Heuristics from Modeling of Spectral Overlap in FÃ¶rster Resonance

Article pubs.acs.org/jcim

Cite This: J. Chem. Inf. Model. XXXX, XXX, XXX−XXX

Heuristics from Modeling of Spectral Overlap in Förster Resonance Energy Transfer (FRET) Qi Qi, Masahiko Taniguchi,* and Jonathan S. Lindsey* Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695-8204, United States

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S Supporting Information *

ABSTRACT: Among the photophysical parameters that underpin Förster resonance energy transfer (FRET), perhaps the least explored is the spectral overlap term (J). While by definition J increases linearly with acceptor molar absorption coefficient (ε(A) in M−1 cm−1), is proportional to wavelength (λ4), and depends on the degree of overlap of the donor fluorescence and acceptor absorption spectra, the question arose as to the value of J for the case of perfect spectral overlap versus that for representative fluorophores with incomplete spectral overlap. Here, Gaussian distributions of absorption and fluorescent spectra have been modeled that encompass varying degrees of overlap, full-width-at-halfmaximum (fwhm), and Stokes shift. For ε(A) = 105 M−1 cm−1 and perfect overlap, the J value (in M−1 cm−1 nm4) ranges from 1.15 × 1014 (200 nm) to 7.07 × 1016 (1000 nm), is almost linear with λ4 (average of λabs and λflu), and is nearly independent of fwhm. For visible-region fluorophores with perfectly overlapped Gaussian spectra, the resulting value of J (JG−0) is ∼0.71 ε(A)λ4 (M−1 cm−1 nm4). The experimental J values for homotransfer, as occurs in light-harvesting antennas, were calculated with spectra from a static database of 60 representative compounds (12 groups, 5 compounds each) and found to range from 4.2 × 1010 (o-xylene) to 5.3 × 1016 M−1 cm−1 nm4 (a naphthalocyanine). The degree of overlap, defined by the ratio of the experimental J to the model JG−0 for perfectly overlapped spectra, ranges from ∼0.5% (coumarin 151) to 77% (bacteriochlorophyll a). The results provide insights into how a variety of factors affect the resulting J values. The high degree of spectral overlap for (bacterio)chlorophylls prompts brief conjecture concerning the relevance of energy transfer to the question “why chlorophyll”.

1. INTRODUCTION Transfer of excited-state energy from one chromophore to another chromophore constitutes a widespread molecular process. For cases where the distance between the donor chromophore and the acceptor chromophore is sufficiently large that the electronic (i.e., absorption, fluorescence) spectra are essentially unaltered, the energy-transfer process is accurately described as a Förster process.1−4 Förster energy transfer entails the through-space coupling of the transition dipole moments of the respective donor and acceptor.5,6 Förster energy transfer7−22 is rich in scope,23−44 undergirding diverse processes ranging from photosynthetic antenna function to myriad biological assays. Nearly 20 descriptive names have been employed,45 of which Förster resonance energy transfer (FRET) may be the most popular across the life sciences. The equation for Förster energy transfer affords the rate constant for energy transfer, given knowledge of the donor excited-state lifetime in the absence of the acceptor (τ(D)), the donor fluorescence quantum yield in the absence of the acceptor (Φf(D)), and the center-to-center distance (R) between donor and acceptor (eq 1).46,47 A reparameterized version affords the distance at which energy transfer is 50% © XXXX American Chemical Society

efficient, the so-called Förster distance (R0), sidestepping the requirement for knowledge of τ(D) (eq 2).46 The equation includes four variable parameters: an orientation term (κ2),45 the refractive index of the intervening medium (n),48 Φf(D), and the spectral overlap term (J). The spectral overlap term measures in part the energy matching of the donor and acceptor chromophores and is calculated by integrating the overlap of the donor fluorescence spectrum and the acceptor absorption spectrum. The J term has been less explored until recently,49−51 and the longstanding method of calculation by physically overlaying spectra has often been a source of confusion.52 k trans =

9000 ln(10)κ 2 Φf(D) 128π 5Nn 4τ(D)R6

J

= 8.79 × 10−5(κ 2 Φf(D)Jn−4τ(D)−1R−6) (units of s−1) (1)

R 06 = 8.79 × 10−5(κ 2 Φf(D)Jn−4) (units of Å6)

(2)

