Identifiability of Models for Time-Resolved Fluorescence with
Underlying Distributions of Rate Constants
Noël Boens* and Mark Van der Auweraer
Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200f – bus
02404, 3001 Leuven, Belgium
November 20, 2013
Abstract
The deterministic identifiability analysis of photophysical models for the kinetics of
excited-state processes, assuming errorless time-resolved fluorescence data, can verify
whether the model parameters can be determined unambiguously. In this work, we have
investigated the identifiability of several uncommon models for time-resolved
fluorescence with underlying distributions of rate constants which lead to nonexponential decays. The mathematical functions used here for the description of nonexponential fluorescence decays are the stretched exponential or Kohlrausch function, the
Becquerel function, the Förster type energy transfer function, decay functions associated
with exponential, Gaussian and uniform distributions of rate constants, a decay function
with extreme sub-exponential behavior, the Mittag-Leffler function and Heaviside’s
function. It is shown that all the models are uniquely identifiable, which means that for
each specific model there exists a single parameter set that describes its associated
fluorescence -response function.
*
Correspondence author. E-mail: Noel.Boens@chem.kuleuven.be
1
1. Introduction
Fluorescence spectroscopy and imaging are essential tools for the non-invasive study of
matter or living systems at a molecular or supramolecular level.1, 2, 3 Fluorescence is an
ultra-sensitive, globally used method for investigating with high spatiotemporal
resolution the structure and dynamics of organic and inorganic materials, biopolymers,
biological cells and tissues in ensembles and at the single-molecule level. To study the
time evolution (kinetics / dynamics) of excited states, the fluorescent sample is excited
either by modulated light at high frequency (phase-modulation fluorometry = frequency
domain technique) or by a short pulse of light (pulse fluorometry = time domain
technique) and the resulting time-resolved fluorescence response is monitored.4,
5, 6, 7, 8
Time-resolved fluorescence techniques are commonly employed to determine the
parameters characterizing the fluorescence -response function i(t) of a fluorescent
sample (i.e., the temporal fluorescence response to an infinitely short pulse of light
expressed as the Dirac delta function).
Whenever a specific model is proposed for the description of the fluorescence decay of
molecular entities in the excited state, one should investigate first if the fundamental
parameters defining the model can be determined unambiguously from error-free
fluorescence decay data, that is from i(t). This is the topic of the deterministic
identifiability (or identification) analysis.9,
10, 11
Such an analysis informs us on which
information is theoretically accessible from the fluorescence decay data.
Since the first identifiability study of an intermolecular two-state excited-state process,12
identification studies of a broad range of models of intermolecular and intramolecular
two-state and three-state excited-state processes have been reported13, 14, 15, 16, 17, 18, 19, 20, 21,
2
22, 23, 24
(see refs 25 and 26 for supplementary literature data). The issues of controllability
and observability of intermolecular two-state excited-state processes have also been
addressed.27 In addition, several identification studies of models of fluorescence
anisotropy decay have been reported.28, 29, 30, 31, 32 Furthermore, a series of papers dealt
with the identifiability of excited-state processes in the presence of Smoluchowski and
Collins-Kimball transient effects.33, 34,
35, 36, 37, 38
However, only two reports have been
published on the identification of models for fluorescence decays with an underlying
distribution of decay rates as observed for an excited probe quenched by molecules or
ions that are Poisson-distributed over the micelles.39, 40
In this paper we investigate the deterministic identification of several uncommon models
for fluorescence decay with underlying distributions of rate constants which lead to nonexponential fluorescence -response functions i(t). The decay functions considered here
are the stretched exponential or Kohlrausch function, the Becquerel function, the Förster
type energy transfer function, decay functions associated with exponential, Gaussian and
uniform distributions of rate constants, a decay function with extreme sub-exponential
behavior, the Mittag-Leffler function and Heaviside’s function. These decay functions i(t)
usually have a small number of fitting parameters. It is important to know if it is
theoretically (i.e., algebraically) possible to determine these parameters from i(t).
Although for some of the models the functions i(t) were derived a long time ago, up to
now, surprisingly, no study has been devoted to their identifiability.
The present report brings to a close the series (refs 13–40) on the identifiability of
photophysical models. The paper is organized as follows. In Section 2, the mathematical
expressions of the fluorescence decays of models with underlying distributions of rate
3
constants are given. Section 3 deals with the identification of the models presented in
Section 2.
2. Models for fluorescence decays
Time-resolved fluorescence spectroscopy is a powerful tool to obtain information on the
excited-state kinetics of molecular and supramolecular systems.4-8 The traditional – and
most widely used – way of analyzing time-resolved fluorescence traces is to describe the
data with a sum of discrete exponential terms (eq 1).
i(t )    j e
t  j
(1)
j
In eq 1, i(t) represents the fluorescence -response function, that is the decay one would
theoretically observe after an excitation pulse of infinitely short duration described by the
Dirac -function. However, the recovered decay times j (and pre-exponential factors j)
do not necessarily possess clear physical meaning. Moreover, the same fluorescence
decay curves often can be fitted by distributions of decay times41, 42, 43, 44, 45, 46, 47, 48, 49 or
non-exponential functions.50,
51, 52, 53, 54
Therefore, in several cases, this usual multi-
exponential approach to fluorescence decay analysis is inappropriate. Analyzing the
fluorescence decay data in terms of distributions of rate constants is often a better option.
A distribution of decay rate constants (or decay times) must be anticipated to best account
for the observed phenomena in various situations:55, 56, 57, 58 fluorophores incorporated in
micelles, rigid solutions, sol-gel matrices, proteins, vesicles or membranes, biological
tissues, fluorophores adsorbed on surfaces or tethered to surfaces, quenching of
fluorophores in micellar solutions, energy transfer in assemblies of like or unlike
fluorophores, etc.
4
Given a certain photophysical model, the mathematical expression of the fluorescence response function i(t) is a prerequisite for tackling the important problem of deterministic
identifiability, that is to determine whether the parameters which appear in i(t) can be
recovered mathematically from i(t).
The first way of introducing functions i(t) for fluorescence decays resulting from
underlying distributions of rate constants k is via Laplace transform. We define i(t) by the
Laplace transform of f(k):

