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When the Complex Makes It Easy: Phasor Plotting as a Model Independent Representation of Fluorescence Decay in Flow Cytometry

2020; Wiley; Volume: 97; Issue: 12 Linguagem: Inglês

10.1002/cyto.a.24223

1552-4930

László Bene, László Damjanovich,

Neural dynamics and brain function

Fluorescence lifetime is an important contrast modality of selectively detecting different types of dyes and their interactions, where possible contaminating effects of hardly controllable factors such as exciting light flux and local exciting dye concentration are absent (1, 2). Time- and frequency-domain detections of fluorescence lifetime have been elaborated both in microscopy and flow cytometry (2-5). Phasor plot is a graphical replacement of the fluorescence decay curve of a fluorescent system. Although excellent accounts of phasor plots are existing in the literature (6, 7), we recast it now from the viewpoint of the general systems and communication theory. The approach of systems theory is that the examined object is treated as a black box, and the structure of the black box is inferred from the distortion of shapes of signals introduced into the black box (8) (Fig. 1). The shape of the input signal is distorted upon interactions with the elements of black box. The types and strengths of interactions can be found via a careful analysis of the shape distortions of the output signal in a noninvasive manner, not necessitating structural changes of the black box. The black box can also be conceived as an information channel influencing the transmitted information meant by the shape of input signal and the received information represented by the shape of the output signal. For the sake of concreteness, let us first mention an LR-circuit (9) (first row of Fig. 1). Upon excitation of the circuit with a square pulse, the elicited current will show a degree of smoothing of both the front and rear of the input signal, dictated by the relaxation time constant, τ. The relaxation time expresses the degree of inertia of the system against the perturbation and here is dictated by the inductivity to resistance ratio (L/R) (9). As a second example, let us take a microscope objective (second row of Fig. 1). The image of a rectangular brightness distribution of an object placed in the focal plane will no more be rectangular in the image plane, but more-or-less blurred at the edges. The degree of blurring is characterized by the critical distance (d)—or reciprocal resolving power—of microscope resolution, which is dictated by the ratio of the detected wavelength and the objective's numerical aperture (10, 11). As the third example, let us take a fluorescent dye solution in a cuvette, excited by a square light pulse at a suitable wavelength for eliciting fluorescence (third row of Fig. 1). Then, the shape of the evoked fluorescence signal will also show distortions characteristic of the dye solution. At the front of the signal, there is no distortion because of the instantaneous absorption of photons (10−15 s). However, at the rear, some lengthening of emission is seen governed by the storage time of excitation, the fluorescence lifetime, τ, which in turn is defined as the reciprocal of the sum of radiative and nonradiative rate constants, kf and knr, respectively. A key to understanding of phasor-plots is the observation that the output signal of a black box can be forecasted for any input signal shape if the systems response to an infinitely narrow excitation pulse (Dirac- or δ-pulse) is known (Fig. 2A). This function is called impulse response function or weight function h(t), and considered as a fingerprint of the system (8-11). Its significance lies in that, in the knowledge of it, the response of the system mathematically can be found out for any input signal shape via the operation called convolution. During convolution, each particular g(t0) value of a g(t) function is replaced by a weighted average of the whole g(t) function with a weighting function h(t) centered on the particular t0 time point. This operation is analogous to smoothing clothes with an iron: where g(t) is the cloth with the creases to be removed, and the weight function h(t) is the iron. An important mathematical theorem is that of convolutions. It states that any convolution can be transformed into the product of two complex numbers, the so-called Fourier-transforms of the convolved quantities, with the consequence that any time function of the black box can be rendered visible by appropriate geometric features in the two-dimensional plane (8-11). A commonplace example for Fourier-transform is splitting up into different color beamlet components of a white light beam by a prism, or a drop of water. We saw above that the impulse response or weight function represents the system's structure as a finger print. Its Fourier-transform is called the transfer function. Because this is a complex number, it can be written as a product of a modulation m and complex exponential term containing a phase ϕ (Fig. 2B). The traditional way of representing lifetime measurement data is separately representing the modulation and phase in a common frame as monotonously decreasing and increasing functions of the circular frequency (12). However, by expanding the complex exponential of the phase via Euler's rule into the sum of a cosine (real) and a sine (imaginary) term, and by multiplying them with the modulation term, the transfer function can be resolved into the sum of a real and an imaginary term, called B and A, respectively (Fig. 2C). This second alternative representation of the transfer