The diffraction limit in fluorescence microscopy can be overcome the use of molecules that can switch between a fluorescent “activated” state and a non-fluorescent “dark” state [1, 2, 3, 4]. In these techniques, only a small fraction of the molecules are fluorescing at any given time, producing a sparse image consisting of (usually) non-overlapping blurs (the shapes of which are related to the point spread function (PSF) of the imaging system). A molecular position can be determined from each blur, by fitting the intensity profile to the PSF. By repeatedly activating and localizing different subsets of the molecules, one can thus build a complete map of the fluorescently-labeled structure. The precision of the fitting procedure is determined by the number of photons collected [5, 6] and if the sample is labeled at sufficiently high density [7] then the resolution of the final image can be significantly better than λ/10 for photon counts in excess of a few hundred per molecule. These “localization microscopy” techniques are now being used to study a wide range of topics of biological significance, including the organization of chromatin during mitosis [8], the organization of proteins involved in bacterial chemotaxis [9], actin dynamics [10], clustering of membrane proteins [11], localization of proteins in mitochondria [12], and interactions between mitochondria and the cytoskeleton [13].

In initial localization microscopy experiments, the fluorophores were usually controlled by two separate wavelengths, with one bringing the molecule from the dark state to the activated state and the other causing the activated state to fluoresce [2, 3, 4]. In that case, the probability 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, 15, 16, 17, 18, 19], the switching between dark and activated states is controlled by the same wavelength that excites fluorescence from the activated molecules. Such approaches have the advantage of greater simplicity in hardware, at the expense of reduced latitude of control.

During a localization microscopy experiment, the number of fluorophores 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].) The effect of irreversible bleaching is to change the tolerances for controlling the activation probability 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.

For an experimenter seeking a “snapshot” image of a short-lived structural feature, this speed effect of bleaching may be advantageous and hence worth optimizing. Conversely, for studies of long-term dynamics (in which case bleaching limits the durations of processes that can be studied), quantitative studies of image acquisition in the presence of bleaching are necessary to at least minimize the negative effects of bleaching. Even for fluorophores that can go through a very large number activation/deactivation cycles before irreversibly bleaching [20], maximizing the number of usable (i.e. single-molecule) images obtained is still desirable if monitoring small structures during a very long process.

Given that bleaching affects the tolerances on the activation probability per molecule and thus the image acquisition rate, the question that we study here is how to control 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]. In this acquisition scheme, the 2-molecule error rate E₂, 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], is also constant. Deviations from the optimal scheme (which manifest as perturbations of the linear dependence of n(t) on time) cause the number of single-molecule images to decrease. Interestingly, for fast acquisition (corresponding to larger p and E₂), 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.

In this work, we assume that the bleaching rate β and activation probability p are both functions of the same excitation intensity I, and are hence no longer independent quantities. We consider 4 related scenarios, based on plausible kinetic models of bleaching in switchable fluorophores used for superresolution microscopy. We show that the optimal data acquisition scheme depends very sensitively on the bleaching mechanism, highlighting the critical need for a detailed understanding of bleaching mechanisms in fluorophores used for superresolution localization microscopy. We will use a quasi-steady state model for the activation probability, assuming that at any instant in time the probability of a fluorophore being in a given state depends only on the excitation intensity I and the rate constants for different upward and downward transitions. We will assume that the fluorophores are independent of each other (i.e. we are not considering processes such as Forster Resonance Energy Transfer).

Our focus will be on maximizing the number of single-fluorophore images while minimizing multi-fluorophore images. Recent work has shown that multi-fluorophore images can also be analyzed to obtain fluorophore position information [23], in which case one would want to maximize the number of images with 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

If we assume that the molecules are independent of each other, the probability of m molecules being simultaneously activated in a region of size λ is given by the binomial distribution:

We can set a bound on the activation probability p and derive two useful results for this work, by invoking a result derived previously [22]:

We also showed previously that maximizing the number of single-fluorophore images in a single cycle requires that p be less than 1/n [22]. Increasing p above this level actually decreases the number of 1-molecule images obtained (which can be shown by differentiating p₁ 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 f₂/2f₁ [22], while in the case of bleachable fluorophores it is f₂/f₁ [21]

We will normalize the error rate to simplify our notation for p:

With these results, it is possible to further simplify Eq. (2). Using p = Ẽ/n, and the identity ${\left(1+\frac{x}{n}\right)}^n\to e^x$ for large n and fixed x, we get:

1.2. Expected times

We will be computing the expected amount of time in which exactly m molecules are activated. We thus consider the integral:

We will express t_m in terms of 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.

