Breaking the Low Concentration Barrier of Single‐Molecule Fluorescence Quantification to the Sub‐Picomolar Range

Abstract

Single‐molecule fluorescence techniques provide exceptional sensitivity to probe biomolecular interactions. However, their application to accurately quantify analytes at the picomolar concentrations relevant for biosensing remains challenged by a severe degradation in the signal‐to‐background ratio. This so‐called “low concentration barrier” is a major factor hindering the broad application of single‐molecule fluorescence to biosensing. Here, the low concentration limit is broken into while keeping intact the confocal microscope architecture and without requiring complex microfluidics or preconcentration stages. Using fluorescence lifetime correlation spectroscopy (FLCS) and adding a diaphragm to the laser excitation beam, a limit of quantitation (LOQ) down to 0.1 pM is achieved, significantly below the state‐of‐the‐art. The physical parameters setting the LOQ and introduce a broadly applicable figure of merit (FoM) is identified that determines the LOQ and allows for a clear comparison between experimental configurations. The approach preserves the ability to monitor dynamic interactions, and diffusion times, and distinguish species in complex mixtures. This feature is illustrated by measuring the biotin‐streptavidin association rate constant which is highly challenging to assess quantitatively due to the strong affinity of the biotin‐streptavidin interaction. These findings push the boundaries of single‐molecule fluorescence detection for biosensing applications at sub‐picomolar concentrations with high accuracy and simplified systems.

Single‐molecule fluorescence detection can become highly challenging for diluted concentrations in the picomolar range. This “low concentration barrier” is broken into and achieves quantitative detection down to 0.1 pM. Introducing a diaphragm into a standard confocal microscope and implementing fluorescence lifetime correlation spectroscopy (FLCS) achieve superior performance, enabling precise analysis of biomolecular dynamics, as illustrated for biotin‐streptavidin binding.

1. Introduction

Fluorescence techniques have become essential tools to investigate the molecular mechanisms involved in biology, notable thanks to their exceptional sensitivity down to the single‐molecule level.^{[ 1} , ² , ³ , ^{4 ]} A broad class of fluorescence‐based methods is currently available to monitor molecular dynamics with high sensitivity and temporal resolution, including fluorescence time trace analysis,^{[ 5 ]} fluorescence correlation spectroscopy (FCS),^{[ 6} , ⁷ , ^{8 ]} Förster resonance energy transfer (FRET),^{[ 9} , ^{10 ]} single particle tracking (SPT),^{[ 11 ]} or multiparameter fluorescence detection (MFD).^{[ 12} , ^{13 ]} However, the applications of single‐molecule fluorescence techniques to biosensing and diagnostics at sub‐picomolar concentrations remain scarce.^{[ 14} , ^{15 ]} Enzyme‐linked immunosorbent assay (ELISA)^{[ 16} , ^{17 ]} and polymerase chain reaction (PCR),^{[ 18 ]} the two gold standards for protein and DNA detection, do not rely on single‐molecule fluorescence detection. As we discuss below, there are many different technical challenges currently preventing the deployment of the rich single‐molecule fluorescence toolbox to the quantitative detection of analytes at sub‐picomolar concentrations relevant for biosensing.

A confocal microscope equipped with a high numerical aperture (NA) objective is the ubiquitous workhorse for all the different single‐molecule fluorescence techniques (Figure 1a). The rationale behind this choice is to maximize the detected fluorescence brightness per molecule while simultaneously minimizing the background from Rayleigh and Raman scattering.^{[ 19 ]} However, a direct consequence of the diffraction limit in microscopy is that the detection of single fluorescent molecules is only achievable in a specific concentration range (Figure 1b).^{[ 14} , ^{20 ]} For a 1 fL confocal volume, the concentration to isolate a single‐molecule is 2 nM. Concentrations significantly deviating from the nanomolar range will result either in ensemble‐averaging effects where the single‐molecule resolution is lost (high concentration limit, micromolar range) or in severe degradation of the signal‐to‐background ratio (low concentration limit, picomolar range). Many different strategies (involving mainly nanophotonics and plasmonics) have been devoted to extending the high concentration limit and overcoming the diffraction limitations in confocal microscopy.^{[ 14} , ²⁰ , ²¹ , ^{22 ]} On the contrary, the low concentration limit in the picomolar range remains significantly less documented when it comes to confocal single‐molecule fluorescence.^{[ 23 ]} The parameters defining the limit of detection (LoD) for confocal microscopy are not clearly stated, nor is the microscope configuration yielding optimum results. Since the infancy of single‐molecule fluorescence detection, it is commonly believed that the higher the objective NA the better, yet this common belief can be questioned as high NA objectives also lead to low sensing volumes.

Figure 1.

Challenges of using confocal microscopy for detecting single‐molecules at picomolar concentrations. a) Sketch of the confocal diffraction‐limited volume. R is the average distance between individual molecules. b) Comparison of the practical ranges for single‐molecule detection between confocal and nanophotonic approaches (including FCS‐related methods). c–e) Comparison between the different confocal microscope configurations used in this work: (c) high NA microscope objective, (d) low NA objective, and (e) high NA objective with a diaphragm on the excitation beam. The time traces at the bottom were recorded experimentally with a 2 pM solution of CF640R fluorescent molecule.

