Chapter 4 – Making Light Work Harder in Biology  125

it has marginally different focal lengths corresponding to the x-​axis and y-​axis. This results

in fluorophores above or below the xy focal plane having an asymmetric PSF image on the

camera, such that fluorescence intensity appears to be stretched parallel to either the x-​ or

y-​axis depending upon whether the fluorophore is above or below the focal plane and the

relative geometry of the cylindrical lens and the camera.

Measuring the separate Gaussian widths in x and y of such a fluorophore image can thus

be used as a sensitive metric for z, if employed in combination with prior calibration data

from surface-​immobilized fluorophores at well-​defined heights from the focal plane. The rate

of change of each Gaussian width with respect to changes in z when the Gaussian width is

minimum is zero (this is the condition when the fluorophore image is in focus with respect

that the appropriate axis of x or y). What is normally done therefore is to toggle between

using the x and y widths for the best metric of z, at different z, in order to span the largest

possible z range for an accurate output prediction of z. The main issues with the astigmatism

method are that the localization precision in z is worse than that in x and y by a factor of ~1.5

and also that the maximum range in z, when using a typical high-​magnification microscope

optimized for localization microscopy, is roughly ±1 μm.

Corkscrew PSF methods, the most common of which is the double-​helical PSF (DH-​

PSF) approach, can be used to generate z information for fluorophore localization. These

techniques use phase modulation optics to generate helical, or in the case of the DH-​PSF

method, double-​helical-​shaped PSFvolumes in the vicinity of the sample plane. The helical

axis is set parallel to the optic (z) axis such that when a fluorophore is above or below the focal

plane, the fluorophore image rotates around this central axis. In the case of DH-​PSF imaging,

there appear to be two fluorescent spots per fluorophore, which rotate around the central

axis with changes in z. In this instance, x and y can also be determined for the fluorophore

localization as the mean from the two separate intensity centroid values determined for each

separate spot in a pair.

This method has a downside of requiring more expensive phase modulation optics in

the form of either a fixed phase modulation plate placed in a conjugate image plane to the

back aperture of the objective lens (conjugate to the Fourier transformation plane of the

sample image) or a spatial light modulator (SLM) consisting of an array of electrically pro­

grammable LCD crystals, which can induce controllable levels of phase retardation across a

beam profile. However, the precision in z localization is more than twice as good as the other

two competing methods of multiplane and astigmatism imaging (see Badieirostami et al.,

2010). A downside is that all multilobe-​type methods have a larger in-​plane extent, which

further reduces the density of active markers that can be imaged without overlap, which can

present a real issue in the case of intermediate-​high copy number systems (i.e., where the

concentrations of fluorescently labeled biomolecules is reasonably high).

Worked Case Example 4.1: Super-​Resolution Imaging

Video-​rate imaging epifluorescence microscopy with a water immersion objective lens NA

1.2 was performed on a low density of surface-​immobilized GFP using a 473 nm laser to

reveal fluorescent spots that bleached after an average of ~30 consecutive image frames.

Spherical bacterial cells of diameter ~2 μm were engineered to contain a GFP-​labeled pro­

tein X that under the conditions of the experiment was known not to aggregate or oligo­

merize and was expressed in the cytoplasm with typically ~120 molecules present per cell

at any one time. Narrow-​field epifluorescence was used on these cells immobilized to a

microscope coverslip to sample at 3 ms per frame, which initially indicated bright slightly

grainy fluorescence intensity toward the outer perimeter of the cell when focusing at a

midcell height, much dimmer toward the cell center, but after some continuous illumin­

ation, the outer region also became dimmer until eventually distinct diffusing fluorescent

spots could be seen, ~15% of which had similar integrated intensity to the epifluorescence

data, with the remaining spots having integrated intensity values of ~2, ~3, or more rarely

~4+​ times within relative proportions of ~50%, ~25%, and ~10% that of the epifluores­

cence data. Preliminary tracking data suggest that a typical spot diffused its own PSF

KEY BIOLOGICAL

APPLICATIONS: SUPER-​

RESOLUTION METHODS

Imaging subcellular architectures

with nanoscale spatial preci­

sion; Quantifying molecular

mobility; Determining molecular

colocalization.