INTRODUCTION

The fluorescence decay kinetics of molecules are often impossible to approximate with one exponentially decaying curve [1, 2]. Numerous methods of multi-exponential approximation of the decay kinetics have consequently been proposed to obtain the amplitudes and real factors of exponentials [1–3]. To employ any of the methods, it is necessary to preset the sum N of exponentials that form the kinetics. Because the sum N is invariably unknown, several tentative N values are usually selected and approximation is performed for each of them. The next step is selecting the value N* such that it ensures the deepest minimum for squared deviations of the theoretical curve from the experimental one (e.g., see [3, 4]). However, when iteration methods of nonlinear optimization are used with a preset N value, estimates obtained for the parameters fail to achieve a global minimum of approximation. The reliability of the procedure of seeking N* is therefore highly questionable. N has been included in the set of independent variables in a recent study with an iterative method used to solve a nonlinear least squares problem [5]. A global minimum of approximation has still not been achieved. Estimates of N* can be obtained together with the parameters by approximating the kinetics using certain noniterative methods [6, 7], which lead to the global minimum of approximation. However, an extremely low reliability is characteristic of noniterative methods used to approximate kinetic curves [1]; the methods fail to work with experimental data more often than otherwise, although the causes are still unknown. The question arises as to whether the sum N* of approximating exponentials can be found without using any approximation procedure. This parameter often reflects the processes that are necessary to study in more detail. Several examples are given below.

(1) A study of charge separation in bacterial photosynthetic reaction centers has shown that bacteriopheophytin acts to transfer an electron from a primary donor to a “primary” electron acceptor [8]. More detailed studies were prompted by the following observation. When reaction centers were excited by exposure to light and were in a state with a reduced primary electron acceptor, the fluorescence decay kinetics of the primary donor included not only a component with picosecond-scale relaxation, but also a component with a life of approximately 10 ns [8, 9]. Further studies revealed a recombination nature of this radiation and identified the molecules involved in the recombination process [8].

(2) Once pheophytin was implicated in charge separation in photosystem II reaction centers of higher plants [10, 11], it became clear that a respective (recombination) component must be identified in the fluorescence decay kinetics of the primary electron donor. The component was identified [12, 13].

(3) Electron transfer from a primary donor to a bacteriopheophytin molecule in the reaction centers of photosynthesizing bacteria is preceded by reversible electron transfer from the primary donor (a dimer of bacteriochlorophyll molecules) to a bacteriochlorophyll molecule [14–16]. Hence, a respective recombination component was assumed to occur in the induced fluorescence kinetics of the primary electron donor in reaction centers. The component was identified [17]. A detailed analysis of the induced fluorescence kinetics showed that only one exponential was present in the kinetics at temperatures of 5–25 K [18, 19]. The factor of this exponential gives the rate constant for electron abstraction from the primary donor. Two exponentials were detectable in the induced fluorescence kinetics of the primary donor in a temperature range of 25–200 K. While the first exponential described the charge separation process, the other described charge recombination. Three decaying exponentials were observed in the induced fluorescence kinetics at environmental temperatures higher than 200 K. The first exponential described the charge separation process, as above, while the second and third ones described charge recombination with traps of different depths. Hence, it is important for a researcher to have a means to obtain, in a timely manner (during an experiment), data on the sum of the exponential components that form the decay kinetics, for example, that of fluorescence.