Counting the number of molecules in a highly diluted solution resembles “finding a needle in a haystack” problem. For concentrations below 1 picomolar, the average distance between two fluorescent emitters becomes on the order of several micrometers (Figure S1a, Supporting Information), far exceeding the microscope resolution. With such large intermolecular distances, the time between consecutive detection events becomes excessively large, on the order of several seconds in the case of free Brownian diffusion (Figure S1b, Supporting Information). In this scenario, the background contribution can quickly overcome the signal as most of the time the detector accumulates scattered background photons or dark counts.^{[ 24} , ^{25 ]} To overcome these long waiting times, a possibility involves stirring the sample to accelerate the detection rate.^{[ 26 ]} However, a major drawback of this approach is that the burst duration of each single‐molecule event is also significantly reduced, thereby limiting the amount of collected photons per molecule and leaving the signal‐to‐background SBR nearly unchanged. Another possibility would be to use a microscope objective with a lower numerical aperture so as to increase the size of the detection volume and detect more easily individual molecules in a diluted solution (Figure S1c, Supporting Information). Here the limitation is that the SBR gets strongly degraded for low NA objectives, with a typical SBR value below 0.5 for NAs lower than 0.4 (Figure S1d, Supporting Information). Lastly, a major challenge that is very often overlooked is that the background level itself also encounters intensity noise fluctuations and can show some temporal correlation. If the background was perfectly known and perfectly constant, it could be subtracted from the detected intensity without much issue. However, because of the Poissonian process of single photon detection, the background level B experiences shot noise fluctuations alike the signal, with a standard deviation equivalent to $\sqrt{B}$. This background noise prevents a basic subtraction of the baseline from the signal to achieve an accurate quantification of the number of molecules at ultralow concentrations.

Here we aim to investigate the limit of quantitation (LOQ also sometimes referred to as the limit of quantification)^{[ 27 ]} for single‐molecule fluorescence counting with fluorescence lifetime correlation spectroscopy (FLCS) at low concentrations in the subpicomolar range. In this work, we define the LOQ as the minimal concentration where FLCS can be used to precisely determine the local number of fluorescent molecules with a relative error equal to 1. Our approach to experimentally characterize the relative errors differs from the classical LOQ definition as the background level plus 9 times the standard deviation.^{[ 15 ]} However, our method ensures a comprehensive assessment is provided with all the relevant information available for accurate quantification depending on the end‐user needs. This value also differs from the standard definition of the limit of detection (LOD) generally used in biosensing (background level plus 3 times the standard deviation) which assesses the statistical reliability of the measurement against the background.^{[ 28} , ²⁹ , ^{30 ]} We introduce a simple figure of merit (FoM) which is shown to determine the relative error and is broadly applicable to compare between experimental configurations and figure out the best microscope configuration. We also discuss a simple modification of the ubiquitous confocal microscope (introducing a diaphragm on the laser excitation beam) and demonstrate the superior sensing performance achieved using this configuration.

FLCS is a powerful variation of fluorescence correlation spectroscopy (FCS) that uses lifetime information to improve measurement accuracy.^{[ 31} , ³² , ³³ , ^{34 ]} In any single‐molecule fluorescence experiment, determining the background intensity level is not completely straightforward as this level may fluctuate over time and/or include luminescence emission from contaminants. In FLCS, the known fluorescence lifetime of the target molecule can be used to separate the signal from the background,^{[ 32} , ³⁵ , ³⁶ , ^{37 ]} allowing for the precise measurement of local concentrations, diffusion coefficients, and biochemical reaction rates. Other applications of FLCS include distinguishing separate species in a complex mixture,^{[ 37} , ³⁸ , ^{39 ]} and removing afterpulsing artifacts.^{[ 40 ]} However, while FLCS has been established as a quantitative method to count single‐molecules,^{[ 31} , ³⁷ , ³⁸ , ^{41 ]} the factors determining the LOQ in FLCS remain unknown like the approaches to improve the LOQ and the definition of a broadly applicable FoM. In this work, we address all these challenges at once. We illustrate the effectiveness of our technique by monitoring the interaction dynamics of biotin binding to streptavidin. The biotin‐streptavidin interaction is a prime example of a biochemical reaction with a very high affinity and has major applications in biotechnology.^{[ 42} , ^{43 ]} However, its high association rate constant in the range 3.0 × 10⁶ to 4.5 × 10⁷ M⁻¹.s^{−1[ 44 ]} makes this interaction highly challenging to monitor using standard FCS technique.^{[ 6} , ^{45 ]} Altogether, our findings open up new possibilities to extend the sensitivity of FLCS in biosensing applications, paving the way for detecting subpicomolar concentrations with high accuracy.

2. Results and Discussion

Figure 1c–e summarizes our different experimental configurations to explore the limits of FLCS at picomolar concentrations. As a fluorescent molecule, we select CF640R for its high brightness, good photostability, and relatively long 4.4 ns fluorescence lifetime.^{[ 39} , ^{46 ]} We also perform our characterization experiments (Figures 2 and  3 ) in D₂O buffer to benefit from a 30% increase in brightness thanks to the reduction of the quenching losses induced by water in the red spectral range.^{[ 47} , ^{48 ]} However, our approach to determining the LOQ and the FoM remains fully general and can be applied to other fluorescent emitters and spectral ranges without problem. While FLCS should be the preferred approach to determine (and correct for) the background on each experimental run, our data will also discuss the use of classical FCS without using lifetime filtering.

Figure 2.

Determining the best‐performing microscope configuration and the limit of detection for FLCS at low concentrations. a) Evolution of the average number of molecules measured by FLCS as a function of the CF640R concentration. Throughout this figure, markers are experimental data, and the error bars represent twice the standard deviation (the number of experiments and integration times are detailed in Table S1, Supporting Information). The black lines are the expected values based on the detection volumes measured at concentrations above 10 pM (serving as ground truth information here). When the experimental data deviates significantly from the black line prediction, the FLCS method becomes inaccurate. b) The absolute value of the relative error in the FLCS number of molecules as a function of the concentration for the different configurations, determined from the data in (a). The solid color lines are a numerical fit to the data, while the dashed lines correspond to a theoretical model based on Equation S4 (Supporting Information). The configuration with the 1.2NA objective with the diaphragm is the most accurate (lowest relative error) for low‐concentration detection. c) Product of the number of molecules in the detection volume N times their fluorescence brightness CRM as function of the concentration for the different configurations in Figure 2. The points where this quantity corresponds to the background level B and the square root of B are indicated by markers. As seen in (a,b), FLCS is still accurate when N*CRM < B (empty markers, dotted lines) but the relative error increases dramatically when $N\ast CRM<\sqrt{B}$ (filled markers, dashed lines). d) Relative error in the FLCS number of molecules (similar to a) as a function of the quantity $N\ast CRM/\sqrt{B}$. When the errors are plotted as function of this term, all the three cases collapse (within experimental uncertainties) into a single trend. This observation demonstrates that the quantity $N\ast CRM/\sqrt{B}$ is the key factor defining the accuracy in FLCS detection of the number of molecules, leading us to use it for our definition of the FoM.

