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Energy Transfer to Multiple Acceptors in One,Two, or Three Dimensions

  • Chapter
Principles of Fluorescence Spectroscopy

Abstract

In the previous two chapters on energy transfer we considered primarily covalently linked donor—acceptor pairs, or situations in which there was a single acceptor near each donor. However, there are numerous situations in which there exist multiple acceptors, such as donors and acceptors dissolved in homogeneous solutions. More interesting examples of the multiple-acceptor case occur in membranes and nucleic acids. Suppose one has a lipid bilayer that contains both donors and acceptors (Figure 15.1). Each donor will be surrounded by acceptors in two dimensions. Since the acceptor distribution is random, each donor sees a different acceptor population. Hence, the intensity decay is an ensemble average and is typically non-exponential. A similar situation exists for donors and acceptors that are intercalated into double-helical DNA (Figure 15.1), except that in this case the acceptors are distributed in one-dimension along the DNA helix.

The theory for these multiple-acceptor cases is complex, even for random distributions in three dimensions. For a completely homogeneous and random solution, with no excluded volume, the form of the donor intensity decay and donor quantum yield is known, and was described by Förster.1,2 However, consider a protein with a buried fluo-rophore that serves as the donor. The exact form of the intensity decay will depend on the acceptor concentration, and on the distance of closest approach (r C) between the donor and acceptor, which could be approximated by the radius of the protein (Figure 15.1, left). The concept of a minimum D—A distance becomes particularly important for membrane-bound proteins, where r C may reflect the size of a membrane-bound protein, the presence of boundary lipid which excludes the acceptor, or the distance of the donor above the membrane surface. The theory for resonance energy transfer (RET) under these conditions is complex.

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References

  1. Förster Th. 1949. Experimentelle und theoretische Untersuchung des zwischenmolekularen Ubergangs von Elektronenanregungsenergie. Z Naturforsch A 4:321–327.

    Google Scholar 

  2. Förster Th. 1959. 10th Spiers memorial lecture, transfer mechanisms of electronic excitation. Discuss Faraday Soc 27:7–17.

    Article  Google Scholar 

  3. Bojarski C, Sienicki K. 1990. Energy transfer and migration in fluorescent solutions. In Photochemistry and photophysics, Vol. I, pp. 1–57. Ed JF Rabek. CRC Press, Boca Raton, GL.

    Google Scholar 

  4. Galanin MD. 1955. The influence of concentration on luminescence in solutions. Sov Phys JETP 1:317–325.

    Google Scholar 

  5. Maksimov MA, Rozman IM. 1962. On the energy transfer in rigid solutions. Opt Spectrosc 12:337–338.

    Google Scholar 

  6. Elkana Y, Feitelson J, Katchalski E. 1968. Effect of diffusion on transfer of electronic excitation energy. J Chem Phys 48:2399–2404.

    Article  CAS  Google Scholar 

  7. Steinberg IZ, Katchalski E. 1968. Theoretical analysis of the role of diffusion in chemical reactions, fluorescence quenching, and nonra-diative energy transfer. J Chem Phys 48(6):2404–2410.

    Article  CAS  Google Scholar 

  8. Kusba J. 1987. Long-range energy transfer in the case of material diffusion. J Luminesc 37:287–291.

    Article  CAS  Google Scholar 

  9. Yokota M, Tanimato O. 1967. Effects of diffusion on energy transfer by resonance. J Phys Soc Jpn 22(3):779–784.

    Article  CAS  Google Scholar 

  10. Gösele U, Hauser M, Klein UKA, Frey R. 1975. Diffusion and longrange energy transfer. Chem Phys Lett 34(3):519–522.

    Article  Google Scholar 

  11. Faulkner LR. 1976. Effects of diffusion on resonance energy transfer: comparisons of theory and experiment. Chem Phys Lett 43(6):552–556.

    Article  CAS  Google Scholar 

  12. Millar DP, Robbins RJ, Zewail AH. 1981. Picosecond dynamics of electronic energy transfer in condensed phases. J Chem Phys 75(8):3649–3659.

