All-Optical Sensing of the Components of the Internal Local Electric Field in Proteins

Abstract

Here, we present a new all-optical method of interrogation of the internal electric field vector inside proteins. The method is based on experimental evaluation of the permanent dipole moment change upon excitation and the pure electronic transition frequency of a fluorophore embedded in a protein matrix. The permanent dipole moment change can be obtained from two-photon absorption measurements. In addition, permanent dipole moment change, tensor of polarizability change, and transition frequency for the free chromophore should be calculated quantum–mechanically. This allows obtaining the components of the electric field by considering the second-order Stark shift. We use the fluorescent protein mCherry as an example to demonstrate the applicability of the method.

1. Introduction

Knowledge of the local electric fields inside proteins can shed light on different biological problems, such as functioning of ionic channels [1], color vision [2], and molecular mechanics of enzymes [3], to name a few. Internal electric fields in proteins were probed by Stark-effect-modulated hole burning [4] and electric-field-induced splitting of absorption bands of otherwise degenerate porphyrin transitions using either low-temperature absorption [5] or electronic circular dichroism [6] spectroscopy. The Stark-shift-induced relative change of fluorescence intensity, with a preliminary calibration of the effect with respect to known externally applied field, was used to estimate the field at different locations of a voltage-gated ion channel [1]. The incremental relative change of electric field upon substitution of a neutral to charged amino acid residue at a selected location inside a protein can also be tracked with the vibrational Stark effect [7] or NMR [8]. Using molecular dynamics simulations, Callis and coauthors calculated electric field in a large number of different proteins with the aim of elucidating its role in the spectral shifts of tryptophan fluorescence [9], [10]. Most of these experimental and theoretical results suggest very large internal electric fields, i.e., on the order of 10⁶ − 10⁸ V/cm.

Yet, because of the complexity of the protein systems, reliable and efficient methods of measurement of the absolute values and directions of the field in physiological conditions are lacking. For example, hole burning and high-resolution spectroscopy methods require cryogenic temperatures and, in the former case, application of an external electric field. The circular dichroism method is limited to chromophores with degenerate electronic states. The relative changes of fluorescence due to the internal Stark effect provide an effective projection of the field on the vector corresponding to the difference between the permanent dipole moments in the ground and excited states, i.e., Δμ.

We have recently shown that the quantitative two-photon absorption (2PA) spectroscopy in combination with the one-photon absorption (1PA) spectroscopy can provide a new alternative method of measuring electric fields inside proteins at physiological conditions [11]. We used a strongly simplified 1-D model of the chromophore π-electronic system to estimate the local field projections on the chromophore Δμ vector in a series of red fluorescent proteins. In this simple model, all zero-field (vacuum) chromophore parameters, which are required for the field estimations (permanent dipole moment change, polarizability change, and transition frequency), were obtained solely from the experiment.

Here, we further develop this approach and show that from the all-optical measurements, including 1PA and 2PA spectra, and from the calculated vacuum chromophore parameters, one can quantitatively determine the two projections of the local electric field (E_x and E_y) on the x-axis and y-axis of the planar probe molecular frame. The thus obtained electric field vector (E_x, E_y) is nothing but the projection of the 3-D E vector on the molecular plane. Since the typical exogenous (such as Alexas, FITC, PE, Cys, APC, or Rhodamines) or endogenous (such as tryptophan, green or red fluorescent protein chromophores) fluorescent probes are planar molecules, this approach is quite general. Here, we use the fluorescent protein mCherry as an example to show the applicability of the approach.

2. Model

Our method is based on the measurement of the two key parameters of a probe inside a protein: 1) 1PA frequency ν and 2) change of the permanent electric dipole moment upon S₀ → S₁ excitation, i.e., |Δμ|. While the first value can be obtained from the 1PA (or fluorescence excitation) spectrum, the second requires measurements of the 2PA spectrum, 1PA and 2PA cross sections, as well as polarization ratio (i.e., the ratio of the fluorescence signals obtained upon linear and circular two-photon excitation) in the region of the first electronic 0-0 transition [11]. Several molecular parameters of the (planar) fluorophore in vacuum should be calculated quantum–mechanically. These include: 1) the components of the tensor of polarizability difference between the S₁ and S₀ states, i.e., Δα_xx, Δα_yy, where x, y is the system of coordinates in which the Δα tensor is diagonal; 2) vacuum Δμ components in the same frame, i.e., Δμ_0,x and Δμ_0,y; and (3) vacuum transition frequency, i.e., ν₀.