function is called a phasor-plot (or AB-, SG-, ND-plot) (6, 13). In the case of image formation in a microscope, the δ-pulse corresponds to a point in the object plane. The impulse response of the microscope is the point-spread function (PSF) whose width is proportional to the critical distance given by the Abbe's formula. The H(ω) transfer function for the microscope is the Fourier-transform of the PSF, called now optical transfer function (OTF). Imaging quality of microscopes can be improved by purposeful alteration of the OTF, called spatial filtering (9-11). We introduced phasor plots above as a graphical representation of the transfer function of the system, which in turn has been obtained by Fourier-transformation of the system's impulse response function. However, it can be obtained without any assumption on the functional form of the impulse response by a harmonic excitation of the system, that is, carrying out the Fourier-transform at the hardware level, in a phase-modulation measurement (Fig. 3A). Here, both the demodulation and phase shift can be determined from either the fluorescence response curve or the associated demodulation-phase curve pair (Fig. 3B) and AB-plot (Fig. 3C) at the circular frequency of excitation (ω0), from which the apparent phase and demodulation lifetimes can also be determined. In contrast, the single δ-pulse excitation above only fixed the shapes of these curves and did not fix the actual values of phase, demodulation, and associated lifetimes. However, this can be made with a trick. If the single δ-pulse is repeated many times, resulting in a “δ-comb,” then the pulse series acts similarly to a harmonic excitation, with a circular frequency dictated by the pulse repetition rate. At this circular frequency, the phase and modulation lifetimes can be read off analogously to the harmonic case from the phase-demodulation curve pair or from the AB-plot (14). We saw above that the A and B quantities are quadrature components of the complex transfer function, which is the Fourier-transform of the impulse response or weight function of the dye solution. If A is plotted against B, then the complex plain can be divided into three regions (Fig. 4A). For a single component solution having a single fluorescence lifetime there is a strict relationship between the demodulation and the phase: m = cos(ϕ). Consequently, the corresponding (m, ϕ) pairs constitute a circle, the “universal circle” of radius 0.5, and origin of (0.5, 0), on which fluorescence lifetime increases by proceeding with ϕ anti-clockwise from horizontal direction (ϕ = 0). The m = cos(ϕ) relationship for the “universal circle” conveniently understandable, for example, because the horizontal diameter of the circle is seen at 90° from all points of the circle, according to Thales' theorem. Points inside the “universal circle” obey the m < cos(ϕ) relationship, referring to ground-state heterogeneity or the presence of Förster resonance energy transfer (FRET) for the donor (measured at the donor side). In these cases, the impulse response function of the system is composed only of positive amplitude components. However for excited state reactions such as solvent relaxation and FRET toward an acceptor (measured at the acceptor side), phasor points can also get outside of the circle, obeying the m > cos(ϕ) relationship (6, 7, 15). In the case of excited state reactions, the impulse response function contains also negative amplitude (pre-exponential factor) component(s), representing indirect excitation by the molecular environment. Physical rotation can also be conceived as a special energy transfer process between the different orientations of the oscillating molecular dipole. The parallel polarized component of fluorescence decays via both the natural decay of the excited state, and via the molecule rotating out from the orientation of parallel emission. The corresponding phasor point falls inside the circle mimicking ground state heterogeneity. In contrast, the perpendicularly polarized component of fluorescence decays via the natural decay of the excited state, and via the molecule rotating into the orientation of perpendicular emission, the latter described by a negative amplitude, rendering the corresponding phasor point to fall outside of the circle, like for FRET toward the acceptor (16). Ground state heterogeneity and FRET from the donor can be distinguished via the geometrical shapes defined by the phasor points. For ground state heterogeneity, they constitute straight line intervals with the cutting points of their elongations with the circle giving the lifetimes of the mixed components (12, 13) (Fig. 4B). For FRET from the donor, phasor points constitute banana-like curved geometrical shapes (18) (Fig. 4C). For an explanation of the latter, we should think of that, in the absence of FRET the phasor point is the mixture of pure donor and pure background phasors locating on the circle to left and to the right, dividing the connecting bisector according to the lever rule: the distance of the mixture point from one end point of the bisector is proportional to the fluorescence intensity for the opposite point. Because of this, the starting point is located to the left inside the circle, but close to the perimeter. The endpoint of the FRET trajectory is the mixture of the close to 100%-quenched donor and background. The phasor point representing this mixture locates to the right, inside the circle, close to the perimeter. Besides, the starting and the end points are located such a way that their connecting intersection cuts the circle at the pure donor and background points. At