Second-Harmonic Generation Imaging of Membrane Potential with Retinal Analogues

Abstract

Second-harmonic generation (SHG) by membrane-incorporated probes is a nonlinear optical signal that is voltage-sensitive and the basis of a sensitive method for imaging membrane potential. The voltage dependence of SHG by four different probes, three retinoids (all-trans retinal), and two new retinal analogs, 3-methyl-7-(4′-dimethylamino-phenyl)-2,4,6-heptatrienal (AR-3) and 3,7-dimethyl-9-(4′-dimethylamino-phenyl)-2,4,6,8-nonatetraenal (AR-4), and a styryl dye (FM4-64), were compared in HEK-293 cells. Results were analyzed by fitting data with an expression based on an electrooptic mechanism for SHG, which depends on the complex-valued first- and second-order nonlinear electric susceptibilities (χ₂ and χ₃) of the probe. This gave values for the voltage sensitivity at the cell's resting potential, the voltage where the SHG is minimal, and the amplitude of the signal at that voltage for each of the four compounds. These measures show that χ₂ and χ₃ are complex numbers for all compounds except all-trans retinal, consistent with the proximities of excitation and/or emission wavelengths to molecular resonances. Estimates of probe orientation and location in the membrane electric field show that, for the far-from-resonance case, the shot noise-limited signal/noise ratio depends on the location of the probe in the membrane, and on χ₃ but not on χ₂.

Introduction

Membrane potential is a hallmark of all living cells and is the basis for the generation of voltage signals in excitable cells. Potential changes can be recorded either electrically, using various kinds of electrodes or optically, using voltage-sensitive indicators. An important advantage of optical recording is that it allows simultaneous multisite recording of local electrical signals from identified subcellular compartments that are too small to be recorded from with electrodes. Most voltage-sensitive indicators are fluorescent molecules that respond to membrane-potential changes by a fast electrochromic mechanism whereby the displacement of intramolecular charges changes the energy necessary to switch a molecule from its ground to an excited state (see below) and thereby its spectral absorbance (1).

Our report deals with a different class of potentiometric probes that base their response to changes in trans-membrane potential on the modulation of optical second-harmonic generation (SHG). This is a nonlinear optical phenomenon whereby two long-wavelength photons are converted into one photon with half the wavelength, i.e., twice the energy. The nonlinear (quadratic) dependence of this process on the excitation intensity is similar to that of two-photon absorption-based fluorescence excitation in which the energy needed to elevate a fluorophore from its ground to an excited state is provided by the simultaneous absorption of two long-wavelength photons.

The electromagnetic field of light propagating through a material distorts the electronic structure of the atoms and molecules that it is made of. This causes charge displacements and thus induces oscillating dipole moments that reradiate light, which, in the case of SHG, is at twice the frequency of the excitation light. This process occurs without delay and generates coherent light. As discussed in more detail in the following section, the static electric field of the membrane potential can influence SHG by affecting the steady-state molecular polarization of the probe and thus the induction of an oscillating dipole by the electric field of the driving light.

Optical second harmonics are not generated in materials that have a center of inversion symmetry (centrosymmetry), where equal-magnitude dipoles are induced in opposite directions and the emitted radiation cancels out. The requirement for center asymmetry in cellular membrane studies using SHG probes is satisfied if they populate preferentially one of the membrane leaflets or if there is a static electric field. Otherwise the coherent waves of second harmonic light from each leaflet are equal but of opposite phase and sum to zero by destructive interference (2).

The use of SHG probes for potentiometry has several potential advantages over the use of voltage-dependent fluorescent dyes.

Firstly, the need for center asymmetry rules out the generation of second-harmonic signals from probe molecules randomly oriented in solution or nonspecifically bound to elements in either the intra- or extracellular space. Thus, SHG recordings are less contaminated by noise and background fluorescence than are fluorescence measurements, which include membrane-potential independent signals from fluorophores bound nonspecifically to tissue other than the cell's surface membrane.

Secondly, the nonlinearity of the SHG mechanism, in common with two-photon microscopy, permits optical sectioning.

Thirdly, the use of long-wavelength light for second-harmonic excitation allows imaging at greater tissue depths than is possible with conventional (one photon) fluorescence and potentially with less photodamage (3). Although the SHG signals generated by styryl dyes are weaker than their fluorescence signals, they can be as or more sensitive to membrane voltage (4,5).

The SHG probes used for fast potentiometry are most commonly styryl dyes (3,5–11) but also include all-trans retinal (12). Here we introduce two new retinal analogs (Fig. S1 in the Supporting Material), 3-methyl-7-(4′-dimethylamino-phenyl)-2,4,6-heptatrienal (AR-3) and 3,7-dimethul-9-(4′-dimethylamino-phenyl)-2,4,6,8-nonatetraenal (AR-4) to serve as voltage-dependent SHG probes and compare them on the basis of their voltage sensitivity and response kinetics to two established probes, FM4-64, a styryl dye, and all-trans retinal (ATR). The comparison is guided by the derivation of a concise expression for the relationship between the SHG signal strength and membrane potential. Moreover, the analysis also provides information about the probe's location and orientation inside the membrane and should be useful for evaluating SHG probes in general. The results also highlight an important distinction between sensitivity and signal strength that provides a rational basis for fine-tuning molecular structure to improve probe performance.

