Open access peer-reviewed chapter

Diffusion Theory for Cell Membrane Fluorescence Microscopy

By Minchul Kang

Submitted: November 18th 2019Reviewed: February 20th 2020Published: September 30th 2020

DOI: 10.5772/intechopen.91845

Downloaded: 91

Abstract

Since the discovery of fluorescent proteins and the development of DNA recombinant techniques, various fluorescence methods have significantly improved our understanding of cell biology at a molecular level. In particular, thanks, in large part, to technological advances in these fields, fluorescence techniques such as fluorescence recovery after photobleaching (FRAP), fluorescence correlation spectroscopy (FCS), and single-particle tracking (SPT) have become standard tools in studying cell membrane structure as well as the diffusion and interaction of biomolecules in the cell membrane. In this chapter, we will review some topics of the diffusion theory from both deterministic and probabilistic approaches, which are relevant to cell membrane fluorescence microscopy. Additionally, we will derive some basic equations for FARP and FCS based on the diffusion theory.

Keywords

  • diffusion theory
  • fluorescence recovery after photobleaching
  • fluorescence correlation spectroscopy
  • cell membranes

1. Introduction

Diffusion is an idealization of the random motion of one or more particles in space. Since diffusion is a dominant way for biological organisms to transport various molecules to desirable locations for cell signaling, the role of diffusion within biological systems is critical [1, 2, 3]. Therefore, to quantify the diffusion coefficient, a measure of diffusion rates, is essential to understand both the physiology and pathology of cells in terms of cell signaling time scales [1, 2, 3]. Moreover, the diffusion coefficients of proteins may also provide information on the landscape of the membrane environment where diffusion occurs [4, 5, 6]. However, quantifying the diffusion especially in live cell membranes is still challenging although a couple of tools are available including fluorescence recovery after photobleaching (FRAP) and fluorescence correlation spectroscopy (FCS) [7, 8]. Diffusion is quantified by a diffusion coefficient, D, which characterizes the proportionality in a linear relationship between mean squared displacement (MSD, x2)of a Brownian particle and time [9, 10]. To determine the diffusion coefficients of biomolecules of interests, mathematical models for the diffusion process are compared with experimental data in FRAP and FCS analysis. In this chapter, we bridge the gap between experimental and theoretical aspects of FRAP and FCS by reviewing mathematical theories for FRAP and FCS.

2. Diffusion equation

2.1 Diffusion equation from the deterministic point of view

In 1855, Fick [11] published two cornerstone papers on diffusion, in which he proposed the fundamental laws describing the transport of mass due to the concentration gradient and an associated mathematical model. According to Fick’s first law, the diffusive flux (J) is proportional to the concentration gradient of diffusants (du/dx) with a proportionality constant called a diffusion coefficient, D. In one-dimensional spatial dimension (R1), Fick’s law can be represented as

J=DdudxE1

where Jxtis the diffusion flux and uxtis the concentration of diffusants at the location xat time t. The diffusion coefficient can be calculated by the Stokes-Einstein equation [12, 13]:

D=kBT6πηrE2

where kBis Boltzmann’s constant, Tis the absolute temperature, ηis the dynamic viscosity, and ris the radius of the spherical particle. Assuming the conservation of mass in an infinitesimal interval xx+Δx, we obtain

tuxtΔx=JxtJx+Δxt;
ut=JxtJx+ΔxtΔxE3

where uxtΔxis the total number of molecules in the interval xx+Δxand JxtJx+Δxtis the difference of influx and efflux in and out of the interval (i.e., net change in the total number of molecules in the interval) as shown in Figure 1.

Figure 1.

The change in the number of molecules in an intestinal interval due to diffusion.

By combining Eqs. (1) and (3) and by taking the limit in Δx0, we have Fick’s second law that describes the diffusion process in a form of partial differential equation:

ut=D2ux2E4

Eq. (4) is often referred to as the one-dimensional diffusion equation or heat equation. Similarly, two-dimensional (R2) and three-dimensional (R3) can be derived as

ut=D2ux2+2uy2
ut=D2ux2+2uy2+2uz2E5

In a more compact form, the diffusion equations are written using the Laplace operator, Δ:

ut=DΔuE6

where Δu=2ux2in R1, Δu=2ux2+2uy2in R2, and Δu=2ux2+2uy2+2uz2in R3.

