Two-Photon Absorption Properties of Gold Fluorescent Protein: A

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B: Biophysics; Physical Chemistry of Biological Systems and Biomolecules

Two-Photon Absorption Properties of Gold Fluorescent Protein: A Combined Molecular Dynamics and Quantum Chemistry Study Yusuf Simsek, and Alex Brown J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b00885 • Publication Date (Web): 09 May 2018 Downloaded from http://pubs.acs.org on May 10, 2018

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The Journal of Physical Chemistry

Two-photon Absorption Properties of Gold Fluorescent Protein: A Combined Molecular Dynamics and Quantum Chemistry Study Yusuf Simsek†,‡ and Alex Brown∗,‡ Vocational School of Health Services, Gazi University, Ankara, Turkey, and Department of Chemistry, University of Alberta, Edmonton, AB, Canada, T6G 2G2 E-mail: [email protected] Phone: +1(780)-492-1854

∗

To whom correspondence should be addressed Gazi University ‡ University of Alberta †

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Abstract Molecular dynamic (MD) simulations were carried out to obtain the conformational changes of the chromophore in the gold fluorescent protein (PDB ID: 1OXF). To obtain two-photon absorption (TPA) cross-sections, time dependent density functional theory (TD-DFT) computations were performed for chromophore geometries sampled along the trajectory. The TD-DFT computations used the CAM-B3LYP functional and 631+G(d) basis set. Results showed that two dihedral angles change remarkably over the simulation time. TPA cross-sections were found to average 13.82 GM for the excitation to S1 computed from the equilibrium geometries; however, extending the structures with water molecule and GLU residue, which make H bonding with the chromophore molecule, increased excitation energies and TPA cross-sections significantly. Besides the effects of surrounding residues and the dihedrals on the spectroscopic properties, some bond lengths affected the excitation energies and the TPA cross-sections significantly (up to ±25–30%) while the effects of bond angles were smaller (±5%). Overall the present results provide insight in the effects of conformational flexibility on TPA (with gold fluorescent protein as a specific example) and suggest that further experimental measurements of TPA for gold fluorescent protein should be undertaken.

Introduction Two-photon absorption (TPA) is involves the simultaneous absorption of two photons, which may have equal or different frequencies, to excite a molecule from its ground state to a higher energy electronic state. TPA was first investigated theoretically by Goeppert Mayer in 1931. 1 After the invention of lasers, TPA was first experimentally observed by Kaiser and Garrett in 1960. 2 As seen in Figure 1, for the case of degenerate (or one-color) TPA, two photons, which each have half of the energy (twice the wavelength) required for one photon absorption (OPA), can promote the excitation, and thus absorption occurs in the molecule. For nondegenerate TPA, the energies of each photon are different. 3 All of the photons generated

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from a laser source have the same energies. When a fluorescent material is illuminated with a single laser source, one is interested in the degenerate TPA (see Figure 1 b) rather than the non-degenerate TPA (see Figure 1 c). Although, the TPA process is weak relative to the OPA, the longer wavelength for TPA can penetrate more deeply into the absorbing material. 4–6

Figure 1: Schematic diagrams of (a) one photon absorption (OPA), (b) degenerate two photon absorption (TPA), and (c) non-degenerate TPA, along with the corresponding fluorescence.

The distinct properties of TPA versus OPA provide a wide range of potential applications such as TPA fluorescence imaging, 7–9 optical data storage, 10 phototherapy, 11,12 optical power limiting and sensors. 13 Fluorescence proteins having a good quality of two-photon excitation properties are useful for imaging of living systems. 14,15 To visualize the dynamics of biological process such as signal transmission activity in a neuron cell 16 or meiosis, 17 the imaging technique needs to be less hazardous. Infrared (IR) excitation causes less damage than UV-Vis excitation and due to its deeper penetration in biological tissues, 18,19 IR excitation gives imaging from thick samples, which is not possible at shorter wavelength due to strong absorption and diffusion of surrounding tissues elements. 20 Similarly, two-photon excitation can penetrate deeper in a tissue 21 and gives a better resolution of three-dimensional images. 22 Additionally, two-photon excitation provides reduced out of focus bleaching and less autofluorescence. 23 Therefore, TPA has been extensively studied both theoretically 24–27