Background and analysis of voltage-dependent SHG

Charge polarization and the induction of dipoles underlie the propagation of light in any medium (including vacuum). The relationship between the induced polarization vector (P_i) and the electric field of the excitation light can be expressed as a Taylor series expansion of the form (13)

$$P_i=\chi E\left(1+\frac{E}{E_1}+{\left(\frac{E}{E_2}\right)}^2+\dots .\right),$$ (1)

where χ is the electrical susceptibility, which provides a measure of how easily the dielectric medium is polarized by the light's electric field (E), and E₁, E₂, … E_n are coefficients that are on the order of the electric field strengths that bind atoms together (∼100 V/nm). When the electric field of the incident light is much smaller than the internal molecular fields, charge polarization is an approximately linear function (P_i = χE) of the electric field of the applied light. When the field strength of the incident light is significant relative to the atomic fields of the material (such as the electrical field strengths reached by focused and pulsed laser light sources), the other terms in the expansion are no longer insignificant and the polarization is given by

$$P_i=\chi E+{\chi }_2{E}^2+{\chi }_3{E}^3+\dots {\chi }_n{E}^n,$$ (2)

where χ_n is the n^th-order nonlinear electric susceptibility and is a complex-valued n^th-order tensor (14). If the electric field is a sinusoidal function of time such as E = E₀ exp(iωt), quadratic and higher terms in Eq. 2 cause P_i to contain harmonics of the fundamental frequency (ω) of the incident light that drives the process. The induced polarization can then be rewritten as

$$P_i=\chi E_0\exp \left(i\omega t\right)+\frac{{\chi }_2{E}_0^2}{2}\left[1-i\exp \left(i2\omega t\right)\right]+\frac{{\chi }_3{E}_0^3}{4}\left[3\exp \left(i\omega t\right)-i\exp \left(i3\omega t\right)\right]+\dots .$$ (3)

In this expression, the exp(i2ωt) term represents charge oscillation at twice the fundamental frequency resulting in the reradiation of coherent light at the second harmonic frequency (2 ω) with a field strength that is proportional to the incident light intensity, making the SHG intensity proportional to the square of the incident intensity (I ∝ E²₀).

The electric field of the membrane potential can, in principle, influence SHG in two ways.

One is by affecting the probe alignment in the membrane relative to the optical polarization (direction of the electric field) of the incident light. The response time of SHG probes that base their voltage sensitivity on molecular realignment depends on the scale of the underlying readjustment. The large-scale electrophoretic redistribution (flip-flop) of amphipathic probe molecules from one membrane leaflet to the other generates responses on a timescale of seconds (15). Voltage-evoked molecular realignments on a finer scale, as can be monitored by fluorescence resonance energy transfer, have been shown to generate submillisecond responses (16). However, the interacting molecules used in this study (16) were within 0.7 nm of each other, suggesting that, for molecular realignment to be the source of SHG signals on the submillisecond timescale, the realignment motion would have to take place on a similar or finer length-scale.

The second way the membrane field can influence SHG is by an electrooptical process called electric field-induced second-harmonic generation (EFISH) (17), which depends on intramolecular charge transfer and follows changes in membrane voltage (V) with a picosecond response time. EFISH involves the third-order term in Eq. 3 as

$$P_i=\dots {\chi }_3{E}_m{E}_{\omega }^2,$$

where E_m is the membrane electric field at the SHG-probe's position and E_ω = E₀ exp(iωt) is the electric field of the incident light. The term

$${\chi }_3{E}_{\omega }^2{E}_m$$

oscillates at the second harmonic because

$$E_{\omega }^2={\left[E_0\exp \left(i\omega t\right)\right]}^2=E_0^2\exp \left(i2\omega t\right).$$

To get a better understanding of voltage-dependent SHG signals we consider the underlying electrooptic mechanism in more detail. The polarization (P_i2ω) that gives rise to the voltage sensitivity of an electrooptic SHG mechanism depends on the second and third terms from the Taylor series (Eq. 2) and is given by (18,19)

$$P_{i2\omega }=\left({\chi }_2+{\chi }_3{E}_m\right)E_{\omega }^2.$$ (4)

Whereas, in most materials,

$${\chi }_3{E}_m\ll {\chi }_2,$$

for a cell with a typical resting membrane potential of −60 mV the average electric field across a 7.5-nm thick bilayer (20) is in the range of 10 mV/nm (10⁷ V/m) and thus, χ₃E_m can become comparable to χ₂. As a result, the induced polarization can be quite sensitive to changes in the membrane electrostatic field.