Figure 3.

Background correction in FLCS to improve the accuracy at low concentrations. All these data were recorded on the 1.2NA objective with the diaphragm, as this configuration was determined in Figure 2 to provide a superior performance. a) Evolution of the average number of molecules measured with different techniques (FCS and FLCS, with and without background correction, see text for details) as a function of the CF640R concentration. Markers are experimental data, error bars correspond to twice the standard deviation. The number of experiments and integration times are detailed in Table S1 (Supporting Information). The black line is the expected trend (ground truth) based on the detection volumes measured by FLCS at concentrations above 10 pM. The color lines are a guide to the eye to better visualize the deviation from the linear prediction. b) Evolution of the relative error at 0.2 pM concentration when different modifications are applied to the experimental protocol. c) The absolute value of the relative error in the number of molecules as a function of the concentration for the different configurations, determined from the data in (a). Lines are a numerical fit to the data. d) Accuracy gain as compared to regular (uncorrected) FCS as a function of the concentration, determined from the ratios of the relative errors in (c).

To represent a typical configuration found on confocal microscopes, we use a 1.2 NA water immersion objective (Figure 1c). This configuration presents the highest fluorescence brightness (count rate per molecule CRM) and the lowest background intensity B, but also the smallest detection volume. Clear peaks associated with single‐molecule diffusion events are visible on the fluorescence intensity time trace at 2 pM CF640R concentration (Figure 1c) but at a relatively low frequency of ≈1 event per second. As an alternative for a moderate (lower) NA objective, we use a 0.5 NA water immersion objective (Figure 1d, quite similar results were obtained while testing a 0.6 NA air objective; they will be not represented here for clarity). The 0.5 NA features the lowest brightness but allows for the largest detection volume, being 120 × larger than the 1.27 fL diffraction‐limited volume found with the 1.2 NA objective.

The third and last configuration is represented in Figure 1e: a diaphragm is installed on the laser excitation beam to reduce its diameter to 1.35 mm and underfill the 6.5 mm objective back aperture. The rationale behind this choice is to enlarge the confocal detection volume and voluntarily deviate from the diffraction limit set by the objective's NA. As a consequence of the diaphragm presence, the laser spot size is increased allowing to excite molecules on a larger volume.^{[ 49 ]} To account for this, the confocal pinhole has to be increased from 50 to 200 µm. A detailed characterization of the influence of the diaphragm diameter is presented in Figure S2 (Supporting Information) together with guidelines to set the experimental parameters. Technically, this configuration merges the advantages of the previous two setups (Figure 1c,d): the excitation is virtually performed as if we were using a low NA objective (large excitation volume), but the collection still takes advantage of the high collection NA (high detection efficiency for high fluorescence brightness). One drawback is that the laser power has to be increased to compensate for the lower excitation intensity due to the larger spot size. This increased power also leads to an increase in the background level, which scales proportional to the volume and the laser power. Currently, the brightness CRM for the 1.2 NA with diaphragm case is limited by the maximum power available from our fiber‐coupled 635 nm pulsed laser diode (PicoQuant LDH series). We have also calibrated the brightness CRM as a function of the excitation power for all the 3 configurations to ensure our results avoid saturation and are representative of the linear regime of fluorescence.

With the 3 microscope configurations (Figure 1c–e), we have a set of cases where the count rate per molecule CRM and the detection volume vary by more than 30 × and 120 × respectively. The questions are now to determine what is the FLCS limit of quantitation (LOQ) for each case, what are the key parameters determining the LOQ and which is the best‐performing setup. To this end, we perform a series of dilution experiments,^{[ 37 ]} starting from concentrations in the nanomolar range and gradually decreasing the CF640R concentration to monitor when the number of molecules N measured with FLCS deviates from the expected linear trend. Figure 2a summarizes our main results. Each experimental data point results from a complete FLCS analysis (see Experimental section for details and Figure S3, Supporting Information for representative raw datasets). Above 10 pM concentrations, the results for all the 3 configurations follow a linear trend with the concentration, where the detection volume determines the slope. Even the 0.5 NA objective allows a quantitative assessment of the number of molecules in this regime, despite the prevailing belief in the field that high NA objectives are essential.

We are primarily interested in noting when the FLCS number of molecules deviates from the linear trend. The deviation is better visualized while plotting the relative error (N_measured − N_expected )/N_expected as a function of the concentration (Figure 2b). Empirically, we find that the relative error follows a power‐law dependence on concentration, with a −1.5 exponent across all three configurations (color lines in Figure 2b). We have derived a theoretical model (dashed black lines in Figure 2b) detailed in Section S6, Supporting Information, though the reader may easily omit this demonstration which is not central to the rest of our article. The case with the 1.2 NA objective is the first to deviate at concentrations below 10 pM, followed by the 0.5 NA objective below 2 pM. The configuration using the 1.2 NA objective with a diaphragm is the most accurate at low concentrations, with the deviation being only visible at concentrations below 0.5 pM (Figure 2a,b).

As briefly mentioned in the introduction, two different definitions come into play while discussing the limit of detection. Generally in biosensing, the limit of detection LOD is defined as the minimal concentration where the signal exceeds the background plus three times the standard deviation.^{[ 29} , ^{30 ]} This LOD definition ensures the reliability of the measurement against the background ensuring that the probability of a false negative is typically below 1%. However, determining the presence or absence of fluorescent molecules against a certain background is generally not a question of interest for FCS users. Instead, the question focuses on precisely determining the local number of fluorescent molecules with a chosen level of accuracy. Determining the maximum acceptable error a priori depends on the targeted application. Here we arbitrarily consider the concentration where the relative error in N amounts to 1, and use this value to define the so‐called limit of quantitation LOQ (also known as limit of quantification). The LOQ is more demanding and generally higher than the LOD, which is expected to be found in the tens of femtomolar range (see a discussion in Figure S4, Supporting Information together with background correlation data).

