# Theory

## Photon Budget

To assess the feasibility of this project, we performed a detailed photon budget, so as to determine if enough photons would be detected by the camera in each photoswitching cycle. This calculation is based on the work of Thompson et al. where a detailed analysis of single molecule localization is performed. This same method is used in STORM literature and it proved to be very close to the actual results.

In short, and as shown below, the power delivered at the sample was calculated by determining the approximate area of the beam at the sample and then estimating the amount of power per area on the sample plane (a 0.5 factor was included to account for power losses in the laser path) (Eq 1). Then this energy flux was converted to a photon flux by using Planck’s constant (Eq. 2). The incoming photon flux was conv.erted to an effective photon flux hitting the sample by using the sample’s cross section (obtained from the dye’s quantum yield, Eq. 3). To calculate the approximate photon emission rate we used the quantum efficiency of the sample (Eq. 4).

$(1)(\frac{P}{A})_{sample}=\frac{P_{laser}}{A_{sample}}\,$

$(2)E_{photon}=\frac{hc}{\lambda}$

$(3)\sigma=1000ln(10)\frac{\epsilon}{N_a}$

$(4)Ph_{flux}=\frac{(\frac{P}{A})_{sample}Q_{e,dye}}{\sigma}$

In Eq. 1 $(\frac{P}{A})_{sample}$ is the power area density at the sample plane, $P_{laser}$ is the overall laser power and $A_{sample}$ is the area at the sample plane. In Eq. 2 $E_{photon}$ is the energy of a photon, $h$ is Planck's constant, $c$ is the speed of light and $\lambda$ is the wavelength of light used. In Eq. 3 $\sigma$ is the photon cross section of the dye and $\epsilon$ is its extinction coefficient. In Eq. 4 $Ph_{flux}$ is the emitted photon flux from each dye molecule and $Q_{e,dye}$ is the quantum efficiency of the dye. All the parameters are at standard conditions of temperature and concentration.

Using the latter number, we took the reported parameters for 4 different cameras: the AVT Manta 032-B, the Hamamatsu Orca Flash 2.8, the Andor Neo-sCMOS and the Andor iXon3. These were picked since the first one is the standard camera in this lab, the third and fourth would be the ideal components and the second one seems to be the best price-quality compromise we could go with in this project.

To approximate the number of photons actually detected with each camera, we estimated the light losses as 70% due to detection angle, 20% due to filters and 50% due to the objective (Thompson et al, 2002). Additionally, we accounted for the lower N.A. of our objective as 26% additional losses (Equation 5). With this, the quantum efficiency of the detector and the integration time we determined the effective amount of photons collected per cycle at the detector.

$(5)Ph_{collected}=\frac{0.3*0.8*0.5*0.74*Q_{e,detector}}{f_{detection}}$

Here $Ph_{collected}$ is the number of photons collected per cycle, $Q_{e,detector}$ is the quantum efficiency of the detector and $f_{detection}$ is the frequency of acquisition, which in our case translates as the inverse of the integration time.

To estimate the noise sources, we arbitrarily set a range of 1 to 10 electrons/(pixel second) as spurious light, and used the specified values for read and dark noise (the first and last weighted by the integration time) (Eq. 6).

$(6)b=N_r+Bt+Dt\,$

In Eq. 6, $b$ is the total background, $N_r$ is the readout noise, $B$ is the spurious light, $D$ is the dark current and $t$ is the time .

With the collected photons, the pixel sizes at the sample plane and the noise estimations we calculated the expected uncertainty in the localization of a single molecule per cycle, aside from the amount of photons counted and the uncertainty of this count. Finally, we calculated the Full Width at Half Maximum of the localization uncertainty as our resolution, and as a function of the number of cycles (1, 10 and 100). This is all summarized in the table below. It will be noted here though that the actual number of switching cycles is limited by the total number of photons that a dye molecule can emit before photobleaching, usually in the range of 250000 for an EM-CCD detector (less numbers for the less efficient detectors). (Thopmson et al., 2002)(Eqs. 7, 8, 9 and 10).

