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12.1 Introduction

Biophotonics is a branch of biophysics that analyzes changes in light to study biological processes [19]. Typically, the intensity or amplitude of light changes when interacting with a biological specimen as a result of absorption or scattering by molecules and molecular assemblies within the sample. For molecules that are intrinsically fluorescent, or fluorescent as a result of being labelled with a fluorophore, excitation light at one wavelength can be converted to another wavelength (longer wavelength for one photon excitation, and shorter wavelength for multi-photon excitation) [14]. This shift in the wavelength of light allows for very precise localization and quantification of fluorescent molecules, often with single molecule sensitivity. Biosensors have also been developed that show changes in their fluorescence intensity when bound to a specific ligand of biological interest. Typically, the intensity of either the bound or unbound state of the fluorophore is reduced or quenched. Ligand binding induces the molecule to switch to its alternate state thus transducing the binding of ligand into a change in fluorescence intensity (and often, fluorescence lifetime). The experimentalist can monitor this change to visualize the extent of ligand binding or the free concentration of ligand. In addition to the intensity (amplitude) and colour (wavelength) of light, another under-appreciated characteristic of light that can be monitored in biophotonics is the polarization or ‘orientation’ of light. Unlike changes in intensity and colour that are readily detected by most human eyes, the human eye is an extremely poor detector of the polarization of light [31]. Accordingly, even the notion that light has an orientation is often confusing, and the concept that the polarization of light can change with time is at best understood cerebrally rather than viscerally. It is our goal in this chapter to provide a more intuitive explanation of how and why the orientation of light changes when linear polarized light interacts with fluorescent biological samples, and in so doing provide a foundation for understanding how time-resolved fluorescence anisotropy measurements can be applied and interpreted to better understand biological processes.

12.2 The Molecular Basis for Orientation of Fluorescence Emission

A fluorophore is a molecule that can absorb a photon and after a short delay, typically within a few nanoseconds (10−9 s), emit a photon with a slightly longer wavelength (less energy). The absorption of a photon occurs within a specific range of wavelengths resulting in the excitation of one of the fluorophores electrons into an excited state. The wavelengths that can be absorbed by a specific fluorophore, its excitation spectrum, are dictated by the distribution of many discrete energy gaps defining the difference between ground and excited states present in the molecule. The energy differences between these states can be explained using quantum mechanics by accounting primarily for permitted electronic, vibrational and rotational states of the molecule. The absorption of a photon by a fluorophore is fast, occurring typically within a few femtoseconds (10−15 s). Over a period of a few hundred femtoseconds, the large selection of potential rotational and vibrational excited-state sublevels will decay into the lowest-energy singlet excited state. This consolidation results from rotational- and vibrational-energy loss due to kinetic interactions of the excited fluorophore with surrounding molecules, and is in part responsible for the Stokes shift between the wavelengths of excitation and emission. An excited fluorophore may remain in the singlet state for a period lasting from picoseconds (10−12 s) to tens of nanoseconds (10−9 s). The amount of time any individual fluorophore and excitation event remains in the singlet state is stochastic. Nonetheless, fluorophores do have characteristic fluorescence lifetimes that describe the average time they will remain in the singlet-state. With time, a singlet state fluorophore will eventually decay to its ground state by either emitting a photon (fluorescence emission), by transferring energy to a nearby ground state fluorophore (Förster Resonance Energy Transfer, FRET), or the energy can be lost by the formation of a long-lived triplet state, collisional quenching, or some other non-radiative excited state reaction.

In polarization and anisotropy experiments, the orientation of photons emitted from a fluorophore is characterized relative to the orientation of the electric field of the excitation light. To understand the basis of fluorescence polarization we must appreciate that individual fluorophores have an ‘orientation’ in space that can change with time. Most importantly, their ‘orientation’, relative to the orientation of the excitation light electric field, can dramatically influence the probability that they absorb a photon to transition into an excited state. When a photon is absorbed, a ground-state electron in the fluorophore is boosted to a higher orbital. At that instant, the balance between lower-mass, fast-moving, negative charged electrons and higher-mass, slow-moving, positive charged protons in the molecule is perturbed establishing an oscillating dipole (a vector separation of positive and negative charges). This oscillating dipole is called a Transition Dipole or more specifically an Absorption Dipole, and has both magnitude and orientation. The highest probability for excitation is achieved when a fluorophore is illuminated with light at a wavelength appropriate for excitation, and if the electric field of the excitation light is parallel to the orientation of the fluorophores absorption dipole (θ = 0°). In contrast, excitation does not occur if the electric field is perpendicular to the absorption dipole orientation (θ = 90°). At intermediate orientations (0° < θ < 90°) the probability for excitation can be calculated from the angle between the electric field orientation and that of the absorption dipole:

$$ p_{excitation} \propto \cos^{x} \theta $$
(12.1)

where, for linearly polarized light, x is 2 for one-photon excitation and 4 for two-photon excitation. This preference for fluorophore excitation at low values of θ is called photoselection (see Fig. 12.1).

Fig. 12.1
figure 1

Photoselection. Electric field orientation of linearly polarized light (black arrow) is depicted by a double blue arrow (a). Linear polarized light will selectively excite fluorophores whose absorption dipole orientation (depicted as brown bars) is similar (light brown) to that of the excitation lights electric field. Only these orientation ‘selected’ excited fluorophores will emit fluorescent light (a). The orientation of this emitted light (green double arrows), is correlated with the electric field orientation of the excitation light (solid double arrow), but is degenerate (dashed green double arrows) primarily because of the dependence of photoselection where θ is the angle between the electric field orientation of the excitation light and the absorption dipole orientation. Panel b depicts radial probability plots of the dependence of photoselection on θ (\( p \propto \cos^{x} \theta \sin \theta \)) where x = 2 for 1-photon excitation (left) or x = 4 for 2-photon excitation (right). In these plots the integrated area of each 3-dimentional plot (depicted in pink) is set to 1, and the length of a double black arrow (for a particular value of θ is proportional to its probability for excitation. Note that excitation is not possible when θ = 90°, and that 2-photon excitation will preferentially select fluorophores with θ values closer to 0°