The magnitudes of the local electric fields in proteins are usually so strong (10⁷ − 10⁸) V/cm) [4]-[6], [9], [10] that |Δμ| acquires a significant induced part, through the molecular polarizability:

$$\mathrm{\Delta }{\mu }_x=\mathrm{\Delta }{\mu }_{x,0}+\frac{1}{2}\mathrm{\Delta }{\alpha }_{\mathit{xx}}{E}_x$$ (1)

$$\mathrm{\Delta }{\mu }_y=\mathrm{\Delta }{\mu }_{y,0}+\frac{1}{2}\mathrm{\Delta }{\alpha }_{\mathit{yy}}{E}_y.$$ (2)

If Δμ_x and Δμ_y were known, it would be possible to obtain E_x and E_y, by solving back (1) and (2):

$$E_x=2(\mathrm{\Delta }{\mu }_x-\mathrm{\Delta }{\mu }_{0,x})/\mathrm{\Delta }{\alpha }_{\mathit{xx}};E_y=2(\mathrm{\Delta }{\mu }_y-\mathrm{\Delta }{\mu }_{0,y})/\mathrm{\Delta }{\alpha }_{\mathit{yy}}.$$ (3)

The system of the two equations needed to find Δμ_x and Δμ_y is as follows:

$$\mathrm{\Delta }{\mu }_x^2+\mathrm{\Delta }{\mu }_y^2={|\mathrm{\Delta }\mu |}^2$$ (4)

$$\mathit{h\nu }=h{\nu }_0-\mathrm{\Delta }{\mu }_x{E}_x-\mathrm{\Delta }{\mu }_y{E}_y,$$ (5)

where the second equation is a standard Stark shift expression in the point dipole approximation. Substituting the field components from (3) into (5) and regrouping it, we obtain

$$\frac{2}{\mathrm{\Delta }{\alpha }_{\mathit{xx}}}{\left(\mathrm{\Delta }{\mu }_x-\frac{1}{2}\mathrm{\Delta }{\mu }_{0,x}\right)}^2+\frac{2}{\mathrm{\Delta }{\alpha }_{\mathit{yy}}}{\left(\mathrm{\Delta }{\mu }_y-\frac{1}{2}\mathrm{\Delta }{\mu }_{0,y}\right)}^2=h({\nu }_0-\nu )+\frac{\mathrm{\Delta }{\mu }_{0,x}^2}{2\mathrm{\Delta }{\alpha }_{\mathit{xx}}}+\frac{\mathrm{\Delta }{\mu }_{0,y}^2}{2\mathrm{\Delta }{\alpha }_{\mathit{yy}}}.$$ (5a)

The systems (4) and (5a) now represent two conical section curves, whose intersections have to be found.

3. Example: Red Fluorescent Protein mCherry

Our measurements for the red anionic chromophore in mCherry fluorescent protein provided |Δμ| = 2.2 D and ν = 16930 cm⁻¹ [11]. To obtain the necessary vacuum chromophore parameters, we started from the available chromophore structure inside the mCherry protein [12] and optimized its geometry using Gaussian 09 [13] and the B3LYP/6-311++G(d,p) density functional and basis set. Since the chromophore in mCherry is slightly twisted, with the dihedral angle between the phenol and imidazolinone rings ~16⁰, we froze the angles and dihedrals involving the positioning of the phenol ring relative to the imidazolinone ring in the optimization process. Using the INDO/S2-CIS method [14], S₀ → S₁ Δμ₀ was then calculated. Its absolute value is |Δμ₀| = 5.4 D, and its direction is depicted in Fig. 1 by a small red arrow (denoted x′) in the left lower corner. Our next step consisted in the calculation of the tensor of polarizability difference between the S₁ and S₀ states (Δα) in the initial coordinate system, shown in the left lower corner in Fig. 1. To this end, we calculated the Δμ components (Δμ_x′ and Δμ_y′) upon application of the uniform electric field along the x′-direction or y′-direction (E_x′ and E_y′, respectively) to a chromophore with the geometry optimized at E = 0, as described above.

Fig. 1.

Chemical structure of the chromophore inside the mCherry fluorescent protein. The initial coordinate frame in the left lower corner (x′, y′, z′) is based on the direction of the calculated Δμ₀ vector (corresponding to x′-axis). The y′-axis is chosen to be perpendicular to x′ lying in the chromophore plane, and the z′-axis is normal to the plane. The large red and green arrows correspond to a new coordinate system (x and y, respectively), which was obtained such that the change of polarizability tensor (Δα) is diagonal in it. This system is used in our model calculations.

By numerically differentiating the functions shown in Fig. 2 and taking the corresponding values at E_x′ = 0 or E_y′ = 0, we obtain the components of the Δα tensor (in ų) in the x′, y′ system of coordinates:

$$\mathrm{\Delta }\alpha =\left(\begin{array}{cc}-25& 26\\ 33& -18\end{array}\right).$$ (6)

Fig. 2.

Quantum–mechanically calculated components of the permanent dipole moment difference vector (Δμ_x′—red line and Δμ_y′—blue line) as function of the uniform electric field applied either along the x′-direction (left plot) or the y′-direction (right plot).