The relative change in SHG intensity I_SHG = |P_i2ω|² from its level at the resting potential (I_SHG(V_R)) to its level after a change (ΔV) in membrane voltage (I_SHG (V_R +ΔV)) is, using Eq. 4, given by

$$\frac{I_{SHG}\left(V_R+\Delta V\right)-I_{SHG}\left(V_R\right)}{I_{SHG}\left(V_R\right)}=\frac{\Delta I_{SHG}\left(\Delta V\right)}{I_{SHG}\left(V_R\right)}=S_0\Delta V\left(1-\frac{\Delta V}{2V_0}\right),$$ (5)

where S₀ is the sensitivity

$$\partial \left(\Delta I_{SHG}/I_{SHG}\right)/\partial \Delta V$$

of SHG to changes in membrane potential at the resting potential (V_R) and given by

$$S_0=-\frac{2\left(d\mathrm{Re}\left[{\chi }_2{\overline{\chi }}_3\right]+V_R{\left|{\chi }_3\right|}^2\right)}{d^2{\left|{\chi }_2\right|}^2+2d{V}_R\mathrm{Re}\left[{\chi }_2{\overline{\chi }}_3\right]+V_R^2{\left|{\chi }_3\right|}^2}$$ (6)

and V₀ is the membrane voltage where SHG is minimal (i.e., parabola minimum) given by

$$V_0=V_R+d\frac{\mathrm{Re}\left[{\chi }_2{\overline{\chi }}_3\right]}{{\left|{\chi }_3\right|}^2},$$ (7)

where d is the thickness of the cell membrane, and ${\overline{\chi }}_3$ is the complex conjugate of χ₃. At the absorption peak of the probe, molecular resonance causes χ₂ and χ₃ to become frequency-dependent and their imaginary parts become significant. Because χ₃ normally changes more rapidly with frequency than χ₂, the induced harmonic oscillation from them is phase-shifted, causing the SHG minimum at ΔV = V₀ to not equal zero (21,22). In the nonresonant case, the imaginary parts of χ₂ and χ₃ are insignificant, and their contributed harmonic oscillations are in phase. In this case, SHG approaches zero at ΔV = V₀ and Eqs. 6 and 7 simplify, respectively, to

$$S_0=-\frac{2{\chi }_3}{d{\chi }_2+V_R{\chi }_3},$$ (8)

$$V_0=V_R+d\frac{{\chi }_2}{{\chi }_3}=-\frac{2}{S_0}.$$ (9)

Materials and Methods

See the Supporting Material.

Results

Voltage sensitivity and minimum SHG voltage

The voltage dependence of the second harmonic signals generated by the four probes was surveyed using trans-cellular electric fields to change the voltage across the cell membrane (Fig. S2). SHG signals were excited and detected as follows. In brief, ∼100 fs pulses of linearly polarized 980 nm light from a Ti:Sapphire laser were focused using a water-immersion microscope objective onto cultured HEK-293 cells adhered to a coverslip that formed the top of a custombuilt recording chamber (Fig. S2). Transmitted SHG light was collected by an oil-immersion substage microscope condenser and detected after separation from the excitation light by a photomultiplier tube with a GaAsP photocathode (see the Supporting Material).

SHG images of HEK cells stained with any of the three retinoid probes or FM4-64 showed second harmonic emission with two maxima and two minima around the cellular circumference, consistent with SHG being maximal when the electric field vector of the incident light is perpendicular and minimal when it is parallel to the membrane (Fig. 1 a). SHG signals, recorded by line-scanning the excitation beam along a path tangential to the surface membrane (Fig. 1 a, dashed line), responded to changes in membrane potential (Fig. 1 b). SHG increased with depolarization for all three retinoid probes and decreased for FM4-64 (Fig. 1 c), consistent with the voltage sensitivity of other SHG potentiometric probes that are styryl derivatives (5,9). This suggests that the retinoid compounds and FM4-64 are oppositely oriented and/or at different places in the cell membrane (see Discussion).

Figure 1.

Optical recording of membrane potential changes with SHG in HEK-293 cells stained with ATR (I), AR-3 (II), AR-4 (III), and FM4-64 (IV). (a) Planar (x-y) SHG image of stained HEK-293 cells (for better visualization of the image background, images have been processed using a nonlinear transfer function i.e., γ = 0.7). (b) Line-scan SHG recordings (average of 25 scans) from a small membrane patch of a single cell (trajectory indicated by dotted line in panel a) during application of a sequence of alternating external electrical field pulses (electric field direction indicated in panel a). Single SHG-time traces were obtained by spatially averaging multiple image rows indicated by green bar (signal). (c) Plot of relative SHG intensity changes ($\Delta I_{SHG}/I_{SH{G}_R}$) as a function of membrane voltage change (ΔV). (Red line) Best fit to the data from four, five, seven, and two cells for ATR, AR-3, AR-4, and FM4-64, respectively, using Eq. 5. Data points are averages of 25 trials and error bars represent standard deviations. Note the inverse relationship between $\Delta I_{SHG}/I_{SH{G}_R}$ and ΔV for FM4-64 (IV, c and d). (d) Normalized SHG intensity plot versus time for 4- (green line) and 25-trial (red line) averages. (Blue line) Anticipated membrane voltage changes (blue scale). Note: a smaller scale for $\Delta I_{SHG}/I_{SH{G}_R}$ was used in panel IV, part (d).