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Once excited, the ‘orientation’ of a fluorophore can also influence the ‘orientation’ of a photon emitted as fluorescence. In biological experiments, there are three primary mechanisms by which the orientation of emitted fluorescence will be influenced. These are, as a result of: (1) conformational changes in the fluorophore while in the excited state, (2) molecular rotation of the fluorophore between the times it is excited and emits a photon, and (3) Förster Resonance Energy Transfer (FRET) to a nearby fluorophore with a different transition dipole orientation. These will be discussed shortly, but first we must explain how polarization and anisotropy are measured, how these measurements differ, and why anisotropy measurements are usually preferred.

12.3 How Do We Measure the Anisotropy of Polarized Fluorescence Emissions?

How can a microscope be used to follow changes in the orientation of molecules in a biological context? Similarly, how can a microscope be used to monitor changes in the proximity of molecules within living cells? Conceptually, if not practically, changes in the orientation, and in some instances changes in the proximity of fluorescent molecules can be determined by measuring the intensity of emitted photons relative to the orientation of the electric vector of a linear polarized light source along three orthogonal axes.

Figure 12.2 depicts a transition dipole (blue double arrow) of a fluorophore (from an isotropic solution) that is excited by linear polarized light (L.P.). The electric field orientation of this light source is shown as a black double arrow. The dipole orientation can be characterized by 2 angles, θ and ϕ. θ is the angle formed between the dipole and the X-axis in the XY-plane. ϕ is the angle formed between the projection of the dipole on the YZ-plane and the Z-axis.

Fig. 12.2
figure 2

Detecting polarization. The transition dipole (blue double arrow) of a fluorophore is excited by a linear polarized light source (L.P.). The electric field vector of the light source is shown as a black double arrow. The three dimensional orientation of the dipole can be characterized by 2 angles, θ and ϕ, where θ is the angle formed between the dipole and the X-axis (pink double cone), and ϕ is the angle formed between the projection of the dipole on the YZ plane (green disk) and the Z-axis. The light intensity emitted from this fluorophore will be proportional to the square of the dipole strength, and the dipole vector can be thought of as being composed of 3 directional components: x, y, and z. A signal proportional to the total intensity of light emitted by our fluorophore can be measured summing the signals detected by light detectors (L.D.) positioned on each of the three axes. The intensity information encoded in the x vector component by convention is called I|| while the intensity information encoded in the y and z vector components is called I⊥. As a result of photoselection and the distribution symmetry of excited molecules formed around the X-axis for an isotropic solution, the y vector component is equal to the z vector component. For an isotropic solution of fluorophores excited with linearly polarized light whose electric vector is parallel to the X-axis, L.D.x will measure a signal whose intensity is proportional to 2·I⊥ (crossed double-headed green arrows), while L.D.y and L.D.z will each measure light signals whose intensity is proportional to I|| + I⊥ (crossed double-headed red and green arrows). The total emission intensity will be proportional to 2·I|| + 4·I⊥ or more simply I|| + 2·I⊥. Note that the xy plane depicted here corresponds to the sample plane on a microscope, and L.D.z corresponds to a detector placed after the microscope condenser (or at an equivalent position on the epi-fluorescence path)

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The X-axis is an axis of symmetry as it is parallel to the electric vector of our light source. The Y and Z-axes are perpendicular to the light source electric vector and thus are not axes of symmetry. The fluorescent light intensity emitted from this fluorophore will be proportional to the square of its dipole length, and the dipole vector can be thought of as being composed of 3 directional components: x, y, and z. By placing light detectors (L.D.) on each of these three axes a signal proportional to the total light intensity emitted can be measured. The emitted light will be proportional to the sum of the three signals (Itotal = Ix + Iy + Iz). L.D.x will detect light related to the yz vector components of the dipole. Similarly, L.D.y will detect light related to the xz vector components, and L.D.z will detect light related to the xy vector components. The intensity information encoded in the x vector component is called I|| while the intensity information encoded in the y and z vector components are called I⊥. As a result of photoselection (the orientationally biased excitation of fluorophores by linearly polarized light) and the distribution symmetry of excited molecules formed around the X-axis for an isotropic (randomly orientated) solution, the y vector component will be equal to the z vector component. Accordingly, when an isotropic solution of fluorophores is excited with linearly polarized light whose electric vector is parallel to the X-axis, the L.D.x detector will measure an un-polarized light signal whose intensity is proportional to 2·I⊥. In contrast, L.D.y and L.D.z will each measure a polarized signal whose intensity is proportional to I|| + I⊥. If we sum the I|| and I⊥ components along all three emission axes, we find that the total emission intensity will be proportional to 2·I|| + 4·I⊥ or more simply I|| + 2·I⊥ [28].

As described above, a light detector placed along the Z-axis of a microscope will collect light proportional to I|| + I⊥. Furthermore, on most light microscopes it is relatively straightforward to position a light detector on this axis to detect changes in polarization. There are two general arrangements commonly used for measuring fluorescence polarization on a microscope, both require separating I|| and I⊥ components of the fluorescence emission along the Z-axis, see Fig. 12.3.

