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Directionality of light absorption and emission in representative fluorescent proteins

Significance Fluorescent proteins have been used extensively in many areas of life sciences. Many of their applications rely on their various biophysical properties, such as excitation and emission wavelengths, excited state lifetimes, or sensitivity to the molecular environment. One aspect of fluorescent proteins that has remained largely neglected is the directionality of their optical properties. In the present work, we describe our experimental determination of the directionality of light absorption and emission in several commonly used fluorescent proteins. Our findings improve our understanding of fundamental properties of fluorescent proteins, and expand the possibilities of development and applications of genetically encoded fluorescent probes.


Supplementary Information Text Supplementary Methods: Calculating Ratios of Extinction Coefficients ( ∥ / ⊥ ).
Based on Lambert-Beer's law we can write for light polarized parallel and perpendicular to the long axis of crystal: The corresponding crystal transmittances can then be described as The above relationships were used to determine the values of extinction coefficients ∥ , ⊥ . However, in order to make our TDM direction determinations as accurate as possible, we wanted to avoid introducing unnecessary experimental errors (from measurements of crystal thickness and protein concentration) Therefore, for determining TDM directions, we used ratios of extinction coefficients ∥ , ⊥ , requiring only measurements of light transmission: Eq. 3 was used to calculate the values of ∥ / ⊥ for measurements with excitation wavelengths of 545 and 593 nm, whose polarization purity was better than 99.8%. For excitation with 405 nm, 458 nm and 488 nm light, we amended our calculations to include contributions of polarizations perpendicular to the one desired, in the following manner.
If the light polarized parallel to the long axis of crystal contains a fraction (1 − ∥ ) of perpendicular polarization, we can write: Similarly, for light polarized perpendicular to the long axis of crystal containing a fraction (1 − ⊥ ) of parallel polarization, we write: By combining equations 4 and 5, we can express the individual extinction coefficients: ( (1, 2) (4) Therefore, if we know the polarization purity ( ∥ , ⊥ ; Table S2), we can calculate ∥ / ⊥ as follows: For light of pure polarization, expression 8 becomes equivalent to expression 3. (8)

Supplementary Methods: Mathematical modeling of light absorption by FP crystals.
At the core of the mathematical model is a simple cos 2 relationship describing the rate of light absorption as a function of the angle between the xTDM vector and the electric field vector of the incoming light. The model takes into account (i) the distinct molecular orientations present within a crystallographic unit cell, (ii) orientation of this unit cell within a crystal, as well as (iii) orientation of the crystal within the laboratory coordinate frame. Here we illustrate the crucial steps of our approach on crystals of the P212121 space group, as implemented in the computational software Mathematica. For other space groups, calculations were carried out analogously.
After storing the list of PDB coordinates of the fluorophore atoms in a variable (fluoAtoms), the fluorophore plane was defined by calculating its normal direction (fluoNormal) using singular value decomposition:

centerLineApprox = ring1center -ring2center;
Since fluorophore atoms do not lie exactly within a plane, neither does this vector. Its projection into the fluorophore plane (centerLine) was therefore calculated and subsequently used to define the angle τ describing the TDM directions:

centerLine = Normalize[Cross[Cross[centerLineApprox, fluoNormal] , fluoNormal]]];
Coordinates of a unit-sized TDM vector within the fluorophore plane were then defined as a function of an angle τ, describing a rotation of the line connecting the centers of the aromatic rings (centerLine) around the fluorophore plane normal (fluoNormal):

pdbTDM[τ_] := RotationMatrix[τ, fluoNormal].centerLine;
TDM vectors for distinct molecules within a crystallographic unit cells can be calculated by applying crystallographic symmetry operations to the coordinates of the TDM for one of the molecules. The symmetry rotations for the P212121 space group are defined by the matrices: The coordinates of the TDMs (in the crystal coordinate system) for the four molecules within a P212121 unit cell were therefore defined as: In order to find the TDM coordinates in the laboratory coordinate system, the orientation of the crystallographic unit cell within the crystal was taken in account. In all the P212121 crystals we investigated, the crystallographic axes B, C coincided with the diagonal directions of the crystal cross-section. Therefore, in crystals positioned flat on the glass cover slip, the crystallographic unit cell was rotated with respect to the xy plane of the microscope by an angle (γ). If we define the long axis of the crystal (which in our P212121 crystals coincides with the crystallographic axis A) to be the laboratory axis x, we can describe the TDM orientations in the laboratory coordinate system: In principle, the TDM orientation within the fluorophore (τ) can then be calculated from known values of γ, polAngle, and absorptionRate. However, the absorption rate in this equation is expressed in units of an unknown size. To bypass this issue, we solved for τ an equation using the unitless ratio (or logratio) of absorption rates of light polarized parallel and perpendicular to the long axis of the crystal (polAngle equal to 0 and π/2, respectively):

Solve[absorptionRate[τ, γ, 0]/absorptionRate[τ, γ, π/2] == observedAbsRateRatio, τ]
This approach invariably yielded two values of τ (Fig 2, Table 1), of which one (designated as τ1) was consistent with results of our measurements of crystals of various tilts along their long axis (angles γ), while the other solution (designated as τ2) varied with crystal tilt, and therefore was not consistent with the physical concept of the transition dipole moment.
Apart from calculating the TDM orientation (angle τ), the above approach also allowed us to generate plots of logratios of extinction coefficients for various TDM orientations (τ) (Fig. S3). and crystal orientations (γ) (Fig. S4)