Research article Received: 17 April 2013,
Accepted: 8 July 2013
Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI 10.1002/bio.2567
Complex temporal decay: a complete analysis R Quintero-Torres,a* HA Castillo-Matadamas,b Jeff F Youngc and RM Bermúdez Cruzd ABSTRACT: Relaxation dynamics is universal in science and engineering; its study serves to parameterize a system’s response and to help identify a microscopic model of the processes involved. When measured data for a phenomenon cannot be fitted using one exponential, the choice of an alternative function to describe the decay becomes nontrivial. Here, we contrast two different, but fundamentally related approaches to fitting nontrivial decay curves; exponential decomposition and the gamma probability density function. Copyright © 2013 John Wiley & Sons, Ltd. Keywords: lifetime; exponential decomposition; function fitting; photoluminescence
Introduction From a signal point of view, two approaches to gathering information from the response of a system after excitation are possible; the time domain studies the response after a short excitation and the frequency domain studies the response after a modulated signal reaches the system. Both options in this generic classification are useful in a wide range of experiments from the life sciences to geological phenomena, including physics, materials and electrical engineering; overlap between the techniques and concepts used is important and unfortunately not always recognized or exchanged between areas. The instrumentation and mathematical tools for the time domain and frequency domain are well defined; gated measurement over time and one form of lock-in filtering or spectral analysis go hand in hand with decaying signals or Fourier transform tools in the frequency domain (1). All time domain studies have in common the possibility of describing the evolution of a system by what is generically referred to as exponential decay. This is because many physical phenomena are described by first-order differential equations (eqn 1) whose solution is an exponential decay. Determining values for the time constant, τ0, and amplitude of exponential decay, A0, from the experimental data is a common task in many areas of experimental research: fluorescence decay analysis in biophysics, radioactive decay in nuclear physics and chemistry, nuclear magnetic resonance, reaction kinetics in chemistry and electrochemistry and medical imaging, as well as the relaxation of active molecules in materials, charge energy in capacitors and settling structures, among others (2–4). If we have a periodic function f(t) defined from t1 to t2 and with a time step defined by Δt, we can pass it through a Fourier processor and obtain information regarding its spectral constitution from 1 1 t2 t1 to Δt. Because of the intrinsic orthogonal property of the complex exponent, the spectral composition given by Fourier is unique. As a different problem, if we have a function that decays, we have two alternatives to describe it, on the one hand, one has to minimize the error between the proposed function and the experimental data, and the amplitudes can be both positive and negative. On the other hand, one may decompose the signal in a unique set of functions, all with a positive amplitude.
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The main difference from the standard technique for spectral decomposition (Fourier transform) is that the real space for a series of exponential functions is nonorthogonal. Thus some degree of interpretation or acceptance is needed to decide on a suitable function. The main purpose of this study is to understand the nature of decay phenomena, limited by single exponential decay and the absence of decay. The region delimited by these borders can be described by a combination of exponentials. Figure 1 illustrates the shaded region where exponential decomposition is possible. If we use the language of photophysics, where a usual scenario that helps to focus the various techniques is fluorescence decay; fluorescence is the spontaneous emission of radiation from an excited molecule, forming a low-energy state of the molecule in the same spin multiplicity as the emitting excited state. For a steady-state absorption–emission system, the important photophysical parameters are the quantum yield of fluorescence (the ratio of emitted to absorbed photons) and the fluorescence decay time (lifetime). Assuming a very simple Poisson process and ignoring the emission spectra, as well as the recombination path, one can assign the probability distribution for photon emission to eqn (1), where τ(t) is the lifetime of the system, in this case, this function of time is a constant, it produces a single exponential decay time.
* Correspondence to: R Quintero-Torres, Centro de Física Aplicada y Tecnología Avanzada, UNAM campus Juriquilla, Boulevard Juriquilla 3001, Juriquilla Querétaro 76230, México. E-mail:
[email protected] a
Centro de Física Aplicada y Tecnología Avanzada, UNAM campus Juriquilla, Boulevard Juriquilla 3001, Juriquilla Querétaro 76230, México
b
Centro Nacional de Metrología, División de óptica y radiometría, El Marques Querétaro C.P. 762461, México
c
Department of Physics and Astronomy, University of British Columbia, Vancouver BC, V6T 1Z1, Canada
d
Department of Genetics and Molecular Biology, Centro de Investigación y Estudios Avanzados del IPN, México City C. P. 07360, México
Copyright © 2013 John Wiley & Sons, Ltd.