THE HANKEL MATRIX OF EXPERIMENTAL DATA

Let I(t) be the decay kinetics of fluorescence (or any other process) and Ij = I(tj) its discrete values measured with the same interval (j = 1, …, n). The intention is to use the resulting kinetic curve to isolate its property that is impossible to measure in a physical experiment. Namely, the objective is to determine the sum N of the exponentials that form the kinetics. For this purpose, m fragments are isolated in the kinetics I(t) and presented as the following vectors: \({\mathbf{\omega }}_{1}^{T}\) = [I1I2 … IL], \({\mathbf{\omega }}_{2}^{T}\) = [I2I3 … IL+ 1], …, \({\mathbf{\omega }}_{m}^{T}\) = [ImIm+ 1 … In], where L = n – m + 1 and T is transposition. If I(t) = I0exp(–t/τ) or Ik = I0exp(–(k – 1)h/τ), then I1 = I0, I2 = ξI1, I3 = ξ2I1, and Im = ξm– 1I1, where ξ = exp(–h/τ). Then ω2 = ξω1, ω3 = ξ2ω1, and ωm = ξm– 1ω1. In other words, all of the vectors ωk are linearly dependent in the case of the monoexponential kinetics I(t). When the kinetics I(t) decays in a multi-exponential manner, the vectors ωk are not all linearly dependent. It is possible to assume that the sum of exponentials forming the kinetics I(t) is the same as the sum of linearly independent vectors of the set ω1, ω2, …, ωm. Following [20], the coordinates of the vectors ωk are used to obtain the matrix Γ:

$${\mathbf{\Gamma }} = \left[ {\begin{array}{*{20}{c}} {{{I}_{1}}}&{{{I}_{2}}}&{{{I}_{3}}}& \cdots &{{{I}_{m}}} \\ {{{I}_{2}}}&{{{I}_{3}}}&{{{I}_{4}}}& \cdots &{{{I}_{{m + 1}}}} \\ {{{I}_{3}}}&{{{I}_{4}}}&{{{I}_{5}}}& \cdots &{{{I}_{{m + 2}}}} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {{{I}_{L}}}&{{{I}_{{L + 1}}}}&{{{I}_{{L + 2}}}}& \cdots &{{{I}_{n}}} \end{array}} \right].$$
((1))

Because antidiagonal elements of the matrix Γ (1) are unchanged, Γ is a Hankel matrix [20]. The sum of linearly independent vectors of the set ω1, ω2, …, ωm is equal to the rank of the Hankel matrix [20] composed of the coordinates of the vectors. Therefore, the sum of exponentials forming the kinetics I(t) is equal to the rank (rank) of the Henkel matrix composed of discrete components of the kinetics I(t). When I(t) is an exponential function, rank(Γ) = 1.

Let ΓΠ = QR be an orthogonal triangular factorization of the rectangular Hankel matrix Γ with a pivot column chosen in the matrix Γ (Π is a permutation matrix) [21]. Because the matrix Π has full rank, then rank(ΓΠ) = min(rank(Γ), rank(Π)) = rank(Γ) [22]. Because the matrix Q is orthogonal, rank(Q) = m and rank(Γ) = min(rank(Q), rank(R)) = rank(R). That is, the rank of Γ is the sum of nonzero diagonal elements of its upper triangular factor R [21]. Therefore, the sum of exponentials contained in the kinetics I(t) is the sum of nonzero diagonal elements in the triangular factor R of the Hankel matrix Γ. The kinetics I(t) includes at least 1024 discrete measurements in modern experiments; therefore, the last statement can be proven, in particular, by performing numerical experiments.

PARAMETERS OF NUMERICAL EXPERIMENTS

We assume that the following sum can be used to describe the fluorescence decay kinetics: I(t) = \(\sum\nolimits_{k = 1}^{k = N} {{{d}_{k}}\exp \left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} {{{\tau }_{k}}}}} \right. \kern-0em} {{{\tau }_{k}}}}} \right).} \) Let

$${{\tau }_{k}} = {{2}^{{k - 1}}}{{\tau }_{{\min }}},$$
((2))

where k = 1, 2, …, N and τmin defines the scale and units of measurement. The relative “distance” δk (δk = τk+ 1/τk– 1) between the time constants that are closest in magnitude to each other does not change in this case. If

$${{\tau }_{k}} = k{{\tau }_{{\min }}},$$
((3))

then δk = 1/k and the relative distance between the closest time constants decreases with the increasing k. If