Our approach to define the LOQ depends on the level of accuracy desired, there is an element of arbitrariness in this decision.^{[ 27} , ^{50 ]} The curves displaying the relative errors in N as a function of the concentration provide all the information and can readily be used to define which concentration gives a maximum tolerable relative error (Figure 2b). Choosing this maximum acceptable error depends on the targeted application. Here we arbitrarily decide to define the LOQ as the concentration where the relative error in N amounts to 1. For the 1.2 NA objective, our results indicate a LOQ of 0.75 ± 0.05 pM, going down to 0.3 ± 0.1 pM for the 0.5 NA objective and 0.15 ± 0.05 pM for the 1.2 NA objective with a diaphragm. These results clearly indicate the superior sensing performance of the configuration using the 1.2 NA objective with a diaphragm. Achieving an intermediate detection volume of a few tens of femtoliter while still preserving the fluorescence brightness per molecule is the key to improving the FLCS accuracy at sub‐picomolar concentrations and pushing its sensitivity.

To further validate our approach, we compare our results with the IUPAC definition of the limit of quantification (LOQ), which sets the threshold as the average background signal plus nine times its standard deviation. Repeated experiments on the blank solution in the absence of fluorescent target molecules assess the average background and its standard deviation (Figure S4a,b, Supporting Information). Considering the mean background plus nine times the standard deviation results in a detection limit in the 10 fM range (Figure S4c,d, Supporting Information), an order of magnitude lower than our reported values. This finding ensures that the false positive and false negative rates remain well below 1%,^{[ 15 ]} highlighting the conservative nature of our claims and reinforcing the robustness of our approach. This analysis is consistent with the discussion identifying systematic errors, likely stemming from residual shot noise fluctuations, as key parameters defining the FLCS accuracy (Figure 3).

While the data in Figure 2a,b indicate when the FLCS deviates from the expected trend and loses its accuracy, the observations do not elaborate on the origins of this effect and what are the key parameters determining the LOQ. We have performed different sets of experiments to further explore the origin of the accuracy deviation. First, theoretical calculations indicate that the deviation is not a statistical problem, as the signal‐to‐noise ratio (SNR) is always greater than 1 for all the 3 microscope configurations at concentrations above 0.05 pM (Figure S5, Supporting Information). This result is further confirmed by the fact that increasing the FLCS integration time from 3 to 18 min does not affect the relative error (Figure S6, Supporting Information). From these observations, we can conclude that the accuracy deviation is a systematic error and is not related to some statistical noise affecting the measurement. In a second set of experiments, we have stirred the solution containing the CF640R molecules while performing FLCS (Figure S7, Supporting Information). The idea is to speed up the diffusion process so as to increase the frequency at which fluorescence events are detected. While the diffusion time is accelerated by more than 10×, again the relative error remains unchanged. This result confirms the error invariance with the integration time: acquiring more single‐molecule bursts surprisingly does not improve the accuracy at subpicomolar concentrations. Last, applying time‐gating^{[ 36 ]} enables an improvement in the accuracy (Figure S8, Supporting Information). Here photons are rejected based on their arrival time before the FLCS filters are applied, allowing to virtually reduce the background level B. The selection of tighter time‐gated intervals reduces the systematic error (Figure S8, Supporting Information). This indicates that the background level B is a key factor determining the FLCS deviation.

To better visualize this effect, we plot in Figure 2c the product of the expected number of molecules N times their fluorescence brightness CRM as function of the concentration. This quantity N*CRM represents the net fluorescence signal collected from the detection volume. For each microscope configuration, we indicate the background intensity level B on the same graph and highlight the concentration where the signal‐to‐background SBR goes below 1 (white markers in Figure 2c). This typically occurs for concentrations ≈5 to 10 pM for the different microscopes, yet FLCS remains accurate for concentrations at least one order of magnitude lower. As the core concept of FLCS is to filter out the photons using the lifetime information to distinguish between the signal and the background, it is thus not surprising that FLCS remains accurate at conditions where SBR < 1. More intriguing is the observation that the LoDs as defined in Figure 2b correspond to the levels where the net signal N*CRM amounts to $\sqrt{B}$ for all the three configurations (filled markers in Figure 2c). To elaborate further on this observation, we reconsider the relative errors in FLCS (similar to Figure 2b) and plot them as a function of the quantity $N\ast CRM/\sqrt{B}$ (Figure 2d). The errors for all the three setups then collapse into a unique trend (within experimental uncertainties), although the configurations differ by more than 100 × in detection volumes and 30 × in molecular brightness. This demonstrates that the quantity $N\ast CRM/\sqrt{B}$ is the key factor defining the accuracy in the FLCS detection of the number of molecules.

We thus define the figure of merit in FLCS as FoM = $N\ast CRM/\sqrt{B}$. The presence of a term in $\sqrt{B}$ instead of the background level, B can be explained as a residual contribution from the shot noise on the background level that remains uncorrected by FLCS. While FLCS is very efficient in filtering signal photons from the background for each experimental run, it does not incorporate the fact that due to the Poissonian nature of the detection event (being signal or background), there is inevitably a shot noise contribution whose uncertainty scales as the square root of the average intensity.^{[ 51 ]} Hence if the total count rate per second is I ₀, the standard deviation of the shot noise will be $\sqrt{I_0}$. We have verified experimentally that this reasoning still holds for the background level: the standard deviation of the noise on the background B indeed scales as $\sqrt{B}$ (Figure S9 and Table S2, Supporting Information) meaning that shot noise is the dominant source of noise in our system and that electrical or computer noise remains negligible.