It is important to note that we are 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]. Rather, we are trying to maximize the total number of single-fluorophore molecules imaged over a fixed time period (generally longer than a single cycle), subject to the constraint that a given number of molecules bleach in that time. The use of the Euler-Lagrange equations contains an implicit assumption that the numbers of unbleached molecules 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

Because single-wavelength superresolution methods are a rapidly evolving area, we will consider several different models that might describe plausible methods and bleaching mechanisms for different fluorophores. In each case, we can write down a kinetic model of one of the following forms:

We now consider four cases:

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

Our first case is similar to initial superresolution experiments, in which the default state of a molecule is the dark state and light is needed to raise the molecule to the activated state. A schematic is given in Fig. 1. While such fluorophores are generally controlled with multiple wavelengths in experiments, in principle a single wavelength could be used (for simplicity) if a fluorophore has strongly overlapping activation and excitation bands. The existence of spontaneous activation [26, 27] when only the longer-wavelength (excitation) beam is turned on suggests that control via a single wavelength may be feasible in some cases. We therefore assume, initially, that the activation probability is given by:

 

Fig. 1 Schematic of states and transitions for a fluorophore in which the dark state is the default state and bleaching occurs from the activated state. We assume multiple vibrational sublevels in the activated and excited states, to account for Stokes shifts of the absorption and emission spectra. The bleaching process depicted occurs from the excited state, and is assumed to not require the absorption of an additional molecule from the excited state.

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It is important to not take Fig. 1 too literally. It is a schematic, and the key point is a sequence of steps: dark → activated → repeated excitation and fluorescence → eventual deactivation or bleaching. We make no assumptions about short-lived or transient intermediate steps; our key assumptions are that (1) these processes have reached a steady state and (2) upward transitions proceed at a rate proportional to the excitation intensity.

After activation, bleaching requires the absorption of a second photon (to go to a more reactive excited state, which may be either the fluorescent single state or a long-lived triplet state), with a rate that is proportional to the intensity, so our kinetic model is:

The Euler-Lagrange equations that must be satisfied to make the integrals in Eq. (6) stationary are:

Rather than using our result in Eq. (16) to derive the Euler-Lagrange equations and then solve them, we will instead use an approach analogous to energy conservation in classical mechanics: Because the Lagrangian in Eq. (6) has no explicit time-dependence (i.e. the time-dependence of p_m is solely due to the time-dependence of 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]:

The requirement of a constant error rate gives us a simple differential equation to solve:

The time-dependent activation probability and illumination intensity are easy to obtain. Because this is a constant error rate scheme, p(t) = Ẽ/n(t) (from Eq. (4)), and from Eq. (10) we know that (for small I) p = I, so we get:

 

Fig. 2 (a) n(t) and (b) p(t) for acquisition at different constant error rates, under scenario 1.

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

In many single-wavelength superresolution experiments, the ground state of a molecule is actually not the dark state; the dark state is reached by the absorption of a photon [14, 15, 16]. Typically, this dark state is a long-lived triplet state. A schematic of this process is shown in Fig. 3. We assume that the rate of transitions from the activated state to the dark state is proportional to the illumination intensity 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 10³ photons per second (a common number in superresolution experiments, e.g. [3, 28, 14, 19] and has an excited state lifetime of order 10⁻⁹ 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, 15]. Putting these assumptions together, we can do some algebra to get the following for the activated state probability:

 

Fig. 3 Schematic of a fluorophore whose default state is activated (i.e. fluorescent). Three plausible bleaching pathways are illustrated, numbered in the order in which they are considered here. Blue upward arrows indicate absorption of a photon, solid diagonal lines indicate bleaching upon the absorption of an additional photon, and diagonal dashed lines indicate bleaching without the absorption of an additional photon.