Importantly, the diffusion equation satisfies the following important properties:

  1. Property 1: Translation invariance. If uxtis a solution of the heat equation, then for any fixed number x0, the function uxx0tis also a solution.

  2. Property 2: Derivatives of solutions. If uxtis a solution of the heat equation, then the partial derivatives of ualso satisfy the heat equation.

  3. Property 3: Integrals and convolutions. If Φxtis a solution of the heat equation, then Φg(the convolution of Swith g) is also a solution where Φgxt=Φxytgydyprovided that this improper integral converges. The improper integral Φgis called the convolution of Φand g.

  4. Property 4: Dilation. Suppose a>0is a constant. If uxtis a solution of the heat equation, then the dilated function v(x,t)=uaxatis also a solution.

Based on these properties, we are now ready to solve the following initial value problem on xR1for0t<:

ut=Duxxux0=HxwhereHx=1,x>00,x0E7

where Hxis often referred to as the Heaviside function.

By Property 4, any solution (uxt) is unaffected by the dilation xaxand tatfor any aR1. Since xtis also unaffected by the dilations (xtaxat=xt), we look for a solution in the form of gαxtfor some constant α. Notice also that gαxtis also invariant under these dilations: αaxat=gαxt. If we let p=αxtand choose α=14D, then by the chain rule, we have

0=utDuxx=p2tgpκ4Dtgp=14tgp+2pgpE8

which reduces to an ordinary differential equation g+2pg=0. This can be solved as.

g=C2+0pC1er2drwhereg0=C2E9

for arbitrary constants C1and C2. Because as t0+, p, for x>0

1=limt0+uxt=C2+0C1er2dr=π2C1+C2E10

where we used a well-known identity (the error function integral):

eax2dx=πa,E11

On the other hand, since as t0, p, for x<0

0=limt0+uxt=C2+0C1er2dr=π2C1+C2E12

which implies that C1=1πand C2=12. Putting together, we have a solution

uxt=12+1π0x/4κter2drE13

Define Φxt=uxxt; then

Φxt=x12+1π0x/4κter2dr
=1πex24κt·14κtE14
=14πκtex24κt

By Property 2, derivatives of solutions, the function Φxt=14πκtex24κtis also a solution to the diffusion equation. Φxtis called the (one-dimensional) heat kernel or the fundamental solution of the heat equation. The graphs of the heat kernel for different tare shown in Figure 2.

Figure 2.

The heat kernel graphs for different t.

From Figure 2, we can see that the heat kernel Φxthas a “bell curve” graph of a normal distribution (Gaussian function) with2Dtas the standard deviation, which sometimes called the Gaussian root mean square width. Also, 14πtmodulates the amplitude of the Gaussian curves, and the amplitude blows up to as t0+and approaches 0as t, i.e.:

limt0+Φxt=0ifx0ifx=0.E15

Also, from the error function integration (Eq. (14))

Φxtdx=1,forallt0Φxytdx=1,forallt0E16

Furthermore, it follows that (i) Φxytsatisfies the heat equation (Property 1: translation invariance) and (ii) Φϕxt=Φxytϕydysatisfies the heat equation (Property 3: integrals and convolutions).

From the definition (Φ=ux), by differentiating Eq. (7) with respect to x, we see that Φxtsatisfies

Φt=DΦxxΦx0=uxx0=ddxHxE17

Even though Hxis not differentiable due to discontinuity at x=0, we can redefine differentiation in a broad sense (weak derivative) and under this weak derivative definition:

ddxHx=δx=0ifx0ifx=0E18

where δxis called the Dirac delta function. The Dirac delta function satisfies a few important properties:

  1. limt0+Φxt=δx

  2. δxdx=1and δxydx=1

  3. δxfxdx=f0and δxyfxdx=fy

The third integration property is sometimes called the sifting property of the Dirac delta function. With these properties, we now can show (heuristically) uxt=Φϕxtsatisfies the following diffusion equation:

ut=Duxxux0=ϕxuxt=ΦϕxtE19

To show Φϕxtsatisfies the initial condition, we apply the sifting property of the Dirac delta function:

ux0=Φϕx0
=Φxy0ϕydyE20
=δxyϕydy
=ϕx

In other words, this result (Eq. (19)) indicates that for any initial value problem, the solution can easily be found as a convolution of the heat kernel and initial data.

