Recent developments in the field of single molecule orientation imaging have led us to devise a simple framework for analyzing fluorescence intensity fluctuations in single molecule polarization sensitive experiments. Based on the new framework, rotational dynamics of individual molecules are quantified, in this paper, from the short time behavior of the time averaged fluorescence intensity fluctuation trajectories. The suggested model can be applied in single molecule fluorescence fluctuations experiments to extract accurate expectation values of photon counts during very short integration time in which rotational diffusion is likely not to be averaged out.
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Single molecule detection techniques have significantly progressed in recent years in the study of dynamic processes such as chemical and biological reaction kinetics and probing molecular motion in heterogeneous materials, without the loss of information that is typically encountered in ensemble averaging . Detection of single molecule rotational motion is an exquisitely sensitive and useful tool in exploring the dynamics in various biological and chemical systems [2–4]. A measure of dipole orientation can be obtained by calculating the reduced linear dichroism signal that is measured by detecting any two orthogonal polarizations in the in-plane of the sample and taking their difference divided by their sum. While in ensemble measurements the rotational dynamics is extracted by fitting the polarization signal to an exponential function , rotational dynamics of individual molecules is extracted by correlating the fluctuations of the reduced dichroism signal . However, most recent experiments have indicated deviations from the expected exponential behavior for the reduced linear dichroism correlation function in different observation times [7, 8]. Various approaches have been presented recently to analyze the origin of the non-exponential behavior for this signal [9–11]. In any event, useful correlations require enhanced temporal resolution of experiment. In typical photon counting experiments, use of emission intensities for polarized light implies time averaging that increases the signal to noise ratio at the cost of temporal resolution. Thus, very fast fluorescence fluctuations may be obscured during integration time. Recently, several approaches have been presented aiming to increase temporal resolution of fluorescence signal [12–14].
This paper introduces a dynamic method to analyze fluorescence fluctuations in single molecule fluorescence polarization sensitive experiments. First principles of non-equilibrium analysis are used to derive a simple framework to extract rotational diffusion parameters from the short time behavior of time averaged fluorescence intensity fluctuations, while properly accounting for the impact of the numerical aperture (NA) value of the microscope objective and the timescale of fluorescence fluctuation.
2. Single molecule orientation imaging
A schematic of the experimental arrangement, defining the azimuthal angle and polar angle , is shown in Fig. 1 . Assuming a plane wave emission polarized along the molecular transition dipole moment, the signals and are proportional to the projections of the transition moment onto the two polarization directions being measured. The cone of light collected by a lens is given by the NA of the collecting objective, such that the collection angle is given by:15], the signals and are related to the transition dipole orientation and the NA. The fluorescence signals that would be seen on the detectors are given by:Eq. (2) and (3) is that for an optical system with a nonzero NA, both polarization channels will detect signals even if the dipole radiation source is oriented parallel to the optical axis. In the simple case where the limit of NA approaches zero; then A~0 and B = C. Additionally, the total signal measured between the two channels, is always dependent.
This dependency is well-known and has been experimentally observed also in bulk measurements [16, 17]. The single fluorophore is modeled by a dipole moment that is rapidly rotating in time. The effect of the polar angle on the measured fluorescence intensity () will drop as the molecule tips out of the sample plane.
3. Fluorescence intensity fluctuations equation
3.1 The influence of rotational diffusion on single molecule fluorescence intensityEq. (5) (below) can be eliminated. Hence, the rotational diffusion equation for the averaged dipole distribution P reads :Eq. (4) into Eq. (5) and using rules of partial derivatives we find:Equation (6) presented here for the first time – the core equation of this manuscript – provides a general framework for treating fluctuation of fluorescence intensity as a result of molecular reorientation. An equation for the averaged intensity is obtained by multiplying Eq. (6) with, performing integration by parts, solving in Laplace space and inverting back to time domain18], is the rotational correlation time and . Consequently, Eq. (7) allows us to extract from an ensemble of fluorescence intensity trajectories that are measured for a single fluorophore. According to Eq. (7), higher values of are characterized by a slow convergence of as expected. Furthermore, in the long time limit , which is a typical behavior. However, convergence of is in the time scale of . Thus, in direct measurements of , the temporal resolution used in experiments must be as low as itself in order to track fluctuations. Typically, using single photon counting devices, such as avalanche photodiodes or photomultiplier tubes used in photon counting experiments, leads to counting loss. Consequently, fast fluorescence fluctuations may be obscured during integration time.
3.2 Time averaged single molecule fluorescence intensity
In order to maximize signal-to-noise ratio for a given temporal resolution, we suggest examining the changes in the total sum of fluctuations during a chosen time interval instead of the amplitude of fluctuations at time t. Consider the time average19] and applied in several cases [20, 21] to get the PDF of time integrals of stochastic variables. The PDF is derived from the PDF at a previous moment t along the straight line for which . Thus, we may writeEq. (9) with respect to the first power of , substitution of Eq. (6) and multiplying with , after performing integration by parts and solving in Laplace space we get:
3.3 Theory & Simulations
To gain further understanding of these analytical results and to test the effect of rotational diffusion, single molecule rotational trajectories at different correlation times were simulated using a random walk on a sphere, assuming . Ten thousand angular trajectories were then used to compute fluorescence signals, and obtaining through Eq. (10). The simulation results (dotted lines) are compared to theory (solid lines) as can be seen in Fig. 2 . In the long time limit t→∞ we get the expected result.