i(t )  i(0)  f (k ) e kt dk
(2)
0

Equation 2 is valid for all f(k) with
 f (k ) dk  1
because f(k) is the inverse Laplace
0
transform of i(t) divided by i(0). f(k) represents the (normalized) probability density
function of rate constants k. i(0) denotes the value of i(t) at time zero, i(t = 0). Equation 2
describes a fluorescence decay with underlying distribution of rate constants k or, more
accurately, a distribution of k described by its probability density function f(k). 59
As an example, let us consider f(k) comprised of a sum of delta functions,
f (k )    j  (k  k j )
(3)
j
The classical multi-exponential expression of i(t) is obtained after substitution of eq 3
into eq 2:
i(t )  i(0)  j e
k j t
(4)
j
with i(t = 0) = i(0) and

j
 1.
j
The second way of introducing the fluorescence decays i(t) of distributions of rate
5
constants is via first-order kinetics of excited species. Let us consider a causal, linear,
time-invariant photophysical system consisting of a single type of ground-state species X,
which is excited with a -pulse of low intensity at time zero, so that the ground-state
species population is not appreciably depleted. The time evolution of the excited-state
species concentration [X*(t)] can be described by the following first-order, linear
differential equation:




d X* (t )
 k (t ) X* (t ) ,
dt
t≥0
(5)
where k(t) is a time-dependent rate coefficient. Integration of eq 5 gives the decay of the
concentration of excited molecular entity X*:
X (t )  b exp    k (u) du  ,
t
t≥0
*


0
(6)
with b the initial (that is, at time zero) concentration of excited species: b = [X*(t = 0)].
The fluorescence -response function i(t) is given by
 t

i (t )  c X* (t )  i (0) exp    k (u ) du 
 0



(7)
It must be emphasized that the pre-exponential factor i(0) ensures that i(t = 0) = i(0)
= b c. In eq 7, c is the emission weighting factor at emission wavelength em given by13
c  kf
  (
em
) d em
(8)
 em
where kf stands for the fluorescence rate constant; em is the emission wavelength
interval around em where the fluorescence signal is monitored; and (em) represents the
emission density of the excited species at em defined by13
 ( em )  F ( em ) /
 F (
em
full emission band
) d em
(9)
6
where the integration extends over the entire steady-state fluorescence spectrum F(em).
Equations 2 and 7 are formally equivalent when the time-dependent rate coefficient k(u)
is given by

