## Model of bleaching and acquisition for superresolution microscopy controlled by a single wavelength |

Biomedical Optics Express, Vol. 2, Issue 11, pp. 2934-2949 (2011)

http://dx.doi.org/10.1364/BOE.2.002934

Acrobat PDF (863 KB)

### Abstract

We consider acquisition schemes that maximize the fraction of images that contain only a single activated molecule (as opposed to multiple activated molecules) in superresolution localization microscopy of fluorescent probes. During a superresolution localization microscopy experiment, irreversible photobleaching destroys fluorescent molecules, limiting the ability to monitor the dynamics of long-lived processes. Here we consider experiments controlled by a single wavelength, so that the bleaching and activation rates are coupled variables. We use variational techniques and kinetic models to demonstrate that this coupling of bleaching and activation leads to very different optimal control schemes, depending on the detailed kinetics of fluorophore activation and bleaching. Likewise, we show that the robustness of the acquisition scheme is strongly dependent on the detailed kinetics of activation and bleaching.

© 2011 OSA

2. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science **313**, 1642–1645 (2006). [CrossRef] [PubMed]

3. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods **3**, 793–795 (2006). [CrossRef] [PubMed]

4. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. **91**, 4258–4272 (2006). [CrossRef] [PubMed]

*p*that a molecule is in the activated state is, to a good approximation, independent of the (irreversible) bleaching rate

*β*per activated molecule. In more recent approaches [14

14. J. Folling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods **5**, 943–945 (2008). [CrossRef] [PubMed]

15. C. Steinhauer, C. Forthmann, J. Vogelsang, and P. Tinnefeld, “Superresolution microscopy on the basis of engineered dark states,” J. Am. Chem. Soc **130**, 16840–16841 (2008). [CrossRef] [PubMed]

16. D. Baddeley, I. D. Jayasinghe, C. Cremer, M. B. Cannell, and C. Soeller, “Light-induced dark states of organic fluochromes enable 30 nm resolution imaging in standard media,” Biophys. J. **96**, L22–L24 (2009). [CrossRef] [PubMed]

17. M. Heilemann, S. van de Linde, A. Mukherjee, and M. Sauer, “Super-resolution imaging with small organic fluorophores,” Angew. Chem. Int. Ed. **48**, 6903–6908 (2009). [CrossRef]

18. I. Testa, C. A. Wurm, R. Medda, E. Rothermel, C. Von Middendorf, J. Flling, S. Jakobs, A. Schnle, S. W. Hell, and C. Eggeling, “Multicolor fluorescence nanoscopy in fixed and living cells by exciting conventional fluorophores with a single wavelength,” Biophys. J. **99**, 2686–2694 (2010). [CrossRef] [PubMed]

19. S. Lee, M. Thompson, M. A. Schwartz, L. Shapiro, and W. E. Moerner, “Super-resolution imaging of the nucleoid-associated protein hu in caulobacter crescentus,” Biophys. J. **100**, L31–L33 (2011). [CrossRef] [PubMed]

*n*in a region of size

*λ*decreases over time if the fluorophores bleach irreversibly. (We distinguish irreversible bleaching, which damages the molecules and permanently renders them non-fluorescent, from the reversible bleaching that is used to temporarily switch fluorophores to dark states in some implementations [16

16. D. Baddeley, I. D. Jayasinghe, C. Cremer, M. B. Cannell, and C. Soeller, “Light-induced dark states of organic fluochromes enable 30 nm resolution imaging in standard media,” Biophys. J. **96**, L22–L24 (2009). [CrossRef] [PubMed]

*p*per molecule: The need for non-overlapping bright spots dictates that, in a region of size

*λ*, on average no more than 1 of the

*n*fluorophores should be activated, so

*p*(

*t*) must be less than or equal to 1/

*n*(

*t*). Due to irreversible bleaching,

*n*(

*t*) is a decreasing function of time, and so

*p*(

*t*) can be an increasing function of time. The result of bleaching is thus to enable faster acquisition: At later times, the activation probability can increase, decreasing the probability that no molecules will be on at any given time.