2.2 Diffusion equation from the stochastic point of view

In many biological systems, passive transports are often described by Brownian motion or diffusion that is observed in random drifting of pollen grains suspended in a fluid. Suppose a Brownian particle located at the position x=0when time t=0has moved randomly on a straight line during time Δt. Since the movement of a Brownian particle is random, the location of the Brownian particle at t=Δtwill be probabilistic. Especially, for smaller Δtelapsed, the Brownian particle will have a higher chance to be found near the starting location x=0similar to a normal (or Gaussian) probability distribution with zero mean and a small standard deviation. For this reason, the Brownian motion is often described mathematically by random variables in time, which is called a stochastic process (time-dependent random variable).

If we let Xtbe a stochastic process in R1describing the position of a fluorescence molecule at time t, i.e., “Xt=x” means that the location of a fluorescence molecule at time tis x, then the probability of the Brownian particle located within the interval 0Δxat time twill be dependent on both Δxand the previous location:

PXt0ΔxX0=0E21

assuming the initial location is the origin (X0=0). Bachelier [14] explicitly calculated this probability as

PXt0ΔxX0=0=0Δx14πDtexpx24DtdxE22

where D(μm/s2) is a diffusion coefficient. The probability density function (the integrand) is the fundamental solution of heat equation (Eq. (14)) that is the normal distribution with standard deviation σ=2Dt.Later, Einstein [12] showed that the probability density function of randomly moving particles (Brownian motion) satisfies the diffusion equation with a solution Φxt(Eq. (17)).

If gyis the probability of a Brownian particle to be found at location ywhen t=0, i.e., PX0=y=gy, then the distribution of the Brownian particles can be determined by solving an initial value problem:

ut=D2ux2ux0=gx,E23

which has the solution

Φgx=14πDtexpxy24Dtgydy.E24

as in Eq. (19).

2.3 Mean squared displacement

The spreading rate of diffusing particles is quantified by a diffusion coefficient, D, which characterizes a linear relationship between mean squared displacement (x2)of a Brownian particle and time, where MSD is defined as

x2=x2PXt0ΔxX0=0dx=x2Φxtdx.E25

For a diffusion process, MSD increases linearly in time with the rate of the diffusion coefficient:

x2=2nDt.E26

where nis the spatial dimension (Rn) for a diffusion process. To derive this relation in 1D (R), we consider tx2

tx2=dtx2Φxtdx
=x2tΦxtdxE27
=Dx22x2Φxtdx

where we used Eq. (17). Notice that by the product rule

Dxx2xΦxtdx=D2xxΦxtdx+Dx22x2ΦxtdxE28

By solving for Dx22x2Φxtdx

Dx22x2Φxtdx=Dxx2xΦxtdx2xxΦxtdx
=Dx2xΦxtD2xxΦxtdxE29
=0D2xxΦxtdx

Next, by integration by parts

D2xxΦxtdx=D2xΦxt+D2ΦxtdxE30
=0+2D

Finally, by putting all together

tx2=2Dx2=2Dt,E31

for R1.

3. Fluorescence recovery after photobleaching

3.1 Principles of FRAP

Fluorescence recovery after photobleaching is a fluorescence-based biophysical tool developed in the 1970s to investigate the diffusion process in membranes of live cells. Discovery of the green fluorescent protein (GFP) and the invention of commercial confocal laser scanning microscopes (CLSMs) have broadened the accessibility of FRAP for many researchers in the field, and the applications of FRAP have become widely extended to the study of intracellular protein dynamics [15, 16, 17, 18]. Over the four decades, there have been considerable advances in microscope technology. However, the basic principle of FRAP remains the same. In FRAP, fluorescently tagged molecules in a small region of interest (ROI) are irreversibly photobleached using a high-intensity laser source for a short period of time, and then the exchange of fluorescence and photobleached molecules in and out of the bleached region is monitored using low-intensity laser excitation to track fluorescence recovery (Figure 3A). Due to the artifacts such as the diffusion during the photobleaching step (Figure 3B) and the photofading during the imaging step, FRAP data requires some corrections (Figure 3C). The diffusion during the photobleaching step can be corrected by using the experimentally measured postbleach profile as an initial condition for the FRAP model [19, 20, 21]. On the other hand, the photofading during the imaging step can be corrected by diving the raw FRAP data (FDatat) by the fluorescence intensity from the whole image (FDatat) (Figure 3D) [19, 20, 21]. Since different transport and reaction mechanisms may affect the curvature and the mobile fraction of a FRAP curve in various manners, kinetic parameters for underlying mechanisms can be obtained by comparing the FRAP curve to the corresponding theoretical FRAP models. For example, Dcan be measured by comparing a diffusion FRAP model with FRAP data for the best fitting D[19, 20].

