A Simple Model for Interpreting the Basics of Time Dependent Fluorescence Spectroscopy | PHYS 552, Study Guides, Projects, Research of Optics

Download A Simple Model for Interpreting the Basics of Time Dependent Fluorescence Spectroscopy | PHYS 552 and more Study Guides, Projects, Research Optics in PDF only on Docsity! A simple model for interpreting the basics of time dependent fluorescence anisotropy Athanasius Kircher Athanasius Kircher (1602-1680) beschreibt in seinem Buch »Ars Magna Lucis et Umbme* die Wirkung eines Holz extrakts von Lignum ne- phriticum in Wasser und endrtert die Nutzung von Glidhwitemchen als Haws belewchrung. Johann Wilhelm Ritter (1776-1810) entdeckte bein Experimentionn mit Silberchloridan der Universitit Jena 1801 das ultraviolette Ende des Spektrums, Johann Wilhelm Ritter Inseiner »Farberlehre® beschreit auch Johann von Goethe (1749-1832) die fluoreszie- renden Lichterscheinungen. Er fordert die Laser auf»... tauche ein frisches Stiick Baumirinde der Rosskasta- nie in ein Glas mit Wasser, sofort erscheint eine rein himmelblaue Fare.” Johann Wolfgang v. Goethe 1602-1680 1776-1810 1749-1832 Der Erfinder des Kaleido- shops, Sir David Brewster (1781-1868), beschreibt 1833 die rote Strahlung von Chlorophyll. Sir David Brewster 1781-1868 1845 entdeckte Sir John Frederick William Herschel (1792-1871) eee Fluoreszenz in einer Chininlisung. Sir John Frederick William Herschel 1792-1871 Sir George Gabriel Stokes Sir George Gabriel Stokes (1819-1903) bezeichnet 1852 die Leuchterschei- nung des Minerals Fluss- spat (Calciumfluorid) als Fluoreszenz, 1819 -1903 Max Haitinger Max Haitinger (1868-1946) priigte 1911 den Begriff Fluorochrom und gilt als Begyiimder der modermen Pluores- zenzinikroskopie und der Flucrochromierungs- technik, 1868-1946 Alexander Jablonski (1898-1980) stellt 1935 sein Modell zur Erkiérung der Fluoveszenz vor: Ein Fluorochrom besitzt unterschiedliche enengeti- sche Formen, sogenanate Singuleit-Zusttinde (SO, S1, S2...). Alexander Jablonski 1898-1980 Albert Hewett Coons Albert Hewett Coons (1912-1978) und Melvin Kaplan entwickeln ab 1950 dic Immunfluoreszenz- techuik, die im den Natur- wissenschaften zu Auf- sehen erregenden Ergeb= nnissen auf dem Gebiet der Zellforschung filrte. 1912-1978 General solution for all weird bodies A Wigner Rotation Matrix here, a Spherical Harmonic there, and a Legendre Polynomial everywhere, and you have the answer. Got it? Integrate a bit, expand in these lovely functions, orthogonalize everything, and so on. It’s easy – no sweat. A theoretician solving for the rotational anisotropy of a completely anisotropic body PHYSICAL REVIEW VOLUME 107, NUMBER 1 JULY 1, 1957 Isotropic Rotational Brownian Motion W. H. Forey Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts . (Received March 8, 1957) The Brownian motion of the orientation of any rigid body (sphere, set of rectangular axes, etc.) is calcu- lated for the case of individual random infinitesimal rotations whose probabilities are independent of the direction of the axis of rotation (isotropic case). The calculations are carried out by means of quaternions; the more important formulas are also given in terms of the rotation angle and axis direction. The unit solu- tion, or distribution of orientations arising from a specified initial orientation, is found as a series converging rapidly for not-too-small times. Tr turns out to be expressible in terms of a theta function, and thus there is