Burst duration and FCS diffusion time

Diffusion, flow, burst selection, and photobleaching in confocal single-molecule fluorescence

In confocal single-molecule fluorescence experiments, molecular transport through the detection volume is commonly characterized using two different observables: the diffusion time extracted from fluorescence correlation spectroscopy (FCS) and the burst duration obtained from single-molecule detection (SMD) or burst analysis. Both quantities probe molecular motion through the same optical volume, yet they originate from fundamentally different experimental procedures and statistical definitions.

The apparent similarity of their numerical values often suggests a direct equivalence. However, this impression can be misleading, because each observable responds differently to transport mechanisms, detection geometry, selection criteria, and photophysical processes. As a consequence, diffusion time and burst duration must be treated as distinct quantities whose relationship depends on the underlying physical assumptions and on how the experiment is performed.

In this article, we develop a unified theoretical framework to analyze the diffusion time and burst duration within a three-dimensional Gaussian detection model. We first derive both quantities independently using their full mathematical formalism. We then compare the two observables and establish their relationship under ideal diffusion conditions. Finally, we investigate how burst-selection procedures, directed flow, and photobleaching modify each observable and how these effects alter their mutual relation.

Both FCS and burst-based single-molecule experiments rely on a confocal detection volume that is well approximated by a three-dimensional Gaussian molecular detection function (MDF):

$$ W(\mathbf r) = \exp\left( -\frac{2(x^2+y^2)}{w_0^2} -\frac{2z^2}{z_0^2} \right) $$

where $w_0$ and $z_0$ denote the lateral and axial $1/e^2$ radii of the detection volume.

This Gaussian MDF constitutes the standard model underlying classical FCS theory and most analytical treatments of single-molecule residence times. It provides a consistent geometric framework for comparing correlation-based and trajectory-based observables.

For freely diffusing molecules, transport is governed by the three-dimensional diffusion propagator

$$ G(\mathbf r,t \mid \mathbf r_0,0) = \frac{1}{(4\pi Dt)^{3/2}} \exp\left( -\frac{|\mathbf r-\mathbf r_0|^2}{4Dt} \right) $$

where $D$ is the diffusion coefficient.

A crucial point is that the time a molecule spends in the detection volume is not a single well-defined quantity but a stochastic variable with a broad distribution. This distribution depends on the diffusion coefficient, the entry position of the molecule, and the spatial weighting imposed by the MDF. Any experimentally extracted time scale therefore corresponds to a specific statistical functional of this distribution rather than to a unique physical residence time.

In FCS, the normalized fluorescence intensity autocorrelation function is defined as

$$ G(\tau) = \frac{\langle \delta I(t)\,\delta I(t+\tau)\rangle}{\langle I\rangle^2} $$

For pure diffusion, this can be expressed as

$$ G(\tau) \propto \int d^3r \int d^3r_0\, W(\mathbf r)\,W(\mathbf r_0)\, G(\mathbf r,\tau \mid \mathbf r_0,0) $$

For a three-dimensional Gaussian MDF, this integral yields

$$ G(\tau) = \frac{1}{N} \frac{1}{(1+\tau/t_D)\sqrt{1+\tau/(\kappa^2 t_D)}} $$

with

$$ t_D = \frac{w_0^2}{4D},\qquad \kappa = \frac{z_0}{w_0} $$

The diffusion time $t_D$ is not the mean residence time of an individual molecule. Instead, it is a correlation-derived geometry time scale governed by the quadratic spatial weighting of the MDF in the autocorrelation function.

Formally,

$$ t_D \approx \frac{\langle r^2\rangle_{W^2}}{6D},\qquad \langle r^2\rangle_{W^2} = \frac{\int r^2 W(\mathbf r)^2\,d^3r} {\int W(\mathbf r)^2\,d^3r} $$

For a Gaussian MDF, the lateral contribution evaluates to

$$ \langle r^2\rangle_{W^2} = \frac{3}{4}w_0^2 $$

which directly leads to the expression $t_D = w_0^2/(4D)$.

In burst analysis, individual molecular events are identified by applying an intensity threshold $W_{th}$. A burst is defined as

$$ t_B = t_{exit} - t_{entry},\qquad W[\mathbf r(t)] > W_{th} $$

Bursts therefore begin and end on iso-intensity surfaces of the MDF rather than at a geometric boundary of the confocal volume.

For a Gaussian detection function, the effective threshold radius scales as

$$ r_{th} \propto w_0\sqrt{\ln(1/W_{th})} $$

The mean burst duration is

$$ \langle t_B\rangle = \int_0^\infty t\,p_{burst}(t)\,dt $$

where $p_{burst}(t)$ denotes the threshold-conditioned residence-time distribution.

Burst analysis measures a first moment of residence times with linear spatial weighting, modified by explicit event selection. Short, peripheral trajectories often do not exceed the detection threshold and therefore do not contribute to the measured distribution.

In practice, burst detection is a multi-step selection process.

Intensity thresholding. Photon arrival times are binned into short intervals $\Delta t$ or processed using a sliding window. A burst is initiated when the photon count rate exceeds a predefined threshold

$$ R(t)=\frac{N_\gamma(t,\Delta t)}{\Delta t}>R_{th} $$

which is equivalent, up to proportionality factors, to $W[\mathbf r(t)]>W_{th}$.