Fig. 12.3
figure 3

Separating I|| and I⊥. When linearly polarized light (L.P.) with electronic vector E is used to excite a sample (s) on a microscope the fluorescence emission along the z-axis (yellow arrow) will be comprised of I|| and I⊥. The magnitude of these two intensity components can be measured either sequentially (panel a) or in parallel (panel b). The sequential configuration uses a single light detector (L.D.) and a linear polarizing filter (Pol.) that is first positioned parallel (panel a top; 0°) to the electronic vector of the light source to measure I|| and then positioned perpendicular (panel a bottom; 90°) to measure I⊥. In the parallel detection configuration (panel b) a polarizing beam splitter (P.B.S.) is used in conjunction with two linear polarizing filters and two light detectors to measure I|| and I⊥ simultaneously

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The first design measures I|| and I⊥ sequentially (panels a), while the second strategy measures I|| and I⊥ at the same time (panel b). To measure I|| and I⊥ sequentially, a linear polarizer (Pol) is positioned between the sample (s) and the light detector. When the linear polarizer is oriented with the electric field of the excitation source (0°), the detector will measure a signal proportional to I|| (Fig. 12.3a). Alternatively, when the polarizer is oriented orthogonal to the electric field of the excitation source (90°), the detector will measure a signal proportional to I⊥. On a simple and inexpensive system the orientation of the linear polarizing filter can be changed manually. Alternatively to minimize the time interval between measuring I|| and I⊥ signals, the rotation of the polarizing filter can be automated using either a mechanized rotating filter mount or by using a filter wheel to switch between two orthogonally oriented linear polarizing filters.

To accurately measure polarization in the schemes outlined above it is essential to accurately align the polarizing filters at 0° and 90°. In our laboratory, fluorescence emission filters are removed to allow linearly polarized excitation light to project directly onto the light detector. Next the orientation of the polarizing filter is rotated until the weakest transmitted signal is found (90°). The 0° orientation is then a 90° rotational offset from the 90° position. This alignment procedure assumes that the detectors are insensitive to the polarization of the light. Side-on photomultipliers are often very sensitive to polarization while end-on photomultipliers typically are not.

To measure I|| and I⊥ signals in parallel, a polarizing beam splitter (P.B.S.) can be used to separate these signals simultaneously (Fig. 12.3b). For live cell measurements this scheme is preferable to the sequential arrangement because it eliminates artefacts caused by the motion of polarized cellular components during the time interval between measuring I|| and I⊥ signals. By placing a polarizing beam splitter after the sample, I|| and I⊥ fluorescent signals can be separated and measured by two separate dedicated light detectors. Typically, the I⊥ signal is reflected by the beam splitter while the I|| signal is transmitted. Polarizing beam splitters can be wavelength dependent. Consequently, when adapting a microscope for polarization measurements a polarizing beam splitter that has a flat response over a wavelength range that is matched to the emission spectrum of the fluorophore of interest should be selected. It is also worth noting that the contrast ratio (the intensity ratio of the transmitted polarization state vs. the attenuated state) of polarizing beam splitters is reasonable in the transmitted pathway (typically ≥500:1), but typically poor in the reflected pathway (20:1). For this reason, we use two linear polarizing filters whose orientation is matched to the output of the beam splitter to augment the contrast ratio. Linear polarizing filters typically have contrast ratios that are at least 500:1. Finally, when measuring I|| and I⊥ in parallel, each pathway downstream of the polarizing beam splitter will have its own dedicated photo-detector.

A photomultiplier tube would typically be used as the light detector for steady-state polarization measurements in conjunction with laser scanning microscopy such as confocal microscopy or two-photon microscopy. The use of photomultipliers, ideally with a short instrument response function (<200 ps) in conjunction with a pulsed laser light source allows time-resolved polarization measurements using time correlated single photon counting (TCSPC; [3]). Two fluorescence lifetime decay curves are generated, one representing the decay of I||(t) and the other for I⊥(t). Ideally, the inverse of the laser repetition rate should be ≥ to 5 times the lifetime of the fluorophore being studied. For most fluorophores a repetition rate of 20–50 MHz is ideal. With two-photon excitation often a less than ideal laser repetition rate is used (80–90 MHz), and this compromise results in truncated time-resolved decay curves as well as the potential for temporal bleed-through artefacts. The microscope designs shown in panel A or B can also be adapted for steady state polarization imaging using a EMCCD camera as the light detector.

12.4 How Are Polarized Emissions Quantified?

To begin to understand the basis of time-resolved anisotropy measurements we will start with a ‘simple’ population of fluorophores in solution. We will assume that the fluorophores in this sample are randomly oriented (isotropic), that the absorption and emission dipoles of these fluorophores are collinear (have the same orientation), and that these fluorophores do not rotate while in the excited state. Once we understand this simple system we will then consider the impact of having non-collinear dipoles and molecular rotation. With the assumptions of this simple system the orientation of photons emitted from the excited fluorophores in this population will be highly correlated with the orientation of the linear polarized light used to excite them as dictated by the orientational dependence of photoselection (12.1). Essentially, in this population, fluorophores with low values of θ will be preferentially excited while those with θ values near 90° will not be excited. It must also be appreciated that even though the fluorophores in this population are randomly oriented, the number of molecules with θ values near 0° will be much less than those with θ values near 90° (for a more detailed explanation see [31]). Ultimately, the probability of exciting any specific molecule in this isotropic population will be proportional to the product of the probability for excitation at a specific θ values (12.1) and the probability for finding a fluorophore with that θ value in the isotropic population (sin θ):

$$ p \propto \cos^{x} \theta \cdot \sin \theta $$
(12.2)

where x is 2 for one-photon excitation and 4 for two-photon excitation with linearly polarized light. The key concept to understand is that photo selection with linearly polarized light transforms a randomly oriented population of ground-state fluorophores into an ordered population of excited state fluorophores whose dipole orientations are strongly correlated with the orientation of the excitation light electric field. This near instantaneous transformation into a population of ordered excited-state fluorophores as a result of a laser excitation pulse, and our ability to monitor the population orientation using polarization measurements as described above, forms the basis of time-resolved anisotropy measurements. At this point we should note that because this ‘simple’ population is static, it does not rotate during the excited state; polarization measurements should not change as a function of time after the laser pulse. In most biological samples, fluorophores can rotate to some extent during the excited state and so polarization measurements can and will change as a function of time.