20. J. Vogelsang, T. Cordes, C. Forthmann, C. Steinhauer, and P. Tinnefeld, “Controlling the fluorescence of ordinary oxazine dyes for single-molecule switching and superresolution microscopy,” Proc. Natl. Acad. Sci. U.S.A. **106**, 8107–8112 (2009). [CrossRef] [PubMed]

*i.e.*single-molecule) images obtained is still desirable if monitoring small structures during a very long process.

*I*(

*t*) (and hence the activation probability

*p*and bleaching rate

*β*) to optimize the portion of the time in which exactly 1 of the

*n*fluorophores is activated. We previously showed that if

*β*and

*p*can be varied independently (the 2-wavelength case) then the number of single-fluorophore images is maximized by varying the activation probability in such a way that the number of molecules decreases as a linear function of time:

*n*(

*t*) =

*n*(0) –

*ṅt*, where the derivative

*ṅ*is constant in time [21

21. E. Shore and A. Small, “Optimal acquisition scheme for subwavelength localization microscopy of bleachable fluorophores,” Opt. Lett. **36**, 289–291 (2011). [CrossRef] [PubMed]

*E*

_{2}, defined as the ratio of the number of 2-molecule images obtained (and accepted by the analysis software) to the number of 1-molecule images obtained (and accepted by the analysis software) [22

22. A. Small, “Theoretical limits on errors and acquisition rates in localizing switchable fluorophores,” Biophys. J. **96**, L16–L18 (2009). [CrossRef] [PubMed]

*n*(

*t*) on time) cause the number of single-molecule images to decrease. Interestingly, for fast acquisition (corresponding to larger

*p*and

*E*

_{2}), deviations from the optimal scheme also decrease the number of 2-molecule images, partially mitigating the effects of a deviation on the ratio of 1-molecule to 2-molecule images.

23. F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express **2**, 1377–1393 (2011). [CrossRef] [PubMed]

*m*

_{max}or fewer activated fluorophores. While we do not consider this situation directly, we expect that many of the techniques developed here will carry over to the multi-fluorophore case, as one of the key results below (that in many cases the relevant integrals are stationary if the expected number of activated molecules per frame is kept constant) does not require the assumption that we only obtain information from single-fluorophore images.

## 1. Formalism and essential concepts

### 1.1. Activation probabilities

*m*molecules being simultaneously activated in a region of size

*λ*is given by the binomial distribution: where

*n*is the number of molecules in a region of size

*λ*. If the sample is labeled densely enough to resolve features of size

*λ*/10 or smaller [7

7. H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods **5**, 417–423 (2008). [CrossRef] [PubMed]

*n*will be greater than 100 in 2D, or 1000 in 3D. We can thus assume

*n*≫ 1, which simplifies Eq. (1) considerably. The fractional error in approximating

*n*(

*n*– 1)...(

*n*–

*m*+ 1) as

*n*is small for

^{m}*n*≫ 1, so Eq. (1) becomes:

*p*and derive two useful results for this work, by invoking a result derived previously [22

22. A. Small, “Theoretical limits on errors and acquisition rates in localizing switchable fluorophores,” Biophys. J. **96**, L16–L18 (2009). [CrossRef] [PubMed]

*E*

_{2}is the 2-molecule error rate discussed above. The parameter

*f*

_{1}is the probability that the image analysis algorithm being used to process the data will correctly identify an image of a single-molecule and determine its position, while

*f*

_{2}is the probability that the image analysis algorithm will correctly recognize 2-molecule overlaps as such and not analyze them. Consequently,

*p*is bounded, and the upper bound decreases as

*n*increases.