Having a figure of merit for FLCS allows to directly compare between configurations and even predict the LOQ. When the FoM becomes lower than 1, the relative error increases strongly and the FLCS technique becomes inaccurate (see Figure S10, Supporting Information for a plot of this FoM as a function of the concentration). In the conditions used here where the SNR is not an issue, the FoM does not depend on the molecular diffusion time, nor on the number of events per second or on the total integration time (Figures S5–S8, Supporting Information). Achieving the lowest LOQ concentration boils down to maximizing the detection volume and the CRM brightness while minimizing the background level. For this purpose, the configuration in Figure 1e with the 1.2 NA objective and the diaphragm achieves the best performance and highest FoM (Figure S10, Supporting Information), with nearly one order of magnitude gain in FoM as compared to the classical 1.2 NA case. Even more surprising is the observation that the 0.5 NA objective yields a 2.5 × better FoM than the classical 1.2 NA configuration, challenging the usual belief in FCS that the highest NA objective always yields the best results. Sensing at low concentrations outside the usual FCS and FLCS ranges imposes to rethink this postulate.

In regular FCS, the influence from a non‐negligible background is often accounted for by introducing a supplementary correction factor ${(1-\frac{B}{B+N\ast CRM})}^2$.^{[ 6} , ^{52 ]} We now investigate if a similar correction could be applied to FLCS as well. Having demonstrated the superior sensing performance of the 1.2 NA objective with the diaphragm, we consider this case for the data in Figure 3 (similar results can be obtained with the 1.2 NA objective without the diaphragm, see Figure S11, Supporting Information). Since the residual background contribution goes with $\sqrt{B}$ in FLCS, we introduce a correction factor ${(1-\frac{\sqrt{B}}{\sqrt{B}+N\ast CRM})}^2$ for FLCS, where the B term in the classical FCS correction has been replaced by $\sqrt{B}$ (Figure S12, Supporting Information). Figure 3a presents our experimental results and compares between the four different cases of FCS or FLCS with and without correction. Without surprise, regular FCS without background correction is the least accurate method and deviates from the predicted values already at concentrations ≈100 pM. Introducing the background correction term in FCS improves the situation, yet the difficulty to precisely measure the background level B for each experimental set and the variability of B add systematic errors. As a consequence, the data from FCS with correction deviate at concentrations below 10 pM. As we have seen in Figure 2a, uncorrected FLCS performs better than background‐corrected FCS, though the deviation starts at ≈0.5 pM. Finally the ${(1-\frac{\sqrt{B}}{\sqrt{B}+N\ast CRM})}^2$ background correction in FLCS further improves the accuracy by ≈3×, pushing the LoD below 0.1 pM. To the best of our knowledge, this is the first time that a FCS‐related technique is proven accurate down to concentrations of 100 fM. With the molecular mass of CF640R being only 832 Da, this LOQ corresponds to ≈100 fg mL⁻¹. Importantly the detection is not limited to the quantification of the number of molecules, the key FCS abilities to analyze the diffusion time and distinguish between species in a mixture are still preserved.

Figure 3b compares the relative error in the FLCS number of molecules at 0.2 pM concentration after the application of different corrections. As discussed in Figures S6 and S7 (Supporting Information), increasing the integration time and accelerating the diffusion to detect more bursts per second have negligible effects on the relative error. Time gating (Figure S8, Supporting Information) and most importantly FLCS background correction reduces the relative error. The evolution of the relative errors for the different methods is given in Figure 3c as a function of the concentration, allowing to determine the respective LOQ for each technique as in Figure 2b. Alternatively, we can compute the accuracy gain respective to standard FCS by taking the ratio of the relative errors for standard FCS and the technique of interest (Figure 3d). The absolute value of the accuracy gain can reach huge values up to 600×, yet this is primarily because the error for the standard FCS becomes very large. The main interest and message from Figure 3d is that the FLCS background correction outperforms the other approaches (uncorrected FLCS and background‐corrected FCS) in terms of accuracy gain. This highlights the interest in taking into account a supplementary correction for the $\sqrt{B}$ background fluctuations while performing FLCS at ultralow concentrations.

To demonstrate the applicability of the technique beyond accurate concentration measurements in the (sub)picomolar range, we investigate the association dynamics between biotin and streptavidin (Figure 4a). This association is largely used in biotechnology for its high affinity, with its dissociation constant. K_d  ≈4.10⁻¹⁴ m ^{[ 53 ]} making it one of the strongest noncovalent interactions in biology.^{[ 42} , ^{43 ]} However, despite its major role, the measurement of the biotin–streptavidin association rates remains scarce.^{[ 44} , ⁴⁶ , ⁵⁴ , ⁵⁵ , ^{56 ]} While the dissociation rate constant is very slow and practical to measure,^{[ 57} , ⁵⁸ , ^{59 ]} the fast association rate ≈10⁷ M⁻¹ s⁻¹ makes the binding challenging to monitor using conventional methods.^{[ 45} , ^{54 ]} Microfluidics^{[ 44} , ^{56 ]} and silicon nanowires^{[ 55 ]} were used to probe the association dynamics at nanomolar concentrations, while nanophotonics was needed to achieve short enough FCS integration times to probe the interaction.^{[ 46 ]}

Figure 4.

Application of FLCS sensing at low concentrations to determine biotin‐streptavidin association dynamics. a) Sketch of the experiment: biotin labeled with Atto643 is mixed with unlabeled streptavidin to determine the bound fraction of biotin‐streptavidin over the reaction time. Due to the high affinity of the reaction and the fast association rate, picomolar concentrations are needed to enable monitoring of the reaction dynamics with single‐molecule sensitivity. b) Time traces demonstrating the detection of single‐molecule fluorescence bursts at 10 pM of labeled biotin. The burst duration increases over the time of the experiment due to the addition of 125 pM of streptavidin, yet FLCS is better suited to analyze the binding. c) Normalized FLCS correlation curves of Atto643‐biotin mixed with 125 pM streptavidin at T = 0. The shift toward longer diffusion times indicates the binding of biotin to streptavidin. d) Temporal evolution of the free and bound molecular fractions of biotin after adding 125 pM solution of streptavidin. Markers are experimental data, lines are exponential fits with characteristic time τ. e) Comparison of the temporal evolutions of the bound fractions with decreasing streptavidin concentrations. f) Inverse of the association time τ as a function of the streptavidin concentration. Markers indicate the average value, and the error bars represent the fit uncertainty upon processing the data in d,e with Igor Pro 7 (Wavemetrics). The slope determines the association rate constant k_on while the interpolation of the intercept at the origin gives an upper bound for the dissociation rate constant k_off .