Received: October 26, 2018

A

DOI: 10.1021/acs.jcim.8b00753 J. Chem. Inf. Model. XXXX, XXX, XXX−XXX

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Journal of Chemical Information and Modeling Values for n are known, those for Φf are often known, and those for κ2, while often not known, are typically assumed. For J, however, a calculation is invariably required (vide infra). Berlman reported calculations of J values for a matrix of 210 compounds undergoing energy transfer in all possible combinations;53 all nonzero results (∼25 000) were tabulated, but the absence of published spectra, and the absence of modeling of such spectra, precluded insights into the range of calculated J values. To our knowledge, no other tabulations are available nor has modeling been heretofore performed. From three unrelated lines of inquiry, we have long wondered about the typical range of J for various donor− acceptor (i.e., FRET) chromophores and how the degree of spectral overlap affects the value of J. First, as part of the ongoing development of PhotochemCAD,54−57 we have assembled spectra and photophysical parameters (including ε and Φf), but such data alone, encompassing spectra with a range of band widths and a range of Stokes shifts, provide no insights into the J value for any set of donor−acceptor pairs. Second, our interest in the design and construction of lightharvesting antennas58 has prompted considerations of suitable synthetic chromophores for use in light absorption and efficient excited-state energy-transfer processes. Third, our longstanding interest in the possible role of tetrapyrrole macrocycles (i.e., porphyrins, chlorins, bacteriochlorins)59 in the origin of life has prompted us to wonder whether (bacterio)chlorophylls are better suited for energy transfer among collections of identical chromophores (i.e., antenna function) than any other chromophore that could have been selected. The dominant energy-transfer process of interest in antenna function is self-transfer or homotransfer (A* + A ⇆ A + A*), which is inherently reversible. Heterotransfer, the energy-transfer process between nonidentical chromophores (A* + B → A + B*), also can be reversible (depending on the relative energies, Stokes shifts, band widths, and excited-state lifetimes of the respective chromophores);49 a common limiting case accrues where a substantial energy difference causes transfer to be irreversible in practice. The homotransfer process lies at the heart of antenna function in photosynthetic light-harvesting architectures, where a vast majority of chlorophylls or bacteriochlorophylls are deployed.60 Regardless, Förster energy-transfer calculations have been performed predominantly for heterotransfer processes, which are typically encountered in applications outside of photosynthetic antenna systems. In this paper, we first address the question of the interplay of spectral overlap and the J value; for simplicity, we consider transfer using Gaussian distributions. The λ4 dependency and embedded ε(A) of J values preclude a single best value for all chromophores. An alternative measure in this regard is to assess the degree of overlap for given chromophores. We define the degree of overlap as the ratio of the experimental J with a model J; the latter is approximated by Gaussian distributions with perfect overlap for the donor and acceptor spectra. We then calculate the J values for homotransfer and the degree of overlap for 60 compounds ranging from simple arenes to indocyanine green (ICG), thereby spanning the range from the ultraviolet across the visible to the near-infrared region. The 60 compounds are derived from 12 distinct classes of chromophore structures. We finish with a perspective on the effect of various J values across diverse chromophores in energy transfer. The new spectral database57 accompanying PhotochemCAD 356 has made this study possible. The results

may ultimately inform considerations of the design and selection of chromophores for use in light-harvesting architectures and other constructs where FRET is an essential phenomenon.

2. METHODS The spectra and photophysical parameters employed herein for calculations were obtained from PhotochemCAD 3. PhotochemCAD constitutes (1) a database of absorption and fluorescence spectra along with photophysical parameters (ε(A), Φf(D)); (2) a suite of modules for carrying out photophysically relevant calculations; and (3) literature references for the original spectral data and methods of calculation. The program and accompanying databases were freely downloaded at www.photochemcad.com. The calculational modules of relevance here include those for Förster energy transfer and spectral simulations; the latter were employed to generate Gaussian distributions. The reader is referred to citations therein for access to the original literature concerning spectra, photophysical parameters, and methods of calculation. Curve fitting of absorption spectra has been performed over the years using a variety of functions including Gaussian,61 Lorentzian,62,63 Voigt (mixed Gaussian−Lorentzian),64,65 and Lognormal.66,67 For example, Metzler and co-workers reported curve fitting of absorption spectra for ∼160 compounds using Lognormal distributions.68 Presumably all of the same functions could be applied for fitting fluorescence spectra. It warrants emphasis that the shape of a given band (absorption, fluorescence) is typically not symmetric for molecules in condensed media. To create an asymmetric distribution for mimicking an experimental spectrum, one approach is to convert a symmetric distribution on the wavenumber (energy) scale to the corresponding wavelength scale. When more precise curve fitting for an asymmetric spectrum is required, asymmetric (two half-width) Gaussian69 or Lorentzian70 distributions can be used. In practice, an absorption spectrum often contains multiple bands owing to a progression of vibronic transitions and, in some cases, multiple electronic states. In such cases (e.g., chlorophylls in vivo),71,72 multiple Gaussian distributions are employed to simulate experimental absorption spectra. Our objective here is not to accomplish perfect curve fitting of experimental spectra, but to employ simple approximations from which meaningful insights can be gleaned. In this regard, symmetric Gaussian distributions as a function of wavelength appear satisfactory. The calculation of the degree of overlap of Gaussian distributions can be done via a closed expression.49,73 Because our interests extend beyond the overlap to calculation of the J value (and ultimately R0 values), we have generated Gaussian distributions (see the Supporting Information) and used these as spectral files in the calculational module of PhotochemCAD 3. For each fluorophore, the absorption spectrum (and the fluorescence spectrum) often appears as a single peak owing to the coalescence of multiple individual transitions. When the band is split, the highest peak(s) of the absorption spectrum in the overlap region is chosen as λabs, and the corresponding mirror-image transition of the fluorescence spectrum is chosen as λflu, even if a longer-wavelength but lower-intensity band is present. In so doing, the area encompassed by the Gaussian distribution will be maximized. Accordingly, the definition of Stokes shift is slightly different than the typical situation, which generally concerns the respective 0−0 (origin) transitions.74 B


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Figure 1. continued

C


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Figure 1. Structures and names of 60 compounds in 12 groups.

note that the terms pigment, chromophore, and fluorophore are often used here interchangeably.