Temporal continuity and photon-number criteria. To suppress spurious threshold crossings due to shot noise, bursts are often required to satisfy

$$ t_B \ge t_{min},\qquad N_\gamma \ge N_{\gamma,min} $$

Selection functional. The burst-selection procedure can be written formally as

$$ \mathcal S[\mathbf r(t)] = \Theta(W[\mathbf r(t)]-W_{th})\, \Theta(t_B-t_{min})\, \Theta(N_\gamma-N_{\gamma,min}) $$

leading to an observed distribution

$$ p_{burst}(t) \propto \left\langle \delta(t-t_B)\, \mathcal S[\mathbf r(t)] \right\rangle $$

Burst selection introduces intrinsic and controllable biases. The mean burst duration depends explicitly on the chosen thresholds, increasing thresholds bias toward central trajectories and increase $\langle t_B\rangle$, and bursts are defined by iso-intensity surfaces rather than geometric boundaries. Burst statistics are also sensitive to photophysical processes such as blinking and photobleaching.

Although both observables probe molecular motion through the same confocal detection volume, their statistical definitions differ fundamentally. The FCS diffusion time $t_D$ is determined by quadratic spatial weighting of the MDF and reflects the decay of concentration correlations. In contrast, the mean burst duration $\langle t_B\rangle$ is governed by linear spatial weighting combined with explicit selection rules.

Because of burst-selection thresholds, $\langle t_B\rangle$ can be smaller than, comparable to, or larger than $t_D$. There is no universal ordering between the two quantities, whereas $t_D$ is largely insensitive to burst-selection parameters.

Under the assumptions of pure diffusion, Gaussian detection geometry, moderate burst-selection thresholds, and negligible photophysical interruptions, a closed analytical relation can be derived.

For a Gaussian MDF,

$$ \langle r^2\rangle_{W^2} = \frac{3}{4}w_0^2,\qquad \langle r^2\rangle_W = w_0^2 $$

Using $\langle r^2\rangle \approx 6Dt$,

$$ t_D = \frac{w_0^2}{4D},\qquad \langle t_B\rangle = \frac{w_0^2}{3D} $$

Thus,

$$ \frac{\langle t_B\rangle}{t_D} = \frac{4}{3} $$

This ratio represents a diffusion-limited reference case and is not universal.

$$ \frac{\partial c}{\partial t} + \mathbf v\cdot\nabla c = D\nabla^2 c,\qquad \tau_v=\frac{w_0}{v},\qquad \mathrm{Pe}=\frac{vw_0}{D} $$

With advection–diffusion propagator

$$ G(\mathbf r,\tau\mid\mathbf r_0,0) = \frac{1}{(4\pi D\tau)^{3/2}} \exp\left( -\frac{|\mathbf r-\mathbf r_0-\mathbf v\tau|^2}{4D\tau} \right) $$

The autocorrelation function factorizes as

$$ G(\tau) = G_{diff}(\tau) \exp\left( -\frac{(v\tau)^2}{w_0^2(1+\tau/t_D)} \right) $$

When flow is modeled explicitly, $t_D$ remains unchanged.

Burst durations scale approximately as

$$ \langle t_B\rangle \sim \frac{L_{eff}}{v} \qquad (\mathrm{Pe}\gg1) $$

and therefore decrease continuously with increasing flow velocity.

Flow selectively affects burst durations while leaving the FCS diffusion time invariant when modeled correctly.

$$ P_{survival}(t)=e^{-k_{bl}t},\qquad \tau_{bl}=\frac{1}{k_{bl}} $$

$$ G(\tau)=G_{diff}(\tau)e^{-\tau/\tau_{bl}},\qquad \tau_{bl}\gg t_D $$

Bleaching primarily affects amplitude and long-delay behavior.

$$ t_{B,obs}=\min(t_{transport},t_{bleach}),\qquad p_{obs}(t)=p_{trans}(t)e^{-k_{bl}t} $$

Burst durations are systematically shortened.

The diffusion time extracted from fluorescence correlation spectroscopy and the burst duration obtained from single-molecule detection probe molecular transport through the same confocal detection volume, yet they represent fundamentally different observables.

The FCS diffusion time $t_D$ is a correlation-derived geometry time scale governed by quadratic spatial weighting of the molecular detection function, whereas the mean burst duration $\langle t_B\rangle$ is a residence-time observable governed by linear weighting and explicit burst-selection criteria.

Because of these differences, there is no general reason for $t_D$ and $\langle t_B\rangle$ to coincide. Depending on burst-selection thresholds, the mean burst duration can be smaller than, comparable to, or larger than the diffusion time.

Under ideal diffusion conditions–Gaussian detection geometry, moderate thresholding, and negligible photophysical interruptions–a fixed relation can be derived,

$$ \frac{\langle t_B\rangle}{t_D}=\frac{4}{3} $$

This factor is a geometric consequence of Gaussian detection combined with diffusion statistics and specific selection conditions. It is not universal.

Directed flow and photobleaching systematically break this reference relation. Flow introduces an additional transport time scale that leaves the FCS diffusion time unchanged when modeled explicitly, while directly shortening burst durations. Photobleaching truncates bursts and biases burst statistics strongly, whereas its influence on FCS is largely confined to slow non-stationarity.

Diffusion time and burst duration should therefore be regarded as complementary rather than interchangeable observables. Deviations from the diffusion-limited reference ratio provide direct physical insight into burst-selection effects, directed transport, and photophysical limitations in confocal single-molecule fluorescence experiments.

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