The polarization state of a photon emitted by a fluorophore will be correlated with the orientation of the fluorophore (strictly speaking its emission dipole) at the instant of returning to the ground state [33]. As described above, for an isotropic population of static fluorophores with collinear absorption and emission dipoles excited by linearly polarized light, the polarization of emitted photons will also be strongly correlated with the electric field orientation of the polarized light source. Thus, under the conditions described, the polarization of photons emitted by an isotropic population of fluorophores can be used to determine how similar the orientation of the fluorophores are relative to the orientation of the electric field of the excitation source. We need to answer two more fundamental questions before we can more fully understand this correlation quantitatively and then apply it to biological questions; these are: (1) What is the relationship between the orientation of a fluorophore emission dipole and the probability of detecting its emitted photon through parallel or perpendicularly oriented linear polarizers? and (2) How can we parameterize the orientation of the emission from an isotropic population of fluorophores using the measured I|| and I⊥ values?

Figure 12.4 depicts how the emission dipole orientation (double green arrow) of a fluorophore from an isotropic population of fluorophores excited with polarized light will affect the signal intensity measured through a linear polarizer oriented either parallel (Fig. 12.4a) or perpendicular (Fig. 12.4b) to the electric field of the light source.

Fig. 12.4
figure 4

The probability of detection through a polarizing filter. The probability that a photon emitted by a fluorophore will pass through a linear polarizing filter (Pol.) and detected by a light detector (L.D.) is a function of both the orientation of the fluorophores emission dipole (green double arrow), and the orientation of the filter. When the filter is situated along the Z-axis and is oriented at 0° relative to the electric field (E) of the light source the photomultiplier will detect I|| (a). When the filter is rotated to 90° relative to the electric field the photomultiplier will detect I⊥(b). I|| will be proportional to cos2 θ, where θ is the angle formed between electric field of the light source and the emission dipole of the fluorophore (panel c). I⊥ will be proportional to sin2 θ·sin2 ϕ, where ϕ is the angle formed between the emission dipole of the fluorophore and the Z-axis (panel d). For an isotropic distribution of fluorophores excited with linearly polarized light, the distribution of excited state dipole orientations (pink hour-glass shaped cloud) will have a symmetrical distribution of ϕ values around the X-axis (d). Due to this symmetry, the value of sin2 ϕ = ½. Accordingly, for an isotropic distribution of fluorophores I⊥ will be proportional to ½ sin2 θ. Notice that for an isotropic population of fluorophores values of I|| and I⊥ are functions of θ alone

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A representation of the three-dimensional excited state probability distribution (see 12.2) is depicted in pink. The orientation of any single fluorophore from this excited state population can be described by two angles, θ (Fig. 12.4c) and ϕ (4D). When the linear polarizer is oriented parallel to the electric field polarization (0°) the light intensity measured through the filter will be proportional to cos2 θ (where θ is the polar angle of the emitting molecule relative to the electric field polarization). Thus, for a population of fluorophores the measured I|| intensity will be proportional to an average of all the cos2 θ values weighted by their abundance. When the filter polarizer is oriented perpendicular to the orientation of the excitation electric field polarization (90°), the light intensity measured will be proportional to sin2 θ·sin2 ϕ. This equation can be simplified because the excited state distribution is symmetrical around the X-axis, and for the population, the average of sin2 ϕ = ½. Consequently, for the population of fluorophores, I⊥ will be proportional to an abundance-weighted average of all ½ sin2 θ values. This equation illustrates that when a population of randomly oriented fluorophores is excited by linearly polarized light, both I|| and I⊥ will be determined by the value of θ (the polar angle of the emitting molecules relative to the electric field polarization) alone.

Finally we must discus how I|| and I⊥ values can be used to parameterize the orientation of fluorophore populations. There are two main conventions that have been used in the literature, the polarization ratio (p) and emission anisotropy (r). The polarization ratio is simply the intensity difference between I|| and I⊥ divided by the intensity observed by a photo-detector placed along either the Y- or Z-axis (I|| + I⊥):

$$ p = (I_{\parallel } - I_{ \bot } )/(I_{\parallel } + I_{ \bot } ) $$

By setting I⊥ or I|| to 0 the limiting values of polarization can be determined; the polarization ratio can range from −1 to 1. A polarization value of 1 indicates a perfect alignment of emission dipoles with the orientation of the light source electric field. A polarization value of −1 indicates a perfectly orthogonal orientation. Analogous to polarization, the numerator for calculating the emission anisotropy is the intensity difference between I|| and I ⊥. However, the denominator used for calculating the emission anisotropy is the emission intensity with parallel and perpendicular components proportional to the total intensity (I|| + 2·I⊥):

$$ r = (I_{\parallel } - I_{ \bot } )/(I_{\parallel } + 2 \cdot I_{ \bot } ) $$
(12.3)

Substituting 0 for either I⊥ or I|| reveals that the range of possible values for emission anisotropy is from −0.5 to 1. For anisotropy measurements a value of 1 indicates a perfect alignment of emission dipoles with the orientation of the light source and a value of −0.5 indicates a perfectly orthogonal orientation. A population of randomly oriented excited-state fluorophores will have an anisotropy value of 0. Clearly, the polarization ratio and emission anisotropy are just different expressions used to parameterize the same phenomenon, the orientation of light emitted relative to the orientation of the linearly polarized light source electric field. The relationship between r and p is:

$$ r = 2 \cdot p/(3 - p) $$

12.5 Calculating the Time-Resolved Anisotropy Curve

In this chapter we use emission anisotropy rather than polarization because in most biological applications anisotropy is more amenable to analysis. Specifically, we will describe time-resolved emission anisotropy, which monitors how anisotropy values change as a function of time after photo selection/excitation. In Fig. 12.5a, b we display I||(t) and I⊥(t) decay curves for a sample containing the yellow fluorescent protein Venus in an aqueous buffer.