We note that the Stokes shift and fwhm are generally listed in wavenumbers (energy scale), but for convenience we have employed the wavelength scale here. Because the band positions are close, the Stokes shifts are small, the fwhm values are not extremely broad, and the calculations concern the ratio of the Stokes shift/fwhm, no conclusions change here owing to use of the wavelength versus wavenumber scale. In the following text, the term λabs refers to the peak absorption intensity of the overlapped region, not necessarily the most intense absorption band in the entire spectrum. Integrations were performed with step sizes of 1.0 nm. We

3. RESULTS 3.1. Evaluation of Experimental J Values for Representative Compounds. We chose 60 compounds from 12 groups (5 compounds from each group) spanning λabs ∼ 260−790 nm. The classes of compounds represent a rich collection of common fluorophores as follows: (A) aromatic hydrocarbons, (B) polycyclic aromatic hydrocarbons, (C) polyenes/polyynes, (D) coumarins, (E) acridines, (F) D


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Journal of Chemical Information and Modeling Table 1. Spectral Data and J Values for Homotransfera ID

compd

ε(A) (M−1 cm−1)

λabs (nm)

λflu (nm)

λavg (nm)b

fwhm (nm)c

Stokes (nm)

A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 E1 E2 E3 E4 E5 F1 F2 F3 F4 F5 G1 G2 G3 G4 G5 H1 H2 H3 H4 H5 I1 I2 I3 I4 I5 J1 J2 J3 J4 J5 K1 K2 K3 K4 K5 L1 L2 L3 L4 L5

o-xylene phenol durene aniline p-phenylenediamine 1,8-naphthalic anhydride 1,4,5,8-naphthalic dianhydride anthracene 9,10-diphenylanthracene pyranine trans-stilbene 1,4-diphenylbutadiene 1,6-diphenylhexatriene curcumin 4-DMA-4′-nitrostilbene coumarin 1 coumarin 151 coumarin 30 coumarin 343 coumarin 6 phenosafranin thionin oxazine 170 toluidine blue O methylene blue diethyloxacarbocyanine iodide merocyanine 540 diethyloxadicarbocyanine iodide diethyloxatricarbocyanine iodide indocyanine green perylene perylene, PMI perylene, PDI perylene, PMI(OR) perylene, PMI(OR)3 fluorescein rhodamine 123 eosin Y rhodamine 6G sulforhodamine B bis(5-phenyldipyrrinato)Zn bis(5-mesityldipyrrinato)Zn phenyl-BODIPY 4-iodophenyl-BODIPY 4-TMSCCphenyl-BODIPY ZnOEP MgTPP H2OEP protoporphyrin IX DME H2TPP H2N4P(tBu) ZnPc H2Pc H2Pc(OBu) H2Nc(tBu) chlorophyll b chlorophyll a pheophorbide a purpurin 18 bacteriochlorophyll a

254 2340 214 1760 1720 12 100 30 600 8800 14 000 21 600 28 200 33 000 82 400 55 000 27 000 23 500 17 000 54 000 44 300 54 000 35 600 77 600 83 000 74 000 40 700 149 000 138 000 237 000 220 000 194 000 38 500 32 000 50 000 32 000 40 000 92 300 85 700 112 000 116 000 99 000 115 000 115 000 54 000 59 000 48 800 40 700 10 500 6640 6040 4700 72 500 281 800 162 000 134 000 269 000 57 600 86 300 44 500 41 800 92 000

263 271 278 288 324 328 365 375 373 454 294 330 353 425 432 373 384 408 445 459 533 605 613 628 656 485 560 582 688 789 436 479 528 507 536 500 512 525 530 553 485 487 503 516 516 569 604 623 634 649 624 674 699 763 784 643 661 668 702 780

291 292 295 313 390 376 372 375 426 512 345 373 424 542 588 447 484 478 462 500 562 616 641 658 677 498 579 604 712 817 436 569 537 610 578 540 532 544 552 571 501 500 521 526 536 571 609 623 634 654 626 679 701 780 793 644 666 675 708 791

277 281 287 301 357 352 369 375 400 483 319 352 389 484 510 410 434 443 454 480 548 611 627 643 667 492 570 593 700 803 436 524 533 559 557 520 522 535 541 562 493 494 512 521 526 570 607 623 634 652 625 677 700 772 789 644 664 672 705 786

25 26 23 30 43 47 36 9 53 55 52 54 76 96 88 57 70 63 72 64 55 38 61 58 47 41 35 35 45 53 16 94 21 102 86 32 34 40 44 36 37 33 35 30 31 13 21 10 14 20 20 21 17 48 28 17 20 21 23 32

28 21 17 25 66 48 7 0 53 58 51 43 71 117 107 74 100 70 17 41 29 11 28 30 21 13 19 22 24 28 0 90 9 103 42 40 20 19 22 18 16 13 18 10 20 2 5 0 0 5 2 5 2 17 9 1 5 7 6 11