k (u ) 
 k f (k ) e
0

 ku
dk
(10)
 f (k ) e
 ku
dk
0
2.1. Stretched exponential (or Kohlrausch) function59, 60, 61
One function suited to the analysis of fluorescence decays with underlying distributions is
the well-known stretched exponential (or Kohlrausch) function (eq 12). The timedependent rate coefficient k(u) is defined as

k (u ) 

u
 
 
 1
(11)
where 0 <  ≤ 1 and  is a parameter with the dimension of time. The meaning of 
depends on the underlying physical model. Generally, a smaller value of  corresponds to
a wider distribution of rate constants k, suggesting a more disordered system. This
function has been used in various fields (see ref 61 for literature data). Equation 12 is
obtained after substitution of eq 11 in eq 7:
i(t )  i(0) e
t
 
 

(12)
where 0 <  ≤ 1. Note that for  = 1, a mono-exponential i(t) is obtained, while for  = ½,
the Smoluchowski decay law is obtained. To resolve the problems associated with the
unwanted short-time behavior of the time derivative of the Kohlrausch function, the time
origin is shifted to t = . In its modified, generalized form, the Kohlrausch function can
7
be written as60
 t
1 1 
 

i(t )  i(0) e
(13)
with  ≠ 0. The time-dependent rate coefficient k(u) of the generalized Kohlrausch
function is
k (u ) 


 u
1  
 
 1
(14)
and is finite for all times. The derivation of the probability density function f(k) of rate
constants (in eq 2) can be found in papers by Pollard62 and Berberan-Santos et al.60
2.2. Becquerel function61, 63, 64
Another function for describing the fluorescence decays with underlying distributions is
the less-known Becquerel function,
i (t ) 
i (0)
(1  ct /  )1 / c
(15)
where 0 ≤ c ≤ 1 and   1 k is a parameter with the dimension of time ( k denotes the
average rate constant). Equation 15 is a compressed hyperbola for c < 1. For c → 0, i(t)
approaches a single exponential with lifetime : i(t )  i(0) e t  . For c → 1, i(t) becomes
hyperbolic: i(t ) 
i(0)
. In both limiting cases, i(t) is controlled by a single parameter
(1  t /  )
. The probability density function f(k) of rate constants (in eq 2) is the gamma
distribution.61, 63 The time-dependent rate coefficient k(u) for the Becquerel function is63
k (u ) 
1
  cu
(16)
Equation 15 is obtained after substitution of eq 16 in eq 7. An alternative expression of
8
the Becquerel function is:61, 63
i(t ) 
i (0)
(1  at ) p
(17)
2.3. Förster type energy transfer
Electronic energy transfer plays an important role in a number of organized molecular
assemblies. The excitation energy can be transferred from the “donor” molecules to the
“acceptor” traps. Monitoring the fluorescence decay of the donor or the appearance of the
acceptor fluorescence with time can provide information on the energy transfer process.
At room temperature, energy transfer between two chromophores is often well described
by a weak dipole-dipole interaction, i.e., Förster type energy transfer.65 The general
expression for the donor fluorescence decay predicted for Förster energy transfer from an
ensemble of donors to an ensemble of acceptors distributed at random is given by:66
d /6
 t
t 
i(t )  i(0) exp      
   
 
(18)
In eq 18,  denotes the lifetime of the excited donor in the absence of acceptors (i.e., in
the absence of energy transfer);  is a parameter which depends on the density of acceptor
traps, the critical distance R0 for energy transfer (or Förster radius), the type of interaction
(dipole–dipole, dipole–quadrupole) and the Euclidian dimension d (1, 2, or 3) of the
system and the type of transfer. The cause of the non-exponential fluorescence decay of
the donor (expressed by eq 18) is the distribution of distances between donors and
acceptors. The fluorescence described by eq 18 is characterized by an initial nonexponential component followed at longer times by the same decay as found for the
unquenched donor.
9
2.4. Truncated Gaussian (at k0 > 0) distribution of rate constants50
When a Gaussian distribution is assumed for the decay rate constants, the following
expression for the fluorescence decay in the absence of energy transfer can be derived

 2 t 2  2t 

i (t )  i (0) exp   k 0t 
2




 t 


erfc  

2 




  
erfc 

 2 
(19)
where erfc(y) = 1 – erf(y) is the complementary error function. k0 denotes the smallest
decay rate constant (lower bound),  is the average value of the decay rate constants and
 represents the standard uncertainty67 of the normal distribution of the decay rate
constants. The probability density function f(k) is given by
 k   2 
exp 

