Chapter 4 – Making Light Work Harder in Biology  115

directed to later modeling (Mortensen et al., 2010), which includes a more complex but real­

istic treatment.

The spatial precision is dependent on three principal factors: Poisson sampling of photons

from the underlying PSF distribution, a pixelation noise due to an observational uncertainty

as to where inside a given pixel a detected photon actually arrived, and noise associated with

the actual camera detection process. It illustrates that if N is relatively large, then σx varies

roughly as s/​√N. Under these conditions, σx is clearly less than w (a condition for super-​

resolution). Including the effects of pixelation and dark noise indicates that if N is greater

than ~10⁶ photons, then the spatial precision can in principle be at the level of 1 nm to a

few tens of nanometers. A popular application of this method has been called “fluorescence

imaging with one nanometer accuracy” (see Park, 2007).

KEY POINT 4.2

In localization microscopy, the spatial precision scales roughly with the reciprocal of

the square root of the detected integrated intensity from a diffraction-​limited fluores­

cent spot.

To avoid aliasing due to undersampling of the intensity distribution by the camera, Nyquist

theory (also known as Nyquist–​Shannon information theory) indicates that the pixel edge

length multiplied by the image magnification must be less than w. Equation 4.4 can be used

with physically sensible values of s, N, and b to estimate the optimal value of a to minimize

σx in other words, to optimize the image magnification on the camera to generate the best

spatial precision. Typical optimized values of pixel magnification are in the range 50–​100 nm

of the sample plane imaged onto each pixel.

The effective photon collection efficiency of a typical high-​magnification microscope used

for localization microscopy is at best ~10%. Therefore, if one were to achieve a theoretical

precision as good as 1 nm, then a fluorescence point source emitter must emit at least ~10⁷

photons. A bright single organic dye molecule imaged under typical conditions of epifluores­

cence microscopy will emit this number of photons after a duration of ~1 s. This sets a limit

on the speed of biological processes, which can be probed at a precision of 1 nm to a few tens

of nanometers, typical of super-​resolution microscopy. Note, however, that in practice, there

is often a short linker between the dye tag and the biomolecule being tracked, so the true

spatial precision for the location of the biomolecule is a little worse than that expected from

localization fitting theory since it needs to include the additional flexibility of the linker also.

4.2.3  MAKING THE MOST OUT OF A LIMITED PHOTON BUDGET

To investigate faster processes requires dividing up the photon budget of fluorescence emission

into smaller sampling windows, which therefore implies a poorer spatial precision unless the

photon emission flux is increased. For many fluorophores such as bright organic dyes and

QDs, this is feasible, since they operate in a subsaturation regime for photon absorption

and therefore their fluorescence output can be increased by simply increasing the excitation

intensity. However, several less stable fluorophores are used in biophysical investigations,

including fluorescent proteins (FPs), which undergo irreversible photobleaching after

emitting at least an order of magnitude fewer photons compared to organic dyes.

For example, green fluorescent protein (GFP) emits only ~10⁶ photons prior to irreversible

photobleaching for GFP and so can never achieve a spatial precision to the level of 1 nm in

localization microscopy in the typical high-​efficiency photon collection microscopes cur­

rently available. Some variants of FP, such as yellow FP, (YFP), emit in excess of 10⁷ photons

prior to irreversible photobleaching and therefore have potential application for nanoscale

imaging. But similarly, there are less stable FPs (a good example is cyan FP [CFP]) that only

emit ~10⁵ photons before irreversible photobleaching.