While the sign of the voltage dependence of SHG signals was the same for all the retinoid probes, the strength of their baseline signal (I_SHG(V_R)), in the absence of an applied electric field, was different. To obtain comparably bright images for the different probes, acquisition gain settings of 32, 8, 2, and 1 were used for ATR, AR-3, AR-4, and FM4-64, respectively.

The SHG-versus-voltage data (Fig. 1 c and see later in Fig. 3, a and b) were fitted with Eq. 5 to obtain the voltage sensitivity (S₀) at the cells' resting potential (i.e., at ΔV = 0) and the minimum-SHG voltage (V₀). These parameters along with the amplitude at the SHG minima (S₀V₀/2, see Eq. 5) for each of the four probes are presented in Table 1. The measured sensitivities for ATR and FM4-64 are in good agreement with previously published values (3,5,9,12) all obtained from linear fits. For all probes except ATR, the extrapolated minimal SHG amplitudes are above the level of −1, indicating resonant behavior. For ATR, the minimum SHG amplitude (−1.15) corresponds to an impossible (negative intensity) value but is still, within error margins, consistent with a value of ≈−1 (zero intensity), suggestive of nonresonant responsiveness (see Discussion).

Figure 3.

Large-range (a) and close-up (b) reproduction of Fig. 1c; i.e., plot of relative SHG intensity changes as a function of membrane voltage change for ATR, AR-3, AR-4, and FM4-64. Large range (c) and close-up (d) of relative SHG intensity changes as a function of membrane voltage change for ATR with linear (magenta curve) and quadratic (black curve, Eq. 5) best fits and (68%) confidence intervals (dashed-dotted curves) for the quadratic fit. (e) Typical membrane-potential (black) and electric-field (green) profile and possible locations and directions of ATR (depicted in red) as inferred from a best fit to the data in panel c using Eq. 5. (Arrows, top of diagram) Location, direction, and strength of the electric fields due to the three distinct electrostatic potentials, i.e., surface- (E_s), dipole- (E_d), and transmembrane potential (E). (f) Plot of the ratio (R) of the nonresonant second and third-order susceptibilities (χ₂/χ₃) normalized to its narrow distribution value ${\left({\chi }_2/{\chi }_3\right)|}_{\sigma \to 0}$ as a function of the mean (θ₀) and width (σ) of their apparent tilt-angle distribution (P(θ)).

Table 1.

SHG-probe characteristics

SHG probe	n	S0 (1/V)	V0(V)	S0V0/2
ATR	4	2.82 ± 0.08	−0.82 (+0.13/−0.20)	−1.15 (+0.17/−0.26)
AR-3	5	2.17 ± 0.13	−0.57 (+0.11/−0.21)	−0.61 (+0.11/−0.20)
AR-4	7	2.12 ± 0.09	−0.60 (+0.08/−0.12)	−0.63 (+0.08/−0.11)
AR-4 (pc)	5	2.33 ± 0.05	−0.64 (+0.08/−0.10)	−0.73 (+0.08/−0.10)
FM4-64	2	−1.28 ± 0.10	0.69 (+0.49/−0.21)	−0.44 (+0.15/−0.34)

Symbol key: n, number of cells; S₀, average sensitivity at the resting potential; V₀, average minimum SHG voltage; S₀V₀/2, relative SHG minimum; and (pc), patch-clamp stimulation.

To check the accuracy of the estimated voltage changes produced by external field stimulation, the voltage sensitivity of SHG produced by AR-4 was determined using voltage steps applied via whole-cell voltage-clamp (Fig. S3). The estimates of S₀, V₀, and S₀V₀/2 (SHG minimum) obtained in this way agree with the results obtained using external field stimulation (Table 1).

Response time

The response time of the four probes to membrane voltage changes was initially evaluated using the line-scan mode of our imaging system at a scan speed of 1000 lines/s (Fig. 2, a–d). Under these conditions there was no discernable time delay in the response for ATR, AR-3, and AR-4 (Fig. 2, a–c). This was investigated further by increasing the temporal resolution of the measurement to the pixel dwell-time by applying voltage pulses shorter than the time needed for a single membrane pass while scanning along a small (5–10 μm) stretch of the membrane (see Material and Methods in the Supporting Material). Using this approach, the temporal resolution of our system was ∼20 μs (limited by the speed of our pulse generator) and the response of AR-3 and AR-4 to 100-μs voltage pulse showed no significant delay (Fig. 2, e and f). The SHG signal from ATR was too weak to be measured under these conditions. In contrast to the three retinoid probes, the SHG response of FM4-64 to step changes in membrane voltage was substantially slower, with a response time of ∼1.6 ms (Fig. 2 d).