Figure 3.

Example of FRAP data. (A) Representative images from a FRAP experiment on Alexa488-CTxB. (B) A postbleach profile from the image for t=0shows a wider spreading radius (effective radius; re) than the bleaching spot radius (nominal radius; rn) due to diffusion during photobleaching. (C) Mean fluorescence intensity (N = 13) from the bleaching ROI (∘,FDatat), whole image (•, FWholet), and background (□) from a FRAP experiment of Alexa488-CTxB. The image in the inset shows the locations where FDatat (∘) and background (□) were measured. (D) In FRAP analysis, prebleach steady-state, postbleach initial, and postbleach steady-state fluorescence intensities are typically denoted as Fi, F0, and Fi. These parameters can be used to calculate the mobile fraction (Mf) and the immobile fraction (1−Mf) from the corrected FRAP data for photofading (FDatat/FWholet) as indicated in the boxed equation.

3.2 Derivation of diffusion FRAP equation in R1

Quantitative FRAP analysis requires a mathematical description of fluorescence recovery for a given underlying transport/reaction kinetics through two different modes of CLSMs: photobleaching and photo-illumination. Although CLSMs use scanning laser for both photobleaching and photo-illumination, it has been reported for small bleaching spot size (we call this as the nominal radius of the laser); the scanning profile of CLSMs on a confocal plane is well approximated by a Gaussian function:

Irnx=2I02πrn2exp2x2rn2,E32

where rnis the nominal radius, i.e., radius of a bleaching ROI (the half-width at e2laser intensity). Irncan be regarded as a photobleaching mode of CLSMs with a maximal laser intensity I0. A bell-shaped profile of Irnxdefines total laser intensity I0with Irnxdx=I0resulting from the error function integral (Eq. (11)). Since the high-intensity mode of laser (Irnx) causes photobleaching of fluorophores, for illumination, laser intensity has to be attenuated to a lower laser intensity level. Therefore, for an attenuation factor ϵ1, a photo-illumination mode of CLSMs can be described as ϵIrnx. If we let uxtbe the density of fluorophores (or fluorescent proteins) at a location xat time t, then fluorescence intensity at the position xat time twill be proportional to both the illumination laser intensity (ϵIrnx) and fluorophore density (uxt). Assuming the linear proportionality, fxt, the fluorescence intensity at a location xyat time tcan be described as

fxt=q·ϵIrnxuxt,E33

where the proportionality constant, q, is referred to as a quantum yield or quantum efficiency. When a CLSM system is used to photobleach fluorophores, its postbleach profile is not exactly the same as the laser profile in most cases due to diffusion occurring during the photobleaching step. Assuming the first-order photobleaching process with a photobleaching rate α, a governing equation for a photobleaching process of freely diffusing fluorescent proteins can be described as a reaction–diffusion equation:

ut=DΔuαIrnxuux0=u0E34

where u0is the prebleach steady-state fluorescence intensity, which is regarded as a constant. Although the solution to Eq. (34) is hard to find, it is empirically proven [22] that a confocal postbleach profile can be described as a simple Gaussian function (constant minus Gaussian):

φx=Ci1Kexp2x2re2,E35

Note that different underlying kinetics for uyield a different FRAP equation. For free diffusion kinetics, the evolution of uxtcan be described as the diffusion equation subject to the initial condition from a postbleach profile right after photobleaching.

ut=DΔuux0=φxE36

where Dμm2/sis a diffusion coefficient and the Laplacian, Δ=2x2, in R1. The solution of the diffusion equation can be found as (Eq. (19))

uxt=ΦDφ
=ΦDxx¯tφx¯dx¯
=Ci4πDtexpxx¯24Dt1Kexp2x¯2re2dx¯
=Ci4πDtexpxx¯24Dtdx¯CiK4πDtexpxx¯24Dt2x¯2re2dx¯E37
=CiCiK4πDtexpxx¯24Dt2x¯2re2dx¯

by Eq. (11) (error function integration).