Here we take advantage of our high sensitivity at low concentrations to work in the picomolar range where the binding times can be assessed by FLCS. In our experiments, biotin‐labeled with Atto643 is mixed with label‐free streptavidin. The biotin‐Atto643 concentration is kept constant at 10 pM concentration, while the streptavidin concentration varies between 50 and 200 pM (Figure 4a). Under these conditions, the association rate is controlled by the streptavidin concentration as the biotin concentration remains low enough. Moreover, we can assume that each single biotin binds to a different streptavidin, so the fact that each streptavidin holds 4 biotin binding sites can be neglected. The microscope uses the configuration with the 1.2 NA objective with diaphragm to benefit from a better accuracy as demonstrated in Figures 2 and 3. The average number of fluorescent molecules remains ≈0.1 (Figure S13, Supporting Information) so that single‐molecule conditions can be claimed, as exemplified on the fluorescence time traces showing clear single‐molecule fluorescence bursts (Figure 4b).

To separate between the free and bound fractions of biotin, we perform a FLCS analysis with two species, taking advantage of the longer diffusion time of the streptavidin‐bound biotin respective to the free biotin (see Experimental Section for details). Upon binding to streptavidin, the Atto643‐biotin diffusion time gradually evolves from 0.9 ms (free biotin) to 3.7 ms (fully bound biotin) (Figure 4c). This 4.1 × increase in the diffusion time is consistent with the change of molecular weight from 1.26 kDa (free Atto643‐biotin) to 67 kDa (Atto643‐biotin‐streptavidin complex) as $\sqrt[3]{67/1.26}\sim 3.8$.^{[ 45 ]} It is also well beyond the 1.6 × minimum difference in diffusion times needed in FCS to distinguish 2 components.^{[ 60 ]}

The association dynamics are presented in Figure 4d,e for different streptavidin concentrations. The bound fraction increases exponentially with a characteristic time τ  =  1/(k_on [S] + k_off ), where k_on and k_off are the association and dissociation rate constants and [S] is the streptavidin concentration. Finally, the plot of 1/τ as a function of [S] allows to determine the values for the rate constants k_on and k_off (Figure 4f). The slope determines the association rate constant. k_on to be (2.7 ± 0.2) × 10⁷ M⁻¹.s⁻¹ while the intercept at the origin provides an upper bound for the dissociation rate constant k_off to be well below 10⁻⁴ s⁻¹. These values agree well with the data reported in the literature, more details are provided in the Supporting Information Table S3 (Supporting Information). The key aspect of this application is to demonstrate the capability of conducting FLCS studies to distinguish two species for a prime example of a fast association rate, enabled by the extended operational range down to picomolar concentrations.

3. Conclusion

It was largely assumed that single‐molecule fluorescence techniques and FCS were only functional at concentrations above 50 pM, and that detection at lower concentrations was limited by a severe loss in signal‐to‐background ratio.^{[ 19 ]} This so‐called low‐concentration barrier was identified as one of the main limitations preventing the application of the rich single‐molecule fluorescence toolbox to biosensing applications in biology and medicine.^{[ 14 ]} Here we show that the low concentration limit can be overcome with a limit of quantitation LOQ down to 0.1 pM without any major modification to the single‐molecule confocal microscope and without the introduction of complex microfluidics or preconcentration stages. By adding a simple diaphragm on the laser excitation beam, our microscope achieves simultaneously a large detection volume of a few tens of femtoliter and a high fluorescence brightness per molecule. The combination of these two factors enables a 8 × superior sensing performance than the classical confocal microscope and is the key to pushing the FLCS sensitivity toward sub‐picomolar concentrations.

For the first time, we clearly discuss the physical parameters setting the lower limit of quantitation for FLCS and we introduce a universal figure of merit $N\ast CRM/\sqrt{B}$ allowing to compare between experimental configurations and predict the LOQ. Our results establish that the LOQ in FLCS is not a statistical problem determined by the noise on the signal or an issue related to the low frequency of the detection events. Instead, we identify the residual shot noise fluctuations on the background level as a key parameter determining the LOQ. While standard FLCS does not account for this effect, we show that it can be at least partly compensated by introducing a post‐measurement correction factor in the form. ${(1-\frac{\sqrt{B}}{\sqrt{B}+N\ast CRM})}^2$.

Importantly our results are not limited to the measurement of the local concentration, the key abilities to analyze the diffusion time and distinguish between species in a mixture remain preserved, as we demonstrate by monitoring the interaction dynamics of biotin binding to streptavidin. The extended operational range down to picomolar concentrations opens new opportunities to measure fast association rate constants while keeping regular single‐molecule fluorescence resolutions.

The current LOQ of 0.1 pM is found for FLCS with standard commercial optics and fluorescent dyes. Obtaining brighter probes or reducing further the background level would immediately impact the LOQ in a manner that can be predicted using our equations. It should also be stressed that this LOQ applies to FLCS, allowing for analysis of local concentrations, brightness, translational and rotational diffusion as well as distinguishing subpopulations in a mixture.^{[ 33} , ^{34 ]} If only fluorescence bursts are to be measured, then the LOD is expected to be found in the femtomolar range, yet at the expense of a lower amount of information.^{[ 24} , ^{26 ]} Complementary approaches using micro/nanofluidics or preconcentration could be used as well to further improve the sensitivity to femto or attomolar ranges.^{[ 61} , ⁶² , ^{63 ]} In conclusion, our work opens up new directions to extend the sensitivity of FLCS and overcome the low‐concentration barrier in single‐molecule studies while keeping simple instrumentation. This paves the way for future biosensing applications at sub‐picomolar levels with single‐molecule resolution.