Fig. 12.5
figure 5

The lifetime and time-resolved anisotropy of the yellow fluorescent protein Venus. A sample of the fluorescent protein venus was excited using 950 nm light pulses from a Ti:sapphire laser using two-photon photo selection and a fluorescence detection scheme similar to the layout depicted in Fig. 12.3b. The measured I(t) || and I(t) ⊥ decay curves are plotted in panel a. These two traces were globally fit (red dashed lines) to (12.4) and (12.5) and yielded a Venus lifetime of 3.23 ns, a rotational correlation time of 14.31 ± 0.05 ns, and a limiting anisotropy of 0.41. The I(t) || and I(t) ⊥ traces in panel a were next processed using (12.6) to yield the fluorescence lifetime decay, or (12.8) to yield the time-resolved anisotropy decay (panels b and c respectively). The dashed red line in panel b depicts a fit using a single exponential decay model and yielded a lifetime of 3.20 ns. The dashed red line in panel c also depicts a fit using a single exponential decay model and yielded a rotational correlation time for Venus of 14.9 ± 0.1 ns and a limiting anisotropy value of 0.41. The fundamental anisotropy value expected for an isotropic population of fluorophores with collinear absorption and emission dipoles excited by two-photon excitation is 0.57. The difference between this fundamental anisotropy value and the measured limiting anisotropy (0.41) can be accounted for (using 12.12) by multiplying the limiting anisotropy by the product of an instrumental depolarization factor for a NA 1.2 water objective (dNA = 0.81, see Table 12.1) and a depolarization factor for a β dipole angle of 15.4° (dβ = 0.89, see 12.11). We have previously measured an upper limit for the Venus dipole angle β of 16° using low NA objectives [31]

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We will use this sample to introduce how time-resolved anisotropy can be used to analyze fluorescent biological samples. For these measurements the Venus fluorophore was excited using two-photon excitation at 950 nm. I||(t) and I⊥(t) decay curves contain information about the fluorescence lifetime (τ) of Venus as well as the dipole orientation as Venus rotates while in its excited state. The Venus I||(t) and I⊥(t) decay curves are well described by the following two equations:

$$ I_{||} (t) = G\tilde{F}_{0} \cdot e^{{\left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} \tau }} \right. \kern-0pt} \tau }} \right)}} \left( {2r_{0} \cdot e^{{\left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} {\theta_{rot} }}} \right. \kern-0pt} {\theta_{rot} }}} \right)\,}} + 1} \right) $$
(12.4)
$$ I_{ \bot } (t) = \tilde{F}_{0} \cdot e^{{\left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} \tau }} \right. \kern-0pt} \tau }} \right)}} \left( { - r_{0} \cdot e^{{\left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} {\theta_{rot} }}} \right. \kern-0pt} {\theta_{rot} }}} \right)\,}} + 1} \right) $$
(12.5)

where τ is the fluorescence lifetime of Venus the (average amount of time a Venus molecule remains in the excited state), θ rot is the rotational correlation time of Venus in solution, r 0 is the limiting anisotropy (the anisotropy value at time equals zero before any dipole rotation has occurred), G is an experimentally measured constant, particular for a specific microscope configuration, that accounts for differences in the sensitivity between I⊥ or I|| measurements, and F 0 is an intensity-scaling factor, where \( \tilde{F}_{0} = F_{0} \cdot (1 + 2G)/G \). If the sensitivity of both measurements is the same, then G will have a value of 1. In our experience using either the same, or matched photomultiplier tubes, G has a value close to 1 (in this example it had a value of 0.9). Notice, since (12.4) and (12.5) are functions of the same 4 variables, I||(t) and I⊥(t) decay curves shown in Fig. 12.5a can be fit globally to (12.4) and (12.5) respectively to yield more accurate estimates of τ, τ rot and r 0 . Global analysis of the curves depicted in Fig. 12.5a yielded a Venus lifetime of 3.23 ns (±0.00), a rotational correlation time of 14.3 ± 0.1 ns, and a limiting anisotropy of 0.41 (±0.00). Importantly, (12.4) and (12.5) are only applicable if the fluorophore lifetime decays as a single exponential, and if the molecule is spherical, such that its rotational correlation times around its X-, Y-, and Z-axes are identical.

The Venus I||(t) and I⊥(t) decay curves depicted in Fig. 12.5a can be used to calculate a fluorescence lifetime decay curve using (12.6) and is plotted in Fig. 12.5b:

$$ I(t) = I_{||} (t) + 2 \cdot G \cdot I_{ \bot } (t) $$
(12.6)

By substituting (12.4) and (12.5) into (12.6) we observe that I(t) decays as a single exponential with an intensity amplitude of F 0(1 + 2G).

$$ I(t) = F_{0} (1 + 2G) \cdot e^{{{{ - t} \mathord{\left/ {\vphantom {{ - t} \tau }} \right. \kern-0pt} \tau }}} $$
(12.7)

The decay curve in Fig. 12.5b can be modelled as a single exponential with a tau of 3.2 ns (±0.0) in excellent agreement with the lifetime predicted by global fitting of the decay curves in Fig. 12.5a as well as with values in the literature [20].