*p*be less than 1/

*n*[22

22. A. Small, “Theoretical limits on errors and acquisition rates in localizing switchable fluorophores,” Biophys. J. **96**, L16–L18 (2009). [CrossRef] [PubMed]

*p*above this level actually decreases the number of 1-molecule images obtained (which can be shown by differentiating

*p*

_{1}with respect to

*p*in Eq. (1)) while increasing the number of 2-molecule images. The result is that there is a maximum error rate. In the case of non-bleaching fluorophores the maximum error rate is

*f*

_{2}/2

*f*

_{1}[22

**96**, L16–L18 (2009). [CrossRef] [PubMed]

*f*

_{2}/

*f*

_{1}[21

21. E. Shore and A. Small, “Optimal acquisition scheme for subwavelength localization microscopy of bleachable fluorophores,” Opt. Lett. **36**, 289–291 (2011). [CrossRef] [PubMed]

*p*: where

*Ẽ*is the normalized error rate

*Ẽ*= 2

*f*

_{1}

*E*

_{2}/

*f*

_{2}. Note that in this notation,

*np*=

*Ẽ*.

*p*=

*Ẽ*/

*n*, and the identity

*n*and fixed

*x*, we get:

### 1.2. Expected times

*m*molecules are activated. We thus consider the integral: If we wish to pick

*p*(

*t*) in such a way to maximize this integral (for

*m*= 1) or minimize it (for

*m*≠ 1), we have a problem in variational calculus. The most commonly-used tools of variational calculus, the Euler-Lagrange equations [24, 25], require formulating the integral in terms of a time-dependent function and its first derivative, and then varying that function to make the integral an extremum. Note that while satisfaction of the Euler-Lagrange equations makes the integrals in Eq. (6) stationary, this is only a first-order condition that is satisfied by maxima, minima, and saddle points alike. Later, we will consider second-order conditions to determine when

*t*

_{1}is maximized.

*t*in terms of

_{m}*n*(

*t*) and

*ṅ*(

*t*). Physically, it may seem natural to pick

*p*(

*t*) as the function to be varied, since that is the experimentally-controllable parameter. However, the Euler-Lagrange equations apply to problems that are formulated in terms of functions and their derivatives. As we show in the next section, if we have a kinetic model of the bleaching process we can formulate the problem in terms of

*n*(

*t*) and

*ṅ*(

*t*), and use the kinetic model to express

*p*(

*t*) in terms of

*n*and

*ṅ*.

*not*trying to maximize the number of single-fluorophore images obtained in a single activation cycle. As discussed above, the number of single-fluorophore images in a given cycle is maximized when

*Ẽ*= 1 [22

**96**, L16–L18 (2009). [CrossRef] [PubMed]

*n*(

*t*) at the beginning and end of the experiment are fixed. Given that constraint, we are trying to obtain as many single-fluorophore images as possible while bleaching a given number of molecules in a given time. However, a person following the prescriptions given below can pick the time interval and number of molecules bleached in that time interval (

*i.e.*pick the constraints to impose) and then pick the appropriate error rate to bleach the designated number of molecules in the designated time.

## 2. Bleaching models

*β*(

*I*(

*t*)) is the intensity-dependent rate at which molecules bleach. In either case, we can divide both sides by

*n*(

*t*) and get: Because the right hand side depends only on the intensity

*I*in either case, it follows that

*I*can be expressed as a function of −

*ṅ*/

*n*,

*i.e.*there is a one-to-one relationship between the bleaching rate per molecule and the excitation intensity

*I*. Therefore,

*p*is also a function of −

*ṅ*/

*n*. We can thus write our integrals as: Once we have determined the form of

*p*(−

*ṅ*/

*n*) via a model of the bleaching process, we use the Euler-Lagrange equations to obtain a differential equation for

*n*. Our procedure is therefore:

### 2.1. Excitation from the dark to activated state, followed by bleaching from the excited state

26. M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opinion Chem. Biol. **12**, 505–514 (2008). [CrossRef]

27. T. Gould and S. Hess, “Nanoscale biological fluorescence imaging: Breaking the diffraction barrier,” Methods Cell Biol. **89**, 329–358 (2008). [CrossRef]

*I*in units of a saturation intensity chosen so that when

*I*= 1 the probability of being in the higher state is 1/2. Eq. (10) can be derived by setting the rate of upward transitions (proportional to

*I*and (1 –

*p*)) equal to the rate of downward transitions (proportional to

*p*).