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and experimentally. 28–33 One of the interesting applications of TPA is related to bio-imaging with fluorescent proteins. Gold fluorescent protein (GdFP) is one of the engineered fluorescent proteins, which is the most red-shifted Aequorea victoria green fluorescent protein (avGFP) variant. Budisa and coworkers developed this GdFP by incorporating an electron-donating amino group into the chromophore of the enhanced cyan fluorescent protein (ECFP). 33 They claimed that insertion of the amino group into the chromophore increases the red-shift of the chromophore by changing its dipole moment strongly. Their measurements showed that the UV-profile of the GdFP has only one band, which is much broader and has a red-shifted (λmax = 466 nm) absorption maximum compared to the original ECFP. However, the fluorescence emission of the GdFP has two bands. One of them is significantly red-shifted (λmax2 = 574 nm) and the other is a weak shoulder (λmax1 = 480 nm), which is produced by the native ECFP molecules. On the other hand, ECFP has two slightly wavelength-shifted maxima in both its excitation (λmax1 = 434 nm; λmax2 = 452 nm) and emission spectra (λmax1 = 476 nm; λmax2 = 505 nm). Nifosi and coworkers computed TPA cross sections of various models of FP chromophores by using the B3LYP functional and 6-31+G* basis set in their TD-DFT computations. 34 They found remarkably strong two-photon absorption cross-section for blue FPs (BFPs), cyan FPs (CFPs), and GFPs in the range of 500–700 nm. Likewise, zFP538, mOrange, asFP595, and Kaede showed similar TPA cross-section values in the range of 600700 nm. For this reason, they proposed that these spectral windows of TPA may provide advantages for the examination of biological samples, which require more penetration depth to compensate for reduced absorption due to presence of water. In a recent work, 26 Salem and Brown have examined computationally, the OPA and TPA properties of a series of FP chromophores built from GFP or redFP (RFP) with Tyr66 replaced by an non-canonical amino acid. From these studies, a number of chromophore candidates with superior TPA versus the known FPs built from canonical amino acids have been identified; one of the promising chromophores was that from GdFP. The potential for GdFP as a TPA probe

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has prompted the present computational study. To the best of our knowledge, there are no theoretical and experimental studies of the TPA cross-section for the GdFP; as well as no corresponding computational study of OPA and structural properties. In the present work, we present a combined molecular mechanics and quantum mechanics study to investigate the TPA cross-section characteristics of the GdFP. Possible chromophore conformations were obtained by molecular dynamics simulations and their TPA properties were determined via TD-DFT methods.

Computational methods Molecular Dynamics Simulations The protein structure was obtained from the Protein Data Bank service (PDB ID: 1OXF). Water molecules were kept in the structure and missing hydrogen atoms were added via the pdp4amber program with –reduced option in Ambertools 16. 35 As shown in Figure 2, the GdFP structure has two non-standard residues (5ZA and 4IN). Parameters for these residues were generated in two steps. Initially, charges were computed using the restrained electrostatic potential (RESP) fitting procedure. 36 To obtain their partial charges and optimized geometries (see Table S1 and S2 in the Supporting Information (SI)), we used the HartreeFock (HF) method with the 6-31G(d) basis set within the online R.E.D. server development tool. 37–39

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Figure 2: GdFP structure (PDB ID: 1OXF) and its non-standard residues. Atoms colored as carbon=green, nitrogen=blue, oxygen=red, and hydrogen=white.

Then, missing bond angle, torsion and improper torsion angle parameters were generated within the general Amber force field (GAFF) 40 by using the Antechamber program 41 (see Table S3 in the SI). For standard protein residues, the ff14SB force field parameters were used. 42 After parametrization, the topology of the structure was prepared in the xLeap program 43 with explicit water and placed in the center of an octahedral box (15 Å buffer distance) filled with TIP3P type water molecules. Na+ ions were added for neutralization. The system was then minimized and heated in the following fashion. Energy minimizations were completed in two stages. First, positional restraints (see the 2