The total fluorescence intensity from the region of interest can be found by integrating this local fluorescence intensity over the ROI:

Ft=Irnxuxtdx,E38

which is called a FRAP equation. To simplify Eq. (38) by using Eq. (37)

Irnxuxtdx
=2I02πrn2exp2x2rn2CiCiK4πDtexpxx¯24Dt2x¯2re2dx¯dx
=Ci2I02πrn2exp2x2rn2dxCiK2I024π2rn2Dtexp2x2rn2xx¯24Dt2x¯2re2dxdx¯
=CiI0CiK2I024π2rn2Dtexp2x2rn2xx¯24Dt2x¯2re2dxdx¯
=FiFiK12π2rn2Dtexp2x2rn2xx¯24Dt2x¯2re2dxdx¯E39

where Fi=CiI0is the prebleach fluorescence intensity due to fluorophore density Ci. If we let x¯=x+θχwhere θ=4Dt(dx¯=θdχ), then the integral term in Eq. (39) becomes

exp2x2rn2xx¯24Dt2x¯2re2dxdx¯
=θexp2x2rn2θ2θ2χ22x+θχ2re2dxdχ
=θexp2re2x2+rn2x+θχ2rn2re2χ2dxdχ
=θexp2re2+rn2x2+2rn2θxχ+θ2rn2χ2rn2re2χ2dxdχE40
=θexp2re2+rn2x2+2rn2θre2+rn2+θ2rn2re2+rn2χ2rn2re2χ2dxdχ
=θexp2re2+rn2rn2re2x+rn2θre2+rn2χ2rn2θre2+rn22θ2rn2re2+rn2χ2χ2dxdχ
=θexp2re2+rn2rn2re2x+rn2θre2+rn2χ2rn4θ2θ2rn4θ2rn2re2re2+rn22χ2χ2dxdχ
=θexp2re2+rn2rn2re2x+rn2θre2+rn2χ2θ2rn2re2re2+rn22χ2χ2dxdχ
=θexp2re2+rn2x+rn2θre2+rn2χ2rn2re2+2θ2re2+rn2χ2χ2dxdχ
=θexp2re2+rn2x+rn2θre2+rn2χ2rn2re2dxexp2θ2re2+rn2+1χ2
=θexp2re2+rn2rn2re2x2dxexp2θ2re2+rn2+1χ2
=θπrn2re22re2+rn2exp2θ2re2+rn2+1χ2
=θπrn2re22re2+rn2exp2θ2+re2+rn2re2+rn2χ2
=θπrn2re22re2+rn2πre2+rn22θ2+re2+rn2
=π2rn2re24Dt28Dt+re2+rn2

Therefore

Ft=FiFiK12π2rn2Dtπ2rn2re24Dt28Dt+re2+rn2
=FiFiKre28Dt+re2+rn2E41
=Fi1K1+γ2+2t/τD

where γ=rn/reand τD=re2/4D. If we consider the immobile fraction (Figure 3D), the FRAP equation for mobile fluorophores is found as

Ft=Fi1K1+γ2+2t/τDM+1MF0E42

for the mobile fraction, Mis defined as (Figure 3D)

M=FF0FiF0E43

where Fi, F0, and Fare prebleach steady-state fluorescence intensity, postbleach initial fluorescence intensity (F0), and postbleach steady-state fluorescence intensity, respectively. The calculations for the 1D FRAP model can easily be extended to higher-dimensional cases. For example, a diffusion FRAP equation in 2D (R2) and 3D (R3) is found as

Ft=Fi1K1+γ2+2t/τD2D
Ft=Fi1K1+γ2+2t/τD1+γ2+2t/τD3DE44

4. Fluorescence correlation spectroscopy

4.1 Principles of fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy is a standard bioengineering and biophysics technique for the study of molecular movements and interactions [23, 24, 25]. For FCS experiments, a laser beam is focused and stationed at a region of interest in the specimen (usually live cells). The illumination region formed by the focused laser is called a confocal volume, which is generally in the femtoliter range. As fluorescence molecules cross the confocal volume by diffusion or other transporting mechanisms, they emit fluorescence photons responding to the illumination laser (Figure 4A), and the fluctuations in the fluorescence signal, Ft, is monitored as a function of time which is called raw FCS data. Since different FCS measurements from different cells can be quite different depending on the fluorescent protein expression level, the raw FCS data is first standardized by

Figure 4.