*n*≫ 1, in which case

*p*≪ 1, meaning that the

*I*term in the denominator is negligible. We then have the following results: where

*p*is a power law

*p*=

*c*(−

*ṅ*/

*n*)

*, it follows that*

^{a}*Ẽ*=

*np*is a constant if

*n*(

*t*) is chosen to make the integrals in Eq. (6) stationary.

*p*is our Lagrangian. In the case where

_{m}*p*is a power law, we get: where

*p*′

*is the derivative of*

_{m}*p*with respect to its argument

_{m}*np*.

*i.e.*the time-dependence of

*p*is solely due to the time-dependence of

_{m}*n*and

*ṅ*), if we pick

*n*(

*t*) to satisfy the Euler-Lagrange equations then the Hamiltonian

*H*will be a constant (

*i.e.*time-independent)[24]: Because

*H*is time-independent and is a function of a single argument (−

*ṅ*)

*/*

^{a}*n*

^{a−1}, it therefore follows that its argument (−

*ṅ*)

*/*

^{a}*n*

^{a−1}=

*Ẽ*/

*c*is also time-independent, and hence

*Ẽ*is a constant, even as

*n*and

*p*change.

*a*= 1/2 and

21. E. Shore and A. Small, “Optimal acquisition scheme for subwavelength localization microscopy of bleachable fluorophores,” Opt. Lett. **36**, 289–291 (2011). [CrossRef] [PubMed]

### 2.2. Excitation from the activated state to the dark state, followed by photo-induced bleaching

*not*the dark state; the dark state is reached by the absorption of a photon [14

14. J. Folling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods **5**, 943–945 (2008). [CrossRef] [PubMed]

15. C. Steinhauer, C. Forthmann, J. Vogelsang, and P. Tinnefeld, “Superresolution microscopy on the basis of engineered dark states,” J. Am. Chem. Soc **130**, 16840–16841 (2008). [CrossRef] [PubMed]

16. D. Baddeley, I. D. Jayasinghe, C. Cremer, M. B. Cannell, and C. Soeller, “Light-induced dark states of organic fluochromes enable 30 nm resolution imaging in standard media,” Biophys. J. **96**, L22–L24 (2009). [CrossRef] [PubMed]

*I*and the occupation probability

*p*for the activated state, while the rate of transitions from the dark state to the activated state is proportional to 1 –

*p*(the probability of being in the dark state). By setting the dark state probability equal to 1 –

*p*we are implicitly assuming that fluorophores spend a negligible amount of time in the excited state. This assumption is valid if the typical fluorophore yields of order 10

^{3}photons per second (a common number in superresolution experiments,

*e.g.*[3

3. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods **3**, 793–795 (2006). [CrossRef] [PubMed]

28. C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods **7**, 373–375 (2010). [CrossRef] [PubMed]

14. J. Folling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods **5**, 943–945 (2008). [CrossRef] [PubMed]

19. S. Lee, M. Thompson, M. A. Schwartz, L. Shapiro, and W. E. Moerner, “Super-resolution imaging of the nucleoid-associated protein hu in caulobacter crescentus,” Biophys. J. **100**, L31–L33 (2011). [CrossRef] [PubMed]

^{−9}seconds, for a total excited state time of order 10

^{−6}seconds, while the time in the dark state is of order milliseconds to tens of milliseconds [14

**5**, 943–945 (2008). [CrossRef] [PubMed]

15. C. Steinhauer, C. Forthmann, J. Vogelsang, and P. Tinnefeld, “Superresolution microscopy on the basis of engineered dark states,” J. Am. Chem. Soc **130**, 16840–16841 (2008). [CrossRef] [PubMed]

*I*is again normalized so that the probability of being in the higher-energy dark state is 1/2 when

*I*= 1.

