Relative roles of multiple scattering and Fresnel diffraction in the imaging of small molecules using electrons, Part II: Differential Holographic Tomography
Introduction
Cryogenic electron microscopy (cryo-EM) has recently reached the true atomic spatial resolution milestone [1], [2]. As the spatial resolution in high-energy transmission EM (TEM) approaches one ångström, the corresponding depth of focus (also known as the depth of field) tends to become shallower than the longitudinal (along the direction of propagation of the illuminating electron wave) extent of a typical imaged sample, such as a protein molecule. For example, at a spatial resolution of Δ = 1 Å and a wavelength of λ = 0.025 Å (for electrons at 200 keV energy), the depth of focus is equal to [3]. It has been suggested previously that, as a result of the shallow focus, the in-molecule Fresnel diffraction (free-space propagation) becomes a significant factor that cannot be ignored in the reconstruction of the sample from its defocused images [3], [4], [5], [6]. In other words, the propagation of the incident electron wave through the sample can no longer be satisfactorily approximated by projection integrals of the electrostatic potential along straight lines. Instead, the Fresnel diffraction of the electron beam inside the molecule must be explicitly taken into account in order to achieve an accurate 3D reconstruction. The introduction of specially designed aberrations into the EM illumination system can increase the depth of focus [7], [8], but this is still unlikely to fully eliminate the in-molecule Fresnel diffraction effects at atomic resolution. Moreover, as we argue below (see also [9]), attempts to increase the depth of focus in EM tomography may in some cases be counter-productive, since a shallow focus improves the spatial resolution by enhancing the longitudinal localization of individual atoms in the reconstruction of the molecule.
It is known that, in the transmission electron microscopy of sufficiently thin non-crystalline specimens, "diffraction tomography" (DT) [10], [11], [12] can be used to properly account for the Fresnel diffraction inside the sample and correctly reconstruct the three-dimensional (3D) distribution of the complex refractive index, or, equivalently, the spatial distribution of the electrostatic potential in electron imaging. The DT approach can be based on the first Rytov or first Born approximation, instead of the projection approximation utilized in conventional computed tomography (CT). The use of DT for 3D reconstruction of small molecules was discussed in detail in the first part of this work [9]. Here, we develop and test a different method, termed "Differential Holographic Tomography" (DHT), aimed at reconstruction of the 3D structure of small molecules from TEM defocus series collected at a small number of different orientations of the molecule. This method is developed in detail in the next section, and here we only briefly discuss the main physical considerations that underpin the method.
We showed in [9] that, in agreement with the statements found in [3], [4] and elsewhere, in high-resolution electron microscopy (EM) of "sparsely localized" structures, such as small biological molecules, the total contrast distribution in a defocused image can typically be well approximated by an incoherent sum of contrast functions corresponding to individual atoms. This fact can be used as a basis for reconstructing the 3D distribution of the electrostatic potential in a molecule from transmission images collected at different defocus distances and different orientations of the molecule, effectively performing the reconstruction separately for each atom before adding together the individual reconstructed atomic potentials. This general approach is similar to the "independent atoms model" which has been used for crystallographic phase retrieval, sometimes in conjunction with the atomicity constraint, for a long time (see e.g. [13], [14]). However, we note that the independent atoms approximation, which ignores molecular effects, is often applied in a far-field diffraction regime, where the interference of the waves scattered by different atoms is essential. In contrast, the present method is concerned with the Fresnel diffraction region and ignores the interference effects, thus using an incoherent independent atoms approximation. The validity of such an approximation was studied previously in the context of the first Born approximation [9] and is also considered in the next section of this paper.
In order to be able to perform independent localized reconstruction of the potential in the vicinity of each atom, one can collect, for a given orientation of the molecule, one or more defocused images along the propagation direction of the transmitted electron wave. These images can be used to perform computational phase retrieval, e.g. with the help of the Iterative Wave Function Reconstruction (IWFR) method from [15], the contrast transfer function (CTF) based L2-difference minimization method from [16] or the single-image phase retrieval method from [17]. The complex wave amplitude thus reconstructed at a given rotational orientation of the molecule can then be numerically back-propagated to the vicinity of each atom and the contrast function can be evaluated in that vicinity. As the local distribution of the electrostatic potential in the neighborhood of an atomic nucleus is essentially spherically symmetric, its reconstruction from the localized contrast function becomes relatively trivial. In particular, we will show that such a reconstruction can be obtained by summing up slightly defocused local contrast functions obtained at different orientations (that is, averaging them over the illumination directions) and then taking the inverse 3D Laplacian of the sum. The latter operation effectively compensates for the local Transport of Intensity equation (TIE) type differential phase contrast and recovers the potential from its Laplacian [18]. The differential nature of this method for reconstruction of the electrostatic potential is essential in the case of phase objects, such as atoms illuminated by high-energy electron waves, since they do not produce intensity contrast in the exact in-focus position.