Amber parameters in Listing S1 in the SI) (10 kcal/mol/Å ) were applied to the protein atoms to make them frozen and allow only the water molecules and ions to move. Then, energy minimization was continued without without restraints (see the Amber parameters in Listing S2 in the SI). After minimization, the protein structure was heated from 0K to 300K 2

over 200 ps under positional restraint (10 kcal/mol/Å ) in a constant volume (see Listing S3 in the SI). Then, 2ns NTP simulation at 300 K and 1 atm pressure was performed with 2

a smaller restraint (1 kcal/mol/Å ) to eliminate vacuum bubbles in the solvent and small gaps at the edges of the box (see Listing S4 in the SI). After heating, a 45 ns unrestrained NTP simulation was performed for a production run (see Lisitng S5 in the SI for the Amber parameters for the production run). This simulation length is enough to examine the protein 6


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stability and possible changes in the chromophore geometry. Non-bonded cutoff for the van der Waals interaction was set to 12 Å for all simulation runs. Similarly, Langevin dynamics with a collision frequency constant of 1.0 ps−1 was used for temperature regulation throughout heating and production simulations. A Berendsen type barostat was used for controlling the pressure with a pressure relaxation time (taup) of 2.0 ps. The Shake algorithm 44 was applied to constrain bonds involving hydrogen atoms, and thus a 2 fs time step could be used. We analyzed simulation trajectories via the pytraj and cpptraj programs. 45

Input preparations for QM study To see detailed geometrical changes of the chromophore molecule, another simulation with the same parameters but with a very small step length parameter for writing coordinates was continued from an arbitrary selected point, which correspond to 23.0 ns on the trajectory (see circled area in Figure 4). After stabilization of the MD simulations, 1000 continuous snapshots of the chromophore molecule were extracted from the trajectory via the VMD 46 program. Broken bonds were capped with H atoms. A few test computations using TDDFT (CAM-B3LYP/6-31+G(d,p)) showed that atoms close to the connecting residues did not have an effective orbital density and a trimmed structure produced minor differences in both excitation energy (∼1%) and TPA cross-section (∼23%) but required about 75% less computational power with respect to the full structure (see Figure 3). Therefore, we continued with the trimmed structures for the TPA computations keeping in mind that the computed TPA most likely represents a lower-bound. It should be noted that the chromophore structure considered here is similar to the GdFP chromophore examined by Salem and Brown in their TPA screening studies. 26

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Figure 3: HOMO (at the CAM-B3LYP/6-31+G(d,p) level of theory) of the full and trimmed chromophore. Iso surface set at 0.05.

TPA cross-section Singlet-singlet electronic transitions were studied via TD-DFT 47 with the CAM-B3LYP functional 48 and 6-31+G(d) basis set; 49 this computational approach was applied successfully in the previous TPA screening studies. 25,26,50 Water molecules confined in the barrel and hydrogen bonding formation around the chromophore residue were analyzed and it was found that a significant number of H-bonds have been observed (see Figure S1 in SI). As hydrogen bonds have been shown to play important roles in the photophysical properties of fluorescent proteins, 51,52 TD-DFT computations were repeated with extended structures, which include residues that make H-bonds with the chromophore or with explicit water molecules. Previous work 26 has shown that the inclusion of solvent (via PCM) only plays a minor role for excitation energies and TPA properties. The Aug2016 version of the GAMESS-US 53 software package was used for the TD-DFT computations (see an example input in Lisitng S6 in the SI). TD-DFT results computed by GAMESS-US provide TPA transition moments (δ T P A ) and excitation energies (ω) in microscopic units (au). Therefore, we need to calculate the macroscopic TPA cross-section σ T P A in cgs units with the following formula:

TPA σ(GM ) =

N π 3 a50 αω 2 TPA g(2ω, ω0 , Γ)δ(au) c

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(1)

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where α is the fine structure constant (α = 1/137), a0 is the Bohr radius (a0 = 5.2917721067× 10−9 cm), c is the speed of light (c = 29979245800cm/s), ω is the energy of an excitation photon in au, and g(2ω, ω0 , Γ) is the lineshape function that describes the spectral broadening effects. For vacuum based studies, a Lorentzian lineshape function is used generally and it is defined as 54 L(2ω, ω0 , Γ) =

Γ 1 , π (2ω − ω0 )2 + Γ2

(2)

where Γ is an empirical damping parameter, which is used as a broadening factor of the lineshape. In the previous work, 26 Salem and Brown used a Lorentzian line-shape function. To obtain the maximum value of G(2ω, ω0 , Γ), ω must be taken as ω0 /2. Equation (2) then reduces to L(ω0 ) =

1 . πΓ

(3)

Finally, when Equation (3) is introduced in Equation (1), the TPA cross-section in macroscopic units becomes TPA σ(GM )

N π 2 αa0 5 ω 2 T P A = δ(au) . cΓ

(4)

Here, N is 4 and the broadening factor, Γ, is taken as 0.10 eV (Γ = 3.6749326 × 10−3 in au). 26,32,54 The unit of the TPA cross-section is GM and 1 GM corresponds to 10−50 cm4 s photon−1 .