also available a series converging rapidly for small times. The use of the unit solution as a propagation func- tion is discussed briefly, and is illustrated by verification of the iteration property of the unit solution, PHYSICAL REVIEW VOLUME 119, NUMBER 1 JULY 1, 1960 Theory of the Rotational Brownian Motion of a Free Rigid Body* L, Dare Fayrot Department of Physics, Harvard University, Cambridge, Massachuselts (Received February 10, 1960) The orientation of a rigid body is specified by the Cayley-Klein parameters. A system af such bodies subject to small random changes in orientation but not subject to any externally applied torque is then considered in some detail. A diffusion equation is derived with certain linear combinations of the Cayley- Klein parameters as independent variables. This equation is expressed in terms of quantum-mechanical angular momentum operators and a Green's function for the equation is obtained as an expansion in angular momentum eigenfunctions. This expansion can be used to calculate averages of various physical quantities in a nonequilibrium distribution of orientations. It may also be used to calculate the spectral density of fluctuating quantities in an equilibrium distribution. Illustrative examples of both of these applications are given. anisotropic rotational diffusion is given by Favro." He showed that the diffusion equation can be given by af(Q, t) /at= — Hf(Q, 1), (1) where /(Q, ¢) is the probability that the vector of a molecular dipole transition is oriented in the angle (Q) at time / and the Hamiltonian is given by H=Q LDisL;, tea where L is the quantum mechanical angular momentum operator and Dy; is the component of the diffusion tensor. If one is able to find a coordinate system that diagonalizes the diffusion tensor,” then the Hamil- tonian becomes H=E D;L2. (2) It is immediately evident that Eq. (1) is analogous to the Schridinger equation for an asymmetric rigid rotor. It is further shown by Favro that the diffusion Eq. (1) may be solved by the Green’s function method, namely f(,1)= J {O)G(O4|0,4M, (3) where f({p) is the initial probability that the dipole vector is oriented in @) at f=0 and G(Qp| Q, 1) is the Green’s function that describes the rotation of the dipole vector from Q» at =O into Q at time /. The function G(Q)|Q,/) can be expanded in terms of asymmetric rotor wavefunctions. G(Q5 | @, 1) =E WV." (5) ¥a(Q) exp(— Ext), (4) THE JOURNAL OF CHEMICAL PHYSICS VOLUME 57, NUMBER 12 Theory of Fluorescence Depolarization by Anisotropic Rotational Diffusion T. J. Cuvans ano K. B. Eisuxruat IBM Research Laborsory, San Jose, Calijornic 95114 (Received 28 February 1972) with the initial condition G(Qp | 2, 0) = 8(Mo, Q) = Hn* (Qo) Fn (Q), where ¥,(@) and E, are the stationary state eigen- function and eigenvalue for the asymmetric rigid rotor with the substitution #?/2/; by D;. The wavefunction ¥,(Q) may be expanded in terms of symmetric rotor wavefunctions ym“! (@)" ¥,(Q) =F, (Q) = zy Anji? Bim (Q). (5) i=] The coefficients 4,,.9 and eigenvalues E,{ are tabu- lated by Favro and Huntress! for /<2. 15 DECEMBER 1972 Zi (t) = PO) (o-+ (4/15) geguvery exp[—3(D.