$${{\tau }_{k}} = k!{{\tau }_{{\min }}},$$
((4))

then δk = k. In this case, the relative distance between the closest time constants increases. Let h = τmin/4 and T = 8τN. Then the number of measurements is n = 1 + T/h, or n = 1 + 32τN/τmin. If n ≥ 1024, then tj = h(j – 1). If n < 1024, then it is assumed that n = 1024, h = T/(n – 1), and tj = h(j – 1). Measurements of an ideal signal are as follows: \(I_{j}^{{(o)}}\) = \(\sum\nolimits_{k = 1}^{k = N} {{{d}_{k}}\exp \left( {{{ - {{t}_{j}}} \mathord{\left/ {\vphantom {{ - {{t}_{j}}} {{{\tau }_{k}}}}} \right. \kern-0em} {{{\tau }_{k}}}}} \right).} \) Next, white noise samples Sj (ran2.f as in [23]) were specified and discrete measurements of the full signal were calculated as

$${{I}_{j}} = I_{j}^{{(o)}} + \delta ({{S}_{j}} - \bar {S}),$$
((5))

where \(\bar {S} = {{n}^{{ - 1}}}\sum\nolimits_{j = 1}^{j = n} {{{S}_{j}}} \) is the mean noise amplitude and δ is the noise amplitude defined as follows. The signal-to-noise ratio (SNR) in decibels was defined as

$$SNR = 20\log \left( {{{{{I}_{o}}} \mathord{\left/ {\vphantom {{{{I}_{o}}} {{{{\left\langle s \right\rangle }}_{T}}}}} \right. \kern-0em} {{{{\left\langle s \right\rangle }}_{T}}}}} \right),$$
((6))

where Io is the amplitude of the kinetics without a noise contribution and \({{\left\langle s \right\rangle }_{T}}\) = \({{n}^{{ - 1}}}\sum\nolimits_{j = 1}^{j = n} {\left| {{{S}_{j}} - \bar {S}} \right|} \) is the mean amplitude of the absolute noise value. Presetting the necessary SNR value gives δ = 10–SNR/20\({{{{I}_{o}}} \mathord{\left/ {\vphantom {{{{I}_{o}}} {{{{\left\langle s \right\rangle }}_{T}}}}} \right. \kern-0em} {{{{\left\langle s \right\rangle }}_{T}}}}.\)

Discrete measurements were used to fill the matrix Γ, which was then subject to QR factorization with a choice of the pivot column: ΓΠ = QR (dgqpf.f according to [24]). The dgeqpf.f program was slightly modified to process only k (k = m/10) columns of the matrix Γ. Therefore, Q\( \in \) ℜL×k and R\( \in \) ℜk×k. To correct the values of the factor numbers, the small matrix R was subjected to singular value decomposition (dgesvd.f according to [23]): R = \({\mathbf{\tilde {U}\tilde {\mu }}}{{{\mathbf{\tilde {V}}}}^{T}},\)\({\mathbf{\tilde {U}}}\)\( \in \) ℜk×k, \({\mathbf{\tilde {V}}}\) \( \in \) ℜk×k, \({\mathbf{\tilde {\mu }}}\)\( \in \) ℜk×k. When k = m, the corrected factor numbers coincided with the singular numbers of the singular decomposition Γ = UμVT with good accuracy. Because k\( \ll \)m, calculation of the corrected factor numbers was somewhat less accurate, but far quicker. The factor spectrum (j = 1–k) was than calculated: εr(j) = log(max(10–14, \({{\tilde{\mu}}_{j,j}}\))).