Results and discussion The protein structure was simulated for about 45 ns. The root mean square deviation (RMSD) computed for the backbone atoms is shown in Figure 4. The average RMSD of the backbone atoms is 1.05 Å and it is within the acceptable range.

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Figure 4: RMSD of the GdFP. Average = 1.05, standard deviation = 0.09, minimum = 0.0, maximum = 1.42. 1000 snapshots within the each circled areas (starting from 23 ns and 42 ns) were used for QM input preparations.

However, there are some minor fluctuations that can be caused by movements of flexible residues at the loop regions, which do not belong to protein barrel structure, see Figure 2, where loop regions are indicated in blue. To obtain further information about fluctuations, superposition of the structure (see Figure 5) was generated from 10 equally separated snapshots (i.e., consecutive snapshots have 5 ns time intervals).

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Figure 5: The most fluxional parts of GdFP during the MD simulation are presented as superpositions of secondary structures for 10 snapshots along the trajectory. The superposition image of the structure is created from snapshots, which are taken along the trajectory at every 5 ns, with PyMol program. 55 Loop, strand and helix residues are indicated in blue, green, and red, respectively. The chromophore and water molecules are represented in orange and light blue, respectively.

Root mean square fluctuation (RMSF) analysis shows the individual fluctuations of atoms. A plot of the RMSF analysis including the most fluxional residues is shown in Figure 6; also, illustrated is a color bar providing the correlation between atoms and the secondary structure regions. As seen in Figure 6, strand residues always have lower RMSF values. On the other hand, loop residues have the highest RMSF values as they are the most motile regions. Both non-standard residues (56:4IN and 64:5ZA) exhibited less fluctuations than the other residues. Hydrogen bonding analysis showed that these non-standard residues had higher chance of H-bonding with the surrounding water molecules and carboxyl group atoms. Therefore, their freedom is limited by H-bonding (see Figure S1 and S2 in the SI for H-bonding information).

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Figure 6: RMSF of the GdFP.

Figure 7 shows a comparison of bond length changes (violin shape) throughout the simulation. DFT bond lengths, which are computed at the B3LYP/6-31(d) level of theory via the REDServer; and averages of the bond lengths are represented with white circles and horizontal lines, respectively. Overall, the average of the simulation bond lengths and DFT bond lengths coincide quite well with each other. However, the simulation results exhibit small deviations from the DFT values for some of the bond lengths. For example, the C14– N15 bond length (see Figure 8) exhibits the largest deviation with respect to its DFT bond length. This bond lies on the main chain atoms and a small extension on that bond may be caused by the force created by two connecting residues (63:LEU and 65:VAL).

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Figure 7: Changes in bond lengths (population of bond length) for the chromophore structure, see Figure 8 for atom labeling, during the MD simulation and comparison to their theoretical (◦) and average (–) values. Theoretical values from the REDServer as computed at the B3LYP/6-31+G(d) level of theory. The violin symbols have two dimensional information. Their heights show changes in the bond length and thicknesses show the occurrence number, which is normalized to 100 percent. According to the shape of violin, long and slim shapes correspond to the flexible bonds. Oppositely, short and broad shapes correspond to more rigid bonds. Bonds, which have the same amount of percentage in their maximum changes of bond length, are grouped with the same colors. Red=14%, green=15%, blue=16%, magenta=17%, orange=18%, dark green=19%, light blue=21%.