+ D)1] + (4/15) qaersy: expl—3(Ds+D)t] +(4/15) q.qe¥ evs exp[—3(D,+ D)t] + (1/15) (8+a) exp[—(6D+24)¢] 4 + (1/15) (@—e) exp[—(6D—2a)s]}, (12a) am hO=sPO-Ai, (12b) where B=aeye+qiy tqere— (1/3), a= (Do! A) (artery egy t+ y+ 92") +(D,/A) (g2y2+ 92 — Igy 3 +yi+4") + (D./A) (qaPy2? tay — gry e+ ae) —(2D/A), D=4(De+D,+D,), A= (D2+D,-++ D2—D,D,—D,D,—D,D,) "2. The constant D is the average of the three principal diffusion constants and A is related to the anisotropy of the diffusion. The fluorescence polarization anisot- ropy, defined as R(f) = (7) () —fa() /H +20] can be obtained from Eqs. (12a) and (12b), Similarly, the steady-state fluorescence polarization, P, given by P= (K-Ex)/U4Ey), (13) can also be obtained by time averaging over the emis- sion decay. For an exponential decay of the emission = 1 Sauda a¥y Pie =— 3 ae (D,+D) Aqugevy¥s + * i¢3(D-4D)r we find Completely anisotropic rotator From Eqs. (12a) and (12b), we see that in general, aside from P(t), the polarized components of the fluorescence and its anisotropy R(é) have five exponen- tials in their decay. This is expected since both the probability functions of excitation and emission trans- form as the second rank of the rotation matrices which have 2/-+1=5 dimensionality. Only eigenfunctions and eigenvalues for the rigid rotor with /=2 are therefore involved in the evaluation of J)\(¢) and Ji(¢), Hence for a given molecular system, one can see clearly how many exponentials and what type of exponents one would expect to obtain by examining the transforma- tion properties of the asymmetric rotor wavefunctions. Tn(tde, (14a) iu=r f° o her f ” ra(Ddt, (14b) o and 7 Aq.daVe¥2 1+3(D,+D)r B+o + a- ) +(6D+2A)r © 14+(6D—2A)z/" (15) Ellipsoid For symmetric-top molecules, one would generally ex- pect three exponential decay terms, aside from P(1), because there is a twofold degeneracy of eigenvalues in the components of the angular momentum along the symmetric axis for this type of system. The eigen- functions Vim (Q) and Wiam®(Q), and Wen (Q) and W_2.m(@) are degenerate when D.= Dy¥D,. In this case Eq. (12a) is reduced to T(t) = ($+ (1/15) [2 (geet aus)? (1-92) (1-2) ] X exp[—(2D,+-4D,)t}+ (4/15) geye(gevet ayy) X exp[— (5D,+D,)fJ+ (1/15) (39.21) (y2—4) x exp(—6D2)}P(). (17) This corresponds to the rotation of an ellipsoid. prolate or oblate in shape. Sphere . As for the case of isotropic diffusion, only one constant, namely 6D, can characterize the decay in polarization since rotation along any axis has the same diffusion constant, From Egs. (12a) and (12b) we have T(t) = { (1/9) + (2/45) exp(—6D%) (3 cos*’— 1) } P(t), (18a) Fi(t)={ (1/9) — (1/45) exp(—6Dzé) (3 cos*s—1)} P(A), (18b) and R(t) =4(3 cos*k—1) exp(—6Di), (19) Po i= (P—}) (14+-6Dr), (20) where . Po— 4= (10/3) (3 cos®’&— 1), (21) and ) is the angle between the absorption and emission dipoles. We recognize that Eq. (20) is just the expres- sion for steady-state volarization given bv Perrin for spherical molecules." Volume 14, number 5 CHEMICAL PHYSICS LETTERS 1 July 1972 POLARIZED FLUORESCENCE AND ROTATIONAL BROWNIAN MOTION M. EHRENBERG and R. RIGLER institute for Cell Research, Medicinska Nabelinstitutet, Karolinska Mnstitatet, Stackholm, Sweden Received 27 March 1972 b ON Fig. i. The transformation of vectorial components from. one system to the other is achieved with the aid of a matrix M [10]. ix, fo %e\ {cos #3 sin ay 1X4 t a j ty | | ¥, =MiC}, ly l= ual sing, sind |, (4.1) poet ye} ie] \ Z } ls} \f -/ \ \2a/ \L \z0/ cosa with /[ cosy, sin? sing, cosy, sin 2, re i j | sing, cos 0, cosy, sing, sin ca i, Qo cos By { ! \csind Why does a cylinder with rotational diffusion about only its cylinder axis show two correlation times of anisotropy decay? We give up!! WHY? It’s simple – just sines, cosines, and a bit of