By construction, the electrostatic potential reconstructed by the proposed method is localized to narrow regions around the atomic positions along the direction of propagation of the electron wave through the molecule at each orientation. This localization property of the reconstruction algorithm allows one to significantly relax the sampling conditions with respect to the number of required rotational orientations. In the case of conventional CT [19], the usual Nyquist conditions relate the required number of angular steps, na, over 180 degrees of rotation to the number of pixels in the detector rows, nx, via the well-known sampling relationship . In contrast, in our DHT algorithm, the number of required angular positions becomes almost independent of the number of detector pixels, due to the longitudinal localization of the back-propagated contrast function. Indeed, the relevant number of rotational positions here is related to the size of the vicinity of an individual atom inside which its electrostatic potential is significant, instead of being related to the size of the whole molecule. The quality of reconstruction using the DHT method is still likely to depend on the size of the molecule. However, the primary mechanism for the deterioration of the DHT reconstruction quality with the increase in the size of the molecule is in this case related to the increase of the contribution of multiple scattering effects to the contrast function. These effects include the "shading" of atoms by one another along the lines of tomographic projections. The negative influence of these multiple scattering effects on the reconstruction quality can be alleviated by increasing the number of angular orientations in the scan. Several papers have been published recently on the incorporation of multiple scattering into methods for 3D EM reconstruction [20], [21], [22]. Potentially, some of these techniques could be used in conjunction with DHT in the future to take the effects of multiple scattering in the input data into account explicitly and make the resultant algorithm even more robust in this respect.
Apart from its connections with DT [10], [11], [12] stated above, the proposed DHT method has some similarities with the Big Bang Tomography technique developed by D. Van Dyck and co-authors [23], [24], [25], [26]. In that technique, the "depth" position (i.e. the position along the direction of the optical axis) of different atoms in the imaged sample is determined using the CTF, which is similar in principle to the approach used in DHT. However, the reconstruction technique used in [23], [24], [25], [26] is different from that of DHT. Another method, known as holotomography [27], [28], utilizes multiple defocused images to retrieve the complex wave amplitude in the Fresnel region before the subsequent numerical back-propagation and tomographic reconstruction. Unlike the DHT method developed below, however, holotomography employs conventional CT for reconstruction of the 3D distribution of the real and imaginary parts of the refractive index in the imaged sample from the recovered defocused complex amplitudes. The DHT method is also closely related to the problem of the Ewald sphere curvature correction which has attracted significant attention in recent years in the context of high-resolution TEM [4], [5], [6].
Section snippets
Differential holographic tomography algorithm
Consider an imaging setup with a monochromatic plane wave illuminating a weakly scattering object, where is the wave number, is the intensity of the incident wave, is a Cartesian coordinate system in 3D space and z is the optical axis. The complex amplitude U(r) of the wave inside the object satisfies the time-independent Schrödinger equation: , where n(r) is the refractive index. In the case of electron microscopy, one has
Numerical simulations
In this section we describe numerical simulations performed to test the potential of practical reconstruction using the DHT method of sparse atomic structures from TEM defocus image series collected at multiple rotational orientations of the structure. The code used for the calculation of defocused images in these simulations was based on the freely available TEMSIM C++ source code developed by E.J. Kirkland [34]. The TEMSIM programs allow one to simulate defocused TEM images using the
Conclusions
In the first part of this work [9], we argued that in many examples of TEM imaging of small biological molecules or other “sparsely localized” weakly scattering structures, multiple scattering tends to have only a moderate effect and therefore can be safely ignored in the reconstruction procedures, without introducing large errors into the results. Importantly, the in-molecule free-space propagation (Fresnel diffraction) cannot be ignored because of the shallow depth of focus under the imaging
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors wish to thank Dr. Andrew Martin for sharing his software code that was used in the course of this research.
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