Desarrollo y experimentación

ments

A FRAP experiment outputs a sequence of fluorescence intensity values from inside the bleached region with respect to time. When these data are compared to established theoretical models, quantitative information such as the mobility of the fluorescently tagged population, the binding states and binding strength of each state can be extracted. Simple models incorporating diffusion of only one species are disscussed here; extended models are discussed in Appendix C.

1.15.1

Early theoretical FRAP models

The earliest attempts of Axelrod and colleagues [86] and Soumpasis [87] to model photobleaching experiments are purely analytical and predict recovery curves for the idealised cases of pure two-dimensional diffusion monitored by a laser beam of either a Gaussian intensity profile or a uniform circular disc profile. These models assume that the fluorescently labeled membrane component is uniformly distributed in an infinite membrane plane.

It is assumed that the photobleaching of fluorophores to non-fluorescent species follows an irreversible first-order reaction with rate constant αI(r). The concentration of unbleached fluorophore C(r,t) at position r and time t can be calculated from

dC(r,t)

dt = −αI(r)C(r,t) (1.11)

whereI(r) is the bleaching intenisty.

After a bleaching of duration time Tthe fluorophore concentration profile at the start of the recovery phaset=0 is given by

C(r,0) =C0exp (−αTI(r)) (1.12)

whereC0is the initial condition for fluorophore concentration.

The amount of bleaching induced in timeTis expressed by a parameter

K = αTI(0) (1.13)

The equation for lateral diffusion of a single species of fluorophore follows the diffusion equation described in section 1.14. With diffusion coefficientD, the diffusion equation is

∂C(r,t)

∂t = D∇

2_{C(r,t) (1.14)}

where the boundary condition isC(∞,_t)=_C0.

The fluorescence at time t Fk(t) is equal to the sum of the number of the

fluorophores being illuminated and can be expressed, in polar coordinates, as

Fk(t)=

q A

Z

I(r)Ck(r,t)d2r (1.15)

where the parameterqrepresents the product of all quantum efficiencies of laser light absorption, emmission and detection,Ais the attenuation factor of the beam during the fluorescence recovery,I(r) is the intensity profile of the bleaching laser in the membrane plane andCk(r,t) is the concentration of unbleached molecules at

The origin in polar coordinates is taken to be the centre of the bleached area. Fluorescence recovery curves can be transformed into a fractional form, thus making the recovery independent of the absolute intensity, by

fK(t) =

[Fk(t)−Fk(0)]

[Fk(∞)−Fk(0)]

(1.16)

Gaussian intensity profile

For a Gaussian intensity profile,I(r) is given by

I(r)= 2P0 πw2exp − 2r2 w2 ! (1.17)

where w is the half-width at e−₂

of the Gaussian distribution andP0 is the total

laser power.

Using the scheme described earlier in section 1.15, the closed form solution for a Gaussian intensity profile in fractional form is

fK(t) = 1− (tD)2 1+2tK 2+tD/(1+2t) Γ(ν) Γ(tD) P(2K|₂ν₎ P(2K|_2tD) (1.18)

where ν = tD/(1+ 2t), the characteristic diffusion time tD = w2/4D, Γ(ν) is the

gamma function andP(2K|₂ν_{) is the}χ2_{probability distribution.}

Circular intensity profile

For a circular disc profile,I(r) is given by

I(r)=          P0 πw2 for r≤w 0 for r>w

With the assumption of full recovery, the closed form solution for circular intensity profile is f(t) = exp −2tD t I0 _2t D t +I1 _2t D t (1.19) which is independent of the bleaching parameter K.I0andI1are modified Bessel

functions.

If there are two diffusing populations, a fast and a slow one, the recovery curve can be expressed as

both components have identical photobleaching characteristics, diffuse isotropi- cally and independently and recovery is complete.

One of the limitations of the Axelrod and Soumpasis models is the assumption of planar membranes. In reality, biomembranes adopt ‘wavy’ conformations like

h(x,y) = Acos(kx) cos(ky), where Ais the amplitude and kthe spatial frequency, therefore the actual diffusion of resident molecules can be up to twice as large as the predicted from the model [88].

1.15.2

Strip FRAP on spherical membranes

Biological membranes can adopt geometries different to the planar assumptions of the earlier models. Ellenberg et al. proposed the following formula to model the diffusion of lamin B receptor in inner nuclear membranes which is roughly spherical. When a rectangular strip of a cross-section of a spherical membrane is bleached, the fluorescence recovery is given by [89]

f(t)= f(∞₎

r

1− w 2

w2+₄π_Dt (1.21)

which agrees within 5% with the solution of the diffusion equation in one dimen- sion for recovery into an interval of zero intensity, where f(t) is the intensity as a function of time, zero of timetwas taken as the midpoint of the bleach, f(∞_{) is the}

final intensity reached after complete recovery,wis the strip width and Dis the effective one-dimensional diffusion constant. This equation assumes one dimen- sional recovery which is reasonable because the membrane is bleached across its length and full depth.