$(7)<(\Delta x)^2>=\frac{s^2+\frac{a^2}{12}}{N}+\frac{8\pi s^4b^2}{a^2N^2}$

$(8)N_t=\frac{8\pi s^4b^2}{a^2(s^2+\frac{a^2}{12})}$

$(9)<(\Delta N)^2>=N+\frac{4\pi s^2b^2}{a^2}$

$(10)FWHM=\frac{2\sqrt{2ln(2)}<(\Delta x)^2>}{\sqrt{n_{cycles}}}$

Here $<(\Delta x)^2>$ is the uncertainty in particle localization per cycle, $s$ is the standard deviation of the Point Spread Function of the microscope, $a$ is the pixel size at the sample plane and $N$ is the number of photons detected. In Eq. 8, $N_t$ is an estimation on the photons required for the particular localization uncertainty calculated in Eq. 7. In Eq. 9 $<(\Delta N)^2>$ is the uncertainty in the number of photons required. Finally, in Eq. 10 $FWHM$ is the Full Width at Half Maximum of the calculated localization fit and $n_{cycles}$ is the number of imaging cycles per molecule.

The following tables detail the results of our estimations for the FWHM of each fit on the different detectors and with different levels of spurious light (B, in electron/(pixel second)). The number of cycles is indicated at the bottom of each table. It has to be considered though that the number of cycles is limited by the number of photons each dye molecule can give before photobleaching (around 250000 as reported by van de Linde et al., 2008), which is certainly a very real limitation (especially in low quantum yield detectors).

B/Camera Manta Orca Neo iXon
1 12.89 8.34 7.25 12.66
10 84.89 30.96 12.35 57.04
20 168.94 60.44 21.35 112.38
40 337.45 120.13 40.85 223.89
80 674.69 239.87 80.75 447.34
100 843.33 299.79 100.80 559.11
Cycles 1
B/Camera Manta Orca Neo iXon
1 4.07 2.64 2.29 4.00
10 26.84 9.79 3.91 18.04
20 53.42 19.11 6.75 35.54
40 106.71 37.99 12.92 70.80
80 213.35 75.86 25.54 141.46
100 266.68 94.80 31.88 176.81
Cycles 10
B/Camera Manta Orca Neo iXon
1 1.29 0.83 0.72 1.27
10 8.49 3.10 1.24 5.70
20 16.89 6.04 2.14 11.24
40 33.74 12.01 4.09 22.39
80 67.47 23.99 8.08 44.73
100 84.33 29.98 10.08 55.91
Cycles 100

## Fluorophore Photoswitching

There are two main approaches to mediate the fluorophore switching necessary for a STORM set up. The first approach is to use pairs of fluorophores, one of which acts as the reporter and is able to switch to a dark state while other acts as an activator and mediates the recovery of the reporter fluorophore to its fluorescent state. Laser pulses of different wavelengths are alternated to mediate the photoswitching. The original STORM system developed by the Zhuang laboratory at Harvard used this kind of configuration (Rust et al 2006) Red light pulses acts both to activate Cy5 as well as to bring it to its dark state. These red pulses are alternated with green light pulses to activate Cy3, which aids in the recovery of Cy5 from the dark to the emissive state. The second approach, called dSTORM, or “direct STORM,” requires only one fluorophore and a constant laser pulse of one wavelength. In this approach, the fluorophore is able to return to from its dark state to its fluorescent state without the aid of an activator fluorophore.

We chose to create our system using the dSTORM technique (Mike Heilemann et al 2008). It only requires a single laser, instead of two lasers, which decreases the total price of the setup. Futhermore, it does not require code to mediate the pulsing of the laser; instead the laser can be kept continually on. Cy, ATTO, and Alexa fluorophores have all been shown to be able to switch reversibly between an emissive and a dark state without an activator dye. They differ, however, in their brightness, excitation wavelengths, and kinetics of photoswitching.

We decided to focus on ATTO655 and Alexa647. These dyes are able to be excited and switched to its dark state by our 632nm laser. ATTO has a much quicker rate of photoswitching, as it's average time it stays in the off state is on the order of 100s of milliseconds (van de Linde et al). Cy dyes and Alexa dyes, on the other hand, have an average time in the off state of approximately 4s and 19.1s, respectively (Dave et al 2009). Because Alexa647 has relatively slow switching, we compensate by increasing the exposure time and the overall image acquisition time.