E

J (M−1 cm−1 nm4)

log J

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

10.6 11.9 10.9 11.9 11.3 12.9 13.8 13.4 13.6 13.7 13.1 13.5 13.2 13.7 12.9 12.8 12.4 14.2 14.9 14.4 14.6 15.4 15.4 15.5 15.3 15.3 15.6 15.9 16.3 16.6 14.5 14.4 15.0 14.3 14.5 14.1 15.1 15.3 15.3 15.5 15.2 15.2 14.8 15.1 14.9 15.1 14.5 14.4 14.5 14.3 15.6 16.4 16.3 16.3 16.7 15.6 15.8 15.6 15.7 16.3

4.18 7.84 7.99 7.99 2.07 7.54 6.92 2.79 3.92 4.73 1.27 3.30 1.59 4.71 8.56 6.02 2.29 1.57 7.53 2.37 4.14 2.79 2.74 3.05 2.05 1.98 3.67 8.05 1.88 4.12 2.86 2.31 9.24 2.09 3.17 1.29 1.26 2.04 1.96 2.95 1.42 1.66 6.81 1.36 7.47 1.31 3.06 2.55 3.35 2.09 3.80 2.51 1.80 2.23 5.32 4.18 5.76 3.69 4.49 1.91

1010 1011 1010 1011 1011 1012 1013 1013 1013 1013 1013 1013 1013 1013 1012 1012 1012 1014 1014 1014 1014 1015 1015 1015 1015 1015 1015 1015 1016 1016 1014 1014 1014 1014 1014 1014 1015 1015 1015 1015 1015 1015 1014 1015 1014 1015 1014 1014 1014 1014 1015 1016 1016 1016 1016 1015 1015 1015 1015 1016


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Journal of Chemical Information and Modeling Table 1. continued

Structures and compound names are shown in Figure 1. bAverage peak wavelength position of the absorption and fluorescence spectra. cAverage fwhm of the absorption and fluorescence spectra. a

∞

cyanines, (G) perylenes, (H) xanthenes, (I) dipyrrins, (J) porphyrins, (K) phthalocyanines, and (L) chlorins/bacteriochlorins (Figure 1). The classifications are quite general and in some cases overlap. The absorption and fluorescence spectra of each of the 60 compounds are provided in the Supporting Information (Figure S1). The J values for homotransfer were calculated using experimental spectral data from PhotochemCAD (Table 1). The spectral data in Table 1 are provided in nm; the corresponding data in cm−1 are provided in Table S1. The J values for homotransfer for the 60 compounds are plotted on a logarithmic scale for the compounds in each class (Figure 2). The J values range from 4.2 × 1010 (o-xylene) to

J=

∫0 εA (λ)FD(λ)λ 4 dλ ∞

∫0 FD(λ) dλ

=

∫0

∞

εA (λ)FD*(λ)λ 4 dλ (3)

The case of perfect overlap of the absorption and fluorescence spectra generated by Gaussian distributions centered at 600 nm with fwhm = 30 nm and ε(A) = 105 M−1 cm−1 is shown in Figure 3. The y-axis of the absorption

Figure 3. Perfect overlap of donor fluorescence and acceptor absorption spectra represented by Gaussian distributions (λabs and λflu = 600 nm, fwhm = 30 nm, ε(A) = 1 × 105 M−1 cm−1).

spectrum is based on ε(A), and that of the fluorescence spectrum is based on an integrated area = 1. The absorption spectrum has units of M−1 cm−1. The fluorescence spectrum is unitless, and λ4 is in units of nm4; thus, the resulting J value has units of (M−1 cm−1 nm4). Following eq 3, the J value is calculated as 9.18 × 1015 M−1 cm−1 nm4. The J value can be displayed in other units;46,47,75−77 a common interconversion is shown in eq 4a, noting also the equivalent (and commonly employed) units shown in eq 4b. For example, energy transfer from a perylene to a porphyrin is characterized by J = 5.8 × 10−14 mmol−1 cm6, which is identical to 5.8 × 1014 M−1 cm−1 nm4.78

Figure 2. Common logarithm of J values (homotransfer) for 60 compounds in 12 groups. Compounds and photophysical data are provided in Figure 1 and Table 1. The units of J are M−1 cm−1 nm−4.

5.3 × 1016 M−1 cm−1 nm4 (a naphthalocyanine). In general, compounds within a given class exhibit J values within a range of 102, because of little variation for the values of the respective λavg (average of λabs and λflu where the spectra overlap) and ε(A) terms. One class of compounds, the coumarins (group D), is exceptional, with J values spread across a range of ∼103. On the other hand, the calculated J values across all compounds ranged from 1011 to 1016 M−1 cm−1 nm4 (or 10−17−10−12 mmol−1 cm6). The cyanines (group F) and phthalocyanines (group K) have very high J values, whereas the aromatic hydrocarbons (group A) have low J values. The vast range of values, as well as the variation within a class of compounds, prompted a deeper examination of the factors that impact the J value. 3.2. Anatomy of the J Value: Using Gaussian Distributions. In this section, the J value is investigated using Gaussian distributions. The J value is calculated on the basis of wavelength (λ, in nm) using eq 3, where ε(A)(λ) is the molar absorption coefficient of the acceptor; FD(λ) is the fluorescence spectrum; F*D(λ) is the fluorescence spectrum normalized so that the integrated area = 1; and λ is the wavelength over which the integration is performed. Hereafter, the dependent “(λ)” is omitted for simplicity in the term ε(A).

