Principle .E2.80.93 Brownian motion and photoselection Fluorescence anisotropy

brownian motion of nanoparticle

in fluorescence, molecule absorbs photon , gets excited higher energy state. after short delay (the average represented fluorescence lifetime



τ


{\displaystyle \tau }

), comes down lower state losing of energy heat , emitting rest of energy photon. excitation , de-excitation involve redistribution of electrons molecule. hence, excitation photon can occur if electric field of light oriented in particular axis molecule. also, emitted photon have specific polarization respect molecule.

the first concept understand anisotropy measurements concept of brownian motion. although water @ room temperature contained in glass eye may still, on molecular level each water molecule has kinetic energy , there continuous number of collisions between water molecules. nanoparticle (yellow dot in figure) suspended in solution undergo random walk due summation of these underlying collisions. rotational correlation time (that is, time takes molecule rotate 1 radian) dependent on viscosity, temperature, boltzmann constant , volume of nanoparticle:

ϕ

r


=



η
v


k




b



t





{\displaystyle \phi _{r}={{\eta v} \over {k{_{b}}t}}}

the second concept photoselection use of polarized light. when polarized light applied group of randomly oriented fluorophores, of excited molecules oriented within particular range of angles applied polarization. if not move, emitted light polarized within particular range angles applied light.

for single-photon excitation intrinsic anisotropy r0 has maximum theoretical value of 0.4 when excitation , emission dipoles parallel , minimum value of -0.2 when excitation , emission dipoles perpendicular.

r

0



=


2
5



(



3



cos


2



β
−
1

2


)



{\displaystyle {r_{0}}={2 \over 5}\left({{3{{\cos }^{2}}\beta -1} \over 2}\right)}

where β angle between excitation , emission dipoles. steady-state fluorescence measurements measured embedding fluorophore in frozen polyol.

taking idealistic simplest case subset of dye molecules suspended in solution have mono-exponential fluorescence lifetime



τ


{\displaystyle \tau }

, r0=0.4 (rhodamine 6g in ethylene glycol made have absorbance of ~0.05 test sample). if excitation unpolarized measured fluorescence emission should likewise unpolarized. if excitation source vertically polarized using excitation polarizer polarization effects picked in measured fluorescence. these polarization artifacts can combated placing emission polarizer @ magic angle of 54.7º. if emission polarizer vertically polarized there additional loss of fluorescence brownian motion results in dye molecules moving initial vertical polarized configuration unpolarized configuration. on other hand, if emission polarizer horizontally polarized there additional introduction of excited molecules vertically polarized , became depolarized via brownian motion. fluorescence sum , difference can constructed addition of intensities , subtraction of fluorescence intensities respectively:

s
=


i

v
v



+
2
g


i

v
h





{\displaystyle s={i_{vv}}+2g{i_{vh}}}

d
=


i

v
v



−
g


i

v
h





{\displaystyle d={i_{vv}}-g{i_{vh}}}

dividing difference sum gives anisotropy decay:

r
=


d
s




{\displaystyle r={d \over s}}

the grating factor g instrumental preference of emission optics horizontal orientation vertical orientation. can measured moving excitation polarizer horizontal orientation , comparing intensities when emission polarizer vertically , horizontally polarized respectively.

g
=




i

h
h





i

h
v







{\displaystyle g={{i_{hh}} \over {i_{hv}}}}

g emission wavelength dependant. note g in literature defined inverse shown.

the degree of decorrelation in polarization of incident , emitted light depends on how fluorophore orientation gets scrambled ( rotational lifetime



ϕ


{\displaystyle \phi }

) compared fluorescence lifetime (



τ


{\displaystyle \tau }

). scrambling of orientations can occur whole molecule tumbling or rotation of fluorescent part. rate of tumbling related measured anisotropy perrin equation:

r
(
τ
)
=



r

0



1
+
τ

/

ϕ





{\displaystyle r(\tau )={\frac {r_{0}}{1+\tau /\phi }}}

where r observed anisotropy, r0 intrinsic anisotropy of molecule,



τ


{\displaystyle \tau }

fluorescence lifetime ,



ϕ


{\displaystyle \phi }

rotational time constant.

this analysis valid if fluorophores relatively far apart. if close another, can exchange energy fret , because emission can occur 1 of many independently moving (or oriented) molecules results in lower expected anisotropy or greater decorrelation. type of homotransfer förster resonance energy transfer called energy migration fret or emfret.

steady-state fluorescence anisotropy give average anisotropy. more information can obtained time-resolved fluorescence anisotropy decay time, residual anisotropy , rotational correlation time can determined fitting anisotropy decay. typically vertically pulsed laser source used excitation , timing electronics added between start pulses of laser (start) , measurement of fluorescence photons (stop). technique time-correlated single photon counting (tcspc) typically employed.

again using idealistic simplest case subset of dye molecules suspended in solution have mono-exponential fluorescence lifetime



τ


{\displaystyle \tau }

, initial anisotropy r0=0.4. if sample excited pulsed vertically orientated excitation source single decay time



τ


{\displaystyle \tau }

should measured when emission polarizer @ magic angle. if emission polarizer vertically polarized instead 2 decay times measured both positive pre-exponential factors, first decay time should equivalent



τ


{\displaystyle \tau }

measured unpolarized emission set-up , second decay time due loss of fluorescence brownian motion results in dye molecules moving initial vertical polarized configuration unpolarized configuration. on other hand, if emission polarizer horizontally polarized, 2 decay times again recovered first 1 positive pre-exponential factor , equivalent



τ


{\displaystyle \tau }

second 1 have negative pre-exponential factor resulting introduction of excited molecules vertically polarized , became depolarized via brownian motion. fluorescence sum , difference can constructed addition of decays , subtraction of fluorescence decays respectively:

s
(
t
)
=
g


i

v
v



(
t
)
+
2


i

v
h



(
t
)


{\displaystyle s(t)=g{i_{vv}}(t)+2{i_{vh}}(t)}

d
(
t
)
=
g


i

v
v



(
t
)
−


i

v
h



(
t
)


{\displaystyle d(t)=g{i_{vv}}(t)-{i_{vh}}(t)}

dividing difference sum gives anisotropy decay:

r
(
t
)
=



d
(
t
)


s
(
t
)





{\displaystyle r(t)={d(t) \over s(t)}}

in simplest case 1 species of spherical dye:

r
(
t
)
=


r

0



exp
⁡

(

−


t


ϕ

r






)



{\displaystyle r(t)={r_{0}}\exp \left({-{t \over {\phi _{r}}}}\right)}