It is worth noting at this point that Δμ_x depends nonlinearly on E_x even in a narrow range of the fields around 0. This behavior suggests a presence of the higher order (hyperpolarizability) terms in the molecular response to the field. Here, we disregard these terms for simplicity. The diagonalization of the Δα matrix (6) by using the standard procedure yields:

$$\mathrm{\Delta }\alpha =\left(\begin{array}{cc}-51& 0\\ 0& 8\end{array}\right).$$ (7)

The average trace of the resulting tensor 1=2Tr (Δα) = −21.5 ų correlates well with our prior pure experimental determination of Δα = −20.6 ų [11], which was based on a simple 1-D model, thus substantiating the method of quantum–mechanical calculation of Δα used here. The new coordinate system (x, y), obtained as a result of diagonalization of Δα, is presented in Fig. 1 by long red and green arrows with the origin arbitrarily placed near the center of the chromophore. In this new coordinate system, the Δμ₀ vector has the components Δμ_0,x = 3.9 D and Δμ_0,y = 3.8 D. We have also calculated the vertical transition frequency of the chromophore in vacuum, i.e., ν₀ = 18400 cm⁻¹. This value is also in good agreement with the experimental estimation (ν₀ = 19300 cm⁻¹) obtained by extrapolation of the ν versus E dependence to E = 0 in a series of red FPs [11].

Now, substituting the measured values of |Δμ| and ν, as well as calculated Δμ_0,x, Δμ_0,y, Δα_xx, Δα_yy, and ν₀ in the system of (4) and (5a) and solving it graphically (see Fig. 3), we obtain the two sets of solutions: (1) Δμ_x = −2.13 D, Δμ_y = −0.53 D and (2) Δμ_x = 2.20 D, Δμ_y = 0.055 D.

Fig. 3.

Graphical solution of the system of (4) and (5a), presenting two conical sections. The red line is the equation of circle (4), and the blue line is the equation of hyperbola (5a) with shifted origin.

Now, using (3), we find the projections of the field: 1) E_x = 71 MV/cm, E_y = −320 MV/cm and 2) E_x = 20 MV/cm, E_y = −280 MV/cm. Our simple analysis based on the 1-D model [11] suggested Δμ • Δμ₀ > 0, which means that, most probably, only the second set of solutions (i.e., E_x = 20 MV/cm, E_y = −280 MV/cm) is physically meaningful. The resulting vector is presented in Fig. 4 by a blue arrow. It is interesting to compare this result with the calculations based on the knowledge of the molecular structure of the mCherry protein [12]. According to the previous theoretical consideration [15], the main charge redistribution upon optical excitation of this type of chromophore occurs between three carbon atoms constituting the bridge between the two aromatic rings, shown by bold bond lines in Fig. 4. In order to roughly estimate the E_x and E_y components of the field in the bridge region, we make use of the available crystallographic pdb file (2H5Q) and apply a coarse grain approximation, i.e., use Coulomb’s law with dielectric constant equal to 1; consider only point charges on the amino acid residues pointing inside the β-barrel: D59, K70, D81, K84, R95, E148; and assign the whole +1 elementary charge to the NZ atom of lysine and CZ atom of arginine and −1 charge to the CG atom of aspartic acid and CD atom of glutamic acid. The result of this calculation yields E_x = 29 MV/cm and E_y = −95 MV/cm for the field at the central carbon atom of the bridge. These values are in qualitative agreement with the experimental ones (E_x = 20 MV/cm, E_y = −280 MV/cm); the signs and order of magnitude of both projections are the same. The E_x values correlate better than the E_y values, probably because of much larger, and so, better defined, Δμ_x value.

Fig. 4.

Schematic view of the main part of the chromophore found in fluorescent protein mCherry. Coordinate frame is the same as in Fig. 1. The blue arrow depicts the direction of the electric field (projection onto the chromophore plane) found with experimental/quantum calculations approach presented in the text.

The discrepancies between the calculated and measured components of the field can be due to several factors, including intrinsic field nonuniformity inside the protein (making our selection of the bridge region rather arbitrary), higher than second-order molecular response (hyperpolarizability) not taken into account, assuming frozen chromophore geometry upon application of electric field, using coarse grain calculations of the field instead of a more elaborated molecular mechanics approach, uncertainties in the calculated Δα and Δμ₀ values, and others. For example, the correction due to nonlinearity of Δμ_x,y on E_x,y will depend on the range of the relevant fields. For a rather broad range of E_x,y from −5 × 10⁷ to 4 × 10⁷ V/cm, the linearization of the curves in Fig. 2 would provide effective values of Δα_xx = −50 ų and Δα_yy = 4.3 ų (versus Δα_xx = −51 ų and Δα_yy = 8.0 ų for E = 0). This will translate into slightly different components of the calculated field: 1) E_x = 71 MV/cm, E_y = −309 MV/cm or 2) E_x = 20 MV/cm, E_y = −275 MV/cm. More detailed consideration of this and other issues, listed above, will be a subject of our forthcoming publications.

4. Conclusion

We present here a new method of experimental evaluation of the components of electric field in the molecular frame of coordinates. Since this method requires neither low temperatures nor application of electric fields and, moreover, can be efficiently implemented in multiphoton microscopic setups, we believe that it will allow probing electric fields not only in protein solutions but also in live cells and tissues.