _n ,\theta _n ,\varphi _n ,t_{x_n } ,t_{y_n } ,f,\Delta f_n ,q, \ldots ) &\cr \equiv{\textstyle {1 \over 2}}\textstyle\sum\limits_{n = 1}^N {\left\| {s_n ( {\bf{x}}) * e_n ( {\bf{x}} ) * \textstyle\int {f( {{\bf{T}}_n {\bf{r}}} )\,{\rm d}z} - d_n ( {\bf{x}} )} \right\|^2 } . &\cr&&(} ]The factor of ½ is included merely for convenience. The objective function in ([link] is clearly nonlinear due to the coupling between the orientation parameters [\psi _n ,\theta _n ,\varphi _n ,t_{x_n } ,t_{y_n } ] (n = 1, 2, …, N) and the 3D density f.

The parameters in ([link] to be determined can be separated into two groups. (1) The orientation parameters [\psi _n ,\theta _n ,\varphi _n ,t_{x_n } ,t_{y_n } ] that have to be determined entirely by solving ([link] and for which there are no initial guesses, and the structure f itself, for which we may or may not have an initial guess. The number of parameters in this group is very large: n3 + 5m. Note that in single-particle reconstruction, the number of projection data m is far greater than the linear size of the data in pixels, i.e., [m\gg n]. (2) Various parameters which we will broadly call the parameters of the image formation model ([link]–([link]: the defocus settings of the microscope [\Delta f_n ], the amplitude contrast ratio q and, if analytical forms of the envelope function E, the power spectrum of the background noise M, or the structure F are adopted, the parameters of these equations. Some of the parameters in the second group are usually known very accurately or can be estimated from micrograph data before one attempts to solve ([link] (see Section[link]), but they can also be refined during the structure determination process [for the method for correcting the defocus settings, see Mouche et al. (2001[link])].

Owing to the very large number of parameters in ([link] and the nonlinearities present, one almost never attempts to solve the problem directly. Instead, structure determination using the single-particle technique involves several steps. (i) The macromolecular complex is prepared with a purity of at least 90%. (ii) The sample is flash-frozen in liquid ethane. Alternatively, cryo-negative stain techniques or traditional negative stain methods can be used. (iii) Pictures of the macromolecular complexes are taken. (iv) Exhaustive analysis of 2D particle images aimed at increasing the SNR of the data and evaluation of the homogeneity of the sample is performed. (v) An initial low-resolution model of the structure is established using either experimental techniques or computational methods. (vi) The initial structure is refined in order to increase the resolution using an enlarged data set. Only in this step does one attempt to minimize ([link] more or less directly. (vii) Visualization and interpretation of the resulting 3D electron-density map is the last step; it often involves docking of X-ray structures of molecules into EM density maps in order to reveal the arrangement of known molecules within the EM envelope (Fig.[link]). As within the weak-phase-object approximation of the image formation in EM the relation between densities in collected images and the 3D electron density of the imaged macromolecule is linear [([link]], all data-processing methods employed in the structure determination project should be linear, so the densities in the cryo-EM 3D model can be interpreted in terms of the electron density of the protein.


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Typical steps performed in a single-particle cryo-EM structure determination project.

In the actual single-particle project not all the steps have to be executed in the order outlined above. The technique has proved to be particularly useful in studies of functional complexes of proteins whose base state is known to a certain resolution or even of functional complexes whose atomic (X-ray crystallographic) structure is known. In these cases, steps (iv) and (v) can be omitted and the structure of the functional complex (for examples with ligands bound to it) can be relatively easily determined using the native structure as a starting point for step (vi).

In addition to difficulties with obtaining good cryo-EM data, the technique is computationally intensive. The reason is that in order to obtain a sufficient SNR in the 3D structure, processing of hundreds of thousands of EM projection images of the molecule might be necessary. For each, five orientation parameters have to be determined, and this is in addition to determination of the image-formation parameters required for the optimization of correlation searches. In effect, it is not unusual for single-particle projects to consume weeks of the computer time of multiprocessing clusters. This also explains why the knowledge of the base structure simplifies the work to a large degree: when it is known, initial values of the orientation parameters can be easily established, reducing not only the computational time, but also possibilities of errors in the structure-determination process.


Mouche, F., Boisset, N. & Penczek, P. A. (2001). Lumbricus terrestris hemoglobin – the architecture of linker chains and structural variation of the central toroid. J. Struct. Biol. 133, 176–192.
Yang, C., Ng, E. G. & Penczek, P. A. (2005). Unified 3-D structure and projection orientation refinement using quasi-Newton algorithm. J. Struct. Biol. 149, 53–64.

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