*e.g.*a transition from a single ground state

*S*

_{0}to a first singlet excited state

*S*

_{1}, from which some fraction of the molecules are transferred to a triplet state

*T*

_{1}). Because a variety of microscopic models give the same result, it is important to not take Fig. 3 too literally; it is a schematic illustrating that upon absorption of a photon the molecule can either go to a state from which it will fluoresce and return to the ground state (called “activated” here for convenience), or a long-lived state from which it will not fluoresce. The key assumptions are that the dark state is longer-lived than the state producing fluorescence, and that it is reached via photon absorption from the ground state (which we call “activated” here).

29. C. Eggeling, J. Widengren, R. Rigler, and C. A. M. Seidel, “Photobleaching of fluorescent dyes under conditions used for single-molecule detection: evidence of two-step photolysis,” Anal. Chem **70**, 2651–2659 (1998). [CrossRef] [PubMed]

30. G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nat. Methods **4**, 81–86 (2007). [CrossRef]

*n*for

*n*≫ 1, it follows that

*p*≪ 1 and so

*I*≫ 1. We get the following relationship between −

*ṅ*/

*n*and

*I*: The activation probability is then: This is again a power law in −

*ṅ*/

*n*with exponent −1, and so it follows that

*Ẽ*is again constant. Our differential equation is: with solution: Note that in this case, lower error rates actually cause the number of molecules to deplete more rapidly. This is because achieving a low error rate requires a high excitation intensity to place more fluorophores in the dark state. At the same time, increasing the intensity increases the rate at which dark molecules are bleached as well as the number of molecules that are in the dark state and hence available to be bleached.

*n*(

*t*), it is again straightforward to determine

*p*(

*t*) and

*I*(

*t*). For a constant error rate,

*p*=

*Ẽ*/

*n*(Eq. (4)), and for this energy level scheme

*p*= 1/

*I*, so we get:

### 2.3. Photo-induced bleaching from the activated state

*I*= −

*ṅ*/(

*ṅ*+

*k*), so

_{b}n*p*= 1/(1 +

*I*) = 1 +

*ṅ*/

*k*and we get the following for the error rate: Before we solve this model, we will examine one more case, and show that it is equivalent.

_{b}n### 2.4. Bleaching from the dark state without the absorption of a second photon

*I*in terms of

*ṅ*/

*n*, and get

*I*= −

*ṅ*/(

*k*+

_{b}n*ṅ*). This gives

*p*= 1/(1 +

*I*) = 1 +

*ṅ*/

*k*, so we again have for

_{b}n*Ẽ*:

*p*′

*is evaluated with respect to its argument*

_{m}*n*+

*ṅ*/

*k*. The time derivative of Π is: Note that

_{b}*ṅ*+

*n̈*/

*k*is just the time derivative of

_{b}*Ẽ*, so we get that:

*p*

_{1}(given in Eq. (6) as

*Ẽe*

^{−}

*),*

^{Ẽ}*i.e.*we are trying to maximize the number of single-molecule images. The derivatives of

*p*

_{1}are: The time derivative of

*Ẽ*is then: This differential equation has an unstable fixed point at

*Ẽ*= 1, and a singularity at

*Ẽ*= 2. The most interesting cases for our purposes are initial error rates less than 1, for which

*Ẽ*decreases as a function of time.

*Ẽ*from Eq. (38), which can be solved analytically: Because the initial conditions show up additively with the time, changing the initial condition merely shifts the plot in time. Also,

*Ẽ*reaches 0 at a finite time

*t*= (

_{f}*Ẽ*(0)–log 1 –

*Ẽ*(0))/

*k*, which increases as

_{b}*Ẽ*(0) increases. Solutions of Eq. (39) are plotted for different initial errors in Fig. 5.