Figure 2.

Average temporal response of ATR (a), FM4-64 (b), AR-3 (c), and AR-4 (d) to 25 stimuli of 8-ms duration and of AR-3 (e) and AR-4 (f) to five stimuli of 100-μs duration. A best fit to the normalized temporal response of FM4-64 in panel b using 1 − Exp(−t/τ) gives τ = 1.6 ± 0.15 ms (data not shown). Insets in panels e and f are the raw line-scan SHG recordings (nine averages) from which the graphs in panels e and f are obtained.

Discussion

For currently available fast SHG-probes, showing sensitivities (S₀) no larger than ∼10% for physiologically relevant membrane potential changes (≤±100 mV), the relationship between relative SHG intensity change and membrane voltage (Eq. 5) can be treated as a linear function, i.e.,

$$\Delta I_{SHG}/I_{SHG}\approx S_0\Delta V.$$

With higher sensitivities, however, the parabolic relationship between membrane potential and SHG signal strength becomes apparent (see Figs. 1 and 3). It is characterized by two parameters, i.e., the membrane potential (V₀) where the SHG signal is minimal (the vertex of the parabola) and the voltage sensitivity of the SHG signal (S₀), which corresponds to the slope of the parabola at the resting potential (V_R). Estimates of V₀ and S₀ were made by fitting Eq. 5 to the voltage dependence of SHG, as illustrated in Fig. 3 c using data for ATR.

Resonances

The first- and second-order nonlinear susceptibilities χ₂ and χ₃ are complex variables that are frequency-dependent in the vicinity of molecular resonances of the SHG probe. This leads to a phase shift in the components that undergo the induced harmonic oscillation that gives rise to the SHG signal, and as a consequence there is no voltage at which SHG is zero, i.e., S₀V₀/2 > −1.

Such resonance effects are expected for all probes studied except for ATR where one-, as well as two-photon, resonances are at wavelengths well below the excitation used (980 nm) and the corresponding SHG (490 nm) wavelength (21). This is not true for the other probes (see Fig. 3). FM4-64 has its excitation maximum near 515 nm (22) and the two-photon resonances for AR-3 and AR-4 are expected (23) to be significantly red-shifted compared to ATR, due to the protonated Schiff base bound to their ionone ring (Fig. S1).

Orientation in the plasma membrane

S₀ and V₀ depend on the first- and second-order nonlinear susceptibilities, i.e., χ₂ and χ₃ (Eqs. 6 and 7). The values χ₂ and χ₃ are averages of the first- and second-order molecular hyperpolarizability tensors β and γ over the angular spread of the molecule axes such that

$${\chi }_2=N\langle \beta \rangle \text{and}{\chi }_3=N\langle \gamma \rangle ,$$

where N is the number of probe molecules.

For rodlike molecules, such as ATR, β and γ are each dominated by a single axial tensor component and one can assume that all their elements are zero except the β_mmm and γ_mmmm elements, with m being the rod axis. For a thin layer of partially aligned molecules with no preferred orientation in the plane of the bilayer, the molecular distribution function depends only on the inclination (θ) of the rod axis relative to the bilayer normal (z axis; see Fig. S2) and the only nonvanishing components of χ₂ and χ₃ are

$$\begin{array}{l}{\chi }_{2_{zxx}}={\chi }_{2_{xzx}}={\chi }_{2_{xxz}}=1/2N\langle \cos \theta {\sin }^2\theta \rangle {\beta }_{mmm}\\ {\chi }_{2_{zzz}}=N\langle {\cos }^3\theta \rangle {\beta }_{mmm}\end{array}$$

and

$$\begin{array}{l}{\chi }_{3_{xxxx}}={\chi }_{3_{yyyy}}=3/8N\langle {\sin }^4\theta \rangle {\gamma }_{mmmm}\\ {\chi }_{3_{zzzz}}=N\langle {\cos }^4\theta \rangle {\gamma }_{mmmm}\\ {\chi }_{3_{xxzz}}={\chi }_{3_{yyzz}}=1/2N\langle {\sin }^2\theta {\cos }^2\theta \rangle {\gamma }_{mmmm}\\ {\chi }_{3_{xxyy}}=1/8N\langle {\sin }^4\theta \rangle {\gamma }_{mmmm}.\end{array}$$

