Cone Model for Restricted Probe Diffusion

Studies o f probe fluorescence anisotropy in (globally isotropic) biological systems such as membranes are characterised by (nanosecond-picosecond range decay) a non-zero steady state (or very slowly varying millisecond) anisotropy. This was explained in terms o f restricted rotational diffusion within a fixed or very slowly moving environment. The cone model o f Kinosita et al [16] has been widely used to interpret fluorescence anisotropy arising from restricted rotational diffusion o f cylindrically symmetric probe molecules in a locally ordered but globally isotropic environment. Restricted rotational diffusion is modelled as diffusion or ‘w obbling’ restricted to a cone

defined by the angular limits O<0<0c where 0c is the maximum angle o f the cone as shown in Figure 6.1.

The cone model has been applied to a diverse range o f molecular systems including the dielectric relaxation o f molecules in glassy matrixes [17], depolarised light scattering by

a small molecule in an amorphous polymer [18] and segmental motions o f fatty acids in phospholipid bilayers [19].

In the cone model, fluorescence anisotropy can be described using a formalism that relates the emission to the motion o f molecule fixed transition dipole moments within the local environment, which is taken to remain static. The extension o f this approach to

systems where the isotropic diffusion o f the local environment is on a longer timescale is straightforward as long as the two diffusive processes are uncorrelated [20].

Figure 6.1: Restricted rotational motion of a probe in terms of the free rotation set by the limits of the cone angle and defined by the angle ±0c

In this approach, the fluorescence anisotropy as defined by equation (3.2) is given by

[2 1 ].

(6.2)

where P2 is the second order Legendre polynomial relating to the angle between the

initial orientation o f the probe transition dipole moment and its orientation at time t,

w^(Pg) is the (equilibrium) distribution o f probe orientations at t=0 (there being no net

photoselection within the local environment due to the initial isotropic ordering o f the

molecules with respect to the excitation polarisation), g (p g ,0 |p g ,t) is defined as the

probability that the molecule with orientation ji'e at time zero will rotate into a new

orientation pc at time t. Finally, Ro refers to the initial anisotropy for an isotropic

medium that for single photon excitation is given by Rq = R(0) = O.4P2 (cosf^), where 5

is the angle between the emission and absorption dipolar moments o f the probe molecule.

At long times t—»oo the dynamical probability g(pg,01 Pg,oo) becomes identical to

w^(Pg) and (6.2) becomes

Wg(pg) remains symmetric for all directions around the normal (n) to the system and

only the first term remains after integration yielding

R ./R -o = [ J P 2 ( n .^ e ) W s ( n .) d n . J (6.4)

Equation (6.4) is a measure o f the anisotropy o f the stationary distribution ( ), that is,

a maximum when the emission dipole is parallel to the local director n.

At short times approaching the initial photoexcitation (t—>0), the wobbling diffusion constant may be defined as

where 0 denotes the angle between and pe- Essentially, the integral represents the

average o f 0^ at time t over all possible orientations. For an axially symmetric molecule

the wobbling diffusion constant becomes the rotational diffusion constant, whereas a molecule o f arbitrary shape Dw(pe-) corresponds to a weighted average o f the principal diffusion constants. For small t, the angle o f rotation 0, can also be considered small and

therefore = l/2(3cos^ ^ - 1) = 1 -(3/2)^^ and substitution in equation (6.2) can be

written as.

R ( t ) s R

5 R« [ l - 6t ( X ).dX ] (6.6)

where (D^) is the wobbling diffusion constant averaged over the stationary distribution

Wg. The result is equivalent to the small step Debye diffusion model for a cylindrically

symmetric molecule (see section 2.4.3, Chapter 2). From equation (6.6), the quantity

R ^ /Rq represents the degree o f confinement o f the fluorescent probe imposed by the

Assuming therefore that the probe motion is described by the wobbling o f the emission

dipole orientation pe (long axis) in a cone around the director n, fluctuating with a

diffusion coefficient De, the diffusion o f the total distribution probability W g(t) can be

solved. Assuming that the molecule is in a cylindrical reference frame (with 0 and ^ the

polar and azimuthal angle respectively), the diffusion equation is

1 d

sin ^ Ô6 sin ^ Ô0 +

1

sin^ 0 d(j)^ t) with 0 < ^ < 0 ^ ^ (6.7)

with the boundary condition ô ^ J ô 0 = 0 at 0=0max, which for the case o f the emission

dipole moment parallel to the long axis o f the molecule results in an infinite sum o f exponentials for R(t) as follows:

H ‘) = exp(-£>^^/c7,) (6.8)

1 = 1

where A j and Oj are constants which depend solely on 0max- The cone angle is related to

the degree o f constraint Aoo defined as:

(6.9)

For the specific case o f the emission dipole being parallel to the symmetry axis o f the molecules, equation (6.2) can be approximated by the expression

(6.10)

where ^ is the characteristic lifetime with which the initially photoselected distribution

o f orientations approaches the stationary distribution, and A» is the degree o f constraint

defined in (6.4). The closed form approximation is only exact in the limits o f t=0 and

t=oo.

The initial cone angle model uses an orientational distribution function Wg(0) chosen to

be a stationary function with non-zero values for the angular intervals O°<0<0max and

180°- 0max^0^18O° and zero for 0max^0^180°-0max. A more realistic distribution probability was proposed [22] by

w, « e x p ( ^ 7 2 ) e x p ( - ^ ^ ^ 7 4 ) (6.11)

where q is a parameter that determines the width o f the distribution. This form of distribution is similar to that observed for probe alignment in the nematic phase o f liquid crystals (see Chapters 4 and 5). However, a comparison between the models shows that the cone angle extracted from the Kinosita model is close to that for the full width o f the Gaussian model.

The extension o f the Kinosita model to restricted rotational diffusion within an isotropically diffusing environment is straightforward. I f the internal and external motions are uncorrelated, the overall alignment correlation function can be described in

terms o f a product o f the two individual correlation functions [7, 23], giving rise to two

independent rotational relaxation times

R{t) = i?(0)[(l - F ) e x p ( - r / r „ ) + F ] e x p ( - / / r , ) (6.12)

where R(0) represents the global initial anisotropy, Tw is the relaxation time for the probe

rotation within the cone and Zc is the reorientational time for overall motion o f the cone.

F = [0.5cos^;i^(l + cos^;i^)]^ for transition dipoles parallel to the molecular axis. This

approach has been successfully applied to the studies o f TRP-214 in HAS [24] and the

probe fluorescence dynamics in the isotropic phase o f 5CB [3].