2 2 

f (k )  
 k   2 
exp
0  2  2  dk
(20)
2.5. Exponential distribution of rate constants
For the exponential distribution of rate constants:
f (k )  a e  ak
(21)
one obtains i(t) according to eq 2 and after normalization:
i (t ) 
a i ( 0)
ta
(22)
The time-dependent rate coefficient k(u) calculated according to eq 10 is given by:
k (u ) 
1
ua
(23)
10
2.6. Uniform distribution of rate constants
For the uniform distribution of rate constants, the probability density function f(k) is
given by f(k) = 1/h in the interval (m – h/2, m + h/2) and zero outside this interval, so that
i(t) (eq 2) becomes
m
i(t ) 

m
h
2
1 k t
e dk
h h
(24)
2
After normalization one obtains:
i (t ) 
ht
ht

 2 i (0) m t
i (0) m t  2
 ht 
e  e  e 2  
e sinh  
ht
ht
 2


(25)
with sinh y representing the hyperbolic sine of y.
2.7. Fluorescence decay function with extreme sub-exponential behavior63
If one considers a rate coefficient k(u) that decays mono-exponentially to zero,
k (u )  k 0 e  a u
(26)
one obtains from eq 7 after normalization:
 k

i (t )  i (0) exp  0 1  e a t 
 a

(27)
The probability density function f(k) of rate constants has been derived by BerberanSantos et al. 63
2.8. Mittag-Leffler and Heaviside decay functions59, 61, 68
The single exponential fluorescence -response function i(t )  i(0) e  k t can be written as
a power series (eq 28). Two straightforward generalizations of the exponential decay
function are the Mittag-Leffler exponential function (eq 29) and Heaviside’s exponential
function (eq 30).
11

(kt) n
(kt) n
 i(0) 
n!
n 0
n 0 (n  1)

i(t )  i(0) e kt  i(0) 
(28)
(kt) n
n 0 ( n  1)
(29)
(kt) n
n  0 ( n  1   )
(30)

i(t )  i (0) 

i(t )  i(0) 
3. Identifiability
3.1. General concepts
In the deterministic identifiability analysis, one examines as to whether or not the
parameter set  of a given model is uniquely defined assuming error-free observations for
a specific photophysical model.9–11 Therefore, the identification study of models for
excited-state processes investigates whether it is possible to find model parameter sets of
~
the fluorescence -response function i(t), say  and  , so that for all values of t
~ ~
i(t )  i (t )
(31)
~
In other words: is it possible to find different (alternative)  model parameter sets so that
~
the fluorescence -response function i(t) is the same for the true  and alternative 
~
model parameter sets? The tilde on  indicates that this parameter set may or may not be
equal to the true model parameter set . However, both parameter sets should produce the
~
same fluorescence -response function i(t). The tilde on i (t ) indicates that this function
~
is obtained for the alternative model parameter set  .
There are three possible outcomes to the deterministic identifiability analysis. (1) Global
~
(or unique) identifiability is achieved when    , i.e., a unique set of model parameters
12
is obtained. (2) The model is locally identifiable when there is a finite number of
~
alternative  . (3) An unidentifiable model is found when there is an infinite number of
~
alternative  . The aim of the deterministic identifiability analysis is to express all
~
possible alternative model parameter sets  as functions of the true .
Deterministic identifiability is essential because it establishes whether the model
parameters can be extracted from ideal data and can point to the optimal experimental
design. As stated in the Introduction, the aim of the present manuscript is to find out
whether from the non-exponential expressions of the fluorescence -response function
i(t), the different parameters in i(t) can be determined unambiguously from error-free
fluorescence decay data. Imperfect fluorescence decay data resulting from noisy
observations sampled over a finite time window affect the accuracy and precision with
which the model parameters can be estimated. Indeed, real data (e.g., obtained by the
single-photon timing technique) differ from those error-free data by the presence of
Poisson noise (due to photon-counting statistics), systematic errors (e.g., due to reflected
excitation light, dark background) and are deformed by convolution with an instrumental
response function of finite width. Therefore, the analysis of real data will lead a
distribution of each of the parameters recovered from the analysis of the fluorescence
decays. If there is no correlation between the values recovered for the different
parameters, the width of this distribution can be estimated directly by fitting the
experimental data using a non-linear least-squares method.4, 69 The numerical parameter
estimation and the statistical properties of the parameter estimates form the second stage
of any identifiability analysis, called numerical identifiability. However, such a study is
13
beyond the scope of this article. Examples of numerical identifiability studies can be
found in refs 14, 16-20, 51, 70, 71 and 72.
3.2. Identifiability via Taylor (or Maclaurin) series
The Maclaurin series for i(t) is the Taylor series about the origin t = 0:
i(t )  i(0)  t i ' (0) 
t 2 ''
t3
tn
i (0)  i ''' (0)  ...  i n  (0)  ...
2!
3!
n!
(32)
Note that i’(0) means to differentiate i(t) and then put t = 0, etc., i(n)(0) means to find the
nth derivative of i(t) and then put t = 0.
The successive derivatives are – in principle – measurable and they contain information
about the parameters to be identified. To ensure functional equivalence in eq 31 at all
times t, the coefficients of the Maclaurin expansion in eq 32 must be identical for
different model parameter sets of the fluorescence -response function i(t). This results in
the following set of equations:
~
~
i ( n ) (0   )  i ( n ) (0   ) ,
n = 0, 1, …
(33)
~
Solving this set with respect to the alternative model parameter set  in terms of the true
model parameter set  leads to conclusions on deterministic identifiability.
3.3. Stretched exponential (or Kohlrausch) function
The identifiability condition of eq 31 can be explicitly written for i(t) given by eq 12 as
~
  t  
  t  
~
i (0) exp   ~    i(0) exp    
    
