Transmission electron microscopy (TEM) offers direct visualization of fine details of
biological specimen. Recent advancements in sample preparation techniques and
developments in algorithms for sophisticated image processing as well as
availability of computation power pivoted rapid improvements in achievable
resolution of the analysis and widened the range of biological systems that can be
studied (Crowther 2010). In particular, structural analysis of
protein macromolecules by TEM, either in the form of ordered arrays such as protein
two-dimensional (2D) crystals or individual protein macromolecules, has improved
greatly as evidenced by an increasing number of structure determination at
near-atomic resolutions (Armache et al. 2010; Ge and Zhou
2011; Gonen et al. 2005; Yu et al.
2011). In addition, TEM analysis of biological system at
moderate resolution can directly complement high-resolution structures obtained by
X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy, providing
pseudo-atomic resolution structure determination of large, multi-subunit
complexes.

One of the key factors for successful structure determination of a biological
specimen at high resolution is the correction for contrast transfer function (CTF)
of a microscope. While images obtained from the electron microscope suffers from
loss of faithful representation of the true object due to phase and amplitude
modulation derived from imperfect imaging conditions, CTF models how an electron
microscope transfers the actual specimen into a recorded image hence allowing for
distortions present in the micrograph to be estimated (Frank 2006). Under weak-phase approximation, that is, electron scattering and
the subsequent phase shift, is small as in the case for biological specimen, TEM
image, and CTF can be described by the following relationships (Penczek et al. 1997):

I\left(k\right)=H\left(k\right)\mathrm{\Phi}\left(k\right)

(1)

H\left(k\right)=sin\phantom{\rule{0.2em}{0ex}}\gamma \left(k\right)-W\phantom{\rule{2pt}{0ex}}cos\phantom{\rule{0.2em}{0ex}}\gamma \left(k\right)

(2)

where *k* is the spatial frequency vector, *I*(*k*) is Fourier
transform of micrograph, Φ(*k*) is Fourier transform of true object and
*H*(*k*) is CTF. *W* is amplitude contrast ratio, which
denotes contribution of image contrast that result from inelastic scattering in the
image formation that is dominated by elastic electron scattering, and
*γ*(*k*) is phase shift produced by spherical aberration and
defocus that can be described by the Scherzer formula (Williams and Carter 1996),

\gamma \left(k\right)=\frac{\mathrm{\pi}}{2}\left({C}_{s}{\lambda}^{3}{k}^{4}-2\mathrm{\Delta}\mathit{z\lambda}{k}^{2}\right)

(3)

where *k* is the scattering vector, *λ* is the wavelength of
electron beam, *Cs* is the spherical aberration coefficient of a microscope,
and ∆*z* is the defocus value. While other parameters are constant for
a given instrument, defocus is manually adjusted by an operator in order to produce
image with optimal phase contrast. When considering elastic electron scattering
alone, sin*γ*(*k*) is also referred to as phase contrast transfer
function (PCTF). When plotted as a function of spatial frequency, PCTF oscillates
sinusoidally, hence producing alternating negative phase contrast at higher spatial
frequencies (Ruprecht and Nield 2001). If the negative
contrast is left uncorrected, structural features of the object at high resolution
is compromised, and therefore the image fails to represent true features of the
object.

Additional complication with regard to precise CTF estimation comes from continuous
attenuation of amplitude towards higher spatial frequency, termed envelope function,
which is described by a simplified relationship below:

H\left(k\right)=E\left(k\right){H}_{\mathrm{ideal}}\left(k\right)

(4)

where the experimental CTF, *H*(*k*), results from ideal CTF,
*H*
_{ideal}(*k*), multiplied by envelope function,
*E*(*k*). Major contributors of envelope function include beam energy
envelope (*E*
_{spread}), beam coherence envelope (*E*
_{coherence}) and sample drift envelope (*E*
_{drift}). Each envelope function is described by complex formula which
takes account into parameters such as chromatic aberration of the microscope,
semi-angle of aperture, energy spread of emitted beam, lens current stability and
specimen drift (Frank 2006; Sorzano et al. 2007). In addition, the performance of image recorder, as defined by
modulation transfer function, also contributes to the degradation of high-resolution
information in the micrograph.

Due to its importance, there have been a significant number of studies dedicated to
precisely determine CTF from micrographs. These works provided comprehensive
description and algorithms for specific aspects such as defocus determination
(Mindell and Grigorieff 2003), amplitude contrast (Toyoshima
and Unwin 1988; Toyoshima et al. 1993)
and envelope function (Saad et al. 2001; Sander et al. 2003) as well as generation of reliable power spectrum density
(Fernandez et al. 1997; Zhu et al. 1997) from which CTF of experimental data can be modeled from. As a result,
CTF correction is now widely incorporated in various image processing software
packages, and routinely performed for structure determination of vitrified
biological specimen.

In the present work, the effect of precise estimation of image defocus, one of the
most critical parameters required for CTF determination, in the preservation of
structural integrity of specimen is demonstrated. Although the effects of
alterations in critical parameters have been thoroughly investigated for CTF
determination in the past (Sorzano et al. 2009), the main
purpose of this short technical note is to illustrate visually the effect of
appropriate CTF correction. Therefore, for simplicity, detailed theories of image
formation in TEM and algorithms employed in CTF correction are omitted.