Principles of fluorescence correlation spectroscopy analysis. (A) For FCS analysis for free diffusion, a static laser beam is focused on a specific region of interest. As the fluorescence molecules diffuse in and out of a certain domain, commonly called confocal volume (∼1 femtoliter), fluorescence intensities from the confocal volume fluctuate, yielding fluorescence time series. (B) The fluorescence time series data are processed into an autocorrelation curve by taking the average of the original time series data and the shifted time series data by τ to get an autocorrelation function (ACF) in τ. The ACF from the FCS data is next fitted to theoretical autocorrelation functions (ACFs) to determine underlying kinetic parameters, such as a diffusion coefficient.

ΔFtFt=FtFttFtE45

where Ftis the fluorescence fluctuation in the confocal volume and Ft=1T0TFtdtis the time average of the fluorescence fluctuation during observation time T. Notice that the mean of standardized data (ΔFt/Ft) is zero. Next, the autocorrelation function of the standardized data is calculated by multiplying the standardized data, ΔFt/Ft, and the shifted standardized data by τ, ΔFt+τ/Ft, and then taking the average over time:

Gτ=ΔFtFt·ΔFt+τFttE46

Notice that the autocorrelation has the maximum when τ=0and converges to 0 as τincreases as ΔFt/Ftand ΔFt+τ/Ftbecome independent for a large τ.

G0=ΔFtFt2t>0Gτ=ΔFtFt·ΔFt+τFtt=ΔFtFttΔFt+τFtt=0foralargeτE47

An autocorrelation curve carries two crucial information. Since a large molecule will move slower than a light molecule, therefore the correlation decays at a longer time scale. On the other hand, the correlation amplitude is inversely proportional to the concentration of fluorophores due to the denominator for standardization. The information on the diffusion coefficient and concentration of fluorophores can be determined, once a mathematical model for Gτis developed.

Stationarity and ergodicity of the diffusion process play a pivotal role to derive an FCS equation in a closed, yet simple, form. A continuous-time dynamical system such as Brownian motion is called ergodic when all the accessible microstates such as the locations of a Brownian particle are equally probable over a long period, i.e., the statistical properties from the time average at a position are same as the ensemble (spatial) average at any moment. On the other hand, a stationary process is a stochastic process whose probability distribution and parameters are invariant by shifts in time. Stationary and ergodic properties of a diffusion process were proven mathematically [26].

If we let nxtbe the fluorescence molecule density per unit area, the temporal average of nxtat a location x0and the spatial (ensemble) average of nxtcan be defined as

nx0tt=limT0Tnx0tdtnxt0=Enxt0=nxt0PXt0=xdx.E48

Under stationarity and ergodicity of a diffusion process, we assume

nx0tt=limT1T0Tnx0tdt=nxt0PXt0=xdxErgodikicity=nx0PX0=xdxStationarity=nx0xE49

where nxt0can be though as a snapshot of all the positions of Brownian particles at any fixed time t0.

4.2 Derivation of diffusion FCS equation

For the fluorescence molecule density per unit area, nxt, if we let fxtbe the fluorescence intensities due to photons from fluorescent proteins at the location xat the time t, then fxtis proportional to nxt. On the other hand, since more fluorescence photons can be generated under the higher laser intensity, fxtis also proportional to the laser intensity, Ix. Therefore, fxtsatisfies

fxt=QIxnxtE50

where Qis a proportionality constant for the product of the absorption cross section by the fluorescence quantum yield and the efficiency of fluorescence, and Ixis a function describing a Gaussian laser profile:

Ix=2πω2exp2x2ω2E51

where ωis the half-width of the beam at e2, which measures the size of a confocal volume (V).

A bell-shaped profile of Ixdefines a unit confocal volume (V) with V=Ixdx=1, resulting from the error function integral (Eq. (11)). Therefore, the fluorescent intensity (or the number of photons, Ft) from the confocal volume is determined by

Ft=fxtdx=QIxnxtdx=QVIxnxtdxE52

where we used the fact that the Gaussian laser profile defines the confocal volume in the last equality to switch the integration domain from Vto .

Lastly, we will also assume the spatial and temporal independence of fluorescence intensities:

fxtfytt=fxt2tifx=yfxttfyttifxy=fx02ifx=yfx0fy0ifxy.E53

This assumption hypothesizes that fluorescence intensities from different locations are not correlated but independent.