    
(34)
~
Evaluating eq 34 at t = 0 leads to i (0)  i(0) . In other words, the parameter i(0) is
uniquely defined, which is true for all i(t) described in this paper.
14
~
After setting i (0)  i(0) , the identifiability condition of eq 31 is reduced to
~


~
t
t
 ~     . This can only be fulfilled for ~   and    .
 
 
After logarithmic transform of eq 13 (for the modified Kohlrausch function) we have
 t
ln i (t )  ln i (0)  1  1  
 

(35)
Maclaurin expansion of ln i(t) gives
1
!
t 1
t
t
t
ln i (t )  ln i (0)       (   1)    (   1)(   2)   ... 
   ...
3!
(   n)! n!   
   2!
 
 
(36)
2
3
n
~
The identifiability condition of eq 31 at t = 0 gives i (0)  i(0) . The linear t dependence
and the tn (n = 2, 3, …) dependence in eq 36 can never reproduce each other. As a
consequence, eq 31 can be split in a set of eqs 37:
~


t ~t


(37.1)
~
 1 2  1 2
t  ~ t


(37.2)
…
~
  n 1 n   n 1 n
t 
t

~
(37.n)
From eq 37.1 we have
~
  A  with A  ~ / 
(38.1)
For n ≥ 2, eq 37.n can be rewritten as
~
  A   (n  1) A  n  1
(38.2)
15
~
From eqs 38, we obtain A = 1 (i.e., ~   ) and then    . Thus, the model is uniquely
identifiable in terms of i(0),  and .
3.4. Becquerel function
Taking the natural logarithm of eq 15 yields
1
ln i (t )  ln i (0)  ln (1  ct /  )
c
(39)
Power series expansion gives
2
3
4

1  ct 1  ct  1  ct  1  ct 
ln i (t )  ln i(0)             ...
c   2   
3   4  

(40)
~
From the identifiability condition of eq 31 at t = 0, we have i (0)  i(0) . The linear t
dependence and the tn (n = 2, 3, …) dependence in eq 40 can never reproduce each other.
As a consequence, eq 31 can be split in a set of eqs 41:
1
1
t  ~t


(41.1)
c~ 2
2
t

t
2
~ 2
(41.2)
c
etc. From eq 41.1, we have ~   and then from eq. 41.2, we obtain c~  c , resulting in a
uniquely identifiable model.
3.5. Förster type energy transfer
Taking the natural logarithm of eq 18 gives
ln i (t )  ln i (0) 
t
 

 
t
d /6
(42)
~
After setting i (0)  i(0) , the explicit identifiability condition of eq 31 leads to
16
~
d /6
d /6
t
 t   t
t 
~
 ~    ~          . The linear t dependence and the td/6 dependence in eq
    
   

