Fitting models¶

The software includes several models that describe the autocorrelation curves obtained by fluorescence correlation spectroscopy. It also allows the addition of user-defined models. Below, all default models are described.

One-component simple diffusion¶

This model describes the simple diffusion of fluorescent molecules. The model assumes there is only one type of fluorescent molecule, meaning the diffusing molecules’ sizes are pretty much the same. Additionally, the molecules exhibit no so-called triplet states. The model function is given by Equation (1).

where $\tau$ is a lag-time, $N_p$ stands for the number of molecules inside the focal volume, $\tau_{d1}$ is a diffusion time - average time the fluorescent probe spends inside the focal volume. $\kappa$ is the structure parameter defined as the ratio of semi-axes of the ellipsoidal focal volume.

One-component diffusion with triplet states¶

This model describes the simple diffusion of fluorescent molecules and also accounts for molecules in a triplet state. The model assumes that there is only one type (size) of fluorescent molecule. The model function is given by Equation (2).

Where $\tau$ is a lag-time, $N_p$ stands for the number of molecules inside the focal volume, $\tau_{d1}$ is a diffusion time, and the average time the fluorescent probe spends inside the focal volume. $\kappa$ is the structure parameter defined as the ratio of semi-axes of the ellipsoidal focal volume. $T$ stands for the fraction of molecules being in a triplet (dark) state, and $\tau_f$ is the average lifetime for triplet states.

Two-component simple diffusion¶

This model describes the simple diffusion of two types of fluorescent molecules. The model neglects photo-physical processes, such as triplet states. The difference in size of molecules results in two distinct diffusion times $\tau_{d1}$ and $\tau_{d2}$.

Where $\tau$ is a lag-time, $N_p$ stands for the number of molecules inside the focal volume, $\tau_{d1}$ is a diffusion time of component 1, $\tau_{d2}$ is a diffusion time of component 2, $A_1$ stands for the fraction of component 1, and $\kappa$ is the structure parameter defined as the ratio of semi-axes of the ellipsoidal focal volume.

Two-component diffusion with triplet states¶

This model describes the diffusion of fluorescent molecules with two different diffusion coefficients (different sizes). The model also includes information about molecules in a triplet state. The model function is given by Equation (4).

Where $\tau$ is a lag-time, $N_p$ stands for the number of molecules inside the focal volume, $\tau_{d1}$ is a diffusion time of component 1, $\tau_{d2}$ is a diffusion time of component 2, $A_1$ stands for the fraction of component 1, and $\kappa$ is the structure parameter defined as the ratio of semi-axes of the ellipsoidal focal volume. $T$ stands for the fraction of molecules being in a triplet (dark) state, and $\tau_f$ is the average lifetime for triplet states.

Anomalous diffusion¶

This model describes the diffusion of fluorescent probes that cannot be described by simple diffusion. In this model, the diffusion of molecules through the focal volume is described by an algebraic distribution of diffusion times, namely $\omega^2 = 4D\tau_d^\alpha$, where $\alpha < 1$. This model is also convenient to describe the motion of polydisperse molecules. i.e. Fluorescently labelled polymers characterised by a large polydispersity index, PDI. This model is recommended for the analysis of polymer diffusion in the cytoplasm of living cells Kalwarczyk & others (2017). The model is described by equation (5).

In equation (5) $\tau$ is a lag-time, $N_p$ stands for the number of molecules inside the focal volume, $\tau_{d1}$ is a diffusion time - average time the fluorescent probe spends inside the focal volume. $\kappa$ is the structure parameter defined as the ratio of semi-axes of the ellipsoidal focal volume, and $\alpha$ is an anomality exponent.

Three-component simple diffusion¶

This model describes the simple diffusion of three types of fluorescent molecules. The model neglects photo-physical processes, such as triplet states. The difference in size of molecules results in three distinct diffusion times $\tau_{d1}$, $\tau_{d2}$, and $\tau_{d3}$.

Here, $\tau$ is the lag time, $N_p$ is the number of molecules inside the focal volume, and $A_1$ and $A_2$ are the amplitudes of components 1 and 2.

Three-component simple diffusion with triplets¶

This model describes the simple diffusion of three types of fluorescent molecules differing in size. This model includes triplet states.

Here, $\tau$ is the lag time, $N_p$ is the number of molecules inside the focal volume, $\tau_{d1}$, $\tau_{d2}$, and $\tau_{d3}$ are diffusion times of the first, second and third component, and $A_1$ and $A_2$ are the amplitudes of components 1 and 2. $T$ stands for the fraction of molecules being in a triplet (dark) state, and $\tau_f$ is the average lifetime for triplet states.

Two-component diffusion: 1st translational diffusion only, 2nd Translational and rotational diffusion¶

The model describes diffusion in a system composed of two diffusing species, where one species undergoes rotations on a timescale comparable to that of translational diffusion across the focal volume. For details see Michalski et al. (2024)
Parameters description:
$G_\mathrm{inf}$ - offset
$N_p$ - Number of fluorescent molecules
$q$ - amplitude of diffusing species
$A_2$ -Second, non-zero amplitude for rotation
$A_4$ - fourth, non-zero amplitude for rotation
$f$ - fraction relating the rotational to translational diffusion for a given diffusing species.
$A = \pi k_\mathrm{B}$
$T$ - Absolute temperature
$\eta_{R1}$ - Rotational viscosity experienced by the first diffusing species
$\eta_{T1}$ - Translational viscosity experienced by the first diffusing species
$\eta_{T2}$ - Translational viscosity experienced by the second diffusing species
$r_1$ - Hydrodynamic radius of the first diffusing species
$r_2$ - Hydrodynamic radius of the second diffusing species
$\omega_0$- Width of the focal volume
$\kappa$ - Structure parameter

One-component diffusion with rotation¶

The model describes diffusion in a system composed of a single diffusing species with coupled translational and rotational modes. The model also includes triplet states and an anomalous diffusion exponent, which can be set to 1 if normal diffusion is assumed. For details see Michalski et al. (2024)
Parameters description:
$G_\mathrm{inf}$ – offset (baseline of the correlation function for long lag times)
$𝑁_\mathrm{p}$ – number of fluorescent molecules in the observation volume
$f_\mathrm{T}$ – fraction of molecules in the triplet state (triplet amplitude)
$t_\mathrm{T}$ – triplet relaxation time
$f_\mathrm{R1}$ – first non-zero amplitude describing rotational diffusion contribution
$f_\mathrm{R2}$ – second non-zero amplitude describing rotational diffusion contribution
$A$ – constant defined as $A=\pi k_\mathrm{B}$
$T$ – absolute temperature
$\eta_\mathrm{R}$ – rotational viscosity experienced by the diffusing species
$\eta_\mathrm{T}$ – translational viscosity experienced by the diffusing species
$r$ – hydrodynamic radius of the diffusing species
$\omega_0$ – lateral width (radius) of the focal volume
$\kappa$ – structure parameter (ratio of axial to lateral dimensions of the focal volume)
$\alpha$ – anomalous diffusion exponent describing deviation from normal diffusion