4. Experimental Section

Sample Preparation

CF640R and solvent D₂O were purchased from Sigma–Aldrich and used as received. Streptavidin extracted from Streptomyces avidin was purchased from Sigma–Aldrich and it was stored in −20 C. Biotin tagged with Atto 643 (Biotin‐Atto643) dyes was purchased from ATTO‐TEC. For measurements, Streptavidin and Biotin‐Atto643 samples were diluted in phosphate‐buffered saline (PBS) solution. Biotin‐Atto643 in PBS was sonicated for 20–30 min before every experiment to prevent aggregation and form a uniform sample.

To prevent adsorption of streptavidin, the coverslip surface was passivated using polyethylene glycol (PEG) silane (Nanocs PG1‐SL‐IK) of 1000 Da molar mass following the protocol in ref. [64]. Before passivation, the coverslip was cleaned with water and ethanol (96%). Further, the coverslip was dipped in isopropanol and sonicated for 10 min to remove any dirt from the surface. After this, the sample was exposed to UV ozone for 10 min to remove organic impurities. Finally, the samples were put in air plasma cleaner for 10 min, after which the sample was immediately transferred to a sample holder and covered with 1 mm PEG solution prepared on 96% ethanol and 1% acetic acid. The chamber was blown with argon gas and left at room temperature for overnight. The next day sample was rinsed with ethanol and dried using nitrogen. The passivated sample was then stored in a 1% tween20 solution in 96% alcohol and stored in a fridge.

Confocal Microscope Setup

All the FCS measurements were done in a home‐built confocal microscope (Nikon Ti‐U Eclipse) equipped with a 635 nm pulsed laser (LDH series laser diode, PicoQuant) with a pulse duration ≈ 50 ps. To reflect the laser toward the microscope a multiband dichroic mirror (ZT 405/488/561/640rpc, Chroma) was used. Further, Zeiss 63x, 1.2 NA water immersion or Zeiss 16x, 0.5 NA water immersion were used to focus the excitation light as well as collect the emission in the epifluorescence configuration. For the use of the diaphragm (case Figure 2c), a diaphragm of 1.35 mm diameter corresponding to a 35.4% transmission is used to reduce the laser beam diameter and underfill the objective's back aperture (measuring the transmission through the diaphragm is more convenient than measuring its diameter, it is the recommended method to ensure an accurate reproducibility of the data collection). For the rejection of backscattered laser, the emission is passed through the same multiband dichroic mirror (ZT 405/488/561/640rpc, Chroma) and then through two emission filters (ZET405/488/565/640mv2 and ET655, Chroma). A 50 µm (200 µm for diaphragm case) pinhole is used to spatially filer the molecular fluorescence. The signal is collected by a single‐photon avalanche photodiode (APD) (Perkin Elmer SPCM AQR 13) in the range 650–750 nm. In order to block the entering of stray light in the APD and for the rejection of Rayleigh scattered light a supplementary emission filter (FF01‐676/37, Semrock), placed in front of the APD was used. The photodiode output is connected to a time‐correlated single photon counting (TCSPC) module (HydraHarp 400, Picoquant). Table S1 (Supporting Information) summarizes the number of experiments and integration times for the different configurations. The background intensity B was recorded using pure D₂O buffer in the same conditions as for CF640R dyes.

FLCS Analysis

Symphotime64 (Picoquant) is used for the FLCS analysis. The lifetime decay, D(i) of the compound is composed of photons from the sample and background contribution from the scattered excitation light, detector dark counts, detector afterpulsing, and residual room light. At low concentrations (pM range) the background contribution is not negligible. The stray room light, dark counts, and afterpulses are random events. Hence its TCPC pattern is a flat line on 25 ns histogram time scale. The scattered photons form a bump at the beginning of the TCSPC decay curve. Therefore D(i) could be expressed as a linear combination of 3 patterns; pure fluorescence, scattering, and other residual background contributions. Thus D(i) is given by Equation (1):

$$D\left(i\right)={\omega }_F{d}_F\left(i\right)+{\omega }_S{d}_S\left(i\right)+{\omega }_B{d}_B\left(i\right)$$ (1)

where ω _F , ω _S and ω _B  are the photon count amplitude (in a number of photons) of the fluorescence signal, the scattering, and the residual background respectively. d is the normalized TCSPC decay pattern. Here the index i is the channel number for the photons detected within the TCSPC channel. Then a statistical filter function, f(i) is introduced that can satisfy the following requirements.

$$\sum f\left(i\right)\times D\left(i\right)=\omega$$ (2)

$$f^F+f^S+f^B=1$$ (3)

These filter functions can be calculated numerically based on the TCSPC histogram. Here, a pattern‐matching filter was used to calculate the statistical filter function for the sample and background contributions. In this method, the statistical filter function was calculated by appropriately scaling the decay curve of the sample using the decay pattern obtained for the sample at a higher concentration (nM range) with negligible background and the decay pattern of the background recorded in the absence of fluorescent molecules. This gives 3 filter functions; fluorophore, scattered photons, and residual flat background. Once the filter functions have obtained, the filtered correlation can be calculated using Equation (4):

$$G\left(\tau \right)=\frac{⟨{\sum }_i{f}_i{I}_i\left(t\right)\times {\sum }_i{f}_i{I}_i\left(t+\tau \right)⟩}{{⟨{\sum }_i{f}_i{I}_i\left(t\right)⟩}^2}$$ (4)

This correlation function is fitted using a pure diffusion fitting model given by:

$$G\left(\tau \right)=\frac{1}{N}{\left(1+\frac{\tau }{{\tau }_D}\right)}^{-1}\times {\left(1+\frac{\tau }{{\tau }_D{\kappa }^2}\right)}^{-1/2}$$ (5)

Here G(τ) is the autocorrelation function at time τ, N is the total number of CF640R molecules in the observation volume, τ _D is the mean diffusion time and κ denotes the aspect ratio of the axial to the transversal dimension of the detection volume. In this case, κ is taken as 5 based on the confocal measurements done in the past and fits well with experimental correlation data.

