Fluorescence intensity decay models

Excited state decay of organic fluorophores

After electronic excitation the molecule typically decays to the lowest vibrational level of the first excited singlet state, S₁. In far most cases, this happens non-radiatively in the order of pico-seconds by a combination of internal conversion (IC) and vibrational relaxation. From S₁ a number of decay processes can occur to the electronic ground state, S₀. If the oscillator strength of the S₀-S₁ transition is high the molecule may decay to the ground state via emission of radiation (fluorescence) and the molecule is thus termed a fluorophore.

The fluorescence, however, "competes" with non-radiative decay processes such as IC, intersystem crossing (ISC), or a bimolecular quenching process such as FRET and collisional quenching. Each process is associated with a rate constant, k_i. The process with the largest value of k_i dominates the decay. The total rate of decay from the excited state, then, is typically given by that of a unimolecular process:

where the sum is over all decay processes from the excited state. The solution to this 1. order differential equation is

where [S₁] is the concentration of fluorophores in S₁ at time t and [S₁]₀ is the concentration of fluorophores in S₁ at time t = 0. The lifetime of the fluorophore, τ, is defined as the time it takes a population of excited fluorophores to reach 1/e of [S₁]₀. Inserting this in the equation above and isolating t we get

which is called a multi-exponential decay. α_j is the pre-exponential factor and can be used to represent the fraction of fluorophores with lifetime τ_j. The amplitude-weighted lifetime of the fluorophore is then given by:

Fluorescence decay models in DecayFit

In the DecayFit freeware the implemented intensity decay models are listed in the Fit models listbox:

The intensity decay expression of the fit model selected in the listbox is shown below the listbox:

A description of each of the fit models is provided below.

1. single_exp

A single-exponential decay corresponding to a single lifetime. Only the lifetime is provided as fit parameter.

2. double_exp

A double-exponential decay corresponding to two lifetimes. The sum of the pre-exponential factors is constrained to 1 which means that only one pre-factor, a₁, is provided as fitting parameter while the other is set to be 1-a₁.

3. triple_exp

A triple-exponential decay corresponding to three lifetimes. The sum of the pre-exponential factors are constrained to be a₁+a₂+a₃ = 1.

4. four_exp

A four-exponential decay corresponding to four lifetimes. The sum of pre-exponential factors are constrained to be a₁+a₂+a₃+a₄ = 1.

5. FRET

A decay model for analyzing FRET data. The model implements a Gaussian distance distribution between the donor (D) and acceptor (A) with the distribution center (R-mean) and Full Width at Half Maximum (FWHM) as fitting parameters. The critical distance, R₀, must be supplied as input and is not meant to be a fitting parameter (although this is possible by varying the min and max values).

Up to three lifetimes can be specified for the donor in absence of acceptor (tauD₁, tauD₂, tauD₃), each with a pre-exponential factor (a₁, a₂, a₃). These are not meant as fitting parameters but should be determined on a donor-only reference sample. If the donor has a single lifetime only set the min and max values of a2 and a3 to 0 (default).

The fraction of donors not coupled to an acceptor, f, can also be included as a parameter, either constrained or loosely fitted.

An additional component not involved in FRET with a lifetime 'tauB' and relative fraction 'b' can be included in the model. The fraction of the FRET-component in the decay, then, is 1-b.

Note: When using dynamic averaging, the D-A distance is modelled using a single distance - the mean of the Gaussian distribution (R-mean).

6. lifetime_dist

A Gaussian lifetime distribution model. A mixture of two independent Gaussians can be used, each with a lifetime center and a lifetime distribution width (FWHM) as fitting parameters.

7. double_exp_uncon

A double exponential decay in which there is no constraint on the sum of pre-exponential factors.

8. triple_exp_uncon

A triple exponential decay in which there is no constraint on the sum of pre-exponential factors.