When the plane of polarization is perpendicular to the membrane, as in our line-scan experiments, both E_ω and E_m are along z and thus second-harmonic radiance is only generated by

$${\chi }_{2_{zzz}}=N\langle {\cos }^3\theta \rangle {\beta }_{mmm}$$

and voltage-modulated SHG can only result from

$${\chi }_{3_{zzzz}}=N\langle {\cos }^4\theta \rangle {\gamma }_{mmmm}.$$

To calculate 〈cos³ θ〉 and 〈cos⁴ θ〉 the angular distribution of the probe must be known. The amphiphilic and rodlike structure of common SHG probes suggests that their orientation in lipid membrane is parallel to the lipid chains and thus, that they have a narrow molecular tilt angle distribution with a small average tilt (θ₀) from the membrane normal. However, even across the small patch (∼1 μm²) typically sampled by SHG-microscopy, membrane undulations could significantly broaden the apparent tilt-angle distribution (24). For a membrane that is flat on a molecular scale, the tilt-angle distribution of the surface normal (θ_⊥), can be described reasonably well by a modified Gaussian distribution

$$\left(1/{\sigma }_{\perp }^2\right){\theta }_{\perp }\exp \left[-{\theta }_{\perp }^2/\left(2{\sigma }_{\perp }^2\right)\right],$$

where σ_⊥ is its standard deviation (25). Convolving the probes molecular tilt-angle distribution with the surface-normal distribution yields the apparent tilt-angle distribution of the SHG probe (P(θ)). Assuming a Gaussian distribution of width σ_m and mean θ₀ for the molecular tilt angle, we find

$$P\left(\theta \right)=\frac{\sqrt{2\pi }}{\sqrt{{\sigma }_{\perp }^2+{\sigma }_m^2}}{\theta }_{\perp }\exp \left[-\frac{{\left({\theta }_{\perp }-{\theta }_0\right)}^2}{2\left({\sigma }_{\perp }^2+{\sigma }_m^2\right)}\right],$$ (10)

for which the moments 〈cos³ θ〉 and 〈cos⁴ θ〉 can be calculated in closed form. Using this distribution, the nonresonant sensitivity (Eq. 8) as a function of (θ₀), and the apparent width of the tilt-angle distribution,

$$\sigma ={\left({\sigma }_{\perp }^2+{\sigma }_m^2\right)}^{0.5}$$

was evaluated and the χ₂/χ₃ ratio was found (even for relatively broad distributions with large mean tilt angles) to deviate only slightly (see Fig. 3 f) from the narrow distribution value

$${\left({\chi }_2/{\chi }_3\right)|}_{\sigma \to 0}=\left[{\beta }_{zzz}/\left({\gamma }_{zzzz}\cos {\theta }_0\right)\right].$$

The deviations are <20% for σ ≤ 0.7 (∼40°) and θ₀ ≤ 0.9 (∼50°). Note: the sign of β_zzz depends on the direction of the molecule and flips as the probe is flipped within the membrane, whereas the sign for γ_zzzz is the same for either direction.

Location in the plasma membrane

From β, γ, S₀, and V₀ the probe's location (see below) in the plasma membrane can be deduced. The following discussions focus on ATR, because β and γ are unknown for all the other probes.

The first- and second-order molecular hyperpolarizability tensor values, which were reported for ATR (26,27) and appear to be nonresonant values, are

$$\beta =5.2·{10}^{-49}{\text{Cm}}^3{\text{V}}^{-2}\text{and}\langle \gamma \rangle =1.4·{10}^{-57}{\text{Cm}}^4{\text{V}}^{-3},$$

whereby

$${\beta }_{zzz}=\pm \beta \text{and}{\gamma }_{zzzz}=5\langle \gamma \rangle .$$

Thus for an ensemble of perfectly aligned molecules (narrow distribution) with mean tilt angle θ₀ = 0°, the χ₂/χ₃ ratio (Eq. 9) is ±73 mV/nm. With

$$S_0=2.82\pm 0{.08\text{V}}^{-1}\text{and}d=7.5\text{nm},$$

this means the local electric field at the membrane location of the probe has to be (Eq. 9)

$$\begin{array}{l}{E}_m=-\left[2/\left(d{S}_0\right)+{\chi }_2/{\chi }_3\right]\\ =-\left[2/\left(d{S}_0\right)+{\beta }_{zzz}/{\gamma }_{zzzz}\right]\\ =-28.4\pm 2.7·\text{mV}/\text{nm}\end{array}$$

for

$$\text{sign}\left({\chi }_2\right)=-\text{sign}\left({\beta }_{zzz}\right)$$

and

$$E_m=-161\pm 3\text{mV}/\text{nm}$$

for

$$\text{sign}\left({\chi }_2\right)=\text{sign}\left({\beta }_{zzz}\right).$$

For an ensemble of ATR molecules distributed equally between the leaflets, the positive and negative signs of β for oppositely aligned molecules would cancel, making

$${\chi }_2=0\text{and}{E}_m=-94.6\pm 2.7\text{mV}/\text{nm}.$$

An intermediate situation arises when there is some molecular orientation (see below). This situation can be described by introducing a reduced expression

$${\chi }_2^{\ast }=\left(2\alpha -1\right){\chi }_2,$$

where α, which may depend on E_m, is the fraction of molecules oriented in the preferred direction. Then

$$E_m=-\left[2/\left(d{S}_0\right)+{\chi }_2^{\ast }/{\chi }_3\right].$$

All three estimates of the electrostatic field are negative, which means that the transition dipole of ATR is located at a point in the membrane where E_m is negative (i.e., points toward the cytoplasm).