J[M−1 cm−1 nm 4]=J × 1028 [mmol−1 cm 6]

(4a)

mmol−1 cm 6 = M−1 cm 3

(4b)

In eq 3, λ is included in the integration of dλ and cannot be treated as a constant. However, the possible range of λ is limited in common fluorophores, in which case the effect of λ4 can be approximated. J values were calculated as a function of wavelength from 200 to 1000 nm (Figure 4, panel A). The J value increases almost linearly as a factor of λ4, from 2.78 × 1014 to 4.43 × 1015 M−1 cm−1 nm4, a change of 15.9-fold in going from 250 to 500 nm. Plotting the J value divided by λ4 as a function of wavelength (Figure 4, panel B) shows the value of J/λ4 to be almost constant, ranging from 7.126 × 104 M−1 cm−1 (200 nm) to 7.075 × 104 M−1 cm−1 (1000 nm). The slight deviation from linearity stems from the contribution of the tails of the Gaussian as the wavelength increases; if delta functions were used, the relation would be exactly linear. Next, the relationship between J values and the fwhm of perfectly overlapped spectra was explored. As the fwhm of the 4

F


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Figure 4. (A) J value as a function of wavelength (ε = 1 × 105 M−1 cm−1). Gaussian distributions were employed with fwhm = 30 nm and Stokes shift = 0 nm. (B) J values divided by λ4 (nm4) as a function of wavelength; the range is 7.126 × 104 M−1 cm−1 (200 nm) to 7.075 × 104 M−1 cm−1 (1000 nm).

absorption and fluorescence spectrum is increased from 20 to 100 nm (again, centered at 600 nm), the area of the absorption spectrum increases ∼5 times whereas the intensity of the fluorescence spectrum decreases ∼5 times (Figure 5A,B). The intensity of the fluorescence spectrum must be adjusted so that the integrated area equals 1. The calculated J value is almost equal for fwhm = 20 nm (9.17 × 1015 M−1 cm−1 nm4) and fwhm = 100 nm (9.30 × 1015 M−1 cm−1 nm4). Overall, J values are relatively unchanged regardless of fwhm (Figure 5C). In conjunction with the knowledge obtained from Figures 4 and 5, the relationship between J, λ, and fwhm is summarized in a contour plot (Figure 6) and listed in Tables S2 and S3. For the ideal case of perfect overlap of absorption and fluorescence spectra (i.e., Stokes shift = 0) given by Gaussian distributions, the resulting J value (termed JG−0) divided by λ4 is nearly independent of fwhm and wavelength, which ranges from 7.07 × 104 to 7.69 × 104 M−1 cm−1. Within this very narrow range, the value of JG−0/λ4 tends to the upper limit as the wavelength gets shorter and fwhm becomes larger. For typical fluorophores (e.g., λabs and/or λflu > 400 nm, fwhm < 50 nm), the value can be approximated as 7.1 × 104 M−1 cm−1. Furthermore, the JG−0 value divided by λ4 (in nm) and the molar absorption coefficient (in M−1 cm−1) can be approximated as a constant of ∼0.71 (eq 5). The expression, derived for Gaussian spectra, is relatively insensitive to the fwhm value; in the rare situation of Lorentzian-like spectra, the constant is ∼0.51 for fwhm of 20− 30 nm (Tables S4 and S5). Note that, upon application of eq 5 with experimental spectra (where the absorption and

Figure 5. Relationship between J values and fwhm. (A) Gaussian distributions with fwhm of 20 nm (λabs = λflu = 600 nm, ε(A) = 1 × 105 M−1 cm−1). (B) Gaussian distributions with fwhm of 100 nm (λabs = λflu = 600 nm, ε(A) = 1 × 105 M−1 cm−1). (C) J values as a function of fwhm (nm) (Gaussian distribution, λabs = λflu = 600 nm, ε(A) = 1 × 105 M−1 cm−1) are almost constant (9.16 × 1015 to 9.30 × 1015 M−1 cm−1 nm4) regardless of fwhm from 5 to 100 nm.

fluorescence spectra are not perfectly overlapped), the λ should be replaced with λavg. JG − 0 ∼ 0.71ε(A)λ 4 (M−1 cm−1 nm 4)

(5)

3.3. A New Parameter: The Degree of Overlap. During the course of this work, the question arose regarding the extent to which the absorption and fluorescence spectra match. The presentation to this point concerns the absence of any Stokes shift. The effect on the value of J upon variations in the Stokes shift with Gaussian distributions is considered next. For consideration of the Stokes shift, the λavg was fixed to a constant value of 800 nm, and the fwhm for the fluorescence and absorption spectra was set to 80 nm (Figure 7). We define the “degree of overlap” of the fluorescence and absorption spectra as the ratio of the J value for overlap of experimental data compared to that for perfect overlap (Stokes shift = 0). The effect of the ratio of Stokes shift/fwhm (for 80 nm fwhm G


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Figure 8. Relationship between the degree of overlap (%) and the ratio of the Stokes shift/fwhm for six cases (Gaussian distribution, λavg = 400, 600, or 800 nm, fwhm = 20 or 80 nm) with identical results for each. The relation between the percentage degree of overlap (Y) and the Stokes shift/fwhm (X) is given by the following equation: Y (%) = 0.0011267 + 99.998{exp[−(X − 0.000018244)2/0.7213]}.