    Article  CAS  Google Scholar 

  13. Lakowicz JR, Szmacinski H, Gryczynski I, Wiczk W, Johnson ML. 1990. Influence of diffusion on excitation energy transfer in solutions by gigahertz harmonic content frequency-domain fluorometry. J Phys Chem 94:8413–8416.

    Article  CAS  Google Scholar 

  14. Birks JB, Georghiou S. 1968. Energy transfer in organic systems, VII: effect of diffusion on fluorescence decay. J Phys B 1:958–965.

    Article  Google Scholar 

  15. Tweet AO, Bellamy WD, Gains GL. 1964. Fluorescence quenching and energy transfer in monomolecular films containing chlorophyll. J Chem Phys 41:2068–2077.

    Article  CAS  Google Scholar 

  16. Koppel DE, Fleming PJ, Strittmatter P. 1979. Intramembrane positions of membrane-bound chromophores determined by excitation energy transfer. Biochemistry 24:5450–5457.

    Article  Google Scholar 

  17. Szmacinski H. 1998. Personal communication.

    Google Scholar 

  18. Maliwal BP, Kusba J, Lakowicz JR. 1994. Fluorescence energy transfer in one dimension: frequency-domain fluorescence study of DNA–fluorophore complexes. Biopolymers 35:245–255.

    Article  Google Scholar 

  19. Drake JM, Klafter J, Levitz P. 1991. Chemical and biological microstructures as probed by dynamic processes. Science 251:1574–1579.

    Article  CAS  Google Scholar 

  20. Dewey TG. 1991. Excitation energy transport in fractal aggregates. Chem Phys 150:445–451.

    Article  CAS  Google Scholar 

  21. Lianos P, Duportail G. 1993. Time-resolved fluorescence fractal analysis in lipid aggregates. Biophys Chem 48:293–299.

    Article  CAS  Google Scholar 

  22. Loura LMM, Fedorov A, Prieto M. 1996. Resonance energy transfer in a model system of membranes: applications to gel and liquid crystalline phases. Biophys J 71:1823–1836.

    Article  CAS  Google Scholar 

  23. Tamai N, Yamazaki T, Yamazaki I, Mizuma A, Mataga N. 1987. Excitation energy transfer between dye molecules adsorbed on a vesicle surface. J Phys Chem 91:3503–3508.

    Article  CAS  Google Scholar 

  24. Levitz P, Drake JM, Klafter J. 1988. Critical evaluation of the application of direct energy transfer in probing the morphology of porous solids. J Chem Phys 89(8):5224–5236.

    Article  CAS  Google Scholar 

  25. Drake JM, Levitz P, Sinha SK, Klafter J. 1988. Relaxation of excitations in porous silica gels. Chem Phys 128:199–207.

    Article  CAS  Google Scholar 

  26. Dewey TG, Datta MM. 1989. Determination of the fractal dimension of membrane protein aggregates using fluorescence energy transfer. Biophys J 56:415–420.

    Article  CAS  Google Scholar 

  27. Drake JM, Kafter J. 1990. Dynamics of confined molecular systems. Phys Today, May, pp. 46–55.

    Google Scholar 

  28. Pines D, Huppert D. 1987. Time-resolved fluorescence depolarization measurements in mesoporous silicas: the fractal approach. J Phys Chem 91(27):6569–6572.

    Article  CAS  Google Scholar 

  29. Pines D, Huppert D, Avnir D. 1988. The fractal nature of the surfaces of porous silicas as revealed in electronic energy transfer between adsorbates: comparison of three donor/acceptor pairs. J Chem Phys 89:1177–1180.

    Article  CAS  Google Scholar 

  30. Nakashima K, Duhamel J, Winnik MA. 1993. Photophysical processes on a latex surface: electronic energy transfer from rho-damine dyes to malachite green. J Phys Chem 97:10702–10707.

    Article  CAS  Google Scholar 

  31. Schurr JM, Fujimoto BS, Wu P, Song L. 1992. Fluorescence studies of nucleic acids: dynamics, rigidities and structures. In Topics in fluorescence spectroscopy, Vol. 3: Biochemical applications, pp. 137–229. Ed JR Lakowicz. Plenum Press, New York.