The potential across the membrane follows a complicated profile (Fig. 3 e) with three components contributing:

• 1.The overall (trans-) membrane potential (V);
• 2.The surface potential (V_S), due to fixed charges on lipid headgroups causing local differences in the anion and cation concentrations at the membrane-solution interface; and
• 3.The dipole potential (E_d), which arises from dipolar lipid residues in the interior of the membrane.

The sign of the local electric field is negative when the slope of the potential gradient is negative as seen from the outside looking into the cell. The requirement that the ATR transition dipole is located in a region where the local field is negative allows for only two locations (labeled 1 and 2 in Fig. 3 e) or an asymmetric distribution that favors either of those locations. One is between the two dipole layers and the other is in the dipole layer, just beneath the cytoplasmic surface of the membrane. In both cases, the lipophilic β-ionone ring and the induced dipole moment of ATR points away from the aqueous phase. The possibility that ATR is located in the extracellular surface-potential layer (location 3 in Fig. 3 e), where the local electric field is negative, is ruled out by the fact that the field strength there is independent of the cell's membrane potential and the SHG probe at this location should be independent of the membrane voltage.

The local electric field in the dipole layer (location 2 in Fig. 3 e), calculated on the basis of a 240 mV dipole potential (28) across a 1-nm-thick dipole layer (29), is –240 mV/nm. This is close to the electric field estimated from the β/γ ratio and S₀ for sign (χ₂) = sign (β_zzz), i.e., ∼–160 mV/nm, suggesting that ATR is located within the dipole layer. Studies of the insertion and distribution of ATR in biological membranes (30,31) suggest, however, that it is positioned between the dipole layers, where the field strength is expected to be at least an order-of-magnitude smaller, i.e., ∼60 mV/7.5 nm = 8 mV/nm, rather than in them. However, because ATR carries no charge and its polar aldehyde group is only weakly hydrophilic, it may shuttle (32) between the leaflets of the bilayer and equilibrate with a millisecond time constant, as has been reported for retinol, a molecule structurally similar to ATR (33).

As a result, ATR could be expected (30) to experience a wide distribution of insertion depths and increasing motional freedom (wider angular distribution) with depth. In contrast, for strongly amphiphilic dyes such as FM4-64, the polar end will always point toward the aqueous phase—resulting in the orientation being reversed for molecules in different leaflets. As a result, no χ₂-based SHG is generated when a flip-flopping dye is evenly distributed between the leaflets. However, χ₂-based SHG could be produced by hydrophobic or only weakly amphiphilic molecules such as ATR even if they are distributed across the membrane, as long as there is a mechanism that creates a preferred orientation. The most likely candidate for this is the trans-membrane field, which, due to the dipole moment of ATR, would favor parallel over antiparallel ATR alignment but not necessarily a preferential population in either membrane leaflet.

In the case that ATR can shuttle freely between leaflets and asymmetry in the distribution is only determined through orientation in the membrane field, the fraction of molecules oriented in the preferred direction (α) is given by

$$\alpha =\frac{1}{1+\text{Exp}\left[-2\mu E_m/\left(kT\right)\right]},$$

where μ is the dipole moment, k is the Boltzmann constant, and T is the temperature. One can calculate the field strength at the molecules location (given that the position of the molecules is relatively confined, i.e., all molecules see a similar field strength) using

$$\begin{array}{l}{E}_m=-\frac{2}{d{S}_0}-\frac{{\chi }_2^{\ast }}{{\chi }_3}=-\frac{2}{d{S}_0}-\frac{\left(2\alpha -1\right){\chi }_2}{{\chi }_3}\\ =-\frac{2}{d{S}_0}-\left(\frac{2}{1+\text{Exp}\left[-2\mu E_m/\left(kT\right)\right]}-1\right)\frac{{\beta }_{zzz}}{{\gamma }_{zzzz}\cos {\theta }_0}.\end{array}$$

With the permanent dipole moment of ATR (34) μ ≈ 5.3 Debye, T = 292 K, and d = 7.5 nm, we find α = 0.755 and E_m = −128.3 mV nm⁻¹ close to the field estimated from the β/γ ratio for a perfectly aligned ATR ensemble with sign (χ₂) = sign (β_zzz), i.e., ∼–161 mV/nm. Both estimates suggest that ATR is located at least partly in the dipole layer. Of course, α varies across the membrane (for ATR from 0.89 in the dipole layers and 0.52 in between them, assuming field strengths of 240 mV/nm and 8 mV/nm, respectively) and S₀ may be the result of a wide depth distribution of ATR (see above). However, because α ≈ 0.5 in the region between the dipole layers, orientation asymmetry would likely be small and SHG from this region negligible. Only the fraction of ATR molecules located in regions with strong electric fields (close to or in the dipole-layer) are expected to provide for a highly asymmetric distribution necessary for SHG and to dominate the measurement of S₀.