The Venus I||(t) and I⊥(t) decay curves depicted in Fig. 12.5b can also be used to calculate an emission anisotropy decay curve using (12.8) and is plotted in Fig. 12.5c:

$$ r(t) = \frac{{I_{||} (t) - G \cdot I_{ \bot } (t)}}{{I_{||} (t) + 2G \cdot I_{ \bot } (t)}} $$
(12.8)

By substituting (12.4) and (12.5) into (12.8) we can see that r(t) for this example should decay as a single exponential:

$$ r(t) = r_{0} \cdot e^{{ - t/\theta_{rot} }} $$
(12.9)

Analysis of the time resolved anisotropy curve in Fig. 12.5c results in a rotational correlation time of 14.9 ± 0.1 ns, in good agreement with the rotational correlation time previously reported [20] and a limiting anisotropy of 0.41 ± 0.00. These values were also in excellent agreement with the values obtained using global fitting of the decay curves in Fig. 12.5a. The major point is that (12.6) and (12.8) allow the extraction of either the fluorescence lifetime or the time-resolved anisotropy decay from I||(t) and I⊥(t) decay curves. Equations (12.7) and (12.9) are applicable only to fluorophores whose lifetime decays as a single exponential and are spherical, having identical rotational correlation times around their X-, Y-, and Z-axes. In contrast, (12.6) and (12.8) can be applied to extract either the fluorescence lifetime or time-resolved anisotropy from I||(t) and I⊥(t) decay curves obtained from more complex fluorescent samples having multiple lifetimes and/or multiple anisotropy decay components. For this reason I||(t) and I⊥(t) decay curves are obtained from more complicated biological samples and are then processed using (12.8) to generate a time-resolved anisotropy decay curve. The remainder of this chapter will address how to interpret time-resolved anisotropy decay curves.

12.6 The Anatomy of Time-Resolved Anisotropy

A time-resolved anisotropy curve, such as the curve depicted in Fig. 12.5c, describes the ensemble anisotropy value of a population of fluorophores, and follows how this value changes as a function of elapsed time from photo selection with a short pulse of linearly polarized light at t = 0. The time-resolved anisotropy curve has two characteristics that are worth considering, the limiting anisotropy, r 0 , which is the population anisotropy value at the instant of photo selection (t = 0), and a sequence of anisotropy values that describes how the population anisotropy changes as a function of time after photo selection. We will start with interpreting the value of the limiting anisotropy, r 0 , and then discus how to analyze and interpret how anisotropy values can change with time.

The X-axis of time-resolved anisotropy curves represents elapsed time from the instant of photo selection (typically with a short pulse of linearly polarized laser light). The span of the X-axis is primarily dictated by the repetition rate of the pulsed laser used for photo-excitation, and typically ranges from t = 0 to 12.5 ns when an 80 MHz Ti:Sapphire laser is used for two-photon excitation, to 20 ns when a 50 MHz pulsed diode laser is used for one-photon excitation, and to 50 ns when a 20 MHz pulsed diode laser is used. The time resolution of time-correlated single photon counting (TCSPC) electronics used in these types of biophotonics measurements are typically better than a few picosecond, so any value measured for the limiting anisotropy should represent the value of the population anisotropy within ~5 ps of the instant that the population was photo-selected. Equation (12.2), in conjunction with a random number generator to produce values of θ from an isotropic distribution, can be used to run Monte Carlo simulations to predict what the anisotropy value should be for an isotropic population of fluorophores excited with 1-, 2-, or 3-photon excitation. These calculations predict anisotropy values of 0.40, 0.57, and 0.67 respectively [31]. These theoretical predictions represent the highest possible anisotropy values for a randomly oriented population of fluorophores excited by 1-, 2- or 3-photon linearly polarized light, and are called the Fundamental Anisotropy. In practice, the actual anisotropy value measured at t = 0, r 0 , is typically less than the Fundamental Anisotropy and is called the Limiting Anisotropy. Note that r 0 is the anisotropy at t = 0, not to be confused with the anisotropy at infinite time, r ∞ . We can see this attenuation for the Venus anisotropy with two-photon excitation in Fig. 12.5c.

12.7 Depolarization Factors and Soleillet’s Rule

An excellent starting point for learning how to interpret time-resolved anisotropy curves is to consider why the limiting anisotropy is almost always less than the fundamental anisotropy. Depolarization factors (d) are experimental influences that can account for a decrease in the measured anisotropy. One of the prime reasons for using anisotropy rather than polarization ratios is based on Soleillet’s rule [22]. Soleillet’s rule states that the anisotropy measured will be equal to the fundamental anisotropy multiplied by all applicable depolarization factors [13, 28]:

$$ r = r_{f} \cdot \prod\limits_{i} {d_{i} } $$
(12.10)

For anisotropy experiments involving proteins or other cellular components labelled with a fluorophore, there are typically four depolarization factors that should be considered when attempting to account for any discrepancies between theory and a measured anisotropy value. These are: (1) depolarization due to instrumentation, (2) depolarization due to non-collinear absorption and emission dipoles, (3) depolarization due to molecular rotation, and (4) depolarization occurring as a result of Förster Resonance Energy Transfer (FRET). The first two of these factors occur very rapidly and, if applicable, would result in a near immediate drop from the fundamental anisotropy.

12.7.1 Instrumental Depolarization

By definition, the lenses used to build optical microscopes (primarily objectives and condensers) alter the orientation of light. Accordingly, one adverse effect of measuring anisotropy on a microscope (as opposed to a dedicated fluorimeter) is that the orientation of linearly polarized excitation light when it reaches the sample is no longer purely linearly polarized. Similarly, when the fluorescence emission is collected and relayed to the photo-detector, I|| signal can be contaminated with I⊥ components and vice-a-versa. In general, the higher the numerical aperture (NA) of the optics, the greater the instrumental depolarization. The quantitative theory for predicting the impact of lens NA on depolarization (and the value of d NA, see Table 12.1) for a particular microscope setup has a complex dependence on the numerical aperture of each lens used in the excitation and emission light paths, and the refractive index of the medium [1, 2]. It can be visualized empirically by time-resolved anisotropy measurements of the same sample while systematically altering the NA [31]. Simpler schemes for correcting for instrumental depolarization by altering the value of 2 in the denominator of (12.6) to a value ranging from 2 to 1 with increasing NA optics [3, 9, 26, 27] may have value in producing a lifetime that decays as a single exponential, but cannot be used in (12.8) to predict measured anisotropy values as it incorrectly predicts that anisotropy values will increase with higher NA objectives.