*Ẽ*(

*t*), we can solve for

*n*(

*t*) using

*Ẽ*=

*n*+

*ṅ*/

*k*. Because

_{b}*Ẽ*< 1 and

*n*≫ 1, the time dependence of

*n*(

*t*) is, to an excellent approximation, an exponential decay with rate

*k*. The difference between

_{b}*ṅ*and −

*k*is very small. Fortunately, however, the quantity that needs to be controlled with high precision is

_{b}n*I*(

*t*), not

*n*or

*ṅ*. Also, because

*I*≫ 1, there is considerable latitude in the control of

*I*.

*I*(

*t*), we recall that

*p*= 1/(1 +

*I*), and solving Eq. (32) gave

*I*= −

*ṅ*/(

*k*+

_{b}n*ṅ*) = −

*ṅ*/

*k*. The time-dependence of

_{b}Ẽ*n*is approximately

*n*(0)

*e*

^{−kbt}, so −

*ṅ*=

*k*(0)

_{b}n*e*

^{−kbt}, and we get the following for

*I*(

*t*): Solutions to Eq. (40) are plotted in Fig. 6. Because large relative changes in

*I*(

*t*) are required to obtain the optimal scheme, the excitation intensity does not need to be finely-tuned. We show

*I*(

*t*) for 2 pairs of initial error rates, each pair differing by 10%. In each case, the intensity

*vs.*time graphs differ by approximately 10% initially, and the percentage difference in

*I*increases substantially over time. We thus conclude that the optimal acquisition scheme is achievable without delicate fine-tuning. This issue of robustness is further explored in the next section.

## 3. Second-order conditions and robustness

*sufficient*second-order conditions that are straightforward to apply: If the Lagrangian

*p*is everywhere a convex function (second derivative non-negative) of its inputs

_{m}*n*and

*ṅ*then the integral of

*p*is minimized when

_{m}*n*is chosen to satisfy the Euler-Lagrange equations [25]. Conversely, if

*p*is everywhere a concave function (second derivative non-positive) of

_{m}*n*and

*ṅ*then the integral of

*p*is maximized when

_{m}*n*is chosen to satisfy the Euler-Lagrange equations.

### 3.1. Constant Error Rate Schemes

*a*= 1/2 (section 2.1), −1 (section 2.2), and 1 (previous work [21

**36**, 289–291 (2011). [CrossRef] [PubMed]

*c*= 1. The second derivatives of

*p*are:

_{m}*Ẽ*.

#### 3.1.1. The
a = 1 2 case

*p*are functions of

_{m}*n*and

*ṅ*. It is thus only necessary to calculate second derivatives with respect to one of those variables, rather than both.

*t*

_{0}satisfies sufficient conditions for a minimum for all

*Ẽ*; deviations from a constant error rate scheme will increase the number of zero-fluorophore images. This is consistent with our previous findings for two-wavelength acquisition schemes [21

**36**, 289–291 (2011). [CrossRef] [PubMed]

*m*≥ 3,

*t*is minimized for small error rates. This is exactly what we’d expect from an optimal acquisition scheme. It is also not surprising that

_{m}*t*

_{1}is maximized for

*Ẽ*< 1.62, consistent with our goal of getting as many single-fluorophore images as possible.

*t*

_{2}is

*also*maximized for small

*Ẽ*. However, consider the effects of deviations from the optimal scheme: If

*t*

_{1}and

*t*

_{2}are both maximized, then deviations reduce the number of 1-fluorophore images and also the number of 2-fluorophore overlap images. The loss of 2-fluorophore images partially compensates for the loss of 1-fluorophore images, mitigating the effect on the 2-fluorophore error rate (which is the ratio of 2-molecule images to single-molecule imgaes). This is hence a robust acquisition scheme.