Figure 8: Structural representation of the 5ZA residue with atom names. Hydrogen atoms have been removed for clarity. 13



During the simulation, the SHAKE algorithm was used for H atoms, and therefore, all hydrogen related bonds, which are not plotted in Figure 7, have constant bond lengths. According to these results, we conclude that the performed simulation protocol provides a good agreement with the usual values of RMSD and bond length parameters and, thus, these results are consistent with the X-ray structure. 33 The trajectory analysis of the MD simulation showed that two dihedral angles in the chromophore (the tilt and twist angles, see Figure 8) change remarkably over the simulation time. Therefore, the number of geometries that have the same dihedral angles were counted and their counts were plotted with respect to the tilt (θ) and twist (φ) angles in Figure 9. The layer, which is color-coded according to the scale on the right hand side of Figure 9, has counts of the dihedral angles scanned by one degree steps. According to their histogram plots, which are depicted in white color and oriented along their corresponding axis in Figure 9, the tilt and twist angles have averages around -4 and -22 degrees, respectively.

Figure 9: The distribution of counts of the same tilt (θ) and twist (φ) angles. The vertical excitation energy, oscillator strength, and the TPA cross-sections to the lowest singlet excited state, which is the most relevant experimentally, has been computed at the TD-CAM-B3LYP/6-31G+(d) level of theory in vacuum for each of the 1000 continuous snapshots, which were taken from the circled area of the trajectory shown in Figure 4. Second and higher excitations may have larger TPA cross-sections 25,26 but they are of less utility 14


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experimentally. Therefore, only the first excited state (S1 ) was computed. For degenerate two photon excitation, each having half of the energy required for the one photon excitation, must interact with a single molecule or atom at the same time. Therefore, the wavelength of the laser source can be considered to be double the excitation wavelength. Computed wavelengths from the S0 → S1 excitation energies (see Figure 10) range between 400–500 nm and their molar extinctions were computed via the method described in the reference study 56 with a Lorentzian line broadening and intensities related to oscillator strengths. Their molar extinctions are normalized to 1 for easy comparison with the experimental absorptions. Results show that the computed spectrum (λmax = 473 nm) is 7 nm red shifted with respect to the experimental absorption measured in the reference study. 33 However, including a single water molecule associated with the chromophore into the computation caused a 5 nm red shift (λmax = 471 nm) and a reduced intensity.

Figure 10: The normalized molar extinctions obtained via Lorentzian line broadening method 56 from the oscillator strengths versus corresponding wavelengths. Experimental data is obtained via digitizing method of graph image from the reference study. 33

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Figure 11: TPA cross-sections are plotted with respect to their excitation wavelengths. Some different conformers had the same excitation wavelength but had different TPA crosssections.

TPA cross-sections with respect to the excitation wavelength, for the isolated chromophore, are shown in Figure 11. Geometries, which have the same wavelength for excitation, often produced different TPA cross-sections. Excitations, which are selected from eight sampled geometries, excited at 466 nm have TPA cross-sections averaging ∼ 13.82 GM. In a TPA study, 26 Salem and Brown evaluated the TPA cross-sections of various chromophore models. For the optimized structure of the GdFP chromophore, they found a TPA cross-section of 15 GM using a Lorentzian broadening function. Note that we computed the photophysical properties based on a trimmed geometry, see Figure 3. When the TPA cross-section of the trimmed structure of the chromophore is normalized to the full structure, its TPA cross-section may increase to about 17 GM. Figure 12 shows the effect of H-bonding with a single water molecule on excitation energies and TPA cross-sections, see Figure S1 for a snapshot of the position of the water and nearby ARG and GLU residues. Extended molecular structures (chromophore plus one water molecule) produced significantly higher energy excitations and nearly double the TPA cross-sections. Chromophore structures extended with an ARG residue, which makes a H-bond with the oxygen atom of the chromophore, produced very low (< 1.0 eV), and potentially non-physical, excitation energies. On the other hand, extending the chromophore with a trimmed GLU residue, which is also nearby the chromophore, increased the excitation

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energy and TPA cross-section by 11% and 44% respectively.

Figure 12: Comparison of excitations and TPA cross-sections computed from 60 different geometries with and without water residue. Plots marked with square (in red) belong to only chromophore molecules. Other plots marked with circular (in blue) belong to chromophore having H-bonding with water molecule.