~
42 can never reproduce each other. Therefore, from eq 31 we have ~   , d  d and
~   . Hence, this model is uniquely identifiable in terms of i(0), , d and .
3.6. Normal distribution of rate constants
~
Evaluation of eq 19 at t = 0 and the identifiability condition of eq 31 leads to i (0)  i(0) .
Since the exponential function ex and the complementary error function, erfc(y) = 1 –
erf(y), (with x and y defined by eqs 43.1 and 43.2, respectively) can never reproduce each
other, we have the following straightforward identifiability equations: ~
x  x and ~y  y
(eq 43.3 and 43.4).
x  k 0t 

y 
 2 t 2  2t
(43.1)
2
 t
(43.2)
2
~
~ 2 t 2
 2 t2
 (k0  ~)t 
 (k0   )t 
2
2
(43.3)
 ~ ~

 t 
 t
~


(43.4)
~
From eq 43.4 we have ~   and ~   and then from eq 43.3 k0  k0 . This shows that
the model is uniquely identifiable in terms of i(0), k0,  and .
3.7. Exponential distribution of rate constants
~
Evaluation of eq 22 at t = 0 and the identifiability condition of eq 31 leads to i (0)  i(0) .
The identifiability condition of eq 31 reduces to:
17
t
t

~
a a
(44)
so that a~  a . This simple model is uniquely identifiable in terms of i(0) and a.
3.8. Uniform distribution of rate constants
~
Evaluation of eq 25 at t = 0 and the identifiability condition of eq 31 leads to i (0)  i(0) .
 ht 
Since the exponential function e  m t and the hyperbolic sine sinh   can never
 2
~  m and h~  h , making this
reproduce each other, from eq 31, we have directly that m
model uniquely identifiable in terms of i(0), m and h.
3.9. Fluorescence decay function with extreme sub-exponential behavior
~
After the logarithmic transform of eq 27 and setting i (0)  i(0) , the identifiability
condition of eq 31 leads to
~
~
k0
k

1  e a t   0 1  e a t 
~
a
a
(45)
~
~
k0 k0
For t = , eq 45 reduces to ~ 
and hence 1  e  a t  1  e  a t and thus a~  a . This
a
a
~
finally gives k0  k0 . This demonstrates that also this model is uniquely identifiable in
terms of i(0), k0 and a.
3.10. Mittag-Leffler and Heaviside decay functions
Evaluation of eqs 29 and 30 at t = 0 and the identifiability condition of eq 31 leads to
~
i (0)  i(0) for both functions.
For the Mittag-Leffler function (eq 29), the identifiability condition of eq 31 then reduces
to:
18
~
(k t ) n
(kt) n

(~ n  1) ( n  1)
(46)
for n = 0, 1, …, ∞. Because the time sequence (kt, k2t2, k3t3, …) and the gamma function
~
never reproduce each other, we have k  k and ~   , showing that this model is
uniquely identifiable in terms of i(0),  and k.
For Heaviside’s exponential function (eq 30), the identifiability condition of eq 31 leads
to:
~
(k t ) n
(kt) n

(n  1  ~ ) (n  1   )
(47)
for n = 0, 1, …, ∞. Evaluation of eq 47 for n = 0 gives (1  ~ )  (1   ) , which for  >
0 is only possible with ~   . For n = 1 and substitution of ~   in eq 47 leads to
~
~
k t  kt and thus k  k . Hence, also this model is uniquely identifiable in terms of i(0), 
and k.
4. Conclusion
All the tested photophysical models for time-resolved fluorescence with underlying
distributions of rate constants leading to non-exponential functions i(t) given by eqs 12
(Kohlrausch function), 13 (modified Kohlrausch function), 15 and 17 (Becquerel
function), 18 (Förster type energy transfer), 19 (Gaussian distribution of rate constants),
22 (exponential distribution of rate constants), 25 (uniform distribution of rate constants),
27 (function with extreme sub-exponential behavior), 29 (Mittag-Leffler function) and 30
(Heaviside’s function) are uniquely identifiable. This means that for each model with
associated i(t) there is a single set of true parameters  that describe i(t).
19
Acknowledgements
Dr. E. Novikov is thanked for useful scientific discussions.
Graphical abstract
Investigation of the identifiability of several uncommon models for time-resolved
fluorescence with underlying distributions of rate constants leading to non-exponential
decays shows that all the models are uniquely identifiable.
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23