THE RANK OF HANKEL MATRICES IN IDEAL EXPERIMENTS

Figure 1 shows the factor spectra that were calculated using Eq. (5) for the kinetic curves I(o)(t) at various sums of exponentials that form the kinetics I(o)(t); the time constants of the exponentials were distributed according to Eq. (2). The amplitudes of the exponentials were equal. On the one hand, the number shown at a spectral curve (Fig. 1) corresponds to the sum of exponentials forming the kinetic curve I(o)(t). On the other hand, the number shows the number of nonzero diagonal elements in the R factor of the respective Hankel matrix Γ, that is, the rank of the matrix Γ. Thus, the rank of the Hankel matrix (1) fully corresponds to the sum of decaying exponentials forming the kinetics I(o)(t). We note that if N decaying exponentials form the kinetics, then the same sum N of factor numbers is separated from zero factor numbers by a step. A step is observed when the function fr(k) = \({{{{r}_{{k,k}}}} \mathord{\left/ {\vphantom {{{{r}_{{k,k}}}} {\sqrt {{{r}_{{k - 1,k - 1}}}{{r}_{{k + 1,k + 1}}}} }}} \right. \kern-0em} {\sqrt {{{r}_{{k - 1,k - 1}}}{{r}_{{k + 1,k + 1}}}} }}\) has a high maximum at k = N.

Fig. 1.
figure 1

The factor spectra of multi-exponential kinetic curves. The number shown at a curve is the number of the spectrum, the number of the last nonzero factor number in the spectrum (or the rank of the matrix Γ), and the sum of exponentials in the kinetic curve corresponding to the given spectrum.

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Figure 2 shows the factor spectra that were calculated for the kinetic curves I(o)(t) for the case where the time constant distribution follows Eq. (3). Only eight exponentials were detected in this case. When the kinetics I(o)(t) consisted of nine exponentials, then the step separating the nonzero and zero diagonal elements in the R factor disappeared. The effect arose because the time components were too similar and their resolution was consequently unsuccessful. Highly similar time constants were resolvable in the ideal experiment when only two exponentials were present in the kinetics. However, when noise was present in the experiment, then the resolution of the method strongly depended on the noise amplitude. The resolution equation is available from [25].

Fig. 2.
figure 2

The factor spectra of multi-exponential decay kinetic curves with a time constant distribution following Eq. (3). The number shown at a curve is the number of the spectrum, the number of the last nonzero factor number in the spectrum, and the sum of exponentials in the kinetic curve corresponding to the given spectrum.

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Figure 3 shows the factor spectra that were calculated for the kinetic curves I(o)(t) with the time constants distributed following Eq. (4). Insufficient RAM of a PC was the only factor that prevented use of the kinetic curves with more than nine exponentials. However, τ9/τ1 = 362 880 even in that case, which seems to be enough to allow modeling of any experimental data set. Thus, when the sum of exponentials forming I(t) does not exceed eight, there is no cause other than noise that prevents their rapid detection.

Fig. 3.
figure 3

The factor spectra of multiexponential decay kinetic curves with a time constant distribution following Eq. (4). The number shown at a curve is the number of the spectrum, the number of the last nonzero factor number in the spectrum, and the sum of exponentials in the kinetic curve corresponding to the given spectrum.

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THE EFFECT OF NOISE

Let I(o)(t) be a sum of three decaying exponentials (τk = 1, 2, 4 ns; bk = 1) and I(t) be the sum of I(o)(t) and white noise (7). Figure 4 shows the factor spectra calculated for various SNR values. As is seen, factor numbers of one group decrease symbatically with the increasing SNR. Starting from the fourth number, the factor numbers of the group can be aligned with a line that is nearly “parallel” to the abscissa axis. The numbers of the group should therefore be associated with noise. The magnitude of the first factor number is almost independent of the noise level. The first factor number is therefore associated only with the useful signal. Associations of the second and third factor numbers are problematic. Everything is clear in the case of the factor spectrum 100 dB. The factor numbers designated 3 and 4 are separated with a step. This is especially clearly seen in the f spectrum, which is shown with open circles in Fig. 5. The maximum and minimum of the function fr(k) = \({{{{r}_{{k,k}}}} \mathord{\left/ {\vphantom {{{{r}_{{k,k}}}} {\sqrt {{{r}_{{k - 1,k - 1}}}{{r}_{{k + 1,k + 1}}}} }}} \right. \kern-0em} {\sqrt {{{r}_{{k - 1,k - 1}}}{{r}_{{k + 1,k + 1}}}} }}\) correspond to numbers 3 and 4, respectively, and kmin – kmax = 1 and fr(kmax) > 1. These two facts indicate that N* = 3. In the spectrum 80 dB, the factor numbers designated 3 and 4 are separated with a cleft rather than with a step. The maximum and minimum of the function fr(k) (Fig. 5, filled circles) corresponds to numbers 2 and 4. Therefore, it is only possible to assume that N* = 3. In the spectrum 60 dB (Fig. 4), the factor numbers designated 3 and 4 are separated with a cleft, and the maximum and minimum of the function fr(k) (Fig. 5, triangles) correspond to numbers 2 and 3. However, because fr(kmax) < 1, then N* ≠ 2, and SNR must be increased to further accumulate the data in the experiment.