Figure 13 shows the behavior of excitation energies, oscillator strengths, TPA crosssections, and the Λ diagnostic test 57 results with respect to the dihedral angles. Some of the conformers have the same tilt and twist angles and their excitation energies, oscillator strengths, TPA cross-sections, and Λ diagnostic tests are averaged for those tilt and twist angles. According to the density of points plotted in Figures 13 and S3 in the SI, selected portions of the trajectory, which were used for QM computations, have good agreement with the density of the tilt and twist angles plotted in Figure 9. Therefore, we can say that these results reflect the TPA cross-sections of the whole trajectory. The excitation energy histogram shows that the chromophore excites most often between 2.65 eV and 2.80 eV, which corresponds to a wavelength between 442 and 468 nm (see Figure S3-a in SI for the plot of wavelengths). According to the color codes of the excitation energies, they look randomly distributed on the plot in Figure 13-a, i.e., there is not a strong correlation between the structure and excitation energy. On the other hand, we can see slight polarization in the accumulation of the similar color code of the OPA, TPA, and Λ diagnostic plots. Oscillator strengths of most of chromophores are between 0.60 and 0.75, and TPA cross-sections are approximately 9.9 GM. Additionally, results of the Λ diagnostic tests revealed that few excitations have a charge transfer or Rydberg character as Λ > 0.45 for all 17



geometries. Conformers, which have high OPA, have lower TPA cross-sections. Moreover, the TPA cross-sections generally increase along the direction of the virtual diagonal, which runs from the top-left to the bottom-right corner of Figure 13-c.

2.5671

40 50 60 2.3

20 10

2.7 40

3.1 20

0

Tilt angle ( )

20

0 20

50 60 5.1

2.2808

20 10

29.88

14.93

40

34.9 20

0

Tilt angle ( )

20

0.3450 0

20

S0 S1

d)

0 20

0.6960 0.6525

0.5656

30

0.5221

40 60 0.40

4.96

0.2842

0.6090

10

50

9.94 20.0 40

0.4058

40 60 0.30 0.45 0.60 40 20

19.91

30

0.4667

30 50

24.89

10

20

Tilt angle ( )

S0 S1

c)

0.5275

10

2.4239

34.86

0.5883

Oscillator strength

30

0.6491

diagnostic

2.7102

S0 S1

b)

0

Twist angle ( )

20

10

2.9965 2.8534

10

20

Excitation Energy [eV]

Twist angle ( )

0

3.1397

Twist angle ( )

10

S0 S1

a)

TPA-CS [GM]

20

Twist angle ( )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4786 0.55 40

0.70 20

0

Tilt angle ( )

20

0.4351

Figure 13: a) Excitation energies, b) oscillator strengths, c) TPA cross-sections and d) Λ diagnostic tests with their histograms. Values computed at the CAM-B3LYP/6-31G(d) level of theory. To analyze the effects of the bond lengths, bond angles and dihedral angles on the TPA cross-section, Pearson correlation tests (see Figures S4–S6 in the SI) were studied between these geometrical parameters and the TPA cross-sections, which were grouped with respect to wavelength. Pearson correlation analysis showed that some of parameters (bonds: C12–C11, C11–C7, C2–N10, C16–O17; angles: N13–C12–C11, C12–C11–C7, C1–C2–N10; dihedrals: N13–C12–C11–C7, C12–C11–C7–C8 ) are positively or negatively correlated to the TPA cross-sections (see Figures 14-16). To see their individual effects, we selected the "equilibrium geometry" (average geometry over the trajectory) and changed only the targeted parameter 18


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by ±5%. Bond related TPA cross-section behaviors are plotted in Figure 14, see Figure 8 for the atom numbering. The bond between the oxygen atom (O17) and the first five membered ring (C16) has the least positive correlation (m=5.34) between its bond length and the TPA cross-section. However, the C12–C11 double bond (see Figure 8) with the first five membered ring and lies on the tilt dihedral, has a strongly negative correlation (m=42.2) between its bond length and the TPA cross-section. The most positively correlated (m=94.58) bond parameter is the C11–C7 bond, which has a single bonding to the second five membered ring and lies on the twist dihedral. The last bond, C2–N10, represents the NH2 group connected to the six membered ring of the chromophore and it has strongly negative correlation (m=-42.2) to the TPA cross-section.

Figure 14: Behavior of the TPA cross-section with respect to bond lengths. Equilibrium bond lengths are shown with white circles on the markers. The m values are the slopes (GM/Å) of best fit lines. See Figure S7 in the SI for corresponding excitation energies.

Effects of the angle parameters (see Figure 15) on the TPA cross-section are small (

Two-Photon Absorption Properties of Gold Fluorescent Protein: A

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