Table 12.1 Depolarization factor, d NA , for different numerical apertures of the lens and different refractive indexes of the immersion medium
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Depolarization due to non-collinear dipoles: while it is usually assumed that the absorption dipole is identical to the emission dipole of a fluorophore, this is not always true. If we designate the angle difference between the absorption and emission dipole of a fluorophore as β, a depolarization factor that accounts for non-collinear dipoles can be calculated [14]:

$$ d_{\beta } = \frac{3}{2} \cos^{2} \beta - \frac{1}{2} $$
(12.11)

β is thought to be a static intrinsic property of a fluorophore that does not change during the course of a biological experiment. In contrast, some fluorophores do undergo a conformational change in response to absorbing a photon. These dynamic changes in fluorophore structure are typically extremely rapid. Thus, regardless of the mechanism of non-collinearity, non-zero values for β are thought to attenuate the fundamental anisotropy to yield a diminished limiting anisotropy. In most biological applications of time-resolved anisotropy, the objective is to observe how the anisotropy changes with time from the initial limiting anisotropy to explore the fluorophores ability to rotate, as well as if it is in close proximity to other fluorophores (Homo-FRET). Accordingly, when applying Soleillet’s rule it is often assumed that the limiting anisotropy observed, r 0 , is a product of both d NA of the objective and d β (see 12.11) without determining the absolute value of each individual depolarization factor:

$$ r_{0} = r_{f} \cdot d_{NA} \cdot d_{\beta } = r_{f} \cdot d_{NA} \cdot \left( {\frac{3}{2}\cos^{2} \beta - \frac{1}{2}} \right) $$
(12.12)

12.7.2 Depolarization Due to Molecular Rotation

Depolarization caused by the molecular rotation of fluorophores is typically much slower occurring on a time scale of hundreds of picoseconds to hundreds of nanoseconds. Accordingly molecular rotation typically manifests on a time-resolved anisotropy curve as a slow decay of anisotropy from the limiting anisotropy value. We have already shown in (12.9) that a spherical molecule that is free to rotate around its X-, Y-, and Z-axes will decay in time as a single exponential [13, 28]:

$$ r(t) = r_{0} \cdot e^{{ - t/\theta_{rot} }} $$

where r 0 is the limiting anisotropy, and θ rot is the rotational correlation time of the molecule. The rotational correlation time θ rot is related to the rotational diffusion coefficient, D r, and is a measure of how rapidly a molecule can rotate; the smaller the value of θ rot , the faster the fluorophore can rotate. The relationship between θ rot and D r for a molecule in solution is [5, 13]:

$$ \theta_{rot} = \frac{1}{{6D_{r} }} $$
(12.13)

The Stokes-Einstein relationship [13] describes how D r is a function of the absolute temperature (T), the viscosity (η), and the volume per molecule (V):

$$ D_{r} = \frac{{k_{B} T}}{{6{\textit{V}}\eta }} $$
(12.14)

where k B is Boltzmann’s constant. Accordingly:

$$ \theta_{rot} = \frac{{{\textit{V}}\eta }}{{k_{B} T}} $$
(12.15)

The time-resolved anisotropy decay of molecules that are asymmetrical can also be modelled but as a multi-exponential decay (requiring up to 5 decay components):

$$ r(t) = r_{0} \cdot \sum\limits_{i} {a_{i} \cdot e^{{ - t/\theta_{i} }} } $$
(12.16)

where θ i is the ith rotational correlation time of the molecule and a i is the fractional amplitudes of the ith decay component. Often the signal to noise level of time resolved anisotropy measurements, particularly at times much longer than the fluorescence lifetime, preclude accurate fitting to multi-exponential decay models such as in (12.16). In practice, even non-spherical molecules can be fit well to (12.9).

12.7.3 Depolarization Due to Homo-FRET A.K.A. Energy Migration

Another possible reason for observing a time-dependent depolarization when analyzing time-resolved anisotropy decay curves is if energy transfer is occurring between fluorophores in the sample [4, 6, 9, 12, 15, 16, 23–27, 31]. Förster Resonance Energy Transfer (FRET) is a physical phenomenon where excited state energy is transferred by a non-radiative mechanism from a photo-selected fluorophore (the donor) to a nearby ground state chromophore (the acceptor) [7, 8, 10, 11, 17, 18, 32]. There are two general types of FRET, Hetero-FRET (where the excitation and emission spectra of the donor and acceptor are different, and Homo-FRET where the donor and acceptor have identical spectra (they are distinct molecules of the same type of fluorophore). In this chapter we will only consider the impact of Homo-FRET on anisotropy measurements. When Homo-FRET occurs the donor returns to its ground state and the acceptor is concomitantly raised to its excited state. Clearly, if the dipole orientation of the acceptor is different from the dipole orientation of the donor, the emission anisotropy can be altered with Homo-FRET. For Homo-FRET to occur three conditions must me met: (1) The distance between a donor and acceptor fluorophores must be typically less than 10 nm. (2) The fluorophore must have a small Stokes shift such that its absorption spectrum has a large overlap with its emission spectrum. and (3) The acceptors absorption dipole must not be oriented perpendicular to the orientation of the electric field created by the donors oscillating dipole. In time-resolved anisotropy measurements biological samples are typically labelled with a single fluorophore, and the sample concentration is experimentally manipulated to keep it low enough so the mean distance between molecules is much greater than 10 nm. Despite these precautions to avoid Homo-FRET, many biological molecules can form multimers that may result in Homo-FRET. Equation (12.17) describes the anisotropy decay for a spherical complex having two fluorophores in close proximity (a dimer) separated by distance R:

$$ r(t) = r_{0} \cdot \left( {a_{FRET} \cdot e^{ - t/\phi } + a_{Rot} \cdot e^{ - t/\theta } } \right) $$
(12.17)

where \( \phi \) is the Homo-FRET correlation time of the complex, θ is the rotational correlation time, a FRET is the fractional amplitude of the Homo-FRET decay component, a Rot is the fractional amplitude of the rotational decay component. For energy migration between two fluorophores the Homo-FRET correlation time is a function of ω, the FRET transfer rate [21, 24, 29]:

$$ \phi = \frac{1}{2\omega } $$
(12.18)