    Chapter  Google Scholar 

  32. Mergny J-L, Slama-Schwok A, Montenay-Garestier T, Rougee M, Helene C. 1991. Fluorescence energy transfer between dimethyl-diazaperopyrenium dication and ethidium intercalated in poly d(A-T). Photochem Photobiol 53(4):555–558.

    Article  CAS  Google Scholar 

  33. Lee BW, Moon SJ, Youn MR, Kim JH, Jang HG, Kim SK. 2003. DNA mediated resonance energy transfer from 4’,6-diamidino-2-phenylindole to [Ru(1,10-phenanthroline)2L]2+. Biophys J 85:3865–3871.

    Article  CAS  Google Scholar 

  34. Murata S, Kusba J, Piszczek G, Gryczynski I, Lakowicz JR. 2000. Donor fluorescence decay analysis for energy transfer in double-helical DNA with various acceptor concentrations. Biopolymers 57:306–315.

    Article  CAS  Google Scholar 

  35. Kang JS, Lakowicz JR, Piszczek G. 2002. DNA dynamics: a fluorescence resonance energy transfer study using a long-lifetime metal–ligand complex. Arch Pharm Res 25(2):143–150.

    Article  CAS  Google Scholar 

  36. Kim J, Lee M. 2004. Observation of multi-step conformation switching in β-amyloid peptide aggregation by fluorescence resonance energy transfer. Biochem Biophys Res Commun 316:393–397.

    Article  CAS  Google Scholar 

  37. Baneyx G, Baugh L, Vogel V. 2001. Coexisting conformations of fibronectin in cell culture imaged using fluorescence resonance energy transfer. Proc Natl Acad Sci USA 98(25):14454–14468.

    Article  Google Scholar 

  38. Wolber PK, Hudson BS. 1979. An analytic solution to the Förster energy transfer problem in two dimensions. Biophys J 28:197–210.

    Article  CAS  Google Scholar 

  39. Dewey TG, Hammes GG. 1986. Calculation of fluorescence resonance energy transfer on surfaces. Biophys J 32:1023–1036.

    Article  Google Scholar 

  40. Hauser M, Klein UKA, Gosele U. 1976. Extension of Förster's theory for long-range energy transfer to donor–acceptor pairs in systems of molecular dimensions. Z Phys Chem 101:255–266.

    CAS  Google Scholar 

  41. Estep TN, Thompson TE. 1979. Energy transfer in lipid bilayers. Biophys J 26:195–208.

    Article  CAS  Google Scholar 

  42. Dobretsov GE, Kurek NK, Machov VN, Syrejshchikova TI, Yakimenko MN. 1989. Determination of fluorescent probes localization in membranes by nonradiative energy transfer. J Biochem Biophys Methods 19:259–274.

    Article  CAS  Google Scholar 

  43. Blumen A, Klafter J, Zumofen G. 1986. Influence of restricted geometries on the direct energy transfer. J Chem Phys 84(3):1307–1401.

    Article  Google Scholar 

  44. Kellerer H, Blumen A. 1984. Anisotropic excitation transfer to acceptors randomly distributed on surfaces. Biophys J 46:1–8.

    Article  CAS  Google Scholar 

  45. Yguerabide Y. 1994. Theory of establishing proximity relationships in biological membranes by excitation energy transfer measurements. Biophys J 66:683–693.

    Article  CAS  Google Scholar 

  46. Bastiaens P, de Beun A, Lackea M, Somerharja P, Vauhkomer M, Eisinger J. 1990. Resonance energy transfer from a cylindrical distribution of donors to a plan of acceptors: location of apo-B100 protein on the human low-density lipoprotein particle. Biophys J 58:665–675.

    Article  CAS  Google Scholar 

  47. Baumann J, Fayer MD. 1986. Excitation transfer in disordered two-dimensional and anisotropic three-dimensional systems: effects of spatial geometry on time-resolved observables. J Chem Phys 85:4087–4107.

    Article  CAS  Google Scholar 

  48. Zimet DB, Thevenin BJ-M, Verkman AS, Shohet SB, Abney JR. 1995. Calculation of resonance energy transfer in crowded biological membranes. Biophys J 68:1592–1603.