R. Quintero-Torres et al. contained within the data, as independent as possible of the time span available. The discussion shown in Fig. 2 can be used to justify the close relationship between the dynamics of the decay function and a microscopic description of the system, and the idea of an interaction between components in the system can be used to form a model to discuss the meaning of lifetime and exponential decomposition (5). If the interaction between the emitter and the environment can be considered as a small perturbation, then each can be seen as a quasi-independent system and the total description is shown in eqn (2), with positive values of A and τ: n t IðtÞ ¼ ∑ Ai exp (2) τi i¼1
Figure 1. Schematic representation for the decaying phenomena, the semi log scale shows the single exponential decay (SED; continuous black line). Functions faster than a SED cannot be decomposed as a combination of exponential functions (red segmented line). By contrast slow decay functions can be decomposed as a combination of more than one exponential function (blue dotted line).
t Iðt Þ ¼ A0 exp τ0
(1)
This can be used equally in many alternative scenarios for other situations in several areas of engineering and science. This is relevant because the physics behind the behavior may be elucidated by determining the lifetime of the process, which is reasonable only if one exponential defines the behavior. More than one exponential may confuse the explanation rather than explain it; a statistical interpretation may help to guide the interpretation of the physics behind the phenomena. The ultimate goal may be to transform a set of experimental results, remove the influence of the instrument and excitation profile, and determine a function that expresses the dynamics from a statistical point of view or the set of exponentials
However, if the interaction between the environment and the emitter is seen to be strong and to determine the behavior of the emitter, then the lifetime is better described by a probability density function (PDF): ∞ t (3) IðtÞ ¼ ∫ Pðτ Þ exp dτ τ 0 A more appropriate interpretation of eqns (2) and (3) might invoke the segmentation of nature, eqn (2) calls for few independent phenomena, whereas eqn (3) calls for an infinite ensemble of emitters participating in the process. This does not directly invoke the interaction between them, but the details of the interaction should be described by the PDF. The energy relaxation should give information regarding the behavior of the system, as well as the interaction between system constituents; isolated identical emitters will have the least opportunity to interact within a homogeneous environment. Different interacting emitters within an anisotropic and heterogeneous environment will increase the opportunities for interaction and be included in the decay curve (6). This observation will have particular manifestations in different systems in which the decay mechanism is studied; environment,
Figure 2. System, steady-state spectral emission (y vs. E) and excitation decay (y vs. t). (Left box) Independent oscillators that decay independently as a SED and (right box) oscillators with an exchange of energy. (Upper) Three similar oscillators; (right box) the low energy increases in strength at the expense of the high energy and the excitation may decay as a double exponential. (Middle) Very large cluster of oscillators generating a PDF, (right box) the low energy increases strength at the expense of the high energy, the emission looks like a shift at low energy and the excitation may decay as a multiple exponential. (Lower) Very large cluster of oscillators with two clear distributions, (right box) the low energy increases strength at the expense of the high energy, the emission looks as though only one distribution is present (red line) and the excitation may decay as a multiple exponential.
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Complex temporal decay identical emitters, heterogeneities and interactions will have a particular interpretation, Fig. 2 attempts to show the different situations that might arise within the decay process.