If ω, the FRET transfer rate is much faster than the fluorescence emission rate (1/τ), then the amplitude of the Homo-FRET decay component for Homo-FRET between two fluorophores will be approximately 0.5, and:

$$ a_{Rot} = 1 - a_{FRET} $$
(12.19)

In general, aFRET will be approximately 1 − (1/N) where N is the number of fluorophores in a complex that are transferring energy by Homo-FRET. Thus the aFRET amplitude for a trimer transferring energy by Homo-FRET will be greater than for a dimer at the same separation [31]. The rate of energy transfer by FRET has an inverse sixth power dependence on the separation of the donor and acceptor:

$$ \omega = \frac{1}{\tau } \cdot \left( {\frac{{R_{0} }}{R}} \right)^{6} $$
(12.20)

where R is the separation between the donor and acceptor, and R0 is the Förster distance or the distance where the transfer rate (ω) is equal to the emission rate (Γ), the inverse of the lifetime (τ) in the absence of FRET. Equation (12.20) assumes that donors and acceptors in the sample population are randomly oriented and rotate rapidly during the excited state lifetime such that the dipole orientation factor κ2 is equal to 2/3. If it is not valid to assume a κ2 value of 2/3, for example, if fluorophores do not rotate rapidly during the excited state, then the transfer rate of an individual FRET pair can be calculated using (12.21) by specifying the specific value of κ2 [30]:

$$ \omega = \frac{3}{2} \cdot \frac{{\kappa^{2} }}{\tau }\left( {\frac{{R_{0}}}{R}}\right)^{6} $$
(12.21)

Under these circumstances it is possible to observe heterogeneity in the transfer rates of the population of FRET pairs despite having a homogenous separation between donors and acceptors [30]. Vis-à-vis Homo-FRET, having heterogeneous κ2 values can result in an attenuated Homo-FRET decay component amplitude (aFRET) as well as observing multiple Homo-FRET correlation times (as manifest by a multi-exponential Homo-FRET decay component).

12.7.4 Complex Time-Resolved Anisotropy Curves

Time-resolved anisotropy curves typically decay from a limiting anisotropy (r 0) to an anisotropy value of zero if fluorophores are free to rotate in all directions (see 12.16). Similarly, if Homo-FRET is occurring, a faster decay component associated with FRET will be observed in addition to the rotational anisotropy decay (for example see 12.17). For small fluorophores that can rotate rapidly, it may be difficult to differentiate between depolarization caused by fast rotation and depolarization caused by FRET. In some samples the ability of fluorophores to rotate will be hindered. This is often observed for fluorophores that partition into lipidic membranes that align either perpendicular or parallel to the plane of the membrane. The time-resolved anisotropy curve for samples with ‘hindered’ rotation will decay from the limiting anisotropy value to a non-zero asymptote, r ∞ , the anisotropy value at infinite time:

$$ r(t) = (r_{0} - r_{\infty } ) \cdot \left( {\sum\limits_{i} {a_{i} \cdot e^{{ - t/\theta_{i} }} } } \right) + r_{\infty } $$
(12.22)

Even more complicated time-resolved anisotropy curves can be observed if more than one type of fluorophore is present and if the different fluorophores have different lifetimes and time-resolved anisotropies. With multiple fluorophores the average time-resolved anisotropy of a population of fluorophores is [14]:

$$ \bar{r}(t) = \sum\limits_{i} {f(t)_{i} \cdot_r(t)_{i} } $$
(12.23)

where f(t) i is the time-dependent fractional intensity of the ith fluorescent component and r(t) i is the time-resolved anisotropy of the ith fluorescent component. The time-dependent fractional intensity, f(t) i , is the abundance-weighted lifetime of a single fluorophore in the population divided by the sum of the abundance weighted lifetimes of all fluorescent species in the population:

$$ f(t)_{x} = \frac{{a_{x} \cdot I(t)_{x} }}{{\sum\limits_{i} {a_{i} \cdot I(t)_{i} } }} $$
(12.24)

where a x is the abundance of fluorophore x in the population and I(t) x is the lifetime of fluorophore x. Equation (12.24) indicates that the anisotropy of a population of fluorophores is simply the intensity weighted sum of the anisotropy values of the individual fluorophores in the population. Interpretation of ensemble anisotropy values from populations with more than one type of fluorophore (each having unique anisotropy and lifetime values) can be challenging (for example see Fig. 12.6).

Fig. 12.6
figure 6

A complex biphasic time-resolved anisotropy curve. In samples having two or more types of fluorophores having different lifetimes and rotational correlation times, complex multiphasic time-resolved anisotropy curves are possible. To illustrate, we use (12.24) to simulate the ensemble time-resolved anisotropy of a population composed of two fluorophores, A and B. Fluorophore A has a lifetime of 3 ns (panel a) and initially (at t = 0) accounts for 10 % of the fluorescence detected from the sample (panel b). In contrast, fluorophore B has a shorter lifetime of 1 ns (panel a) but accounts for 90 % of the fluorescence at t = 0 (Panel b). Because fluorophore B has a shorter lifetime than fluorophore A, with time its fractional intensity will decline while the fractional intensity of fluorophore B will increase reciprocally (panel b). The rotational correlation time of fluorophore A is 30 ns, while the rotational correlation time of fluorophore B is 1 ns (panel c, the limiting anisotropy was set at 0.4). The black trace in panel c depicts a biphasic anisotropy decay curve. Such a complex time-resolved anisotropy curve could be observed in cells expressing a protein of interest genetically-tagged with a fluorescent protein if the cell has a high level of auto-fluorescence background

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