    Article  CAS  Google Scholar 

  49. Snyder B, Frieri E. 1982. Fluorescence energy transfer in two dimensions. Biophys J 40:137–148.

    Article  CAS  Google Scholar 

  50. Fung B, Stryer L. 1978. Surface density measurements in membranes by fluorescence resonance energy transfer. Biochemistry 17:5241–5248.

    Article  CAS  Google Scholar 

  51. Pedersen S, Jorgensen K, Baekmark TR, Mouritsen OG. 1996. Indirect evidence for lipid-domain formation in the transition region of phospholipid bilayers by two-probe fluorescence energy transfer. Biophys J 71:554–560.

    Article  CAS  Google Scholar 

  52. Wolf DE, Winiski AP, Ting AE, Bocian KM, Pagano RE. 1992. Determination of the transbilayer distribution of fluorescent lipid analogues by nonradiative fluorescence resonance energy transfer. Biochemistry 31:2865–2873.

    Article  CAS  Google Scholar 

  53. Shaklai N, Yguerabide J, Ranney HM. 1977. Interaction of hemoglobin with red blood cell membranes as shown by a fluorescent chro-mophore. Biochemistry 16(25):5585–5592.

    Article  CAS  Google Scholar 

  54. Chigaev Z, Buranda T, Dwyer DC, Prossnitz ER, Sklar LA. 2003. FRET detection of cellular α4-integrin conformational activation. Biophys J 85:3951–3962.

    Article  CAS  Google Scholar 

  55. Kusba J, Piszczek G, Gryczynski I, Johnson ML, Lakowicz JR. 2000. Effects of diffusion on energy transfer in solution using a microsecond decay time rhenium metal—ligand complex as the donor. Chem Phys Letts 319:661–668.

    Article  CAS  Google Scholar 

  56. Kusba J, Li L, Gryczynski I, Piszczek G, Johnson ML, Lakowicz JR. 2002. Lateral diffusion coefficients in membranes measured by resonance energy transfer and a new algorithm for diffusion in two dimensions. Biophys J 82:1358–1372.

    Article  CAS  Google Scholar 

  57. Kusba J, Lakowicz JR. 1994. Diffusion-modulated energy transfer and quenching: analysis by numerical integration of diffusion equation in laplace space. Methods Enzymol 240:216–262.

    Article  CAS  Google Scholar 

  58. Thomas DD, Carlsen WF, Stryer L. 1978. Fluorescence energy transfer in the rapid diffusion limit. Proc Natl Acad Sci USA 75:5746–5750.

    Article  CAS  Google Scholar 

  59. Stryer L, Thomas DD, Meares CF. 1982. Diffusion-enhanced fluorescence energy transfer. Annu Rev Biophys Bioeng 11:203–222.

    Article  CAS  Google Scholar 

  60. Thomas DD, Stryer L. 1982. Transverse location of the retinal chro-mophore of rhodopsin in rod outer segment disc membranes. J Mol Biol 154:145–157.

    Article  Google Scholar 

  61. Yeh SM, Meares CF. 1980. Characterization of transferrin metal-binding sites by diffusion-enhanced energy transfer. Biochemistry 19:5057–5062.

    Article  CAS  Google Scholar 

  62. Wensel TG, Chang C-H, Meares CF. 1985. Diffusion-enhanced lanthanide energy-transfer study of DNA-bound cobalt(III) bleomycins: comparisons of accessibility and electrostatic potential with DNA complexes of ethidium and acridine orange. Biochemistry 24:3060–3069.

    Article  CAS  Google Scholar 

  63. Stryer L, Thomas DD, Carlsen WF. 1982. Fluorescence energy transfer measurements of distances in rhodopsin and the purple membrane protein. Methods Enzymol 81:668–678.

    Article  CAS  Google Scholar 

  64. Duportail G, Merola F, Lianos P. 1995. Fluorescence energy transfer in lipid vesicles: a time-resolved analysis using stretched exponentials. J Photochem Photobiol A Chem 89:135–140.