Fig. 4.
figure 4

The factor spectra of noisy three-exponential decay kinetic curves. The number shown at a curve is SNR for the kinetics used in calculations.

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Fig. 5.
figure 5

The factor f spectra of noisy three-exponential decay kinetic curves: (1) SNR = 40 dB, (2) SNR = 60 dB, (3) SNR = 80 dB, and (4) SNR = 100 dB.

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DISCUSSION

To determine the sum of exponentials that form the kinetic curve, calculating a covariance matrix of experimental data and analyzing its determinants were proposed in quantum chemistry in 1984 [26]. However, calculating a covariance matrix is a time-consuming procedure. Moreover, it is easy to demonstrate that the rank of a Hankel matrix of experimental data is equal to the rank of a covariance matrix of the same data. The two approaches consequently yield equal results; however, the method described above is many times faster. It has been proposed to estimate the sum of the exponentials from the coefficients of the Pad approximation of the Laplace transform of the kinetics [6]. Unfortunately, a far more correct calculation of the coefficients requires far greater processor time expenditures. The greatest agreement of the above method is observed with part of a multi-exponential kinetics approximation (the APM algorithm) [7]. The Hankel matrix Γ = \({{\Re }^{{L \times {{N}_{*}}}}},\) where N* is a preliminary estimate of the sum of exponentials, is subjected to a singular decomposition in the algorithm. However, it has already been shown [27] that to separate all singular values of the signal from singular values of low-level noise the m/N* ratio must be at least 80; i.e., the number of columns in the Hankel matrix must be far greater than the expected sum of exponentials. A recent study has shown that if n is the sum of discrete measurements in the kinetics I(t) then the step is maximal when m = n/2 [28]. As an example, when the intention is to detect eight exponentials, the Hankel matrix should be approximately 640 × 640 in dimensions. The time it takes to calculate a singular decomposition is proportional to n3/8. The same matrix is converted to a orthogonal–triangular form within a time proportional to only to n2/4 [20]. If a step separates the factor numbers of the useful signal from the factor numbers of noise, then the total sum of factor numbers gives the sum of exponentials that form the decay kinetics (Fig. 4, spectrum 100 dB). If the noise level is high and a step is undetectable (Fig. 4, spectra 40 dB and 60 dB), then kinetic data must be accumulated further. When a photon counting technique is employed, it is necessary to accumulate (to average) the tracks recorded by the instrument. This is the only means to reduce the noise level in the signal without distorting the spectra of the useful signal and noise. If a step has still not formed, the analysis does not yield the true sum of exponentials, but rather gives an estimate of the minimum possible sum of exponentials and this estimate should be used when approximating the kinetics. If the average noise level is nonzero in this case, the method detects one extra exponent. It is also expedient to eliminate the apparat function of the instrument by performing deconvolution of the kinetics [29].

COMPLIANCE WITH ETHICAL STANDARDS

The authors declare that they have no conflict of interest. This article does not contain any studies involving animals or human participants performed by any of the authors.