In the case of FM4-64, the situation is somewhat simplified because it is known that it inserts into biological membranes via its lipophilic tail with the pyridinium dicationic head anchored at the membrane surface (22) spanning the dipole layer. The direction of the induced change in its dipole moment upon excitation has not been established. But because the electric field in the dipole layer of the outer membrane leaflet is ∼240 mV/nm (directed toward the extracellular side) and a depolarizing voltage step (a change in trans-membrane field that also points toward the extracellular side) decreases SHG by FM4-64, the induced dipole moment must also point toward the extracellular side (i.e., in the same direction as its permanent dipole moment). Using Eq. 9 with the observed S₀ for FM4-64 (−1.28/V) and the calculated dipole field strength (240 mV/nm) gives an estimated χ₂/χ₃ of 176 mV/nm, ∼2.4 times the value expected for a perfectly aligned ensemble of ATR (73 mV/nm), which is roughly the ratio of the voltage sensitivities of ATR and FM4-64 (∼2.2).

Response time

Although the response of the tested retinoids to changes in the electric field showed no significant delay within the temporal resolution of our measurements (∼20 μs for AR-3 and AR-4 and 1 ms for ATR), the response of FM4-64 was significantly slower (∼1.6 ms; see Fig. 2 d). Because an electrooptic mechanism for SHG voltage sensitivity, which depends on intramolecular charge-transfer, has a much shorter characteristic response time (approximately picoseconds), it is unlikely to be the sole mechanism by which FM4-64 responds to changes in membrane voltage. Molecular realignment of FM4-64 in the cell membrane likely plays a role in the voltage dependence of its SHG response (35). Previous work (3,5,9) using FM4-64 for fast potentiometric measurements did not quantify the response time but some of the data presented show a noticeable (see, for example, Fig. 2c in Dombeck et al. (9)) lag of the SHG response upon rising and falling of the applied voltage.

This agrees with earlier studies of styryl dyes (4,36) indicating that small changes in side chains can increase or decrease the speed of the response and that non-electro-optic mechanisms may play a dominate role. In contrast Jiang et al. (10) reported that, FM4-64 responds to voltage changes on a submillisecond timescale and the optical polarization of the SHG signal is insensitive to changes in the membrane voltage. If, however, small changes in the side chains can increase or decrease the speed of the response, then it is possible that the tilt angle distribution, which does affect SHG, is voltage-dependent. The discrepancy in the response times between Jiang et al. (10) and our study might be due to the difference in the cell types used in the two studies (neocortical pyramidal neurons, and HEK-293 cells, respectively). Differences in the potentiometric performance between cell types, as have been reported for styryl SHG probes (37), also point to a non-electro-optical influence. The performance of fluorescent compounds used for potentiometric measurements have also been reported to vary from one cell type to another, with the least dependence on cell type found for the annine dyes (38).

Signal/noise considerations

Although ATR has one of the highest voltage sensitivities of any of the available potentiometric SHG-probes, it provides for a relatively low signal/noise ratio (∼2 for a single scan of a 100 mV step) which still requires extensive averaging to record membrane potential changes of a few mV. All of the other SHG probes tested here showed smaller voltage sensitivities than ATR but similar (AR-3) or substantially higher (AR-4 and FM4-64) signal/noise ratios, i.e., for single scans, ∼2, ∼4, and ∼5 for AR-3, AR-4, and FM4-64, respectively. This may be due to a lack of resonance enhancement for ATR.

If we take only shot noise into account, the theoretically expected signal/noise ratio (SNR) is given by

$$SNR=\frac{\Delta I_{SHG}\left(\Delta V\right)}{\sqrt{I_{SHG}\left(V_R\right)}}=\frac{E_{\omega }^2\Delta V\left(2d\mathrm{Re}\left[{\chi }_2{\overline{\chi }}_3\right]+\left(2V_R-\Delta V\right){\left|{\chi }_3\right|}^2\right)}{d\sqrt{d^2{\left|{\chi }_2\right|}^2+2d{V}_R\mathrm{Re}\left[{\chi }_2{\overline{\chi }}_3\right]+V_R^2{\left|{\chi }_3\right|}^2}}.$$ (11)

In the nonresonant case (χ₂ and χ₃ are real numbers) and for small changes in the membrane potential (ΔV ≪ |V_R|), Eq. 10 simplifies to

$$SNR=2E_{\omega }^2{\chi }_3\Delta V/d.$$ (12)

Far from resonance, i.e., χ₂ and χ₃ are real numbers, the shot-noise-limited SNR at resting potential depends, as Eq. 12 shows, apart from the excitation intensity (E²_ω), only on χ₃ and ΔV and on neither χ₂ nor V_R. Thus, the appropriate strategy for improving the design of voltage-dependent SHG probes would be to optimize χ₃ and make sure that the probe is exposed fully to the part of the field that depends on the membrane voltage.