    Article  CAS  Google Scholar 

  65. Additional References on Ret Between Unlinked Donor and Acceptor

    Google Scholar 

Experimental

  • Gupta RR, Ramachandra Rao VS, Watkins JJ. 2003. Measurement of probe diffusion in CO2-swollen polystyrene using in situ fluorescence nonradiative energy transfer. Macromolecules 36:1295–1303.

    Article  CAS  Google Scholar 

  • Martin IR, Rodriguez VD, Rodriguez-Mendoza UR, Lavin V. 1999. Energy transfer with migration: generalization of the Yokota-Tanimoto model for any kind of multipole interaction. J Chem Phys 111(3):1191–1194.

    Article  CAS  Google Scholar 

Membranes

  • Barrantes FJ. 2001. Fluorescence studies of the acetylcholine receptor: structure and dynamics in membranes and cells. J Fluoresc 11(4):273–285.

    Article  CAS  Google Scholar 

  • Gorbenko GP, Domanov YA. 2002. Energy transfer method in membrane studies: some theoretical and practical aspects. J Biochem Biophys Methods 52:45–58.

    Article  CAS  Google Scholar 

  • Jones GM, Wofsy C, Aurell C, Sklar LA. 1999. Analysis of vertical fluorescence resonance energy transfer from the surface of a small-diameter sphere. Biophys J 76:517–527.

    Article  CAS  Google Scholar 

  • Loura LMS, Castanho MARB, Fedorov A, Prieto M. 2001. A photophysi-cal study of the polyene antibiotic filipin self-aggregation and filipin-ergosterol interaction. Biochim Biophys Acta 1510:125–135.

    Article  CAS  Google Scholar 

  • Loura LMS, Castanho MARB, Fedorov A, Prieto M. 2001. Fluid–fluid membrane microheterogeneity: a fluorescence resonance energy transfer study. Biophys J 80:776–788.

    Article  CAS  Google Scholar 

  • Loura LMS, de Almeida RFM, Prieto M. 2001. Detection and characterization of membrane microheterogeneity by resonance energy transfer. J Fluoresc 11(3):197–209.

    Article  Google Scholar 

  • Matko J, Edidin M. 1997. Energy transfer methods for detecting molecular clusters on cell surfaces. Methods Enzymol 278:444–462.

    Article  CAS  Google Scholar 

  • Silvius JR, 2003. Fluorescence energy transfer reveals microdomain formation at physiological temperatures in lipid mixtures modeling the outer leaflet of the plasma membrane. Biophys J 85:1034–1045.

    Article  CAS  Google Scholar 

Micelles

  • De S, Girigoswami A. 2004. Fluorescence resonance energy transfer—a spectroscopic probe for organized surfactant media. J Colloid Interface Sci 271:485–495.

    Article  CAS  Google Scholar 

Particles

  • Arnold S, Holler S, Druger SD. 1996. Imaging enhanced energy transfer in a levitated aerosol particle. J Chem Phys 104(19):7741–7748.

    Article  CAS  Google Scholar 

  • Arnold S, 1997. Cavity-enhanced fluorescence decay rates from micro-droplets. J Chem Phys 106(19):8280–8282.

    Article  CAS  Google Scholar 

  • Barnes MD, Whitten WB, Ramsey JM. 1994. Probing femtosecond dynamics in solution on a picosecond time scale: cavity enhancement of spontaneous emission rates in microdroplets. Chem Phys Lett 227:628–632.

    Article  CAS  Google Scholar 

  • Caruso F, Donath E, Möhwald H. 1998. Influence of polyelectrolyte multilayer coatings on Förster resonance energy transfer between 6-car-boxyfluorescein and rhodamine B-labeled particles in aqueous solution. J Phys Chem B 102:2011–2016.

    Article  CAS  Google Scholar 

  • Druger SD, Arnold S, Folan LM. 1987. Theory of enhanced energy transfer between molecules embedded spherical dielectric particles. J Chem Phys 87:2649–2659.

    Article  CAS  Google Scholar 

  • Gan D, Lyon LA. 2001. Interfacial nonradiative energy transfer in responsive core-shell hydrogel nanoparticles. J Am Chem Soc 123:8203–8209.