particularly unsuccessful when exponential decomposition is the goal due to the need for a predetermined function. However, it is possible to solve an over-determined system using linear least squares:
Signal extraction
XA ¼ mðtÞ
Conditioning of the experimental data usually requires removing the influence of the excitation force E(t), when it is different from a Dirac delta and the gating G(t) that produces time and frequency distortion, as well when it is more important than the time between measurements. In a real experiment, information needs to be unraveled, as illlustrated in Fig. 3. The goal is to determine m(t) from S(t) and S0(t). With the usual precautions of the algorithm used for the Fast Fourier Transform (FFT) and clear definition of the measured time, it is possible to obtain m(t). The scatter signal S0(t) from the pumping source E(t) seen by the ∞
instruments G(t), can be expressed as S0 ðt Þ ¼ ∫ E ðτ ÞGðt τ Þdτ ∞
with the Fourier transform F{S0} = F{E}F{G}. The signal after the material M(t), is obtained considering only the pumping source composition E(t) over the material behavior ∞
m(t) and can be expressed as MðtÞ ¼ ∫ E ðτ Þmðt τ Þdτ with the ∞
Fourier transform F{M} = F{E}F{m}. The signal from the material S(t), considering the signal after ∞
the material M(t) seen by the instruments G(t) is Sðt Þ ¼ ∫ Mðτ Þ ∞
Gðt τ Þdτ with the Fourier transform F{S} = F{M}F{G}. Combining these three expressions, we obtain a practical expression to unravel the information from the instrumentation and the pump source. In cases where the gating is very narrow and the excitation is a delta function, m(t) and S(t) are identical. F fSg mðtÞ ¼ F 1 (4) F fS0 g
Exponential decomposition by non-negative least squares Nonlinear least squares can be used to adjust a predetermined function to an ensemble of experimental data, following the objective of minimizing the difference between them; this is
(5)
where A is a vector for all possible amplitudes of the exponential decomposition with an additional constraint; all values of A are positive or 0. min kXA mðtÞk2 with A ≥ 0 A
(6)
For exponential decomposition, the matrix X is constructed using an arbitrary combination of time constants, τ, expected to be present (vector length is identical to the amplitudes to be found) and the time vector used for the collection of information in m(t). 1 X≡ exp ½column of t* row of τ Using all the values of time that we have used to experimentally collect decay values m(t) and the expected values for the time constants involved τ, the system to be solved is represented by eqn (7): 2 t =τ 3 2 3 2 3 A1 m1 e 1 1 ⋯ et1 =τk 6 7 6 7 6 7 (7) 4 ⋮ ⋱ ⋮ 5*4 ⋮ 5 ¼ 4 ⋮ 5 tn =τ 1 tn =τ k Ak mn e ⋯ e Because m(t) has the correct value plus some noise, the amplitudes may require positive and negative values, however, only the positive values are expected in the decomposition and the non-negative least squares will provide the exponential decomposition with an error of ||m-XA||. This may produce unclear results; that is, a signal may generate different results when analyzed for different temporal intervals. This is expected if we consider functions like f(t) = 0.89e t/10 + 0.1e t/30 + 0.01e t/100, the first exponential dominates at early times, the second exponential at middle times and the third at longer times. Therefore, the response from least squares might be different at different time intervals, and
Figure 3. Deconvolution of experimental information, from the excitation signal and gating from the instruments. S0(t) gives information for the excitation and gating, but not the sample, and S(t) gives information for the excitation, gating and sample.
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R. Quintero-Torres et al. unexpected results may occur at some intervals if experimental noise is included. Other procedures have been presented as an alternative to exponential decomposition, Hildebrand (7) suggested an algorithm in many situations, however, it fails to reach the solution because within the core of the algorithm is a polynomial function that has to be solved and its real roots disappear in the presence of slight noise, leading to some missing exponentials and limiting its usefulness. Figure 4 illustrates the exponential decomposition for: f ðtÞ ¼ 0:89et=10 þ 0:1et=30 þ 0:01et=100 þ ε
(8)
with e representing a random noise of 5%, a function span from 0 to 70 and 0 to 200 units of time and a range of time constants τ, from 0.1 to 1000 units of time in a log distribution. This decomposition is quite robust because the non-negative least squares method is able to identify the essential exponential needed to reconstruct the function with minimal error; results can be obtained even with substantial noise. Figure 4 gives the exponential decomposition for the function 0:2 presented in Fig. 1; f 1 ðtÞ ¼ tþ:2 for 10 units of time and a τ universe from 0.1 to 1000 units of time in a log distribution.