Single‐Molecule Burst Analysis

For this work, the Burst Analysis Module of the PIE Analysis with MATLAB (PAM) was used.^{[ 65 ]} All Photon Burst Search function is used for burst detection by putting the minimum photons per burst, the time window of the bursts, and the total photon counts per burst. These values differ for three cases considered 1.2, 0.5, and 1.2 NA with diaphragm due to the difference in diffusion time and background signal. For 1.2 NA (0.5 NA/1.2 NA with diaphragm) configuration, each peak was considered a single‐molecule burst having at least 30 (30/30) photons. A minimum of 2 (2/2) photons per time window of 100 (1000/600) µs was taken.

Streptavidin–Biotin Association Rate Calculation

The association and dissociation dynamics of streptavidin and biotin can be denoted as:

$$S+B\underset{k_{off}}{\stackrel{k_{on}}{⟷}}SB$$ (6)

Here S denotes free streptavidin, B is free biotin and SB is streptavidin–biotin complex. Streptavidin has four biotin‐binding sites. For the measurements, the biotin concentration was taken to be significantly lower than that of streptavidin. Hence it could be safely assumed that only a single biotin binds to a single streptavidin making the kinetic study easier compared to multiple binding to the streptavidin. k_on is the association rate constant to form the complex SB while k_off  is the dissociation rate constant. The time origin t = 0 is set when streptavidin is added to the biotin solution. Differential equations to study the reaction kinetics can be written as:

$$\frac{d\left[B\right]}{dt}=-k_{on}\left[S\right]\left[B\right]+k_{off}\left(\left[B_{tot}\right]-\left[B\right]\right)$$ (7)

$$\frac{d\left[SB\right]}{dt}=k_{on}\left[S\right]\left[B_{tot}\right]-\left(k_{on}\left[S\right]+k_{off}\right)\left[SB\right])$$ (8)

The total biotin concentration is constant and it can be written as [B_tot ] = [B] (t) + [SB](t). Applying the boundary condition that [B] (0) =  [B_tot ] and [SB] (0) =  0 the differential Equations (7) and (8) can be solved yielding,

$$\frac{\left[B\right]\left(t\right)}{[B_{tot}]}=e^{-\left(k_{on}\left[S\right]+k_{off}\right)t}+\frac{k_{off}}{k_{on}\left[S\right]+k_{off}}\left(1-e^{-\left(k_{on}\left[S\right]+k_{off}\right)t}\right)$$ (9)

Dissociation rate constant k_off is much lower than k_on [S]. Hence the second term in the right‐hand side of Equation (9) can easily be neglected.

The bound fraction is:

$$\frac{\left[SB\right]\left(t\right)}{[B_{tot}]}=\frac{k_{on}\left[S\right]}{k_{on}\left[S\right]+k_{off}}\left(1-e^{-\left(k_{on}\left[S\right]+k_{off}\right)t}\right)$$ (10)

Considering a characteristic time $\tau =1/\left(k_{on}\left[S\right]+k_{off}\right)$ free and bound fractions of biotin can be expressed as an exponential temporal evolution $e^{-t/\tau }$.

A pure diffusion model with two species was used to fit the FLCS curve obtained after FLCS filtering to determine the free and bound ratio of biotin. The diffusion times of biotin and streptavidin–biotin complexes were set to be 0.9 and 3.7 ms respectively. These values were obtained by independent measurements at nm concentrations. FLCS correlation amplitude σ₁ and σ₂ corresponds to free and bound biotin respectively based on diffusion time. The free and bound fractions can be written as:

$$\frac{{\sigma }_1}{{\sigma }_1+{\sigma }_2}=\frac{\left[B\right]\left(t\right)}{[B_{tot}]}$$ (11)

$$\frac{{\sigma }_2}{{\sigma }_1+{\sigma }_2}=\frac{\left[SB\right]\left(t\right)}{[B_{tot}]}$$ (12)

These ratios can be fitted using exponential functions as follows:

$$\frac{{\sigma }_1}{{\sigma }_1+{\sigma }_2}=1-a\left[1-e^{\left(-t/\tau \right)}\right]$$ (13)

and

$$\frac{{\sigma }_2}{{\sigma }_1+{\sigma }_2}=a\left[1-e^{\left(-\left(t-t_0\right)/\tau \right)}\right]$$ (14)

Here the term t ₀ is to correct for any delay during the measurement. It the tagging efficiency of biotin to Atto 643 is 100%, the value of a is 1. However, a value of≈0.8 was observed in this case, meaning tagging efficiency is≈80%. The association time τ was obtained by fitting the free and bound ratio of biotin. The value of the association rate constant k_on is obtained from the slope of 1/ τ versus [S].

Statistical Analysis

Pre‐processing: the intensity time trace was filtered by FLCS to compute the lifetime‐specific correlation function, as described in the FLCS Analysis section. No other processing nor filtering was applied to the data. The data presentation corresponds to the average, the error bars represent 2 times the standard deviation. The sample size for each statistical analysis is detailed in Table S1 (Supporting Information). No statistical test was applied, as comparing different populations or samples were made. The statistical analysis was performed using Symphotime64 (Picoquant) and IgorPro 7 (Wavemetrics).

Challenges of detecting single‐molecules at low concentrations; Characterization of the influence of the diaphragm diameter; Examples of FLCS filters and raw correlation data; Number of experiments and integration times for the different configurations; Background correlation in absence of target molecule; Interpolation of the relative error; Signal to noise ratio in FCS; Influence of the integration time on the FLCS error; Influence of the lateral flow on the FLCS error; Influence of time gating on the FLCS error; Shot noise dependence of the background noise; FLCS figure of merit as function of concentration; FLCS background correction applied to the 1.2NA microscope configuration; Background correction prefactors for FCS and FLCS; Number of biotin molecules and brightness per molecule as a function of streptavidin concentration; Literature values for streptavidin‐biotin binding kinetics.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Kayyil Veedu M. and Wenger J., “Breaking the Low Concentration Barrier of Single‐Molecule Fluorescence Quantification to the Sub‐Picomolar Range.” Small Methods 10, no. 2 (2026): 2401695. 10.1002/smtd.202401695

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.