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We obtain an expression of the same form if we assume that the dark state is reached by first passing through the excited state (e.g. a transition from a single ground state S₀ to a first singlet excited state S₁, from which some fraction of the molecules are transferred to a triplet state T₁). 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).

While bleaching mechanisms in different fluorophores are an area of continued investigation, if the dominant bleaching process occurs from the dark state and is induced by the absorption of a second photon [29, 30](pathway 2 in Fig. 3), the bleaching rate per molecule is given by:

If the activation probability per molecule is assumed to be less than 1/n for n ≫ 1, it follows that p ≪ 1 and so I ≫ 1. We get the following relationship between −ṅ/n and I:

Given 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

Next, let us suppose that bleaching can only happen if activated molecules absorb a photon, at a rate proportional to the excitation intensity. (Pathway 3 in Fig. 3) The bleaching rate per molecule is given by:

2.4. Bleaching from the dark state without the absorption of a second photon

Alternatively, let us consider the case where the dark state is reached via absorption of a photon, and bleaching occurs from the dark state without the absorption of a second photon. (Pathway 4 in Fig. 3) Such a scenario would correspond to a first order bleaching process, in which a molecule in the dark state has a constant probability per unit time of undergoing an irreversible bleaching reaction. In this case, the bleaching rate is proportional to the probability of being in the dark state:

 

Fig. 4 (a) n(t) and (b) p(t) for acquisition at constant error rate, under scenario 2. The time at which the number of molecules has decreased by half is shown for each plot in (a).

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Interestingly, in this case the error rate is not constant. We can show, however, that it is a decreasing function of time. To see this, we need to use the Euler-Lagrange equations and some properties of our Lagrangian. The momentum Π for this case is:

To go further, we will assume that our Lagrangian is p₁ (given in Eq. (6) as Ẽe^−Ẽ), i.e. we are trying to maximize the number of single-molecule images. The derivatives of p₁ are:

We can get the time-dependence of Ẽ from Eq. (38), which can be solved analytically:

 

Fig. 5 Solution to Eq. (38) for different initial error rates.

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Once we have Ẽ(t), we can solve for n(t) using Ẽ = n + ṅ/k_b. Because Ẽ < 1 and n ≫ 1, the time dependence of n(t) is, to an excellent approximation, an exponential decay with rate k_b. The difference between ṅ and −k_bn is very small. Fortunately, however, the quantity that needs to be controlled with high precision is I(t), not n or ṅ. Also, because I ≫ 1, there is considerable latitude in the control of I.

To obtain I(t), we recall that p = 1/(1 + I), and solving Eq. (32) gave I = −ṅ/(k_bn + ṅ) = −ṅ/k_bẼ. The time-dependence of n is approximately n(0)e^−k_bt, so −ṅ = k_bn(0)e^−k_bt, and we get the following for I(t):

 

Fig. 6 Excitation intensity I(t) for different initial error rates Ẽ.

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3. Second-order conditions and robustness

The Euler-Lagrange equations that we solved above are only first order conditions. Maxima, minima, and saddle points are distinguished by second-order conditions. Because a variational calculus problem is a calculus problem in infinite dimensions, it is often difficult to formulate necessary second-order conditions without producing an infinite set of equations (one for each direction in function space). However, there are sufficient second-order conditions that are straightforward to apply: If the Lagrangian p_m is everywhere a convex function (second derivative non-negative) of its inputs n and ṅ then the integral of p_m is minimized when n is chosen to satisfy the Euler-Lagrange equations [25]. Conversely, if p_m is everywhere a concave function (second derivative non-positive) of n and ṅ then the integral of p_m is maximized when n is chosen to satisfy the Euler-Lagrange equations.