*Ẽ*, the 2-fluorophore images are more common than images with more activated fluorophores, because for small

*Ẽ p*

_{2}>

*p*(

_{m}*m*≥ 3). Second, the 2-fluorophore images are, in general, more difficult to identify and reject than images with 3 or more fluorophores: 2-fluorophore images generally have fewer photons than images with 3 or more fluorophores, and are larger in cross-section. Also, 2-fluorophore images likely to be only slightly elliptical, while images with more activated fluorophores are more likely to have irregular and large shapes that are easier to identify. Thus, when one deviates from the optimal scheme it is most important that the number of 2-fluorophore images be reduced along with the number of 1-fluorophore images. We therefore conclude that acquisition is optimized for

*Ẽ*< 1.62 in this scenario.

#### 3.1.2. The *a* = −1 case

*maximized*for small error rates (

*Ẽ*< 0.5), while single-fluorophore and multi-fluorophore images are minimized. Specifically,

*t*

_{1}satisfies sufficient conditions for a minimum for

*Ẽ*< 0.219, Acquisition at constant error rate can only be considered optimal for higher error rates (

*Ẽ*> 0.586), in which case the integrand satisfies sufficient conditions for maximizing

*t*

_{1}. At intermediate error rates it is difficult to say whether

*t*

_{1}is a minimum, maximum, or saddle point for the constant error rate acquisition scheme. If one wishes to maximize

*t*

_{2}to make the constant-error scheme more robust, as discussed above, it is necessary to work at

*Ẽ*= 1.27 (a very large

*Ẽ*value).

### 3.2. The exponential case

*n*(

*t*) decays approximately exponentially in the optimal acquisition scheme. In both of these cases,

*Ẽ*=

*n*+

*ṅ*/

*k*, so the second derivatives of

_{b}*p*with respect to

_{m}*n*and

*ṅ*have the same form (up to a factor of

*m*= 0, the right side of Eq. (44) is positive for any

*Ẽ*, so the optimal acquisition scheme minimizes the number of zero-fluorophore frames for any value of the error rate. For

*m*= 1, the right side of Eq. (44) is negative as long as

*Ẽ*< 2, which means that even for very high initial error rates the number of single-fluorophore frames is maximized. Since we established in Eq. (38) that the error rate decreases monotonically if

*Ẽ*< 1, it follows that the bound on the error rate set by the requirement of a decreasing error rate is stronger than the bound set by the second order conditions.

*m*≥ 2, the righthand side of Eq. (44) is always positive at

*Ẽ*= 0 and has zeros at

*Ẽ*< 1 the number of images with 3 or more activated fluorophores is always minimized. The case of

*m*= 2 is interesting: The second derivative of

*p*

_{2}is positive for

*Ẽ*< 0.587 and negative for 0.587 <

*Ẽ*< 2.414, so that for

*Ẽ*< 0.587 the number of 2-fluorophore images is minimized, while for larger

*Ẽ*the number of 2-fluorophore images is maximized.

*a*= 1/2 case is that acquisition here is actually optimized at larger error rates

*Ẽ*> 0.587, while in the other case acquisition is optimized for all

*Ẽ*< 1.62. While working at higher error rates might seem problematic, if one uses good rejection algorithms to remove multi-fluorophore images, a large normalized error rate

*Ẽ*can still correspond to a small absolute error rate

*E*

_{2}= 2

*f*

_{2}

*Ẽ*/

*f*

_{1}.

## 4. Conclusions

23. F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express **2**, 1377–1393 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | K. Lidke, B. Rieger, T. Jovin, and R. Heintzmann, “Superresolution by localization of quantum dots using blinking statistics,” Opt. Express |

2. | E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science |

3. | M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods |

4. | S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. |

5. | R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. |

6. | R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophysical Journal |

7. | H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods |

8. | S. A. Ribeiro, P. Vagnarelli, Y. Dong, T. Hori, B. F. McEwen, T. Fukagawa, C. Flors, and W. C. Earnshaw, “A super-resolution map of the vertebrate kinetochore,” Proc. Natl Acad. Sci. U.S.A. |