intermediate FRET efficiencies the phasor is a mixture of the quenched donor and background points. In their work, Nichani et al. (in this issue, page 1265) developed further the concept of FRET trajectory by sensitizing FRET for indicating the time evolution of a biological process called apoptosis. The development of apoptosis is indicated by the level of caspase-3 enzyme, which in turn is indicated by a FRET probe via the degree of FRET relief achieved by scissoring by caspase-3 the linker domain fusing together the donor and acceptor, the three forming an apoptosis-indicating FRET bioprobe. In the bioprobe, engineered green fluorescent protein serves as FRET donor and Alexa Fluor-546 as the acceptor. For truly indicating caspase-3 levels, prerequisites are the high enough dynamic range ensured by the large starting FRET efficiency, the low enough bioprobe level for avoiding cross-FRET contamination between the nearby intact probes as well as between the intact probes and the liberated acceptors and donors. There are also technical requirements such as the spectral purity of the donor channel. As they demonstrate, phasor plots can be advantageously exploited also in flow conditions at a single excitation frequency for real time tracking a signaling event. However, the true interpretation of the phasor plots requires also the precise knowledge of all components of the donor fluorescence. The multiplexing capability of flow cytometry is well illustrated by the authors via correlating donor fluorescence lifetime with FRET-channel fluorescence, as an indicator for the presence of acceptor in the bioprobes. In steady state flow cytometry and fluorescence microscopy, FRET efficiency is determined generally in a simple donor quenching experiment or in the ratiometric FRET scheme involving two or three signal channels. In the latter case, however, absolute FRET efficiency can only be determined in the knowledge of a calibrating factor called α (or G), responsible for balancing quantum efficiencies and collection efficiencies of fluorescence for the donor and acceptor (19). Direct measurement of FRET efficiency via fluorescence lifetime obviates the need for α. Alternatively, fluorescence lifetime might also aid in determining α, and consequently, instrumental spectroscopic parameters such as fluorescence transmission efficiencies, or absorption coefficient of the donor and acceptor prevailing in the cellular milieu. The α-factor is also an important parameter for determining the donor-acceptor molar ratio of the actual FRET sample in the ratiometric FRET scheme. Besides the role in FRET measurements, fluorescence lifetime has also an important role in converting fluorescence anisotropy values to rotational mobilities. In fluorescence spectroscopy, phasor plots have first been introduced by D. Jameson and his colleagues, but before that, they were also applied in relaxation kinetics, chemical and dielectric, under the name of “Cole-and-Cole plots” (6, 7). Recently these plots have been revitalized by the fact that their usage greatly widens the applicability of single-frequency fluorescence lifetime imaging microscopy (FLIM) and the realization that phasor plots can be applied not only in the frequency domain FLIM, but also in time domain FLIM (14, 18). Theoretically, patch-clamping in electrophysiology might also be a field for a fruitful exploitation of the phasors. Based on the close correspondence between open ionic channels and excited dyes, the decay kinetics of open channels is described by the same formalism as those for fluorescing dyes, by taking over the role of fluorescence lifetime by the mean open time of channels. In addition to the model-independent, that is, free of assumed functional form, characterization of channel kinetics, for example, collective behavior of channels (cooperativity), synchronization of channel openings—the counterparts of excited state dye interactions—as the function of their cluster size might be revealed by the graphical counterparts of the decay curves. Besides, the direct application of the phase-modulation detection of currents, as an analogue of phase-modulation FLIM, might also convey some extra information. Homo-FRET is a FRET process taking place between dyes of identical type having small Stokes' shift. In the limiting case of high excitation intensities close to saturation, homo-FRET might be exchanged for interactions between excited dyes in close proximity, called super/sub-radiance or super/sub-fluorescence. In this case, radiative rate might be either increased or decreased due to emission dipole synchronization by the net effect of the dye local fields, the exciting light, the emitted fluorescence, and the morphology of the dye cluster (20). Even if the mean excited state lifetime of the dyes stays constant, the lifetime heterogeneity might increase manifesting in a broadening of lifetime distributions. Phasor plots might represent these events by an analogue broadening of distributions and falling points outside of the universal circle. Taking together, flow cytometry made a leap-forward with the introduction of spectacular representation of FRET-processes in the complex plane of phasor plots. Financial support to László Bene for this work was provided by TÁMOP-4.2.2.A-11/1/KONV-2012-0045 project co-financed by the European Union and the European Social Fund, OTKA Bridging Fund support OSTRAT/810/213, and science financing support 1G3DBLR0TUDF-247 by the University of Debrecen. We are thanking for Prof. J. Szöllősi for the careful reading our manuscript. László Bene: Conceptualization. Laszlo Damjanovich: Conceptualization.