3.1. Constant Error Rate Schemes

In the constant error rate scenarios considered, the Lagrangians are of the form:

We can use these expressions to determine which integrals are maximized or minimized for small Ẽ.

The second derivatives change sign when they are equal to zero, which occurs when:

Table 1. Summary of second derivatives of integrands for the constant error rate scheme described in section 2.1.

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Table 2. Summary of second derivatives of integrands for the constant error rate scheme described in section 2.2.

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3.1.1. The $a=\frac{1}{2}$ case

In this case, the integrands p_m are functions of $\sqrt{n\dot{n}}$, which is symmetric under exchange of n and ṅ. It is thus only necessary to calculate second derivatives with respect to one of those variables, rather than both.

We find that t₀ 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]. Likewise, for m ≥ 3, t_m is minimized for small error rates. This is exactly what we’d expect from an optimal acquisition scheme. It is also not surprising that t₁ is maximized for Ẽ < 1.62, consistent with our goal of getting as many single-fluorophore images as possible.

It may seem unfortunate that t₂ is also maximized for small Ẽ. However, consider the effects of deviations from the optimal scheme: If t₁ and t₂ 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.

One might wonder whether it would then be even more advantageous to also maximize the number of 3-fluorophore images, 4-fluorophore images, etc. However, when one deviates from the optimal scheme, reducing the number of images with 3 or more activated fluorophores is less important than reducing the number of images with 2 activated and overlapping fluorophores, for two reasons. First, for small Ẽ, the 2-fluorophore images are more common than images with more activated fluorophores, because for small Ẽ p₂ > 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

In this case, zero-fluorophore images are actually maximized for small error rates (Ẽ < 0.5), while single-fluorophore and multi-fluorophore images are minimized. Specifically, t₁ 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 t₁. At intermediate error rates it is difficult to say whether t₁ is a minimum, maximum, or saddle point for the constant error rate acquisition scheme. If one wishes to maximize t₂ 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

In cases 3 and 4 from Fig. 3, we found that n(t) decays approximately exponentially in the optimal acquisition scheme. In both of these cases, Ẽ = n + ṅ/k_b, so the second derivatives of p_m with respect to n and ṅ have the same form (up to a factor of $1/k_b^2$) and the concavity or convexity of the integrand is easy to determine. The quantity that we need to consider is:

For m ≥ 2, the righthand side of Eq. (44) is always positive at Ẽ = 0 and has zeros at $\tilde{E}=m\pm \sqrt{m}$. One consequence is that for 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 p₂ 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.

As discussed above for acquisition at constant error rate, maximizing the number of 2-fluorophore images along with the number of 1-fluorophore images makes the scheme more robust. The key difference between this case and the 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 E₂ = 2f₂Ẽ/f₁.

4. Conclusions

A major goal in any superresolution localization microscopy experiment is to maximize the number of single-fluorophore images obtained. When the activation process is controlled by the same light source as the bleaching process, it is necessary to balance bleaching effects (which reduce the number of usable fluorophores but also reduce the probability of a nearby fluorophore emitting light concurrent with the fluorophore of interest) against activation effects (which determine the relative probabilities of obtaining single-fluorophore and multi-fluorophore images). Some of the details of the bleaching process therefore have significant effects on the optimal acquisition scheme. While short-lived intermediate states do not affect our results, the following aspects of the activation and bleaching kinetics are of critical importance:

We have analyzed 4 plausible models of the bleaching process in different superresolution localization microscopy experiments, and have shown that in each case the optimal acquisition scheme either involves acquisition at constant error rate or with a decreasing error rate. In each case, only two numbers must be known to implement the optimal scheme: a saturation intensity and a bleaching rate constant. In addition, we have shown that the robustness of the scheme (and whether substantial robustness is achieved at low error rates or high error rates) also depends on the details of the bleaching process. Finally, although new fluorophores are being rapidly developed for use in localization microscopy, our methods are general, and can be used to investigate almost any bleaching and activation process controlled by a single wavelength, as well as to predict optimal acquisition schemes for techniques that extract information from multi-fluorophore images [23].