9. | D. Greenfield, A. L. McEvoy, H. Shroff, G. E. Crooks, N. S. Wingreen, E. Betzig, and J. Liphardt, “Self-organization of the escherichia coli chemotaxis network imaged with super-resolution light microscopy,” PLoS Biol. |

10. | N. A. Frost, H. Shroff, H. Kong, E. Betzig, and T. A. Blanpied, “Single-molecule discrimination of discrete perisynaptic and distributed sites of actin filament assembly within dendritic spines,” Neuron |

11. | S. T. Hess, T. J. Gould, M. V. Gudheti, S. A. Maas, K. D. Mills, and J. Zimmerberg, “Dynamic clustered distribution of hemagglutinin resolved at 40 nm in living cell membranes discriminates between raft theories,” Proc. Natl. Acad. Sci. U.S.A. |

12. | T. A. Brown, R. D. Fetter, A. N. Tkachuk, and D. A. Clayton, “Approaches toward super-resolution fluorescence imaging of mitochondrial proteins using palm,” Methods |

13. | B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, “Whole-cell 3d storm reveals interactions between cellular structures with nanometer-scale resolution,” Nat. Methods |

14. | J. Folling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods |

15. | C. Steinhauer, C. Forthmann, J. Vogelsang, and P. Tinnefeld, “Superresolution microscopy on the basis of engineered dark states,” J. Am. Chem. Soc |

16. | D. Baddeley, I. D. Jayasinghe, C. Cremer, M. B. Cannell, and C. Soeller, “Light-induced dark states of organic fluochromes enable 30 nm resolution imaging in standard media,” Biophys. J. |

17. | M. Heilemann, S. van de Linde, A. Mukherjee, and M. Sauer, “Super-resolution imaging with small organic fluorophores,” Angew. Chem. Int. Ed. |

18. | I. Testa, C. A. Wurm, R. Medda, E. Rothermel, C. Von Middendorf, J. Flling, S. Jakobs, A. Schnle, S. W. Hell, and C. Eggeling, “Multicolor fluorescence nanoscopy in fixed and living cells by exciting conventional fluorophores with a single wavelength,” Biophys. J. |

19. | S. Lee, M. Thompson, M. A. Schwartz, L. Shapiro, and W. E. Moerner, “Super-resolution imaging of the nucleoid-associated protein hu in caulobacter crescentus,” Biophys. J. |

20. | J. Vogelsang, T. Cordes, C. Forthmann, C. Steinhauer, and P. Tinnefeld, “Controlling the fluorescence of ordinary oxazine dyes for single-molecule switching and superresolution microscopy,” Proc. Natl. Acad. Sci. U.S.A. |

21. | E. Shore and A. Small, “Optimal acquisition scheme for subwavelength localization microscopy of bleachable fluorophores,” Opt. Lett. |

22. | A. Small, “Theoretical limits on errors and acquisition rates in localizing switchable fluorophores,” Biophys. J. |

23. | F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express |

24. | H. Goldstein, |

25. | B. Chachuat, |

26. | M. Bates, B. Huang, and X. Zhuang, “Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes,” Curr. Opinion Chem. Biol. |

27. | T. Gould and S. Hess, “Nanoscale biological fluorescence imaging: Breaking the diffraction barrier,” Methods Cell Biol. |

28. | C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods |

29. | C. Eggeling, J. Widengren, R. Rigler, and C. A. M. Seidel, “Photobleaching of fluorescent dyes under conditions used for single-molecule detection: evidence of two-step photolysis,” Anal. Chem |

30. | G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nat. Methods |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(180.2520) Microscopy : Fluorescence microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 8, 2011

Revised Manuscript: August 28, 2011

Manuscript Accepted: August 31, 2011

Published: September 30, 2011

**Citation**

Alex Small, "Model of bleaching and acquisition for superresolution microscopy controlled by a single wavelength," Biomed. Opt. Express **2**, 2934-2949 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-11-2934

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### References

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