One can observe in Fig. 2 that for shorter values of , simulation deviates from theory (Eq. (10)) in the scale . This is a consequence of very fast rotational diffusion that yields less predictable value of for time bins of 1nsec used in simulations. Using a sampling time of 0.1 nsec, we get a perfect match (data not shown). That is, the size of the chosen time bins introduces an additional arbitrary time scale which should be short enough to track fluorescence intensity fluctuations. However, for each there is a time limit in which fluctuations become slower and can be tracked by larger time bins, as one can observe in (Fig. 2 – dashed rectangle).
4. The role of photon statistics in single molecule polarization sensitive experiments
In an actual experiment, the time limit in which fluctuations become slower can be quantitatively described by the photon counting distribution. For a stationary dipole, Poisson distribution fully describes the statistics of photon count detection. However, a change in dipole orientation will also cause a change in the mean Poisson quantity. If this change is random, then the arrival of photons to the detector, and hence their subsequent detection, is a doubly stochastic process.
The relation between continuous fluorescence intensity distribution and discrete photoelectron distribution was first derived by Mandel in the context of laser fluctuations :Eq. (11), fluctuations of the emitted fluorescence intensity will cause additional broadening of the photon counting distribution as a result of rotational diffusion. This broadening depends on the integration time T used in experiment. In the limit of long integration times (), fluorescence intensity fluctuations will be completely averaged out in the corresponding fluctuations of . In this case, the probability distribution, approaches a delta function, and the will narrow to a Poissonian. For very short integration times (), fluctuations of will track the fluorescence intensity fluctuations I(t) completely. Thus, the probability distributions of and I(t) are proportional to each other, and in order to capture intensity fluctuations of a particular process of interest characterized by , one must choose an integration time T, shorter than the fluctuation time scale for that particular process. For this case the approximation, can be used [23–25]:Eq. (12).
However, in the time limit in which does not change significantly as result of slower fluctuations, larger sampling intervals can be used. In this extreme case which is dependent, is approximated by Poissonian with mean at time T:Eq. (13) is valid, can be determined by comparing the two approximations above (Eq. (12) and Eq. (13)). Whereas Eq. (12) determines the sampling time in which the fastest fluctuations are still tracked, Eq. (13) must yield the same effectivefor the chosen sampling time in order to continuously track slower fluctuations. For example, in case where n = 1.4, with high NA value of 1.3: ϕmax = 1.1905, A = 0.0781 and B = 0.04. One can observe in Fig. 3(a) that for , the realistic value for the rotational correlation time of fluorophore in water environment  and T1 = T2 = 10 (Eq. (12) for time interval T1 (solid line) and Eq. (13) for time interval T2 (dashed line)), Eq. (13) yields less effective than Eq. (12). Thus in this time limit of T2, fluctuations are not slow enough to be described by Eq. (13). While using T2 = 100 in Eq. (13) and T1 = 10in Eq. (12), both approximations are almost identical (Fig. 3(b)). That is, by choosing different sampling rates in these different time scales of convergence, fluctuations are continuously tracked. However, in the latter case when lower NA value of 0.4 is used (leading to ϕmax = 0.2898, A = 0.000428 and B = 0.0098 values), is decreased as can be seen in Fig. 3(c). In this case, choosing T2 = 400 increases detection probability (Fig. 3(d)). Obviously, for higher values of where convergence is slower, larger sampling intervals needs to be considered.
Variation of out of plane orientation of single dipole during a finite measurement time can alter the statistical properties of the total measured signal in polarization sensitive experiments. We derive the equations that govern the PDF of the measured fluorescence intensity fluctuations for the case of Brownian diffusion of single molecule dipole. While previous works consider expectation values of single molecule fluorescence intensity in the long time limit compared to the rotational correlation time, solution of Eq. (6) yields the short time behavior of fluorescence intensity. However, fast fluctuations as a result of rotational diffusion may be obscured during integration time. Thus, the temporal resolution of experiment must be short enough to allow tracking of fluctuations.
Based on the above model solution, we propose an effective scheme to sample the fluorescence signal for different time scales of fluctuations based on photon counting statistics. We find also that the temporal resolution chosen in experiment must take into account the size of microscope objective NA. This result might be linked to previous results of observed non exponential decay of dichroism fluorescence signal for large NA . This letter discusses only continuous changes of dipole orientation. For discrete molecule reorientation process, one needs to insert a waiting time distribution into the jump process. The case of anisotropic rotation is not explicitly investigated here, however in this case, the diffusion constant would need to be replaced by a tensor.
In fluorescence fluctuation spectroscopy experiments, where it is essential to make sure that the chosen integration time in experiment is sufficiently short to track fluorescence intensity fluctuations, there are cases in which fluctuations, as a result of rotational diffusion, are not averaged out. Applying the suggested model in such cases should yield accurate expectation values of photon counts in very short integration time - as short as the rotational correlation time itself.
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