trigonometry =molecular rotation axis A n <N — ( ) ( ) 2 2D , ,t tt ∂ ∂ ∂∂Φ Ρ Φ = Ρ Φ Φ r D kT fφ = ( ) ( )( ) 2Dcos sin k tk ka k b k e−Φ + Φ ( ) ( )( ) ( )( ) 20 1, cos sin k Dtk kkt a a k b k e ∞ − = Ρ Φ = + Φ + Φ∑ ( ) ( )( ) ( )( )0 1,0 cos sink kka a k b k ∞ = Ρ Φ = + Φ + Φ∑ Ρ Φ,0( ) = an cos nΦ( )+ am cos mΦ( ) Ρ Φ, t( ) = an cos nΦ( )e −n2 Dt + am cos mΦ( )e −m 2Dt ( ) ( ) ( ) ( ) ( ) 0 1 ,0 2 1 ,0 cos 1 ,0 sin k k a d a k d b k d π π π π π π π π π − − − ⎛ ⎞= Ρ Φ Φ⎜ ⎟ ⎝ ⎠ ⎛ ⎞= Ρ Φ Φ Φ⎜ ⎟ ⎝ ⎠ ⎛ ⎞= Ρ Φ Φ Φ⎜ ⎟ ⎝ ⎠ ∫ ∫ ∫ If: Then: The diffusion equation about the cylinder rotation axis (the only free parameter) Specific solution: General solution: t=0 initial distribution Example of initial and time dependent solutions. x2 = 2Dt Just like Einstein told us: Initial angular distribution of an ensemble (from photoselection). Both distribution shapes remain constant with time. Only the amplitudes decrease with their time constants. ( ) 41 2 0 Remember: D t D tr t A e A e A− −= + + ( ) ( ) ( ) 2 2 2 0 0 0 ˆˆThe vertical signal is , , sine et z e d d d π π π ϕ ϕ ϕ θ φ= Ρ Φ ⋅ ⋅∫ ∫ ∫ ( ) ( ) ( ) 2 2 2 0 0 0 ˆ ˆThe horizontal signal is , , sine et x e d d d π π π ϕ ϕ ϕ θ φ= Ρ Φ ⋅ ⋅∫ ∫ ∫ ( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆsin cos cos cos ˆ ˆ ˆ ˆcos sin sin ˆ ˆ ˆ ˆsin cos sin e e e e e e n e n x e x y e y x n x x n e y e x e x y ϕ θ θ ϕ θ θ φ θ φ θ Φ Φ Φ Φ Φ Φ Φ Φ = ⋅ + ⋅ + ⋅ ⋅ = ⋅ = − ⋅ = ⋅ = ⋅ = ⋅ = Now we have to observe the anisotropy ( ) ( ) ˆ ˆ cos sin cos sin cos cos cos sin sin sin cos cos sin sin cos cos sin sin sin e e e e e e e e e e x e θ ϕ θ θ φ θ ϕ θ φ θ θ θ ϕ θ φ ϕ θ φ θ ⋅ = − + = − + Project the distribution of the emission dipole onto the x- and z-axes and integrate over the distribution. Then form the anisotropy expression. ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 2 D 2 2 2 4D 4 2sin sin cos15 15 vert. sig. 4 62cos sin cos15 15 32 cos sin cos sin cos15 8 sin sin cos 215 e a a e a a t e e a a e a t e e e a e e θ θ θ π θ θ θ π θ θ θ θ φ π θ θ φ − − ⎛ ⎞+⎜ ⎟ = ⎜ ⎟ + +⎜ ⎟ ⎝ ⎠ + + ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 2 D 2 2 2 4D 6 8sin sin cos15 15 horiz. sig. 2 8 4cos sin cos15 15 16 cos sin cos sin cos15 4 sin sin cos 215 e a a e a a t e e a a e a t e e e a e e θ θ θ π θ θ θ π θ θ θ θ φ π θ θ φ − − ⎛ ⎞+⎜ ⎟ = ⎜ ⎟ + +⎜ ⎟ ⎝ ⎠ − + ( )( )( ) ( )( ) ( )( ) 2 2 D 2 2 4D 1 3cos 1 3cos 110 6 cos sin cos sin cos5 3 sin sin cos 210 a e t e e a a e a t a e e a r e e θ θ θ θ θ θ φ θ θ φ − − = − − + + ( )( )( ) ( )( ) ( ) ( )( ) 2 2 D 2 2 2 2 4D 1 3cos 1 3cos 110 3 sin 2 sin 2 cos10 3 sin sin cos sin10 a e t e a e a t a e ea ea r e e θ θ θ θ φ θ θ φ φ − − = − − + + − Sometimes given as: ( ) ( )( ) ( ) ( ) 2 2 2 2, sin sin , , ,sin r r D D f t f t f t f t t θ θ θ θθ φ ⎡ ⎤∂ ∂ ∂ ∂ ∇ Ω = Ω + Ω = Ω⎢ ⎥∂ ∂ ∂∂⎣ ⎦ 3 , 8r sphereD kT aπη= ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 , , , , ,11 cos 2 cos 1 2 coscos r f t f t f t f t f t u u D t uu θ θ θθ ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω − − = = − − ∂ ∂ ∂∂∂ ( ) ( ) ( )0, n nnf t P u a t ∞ = Ω =∑ ( ) ( ) ( ) ( )0 0( 1)r n n n nn n dD n n P u a t P u a t dt ∞ ∞ = = − + =∑ ∑ ( ) ( ) ( ) ( )( 1)r n n n n dD n n P u a t P u a t dt − + = ( ) ( ) ( )0 exp ( 1)n n ra t a n n D t= − + ( ) ( ) ( ) ( )0, cos 0 exp ( 1)n n rnf t P a n n D tθ ∞ = Ω = − +∑ ( )( )2 2cos 1 3 2 cos 1Pθ θ= + ( )1 6spherer rDτ = Rotation of a Sphere ( ) ( ) ( ) ( ) 2 2 21 2 ( 1) n n n P u P u u u n n P u uu ∂ ∂ − − = − + ∂∂ Then photoselection