Figure 6. Contour plot of JG−0 values divided by λ4 (nm4) and ε(A) as a function of wavelength and fwhm for Gaussian distributions with perfect overlap. Values of JG−0/ε(A)λ4 are relatively unchanged (0.707 to 0.769).

and variable Stokes shift) on the degree of overlap is displayed in Figure 8. Essentially identical curves are found across a wide range of wavelength (λ = 400, 600, and 800 nm) and fwhm (20 and 80 nm) (see Table S6). The relationship between the degree of overlap (%) and the ratio of Stokes shift/fwhm shown in Figure 8 is found to follow a Gaussian distribution. Thus, while the Stokes shift is a simple barometer for describing the displacement of the fluorescence and absorption spectra, the breadth of the spectra (fwhm) must be taken into account. Even then, the degree of overlap is not a linear

correlation of either the Stokes shift or the Stokes shift/fwhm ratio. The degree of overlap is readily specified for Gaussian distributions. In practice, spectra are often more complex. There are multiple ways to describe the degree of overlap of the donor fluorescence spectrum and the acceptor absorption spectrum. For these real situations, we considered a handful of methods (methods I−V, see the Supporting Information) and evaluated each first with merocyanine 540. We chose

Figure 7. Effect of the Stokes shift and fwhm on J values represented by Gaussian distributions. (A) Perfect overlap, Stokes shift = 0 nm, λabs = λflu = 800 nm. (B) Stokes shift = 40 nm, λabs = 780 nm, λflu = 820 nm. (C) Stokes shift = 80 nm, λabs = 760 nm, λflu = 840 nm. (D) Stokes shift = 120 nm, λabs = 740 nm, λflu = 860 nm. H


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Journal of Chemical Information and Modeling merocyanine 540 because this commonly used visibleabsorbing fluorophore (λabs = 559 nm, fwhm = 34 nm) has a Stokes shift of 20 nm (λflu = 579 nm; fwhm of 35 nm) and modest spectral overlap (Figure S2). Each method was evaluated for homotransfer with merocyanine 540 (Figures S3−S6) and then also with the entire set of 60 representative compounds (Table S7). The advantages and disadvantages of the methods for assessing the degree of overlap for homotransfer with representative fluorophores having various types of spectra are described in the Supporting Information and illustrated in Figure S7; the representative fluorophores include perylene (G1), 1,4-diphenylbutadiene (C2), rhodamine 6G (H4), protoporphyrin IX dimethyl ester (J4), and H2Nc(tBu) (K5). Ultimately, we chose a method that relies on eq 5 to give a model J value for perfectly overlapped Gaussian distributions at λavg. The degree of overlap then is the experimental J value ratioed to the model J value (JG−0) as described in eq 6. The overall results for homotransfer across the 60 compounds are displayed in Figure 9. While the results concern homotransfer, the insights should be applicable to generic spectra. degree of overlap (%) = 100J / ε(A)λ 4 0.71

(6)

4. DISCUSSION Nearly 50 years ago, Berlman published a book containing the energy-transfer parameters for all combinations of 210 members of a set of organic compounds.53 From the 2102 = 44 100 calculations, the results were reported for all donor− acceptor pairs where energy transfer, including heterotransfer and homotransfer, was not zero (∼25 000). For all such cases, the values of R0 and J were reported (as well as C0, the concentration of acceptor molecules in solution such that on average there is one acceptor inside a sphere of radius R0 about the donor).53 While a precious source of data for its time, insights into the results were limited because of the absence of values for ε(A), the Stokes shift, and absorption and fluorescence spectra. Moreover, many compounds of immense present interest were not included (e.g., tetrapyrroles, carotenoids, flavonoids, cyanines). In the modeling reported here, we have examined the factors that impact the magnitude of the Förster spectral overlap term, J. In this section, we first summarize the major heuristics. We then consider the spectral overlap in a set of representative compounds. Next, the J values are evaluated for homotransfer in 60 compounds encompassing 12 classes. The evaluation of the J term is put in perspective with the other parameters required to give a Förster R0 value. Finally, we briefly consider the distinctive spectral features of (bacterio)chlorophylls with regards to homotransfer and antenna function in the context of origin-oflife considerations. 4.1. Heuristics. The findings concerning spectral overlap and J values are as follows: • For perfectly overlapped absorption and fluorescence spectra, the J value is rather independent of the fwhm of the spectra. • The degree of overlap varies in a predictable way on both Stokes shift and fwhm of Gaussian distributions. • For spectra that resemble Gaussian distributions, knowledge of the Stokes shift, average fwhm, λavg, and ε(A) enable a ready estimate of the J value. Specifically, (1) the Stokes shift/fwhm ratio is used to obtain the

Figure 9. Degree of overlap (%, left axis) and J values divided by ε(A) and λ4 (nm4) (right axis) for 60 compounds. (A) Groups A−D. (B) Groups E−H. (C) Groups I−L.