Probability density function From eqn (4), the possibility of using a PDF to describe the decay process is outlined, the core justifies a particular PDF associated with the phenomena at hand. Several candidates have been used to accommodate the data, but the PDF that best connects the system and that is defined at zero is the gamma distribution. To best describe the gamma PDF, it might be convenient to remember the exponential PDF (eqn 9 with a = 1) that describes the probability of failure in an electronic component, this is different from a mechanical component because the former is not affected by wear, as the clock is reset every time the system is off (lack of memory property). The gamma distribution describes the probability of waiting for a independent exponential distributed random variables, each of which has a mean of b. y ¼ f ðxja; bÞ ¼
1 x x a1 eb with x≥0 and a; b > 0 ba ΓðaÞ
(9)
The gamma PDF (eqn 9) substituted in eqn (4) produces: 1 ∞ a1 1 þ t dx (10) I ðt Þ ¼ a exp x ∫x b ΓðaÞ 0 b
∞
Solving eqn (10) requires use of the gamma function,
Þ ∫ x n expðλx Þdx ¼ Γ ðλnþ1 and produces I(t) = I0(1 + bt) a, using nþ1
0
1 we finally get: the definition τ≡ ab t a Iðt Þ ¼ I0 1 þ aτ
(11)
This is again reduced to a single exponential if a reaches infinity, and produces a reasonably good fit in many decay experiments where a is an interpretation of a measure for the local environment and τ is the lifetime average. One aspect that is appealing from this treatment is the property of the PDF eqn (9), as can be seen in Fig. 5 where there is an increase and decrease in the probability with respect to the exponential PDF (a = 1), when a is different from 1. The three-parameter gamma PDF includes another parameter μ, which gives some extra degree of flexibility; rather than using x values from zero to infinity, it collapses all x values from μ to infinity, if μ is zero, eqn (12) returns to eqn (9). y ¼ f ðxja; b; μÞ 1 x μ a1 xμ ¼ e b ; with x ≥ 0; μ ≥ x and a; b > 0 (12) bΓðaÞ b A similar substitution of eqn (12) in eqn (3) produces eqn (13). t a (13) IðtÞ ¼ I0 expðμtÞ 1 þ aτ
Figure 4. Time decay and exponential decomposition of eqn (9). (Left) Details of eqn (9) and the reconstructed function with 5% noise and a time span of 70 and 200 time units. (Right) Results of the exponential decomposition for eqn (9) and 0:2 spanning from 0 to 10 time units; the reconstruction for the function gðtÞ ¼ tþ:2 produces a function indistinguishable from the original.
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Figure 6 shows a shadowed area as the region that can be obtained using eqn (13), this function is not as versatile as the total exponential decomposition, however, if the physics is related to the gamma PDF, it makes more sense to use this model. Interpretation of the gamma parameters is possible when analyzing the photophysics of luminescence for a large ensemble of interacting semiconductor nanoparticles. Continuous
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Complex temporal decay distribution of these particles will behave without interaction if in solution, with a large interaction if in a drop cast or with a selective interaction when in a thin film (8) with a particular dielectric environment in each case. The representative lifetime is 1 , and a value of a = 1 indicates the absence obtained from τ≡ ab of an interaction producing a SED. From Fig. 5, it can be seen that the PDF resembles the decay curves for an ensemble of nanoparticles after an optical narrow filter (9). This may suggest a weighed effect over a cluster of nanoparticles of a particular size.
Conclusions Decay phenomena can be studied using exponential decomposition or PDFs, both routes include a simple single exponential as a limit case. Both methods make use of the least squares algorithm, in one case, non-negative least squares (with restrictions) and in the other case, nonlinear least squares. The extent of the exponential decomposition is shown in Fig. 1 (shadowed) and any function can be decomposed, with the inherent difficulty of finding an interpretation for each exponential. Gamma PDF is more user-friendly in terms of the number of parameters to be identified and the difficult task is determining the number of exponential events (parameter a). These two tools roughly define the possibilities underlying the microscopic causation of the phenomena: multiple discrete entities, continuous distribution of entities and single or noninteracting relaxation as the limit case in both. Exponential decomposition should be a first step in exploring the phenomena because it can give an outline of the richness of the phenomena, however, the gamma function or power fitting may be an indication of the dynamics in the system. Acknowledgement Figure 5. Gamma PDF in a semilog presentation. This highlights the difference with an exponential PDF (continuous black line). a < 1 makes emission at x near 0 more likely and a > 1 makes emission at x > 0 more likely.
This work was partially supported by the Mexican CONACyT (Grant No. 81331).
References
Figure 6. Space of available amplitudes as a function of the fitting parameters, μ and τ. They define the borders of accessible amplitudes, and a defines the values in between these lines. If μ parameter is not used, the horizontal axis is the limit.
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