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ZEISS Microscopy Knowledge Base

Dynamics Profiler Calculations

Calculation of Concentration and Diffusion Coefficient

The diffusion coefficient D is calculated as indicated by the following formula:

Four formulas showing D = w_r^2/(4 τ_d); c = <N>/(V_e N_A); <N>=1/G(0); V_e=π^{3/2} w_r^2 w_z = π^{3/2} S w_r^3

The fluorofore concentration is given as:

Three equations: c = <N>/(V_e N_A); <N> = 1/G(0); V_e = π^{3/2} w_r^2 w_z = π^{3/2} S w_r^3

where

is an average number of molecules and N_A is the Avogadro constant
and
$V_e = pi^{3/2} w_r^2 w_z = pi^{3/2} S w_r^3$
where S is the structural parameter.

Data Quality Criteria (CPM)

The data quality is estimated using CPM values:

Data Quality	CPM Value
Bad (red)	Smaller than 1.
Not bad (yellow)	Between 1 and 3.
Good (green)	Higher than 3.

Filters

Detrending Filter

The detrending filter is applied to remove the trend (e.g. due to the photo bleaching) from the experiment data. The detrended signal I_d(t) is calculated as follows:

I_d(t) = I(t) / Tr(t) detrended signal equation

where the trend, Tr(t) is the gaussian smoothed signal:

Tr(t) = I(t) ⊗ e^{-t^2/(2σ^2)} Gaussian-smoothed trend equation

with σ = FilterWindowParameter / 4 equation

The FilterWindowParameter [ms] is set by the user.

Dust Filter

The dust filter allows you to remove high intensity peaks (e.g. caused by aggregated objects) from the experiment data. Calculations are done on the 500x down sampled signal data. For comparison, intensity data shown in the count rate chart is down sampled with factor 2000x. The data bin is classified as dust (filtered out) if the following criteria is fulfilled:

UserParameter * I(t) > Mean(I(t))

If both the detrending and dust filter are activated, the dust filter is applied after the detrending filter.

Fit Calculation

The unweighted fit of the correlation curve is done using the Levenberg-Marquart algorithm. Reduced chi2 is calculated as follows:

Reduced chi-squared formula and multiple G(τ) autocorrelation model equations

where G_t is the correlation curve value, F_t the fit value, N the number of correlation/fit values and DF the number of fit parameters.

Auto Correlation Fit Models

You can select one of the following models:

One Component 2D

Four equations for G(τ) showing A, Tt, e^{-τ/τt}, τd1, τd2, cf1 and S

One Component 3D

G(τ)=A·(1+T_t·e^{-τ/τ_t}/(1-T_t))·1/((1+τ/τ_d1)·(1+τ/τ_d2/S^2)^0.5)+1

Two Component 2D

G(τ)=A·(1+T_t·e^{-τ/τ_t}/(1-T_t))·(cf1/(1+τ/τ_d1)·(1+τ/τ_d2)+(1-cf1)/(1+τ/τ_d2)^0.5)+1

Two Component 3D

G(τ)=A·(1+T_t·e^{-τ/τ_t}/(1-T_t))·(cf1/(1+τ/τ_d1)·(1+τ/τ_d2)^0.5+(1-cf1)/(1+τ/τ_d2)^0.5)+1

One Component Anomalous 2D

G(τ)=A·(1+Tt·e^{-τ/τt}/(1-Tt))·1/(1+(τ/τd1)^α)+1

One Component Anomalous 3D

G(τ)=A·(1+Tt·e^{-τ/τt}/(1-Tt))·1/((1+(τ/τd1)^α)·(1+(τ/τd1)/s^2)^0.5)+1

Parameters

The parameters used in the calculations are the following:

Parameter	Description
A	The correlation amplitude.
T_t	The triplet fraction.
T_d1	The translational diffusion time.
S	The structural parameter.
τ_t	τ_t = 3µs is the fixed triplet relaxation time.
cf1	The first component fraction.

Normalization of Correlation Curves

Amplitude normalization of the correlation curves allows you to compare different correlation curves and see the differences in diffusion times between them.

If the fit of the correlation curves is already done, activating the amplitude normalization is done using the amplitude fit parameter. Selected correlation curves are rescaled that each selected curve has a correlation amplitude equal to 1 (corresponding number of molecules = 1).

where A is a fitted amplitude.

In case no fit is done, correlation curves are normalized as follows:

where A_mean is the mean amplitude calculated over the first 10 points of the correlation curve:

Pair Correlation of Airyscan Fibers

The pair correlation function (pCF) is a set of cross-correlations between the central fiber and all fibers located at a given distance from the center. For a better signal to noise ratio, cross-correlations are calculated in both directions and then averaged. The pCFs are used and displayed in the polar heatmaps of the Diffusion view.

The Diffusion view can display the data of two different sets of fibers. The group called Outer Elements comprises fibers with a slightly bigger distance from the center (blue element) and is illustrated with orange in the graphic below. The second group is called Inner Elements and comprises fibers with a slightly smaller distance from the center. They are illustrated with the green color in the graphic below.

Hexagonal array of numbered hexagons: center dark blue 1, blue ring 2-7, inner green/orange ring, outer maroon ring to 32

Fiber Group	Fibers
Outer Elements	Comprises fibers 9, 11, 13, 15, 17 and 19.
Inner Elements	Comprises fibers 8, 10, 12, 14, 16 and 18.

ωr Calculation

The ω_r is calculated with the following formula:

During the ω_r calibration, τ_d is calculated in the one-component 3D diffusion model

G(τ)=A·(1+(T_t·e^{−τ/τ_t})/(1−T_t))·1/(1+τ/τ_d)·1/(1+τ/τ_d·1/S^2)^{1/2}+1

and D (diffusion coefficient) needs to be provided and added in the ω_r Calibration tab. The diffusion coefficients of commonly used dyes are already specified in a dropdown list. The following requirements need to be fulfilled in order to generate a suitable calibration file:

Spot measurement duration of at least 60 seconds
Time Resolution of 0.5 µs (ω_r acquisition mode)
The used dye concentration needs to be adjusted to match the expected FCS amplitude range (0.05 – 2)
The emission wavelength of the selected dye should match the selected laser line

Four values are calculated and can be accessed by opening the calibration file with a text editor:

W0Ring3: Effective radius of the observation spot (detector elements 1 to 19 binned together)
W0Ring123: Average radius of the observation spot for a single detector element (average value for detector elements 1 to 19)
R0Ring123: Effective distance between the detector elements for flow measurements
StructuralParameter: Ratio between the observation spot waist in the axial and lateral direction

The generated ω_r calibration file can be used in the Correlation Tools tab to replace the default ω_r calibration values.

Calculation of Autocorrelation Curve

The Dynamics Profiler autocorrelation curve is calculated according to the formula:

Equation G(τ) = <I(t)·I(t+τ)> / <I(t)>^2

Correlation values are calculated for the lag times τ, commonly used in 16/8 multi-tau hardware correlators. This means the lag time τ is increased linearly with the τ =PixelDwellTime for the first 16 values. Then the lag time interval for τ is doubled, and the next eight lag times are calculated. The lag time interval is doubled after each next eight time lag values.

The very first correlation value, corresponding to the one PixelDwellTime, is skipped since it is biased due to correlation of the neighboring data points after low pass filter due to digitalization.