fractional degree of overlap by inspection of Figure 8; (2) the λavg and ε(A) values are used to calculate JG−0 via eq. 5; and (3) the product of the two terms gives the estimated J value. • For a wide variety of fluorophores, the upper limit to J (regardless of homo- or heterotransfer) is given by JG−0 ∼ 0.71ε(A)λ4 (M−1 cm−1 nm4). • The J value for homotransfer across diverse fluorophores ranges from 1011 to 1016 M−1 cm−1 nm4; a significant contribution to the variation is the embedded λ4 dependency and the variation in magnitude of the ε(A) value among fluorophores. Absorption and fluorescence spectra for six representative compounds are shown in Figure 10. Spectra for 60 compounds are provided in Figure S1 along with Gaussian overlays in Figures S3 and S7. The degree of overlap is poor (0.5%) for coumarin 151 (D2) but fair (12%) for coumarin 6 (D5) (Figure 10A,B). Chlorophyll a (L2) exhibits good overlap (48%) whereas bacteriochlorophyll a (L5) exhibits excellent I


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Figure 10. Spectral overlap of the experimental absorption (blue line) and fluorescence (red line) spectra. (A) Poor overlap is represented by coumarin 151 (D2). (B) Fair overlap is represented by coumarin 6 (D5). (C) Good overlap with unutilized fluorescence tail is represented by chlorophyll a (L2). (D) Excellent overlap is represented by bacteriochlorophyll a (L5). (E) Zero Stokes shift but small overall overlap is represented by anthracene (B3). (F) Sizable Stokes shift but substantial overlap is represented by indocyanine green (F5).

absorption spectrum, and a high degree of overlap (72%) is observed. Finally, the spectrum of chlorophyll a illustrates the point that the shape of the absorption spectrum at a shorter wavelength (higher energy) than the region of overlap is immaterial concerning the J value and can be simply ignored. Chlorophyll a and essentially all other tetrapyrrole macrocycles also have additional (ignorable) absorption bands at higher energy, including the vibrational progression associated with the origin band as well as transitions associated with higher electronic energy states (e.g., the Soret band). A direct comparison of J values for classes of compounds that absorb at quite different wavelengths is not meaningful because of the intrinsic λ4 dependency. A comparison that isolates the degree of overlap can be made, however, by dividing the J value by λ4 and ε(A). The results are shown in Figure 9. The value of 0.71 describes the maximum possible value (i.e., for JG−0, for perfect overlap of Gaussian

overlap (77%) (Figure 10C,D). The difference for the latter two compounds illustrates a pitfall of relying on Stokes shift alone to gauge overlap: the Stokes shift of chlorophyll a (5 nm) is smaller than that of bacteriochlorophyll a (11 nm), but the former exhibits considerable tailing of the fluorescence spectrum at a longer wavelength (>700 nm), where no overlap occurs with the absorption spectrum. Anthracene (B3) illustrates further the pitfall of relying solely on Stokes shift to gauge the degree of overlap: the Stokes shift is 0 (for the respective origin transitions for absorption and fluorescence), yet the sharpness of the bands and the size of the vibronic manifold result in an incommensurably low degree of overlap (23%) for homotransfer (Figure 10E). In contrast, indocyanine green (F5) has a Stokes shift of 28 nm, and a more narrow fluorescence spectrum (fwhm 46 nm) than absorption spectrum (fwhm 60 nm) (Figure 10F); consequently the fluorescence spectrum is predominantly covered by the J


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Journal of Chemical Information and Modeling Table 2. J Values and Forster R0 Distances for Homotransfera ID A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 E1 E2 E3 E4 E5 F1 F2 F3 F4 F5 G1 G2 G3 G4 G5 H1 H2 H3 H4 H5 I1 I2 I3 I4 I5 J1 J2 J3 J4 J5 K1 K2 K3 K4 K5 L1 L2 L3 L4 L5

compd

J (M−1 cm−1 nm4)

Φf(D)

R0 (Å)b

o-xylene phenol durene aniline p-phenylenediamine 1,8-naphthalic anhydride 1,4,5,8-naphthalic dianhydride anthracene 9,10-diphenylanthracene pyranine trans-stilbene 1,4-diphenylbutadiene 1,6-diphenylhexatriene curcumin 4-DMA-4′-nitrostilbene coumarin 1 coumarin 151 coumarin 30 coumarin 343 coumarin 6 phenosafranin thionin oxazine 170 toluidine blue O methylene blue diethyloxacarbocyanine iodide merocyanine 540 diethyloxadicarbocyanine iodide diethyloxatricarbocyanine iodide indocyanine green perylene perylene, PMI perylene, PDI perylene, PMI(OR) perylene, PMI(OR)3 fluorescein rhodamine 123 eosin Y rhodamine 6G sulforhodamine B bis(5-phenyldipyrrinato)Zn bis(5-mesityldipyrrinato)Zn phenyl-BODIPY 4-iodophenyl-BODIPY 4-TMSCC-phenyl-BODIPY ZnOEP MgTPP H2OEP protoporphyrin IX DME H2TPP H2N4P(tBu) ZnPc H2Pc H2Pc(OBu) H2Nc(tBu) chlorophyll b chlorophyll a pheophorbide a purpurin 18 bacteriochlorophyll a