    Article  CAS  Google Scholar 

  • Jeuris K, Vanoppen P, De Schryver FC, Hofstraat JW, van der Ven LGJ, van Velde JW. 1998. Fluorescence intensity of dye containing latex particles studied by near-field scanning optical microscopy. Macromolecules 31:8579–8584.

    Article  CAS  Google Scholar 

  • Li. Y, Kuwabara H, Gong Y-K, Takaki Y, Nakashima K. 2003. Resonance energy transfer from dibucaine to acriflavine in polystyrene latex dispersions. J Photochem Photobiol B: Biol 70:171–176.

    Article  CAS  Google Scholar 

Polymers

  • Tcherkasskaya O, Spiro JG, Ni S, Winnik MA. 1996. Energy transfer in restricted geometry: polyisoprene-poly(methyl methacrylate) block copolymer interfaces. J Phys Chem 100:7114–7121.

    Article  CAS  Google Scholar 

  • Tcherkasskaya O, Ni S, Winnik MA. 1996. Direct energy transfer studies of the domain-boundary interface in polyisoprene–poly(methyl methacrylate) block copolymer films. Macromolecules 29:610–616.

    Article  CAS  Google Scholar 

  • Tcherkasskaya O, Ni S, Winnik MA. 1997. Energy transfer studies of binary block copolymer blends, 1: effect of composition on the interface area per chain and the lamellar size. Macromolecules 30:2623–2632.

    Article  CAS  Google Scholar 

  • Yekta A, Spiro JG, Winnik MA. 1998. A critical evaluation of direct energy transfer as a tool for analysis of nanoscale morphologies in polymers: application to block copolymer interfaces. J Phys Chem B 102:7960–7970.

    Article  CAS  Google Scholar 

Theory, Assemblies

  • de Jonge JJ, Ratner MA, de Leeuw SW, Simonis RO. 2004. Molecular dipole chains, III: energy Transfer. J Phys Chem B 108:2666–2675.

    Article  CAS  Google Scholar 

  • Scholes GD, Jordanides XJ, Fleming GR. 2001. Adapting the Förster theory of energy transfer for modeling dynamics in aggregated molecular assemblies. J Phys Chem B 105:1640–1651.

    Article  CAS  Google Scholar 

Theory without Diffusion

  • Rolinski OJ, Birch DJS. 2000. Determination of acceptor distribution from fluorescence resonance energy transfer: theory and simulation. J Chem Phys 112(20):8923–8933.

    Article  CAS  Google Scholar 

  • Yekta A, Winnik MA, Farinha JPS, Martinho JMG. 1997. Dipole–dipole electronic energy transfer: fluorescence decay functions for arbitrary distributions of donors and acceptors, II: systems with spherical symmetry. J Phys Chem A 101:1787–1792.

    Article  CAS  Google Scholar 

Theory With Diffusion

  • Bandyopadhyay T, Ghosh SK. 2003. Diffusion-assisted end-to-end relaxation of a flexible Rouse polymer chain: fluorescence quenching through a model energy transfer. J Chem Phys 119(1):572–584.

    Article  CAS  Google Scholar 

  • Güzntürk KS, Giz AT, Pekcan Ö. 1998. Monte-Carlo simulation of fluorescence decay profiles during interdiffusion of donor–acceptor spheres to mimic latex film formation. Eur Polym J 34(5/6):789–795.

    Google Scholar 

  • Krishna MMG, Das R, Periasamy N, Nityananda R. 2000. Translational diffusion of fluorescent probes on a sphere: Monte Carlo simulations, theory, and fluorescence anisotropy experiments. Chem Phys 112(19):8502–8514.

    CAS  Google Scholar 

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  1. Center for Fluorescence Spectroscopy, University of Maryland School of Medicine, Baltimore, MD, 21201, USA

    Joseph R. Lakowicz

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(2006). Energy Transfer to Multiple Acceptors in One,Two, or Three Dimensions. In: Lakowicz, J.R. (eds) Principles of Fluorescence Spectroscopy. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-46312-4_15

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