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

0.17 0.075 0.3 0.17 0.065 0.32 0.13 0.36 1 1 0.32 0.36 1 1 0.7 0.5 0.53 0.8 0.63 0.78 0.2 0.04 0.63 0.076 0.04 0.05 0.39 0.49 0.49 0.05 0.94 0.91 0.97 0.82 0.86 0.97 0.86 0.67 0.95 0.7 0.006 0.36 0.053 0.23 0.078 0.045 0.15 0.13 0.1 0.12 0.21 0.3 0.6 0.13 0.01 0.117 0.32 0.28 0.08 0.2

7 10 9 12 8 19 23 24 30 31 19 24 30 31 22 20 17 36 45 39 34 36 56 40 34 35 54 65 74 58 41 40 50 38 41 36 52 54 57 58 23 47 30 42 32 32 31 29 29 28 49 72 76 61 46 46 57 52 43 64

4.18 7.84 7.99 7.99 2.07 7.54 6.92 2.79 3.92 4.73 1.27 3.30 1.59 4.71 8.56 6.02 2.29 1.57 7.53 2.37 4.14 2.79 2.74 3.05 2.05 1.98 3.67 8.05 1.88 4.12 2.86 2.31 9.24 2.09 3.17 1.29 1.26 2.04 1.96 2.95 1.42 1.66 6.81 1.36 7.47 1.31 3.06 2.55 3.35 2.09 3.80 2.51 1.80 2.23 5.32 4.18 5.76 3.69 4.49 1.91

K

10

10 1011 1010 1011 1011 1012 1013 1013 1013 1013 1013 1013 1013 1013 1012 1012 1012 1014 1014 1014 1014 1015 1015 1015 1015 1015 1015 1015 1016 1016 1014 1014 1014 1014 1014 1014 1015 1015 1015 1015 1015 1015 1014 1015 1014 1015 1014 1014 1014 1014 1015 1016 1016 1016 1016 1015 1015 1015 1015 1016


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Journal of Chemical Information and Modeling Table 2. continued a

Structures and compound names are shown in Figure 1. bFollowing eq 2, the refractive index (n) = 1.33, and the orientation factor (κ2) = 2/3.

adequately treat many experimental absorption and fluorescence spectra, the framework described here to probe the influence of various factors on the value of the J term should be applicable to the use of other symmetric or unsymmetric79 distributions. The results for Lorentzian distributions are similar (cf., Tables S3 and S5). Moreover, the overlap of such Gaussian distributions (including fwhm and Stokes shift) could be probed by mathematical operations.49 Nonetheless, the graphical displays of Gaussian distributions in the diagrams provide an intuitive perspective concerning the factors that impinge on the value of the J term. A feature of the present work is the focus on homotransfer, which accomplishes two objectives: consideration of features of chromophores relevant for antenna function, and a more general explication of issues of spectral overlap and the value of J within a reasonable set of calculations for a representative set of fluorophores. If all possible transfers (homo- and hetero-) are considered for a set of n fluorophores, a matrix of n2 calculations ensues. In Berlman’s work n = 210, and although the results for those cases where the overlap was essentially zero (e.g., for significantly endergonic transfer) were not displayed, the published tables still contained ∼25 000 entries. By limiting consideration to homotransfer, the set of n fluorophores engenders only n calculations. The use of 60 fluorophores is a manageable set for scrutiny and identification of trends. The results herein provide a facile means for assessing the value of J for a given donor−acceptor pair. The constituents of the pair can be identical, as is the case for homotransfer in light-harvesting antenna function, or nonidentical, as is the common case in most studies of Förster energy transfer. Ultimately, the J value is used in the Forster equation to give a rate constant for energy transfer (eq 1), or, if τ(D) is not known, the Förster R0 value, the distance at which half of the donor− acceptor pairs undergo energy transfer (i.e., the quantum yield of energy transfer is 50%). The J values calculated here for homotransfer for the 60 compounds have been utilized to afford the corresponding R0 values. The results are shown in Table 2. The values calculated here for J range from 4.2 × 1010 to 5.3 × 1016 M−1 cm−1 nm4 whereas the values of R0 range from 50 Å. Selected examples of the latter include F4 (diethyloxatricarbocyanine iodide), K2 (ZnPc), L2 (chlorophyll a), and L5 (bacteriochlorophyll a). It warrants emphasis that there is an inverse sixth-power relationship between J and R0; in other words, R0 α J1/6. Hence, sizable changes in J can cause substantially lesser changes in the value of R0. Said differently, reasonable yields of energy transfer can accrue with only low values of J when other parameters are appropriate, including the donor−acceptor distance of separation and the donor−acceptor orientation (here all calculations were performed assuming the dynamically averaged condition, κ2 = 2/3). The interplay of J and R0 is seen by comparing compounds such as naphthalocyanine K5 (H2Nc(tBu)) with J = 5.3 × 1016 M−1 cm−1 nm4, Φf(D) = 0.01, and R0 = 46 Å versus phthalocyanine K3 (H2Pc), which has a smaller J value (1.8 × 1016 M−1 cm−1 nm4) but larger Φf(D) value (0.6) and thus a greater propensity for homotransfer (R0 = 76 Å). Hence, J is only one of several parameters that enter

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Heuristics from Modeling of Spectral Overlap in FÃ¶rster Resonance

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