In FCS, to analyze the fluorescence fluctuations from the confocal volume (V), an autocorrelation function (ACF) of the variations in Ftis considered. The variations in the number of photons from the mean number of photons in a confocal volume (ΔF) are calculated by ΔFt=FtFtwhere Ftand Ftare the fluorescence intensity in the confocal volume at time tand the mean fluorescence in the confocal volume, respectively. Therefore, by Eq. (52)

ΔFt=FtFt=fxtdxlimT1T0Tfxtdt=QIxnxtdxlimT1T0TQIxnxtdxdt=QIxnxtdxQIxntdx=QIxΔnxtdxE54

where Δnxt=nxtnt, we used the identities Ixdx=1.

Next, the autocorrelation function of the standardized fluorescence fluctuations, ΔF/Ft, is computed by

Gτ=ΔFtFtΔFt+τFtt
Ft2Gτ=limT0TΔFtΔFt+τdt
=ΔFtΔFt+τtE55
=QIxΔnxtdxQIxΔnxt+τdxt
=Q2IxIyΔnxtΔnxt+τtdxdy

where we used Eq. (50).

Notice that nxtsatisfies the diffusion equation (Eq. (19)). Therefore, Δnxt+τalso satisfies a diffusion equation in τand xwith initial time at t(τ=0):

tΔnxt+τ=D2x2Δnxt+τΔnxt=nxtntE56

Consequently, the solution Δnxt+τis found as (Eq. (21))

Δnxt+τ=Δnx¯t14πDτexpxx¯24dx¯E57
=Δnx¯tΦτxx¯dx¯.

Next, we use the ergodicity of a diffusion process to derive some essential properties of the double integral. Because diffusion is an ergodic process, the time average can be replaced by the ensemble average.

ΔnxtΔnyt+τt=ΔnxtΔnx¯tΦτyx¯dx¯t
=ΔnxtΔnx¯ttΦτyx¯dx¯
=Δnx0Δnx¯0Φτyx¯dx¯E58
=Δnx02δxx¯Φτyx¯dx¯
=σ2Φτyx

where σ2=Δnx02is the variance of nx0, or the mean square fluctuations of the fluorescence molecules, and δxx¯is the Dirac delta function defined as Eq. (18). In Eq. (58), the stationary and ergodic assumptions were used in the third line to convert the time average to the spatial average at t=0.

Δnx0Δnx¯0=Δnx02ifx=x¯Δnx0Δnx¯0ifxx¯E59
=σ2ifx=x¯0ifxx¯
=σ2δxx¯

By plugging Eq. (58) back into Eq. (55)

Ft2Gτ=Q2IxIyΔnxtΔnxt+τtdxdy
=Q2IxIyσ2ΦτyxdxdyE60
=Q2σ22πω2exp2x2+y2ω214πDτexpxy24dxdy
=Q2σ22πω2·14πDτexp2x2+y2ω2xy24dxdy

If we substitute y=x+4η(dy=4), then

Ft2Gτ=2Q2σ2πω24πDtexp2x2+x+4η2ω2η24dηdy
=2Q2σ2ππω2exp2x2+x+4η2+ω2η2ω2dηdxE61
=2Q2σ2ππω2exp2x2+x+4η2+ω2η2ω2dηdx

where we used the fact

exp2x2+x+4η2ω2η2dηdy
=exp4x2+44+24η2+ω2η2ω2dηdyE62
=exp4x2+2+η2+4+ω2η2ω2dηdy
=exp4x+Dτη2+4+ω2η2ω2dηdy

Now, we can evaluate the inner integral in Eq. (46) using a substitution z=x+ηfor x

exp4x+Dτη2ω2dx=exp4z2ω2dzE63
=ωπ2

where we used Eq. (11). Back to Eq. (61)

Ft2Gτ=2Q2σ2ππω2exp4x+Dτη2ω2dxexp4+ω2η2ω2
=Q2σ2ωπexp4+ω2η2ω2E64
=Q2σ2ωπω2π4+ω2
=Q2σ2ωπ11+τ/τ_D

by the error function integration (Eq. (11)), where τD=ω2/4D, which is a diffusion time.

If fluorescence molecules undergo Brownian motion, then the number of photons in a confocal volume changes in time due to random movements of fluorescence molecules in and out of the confocal volume. In FCS analysis, the number of photons (or fluorescence molecules) from a confocal volume at any moment tis assumed to follow a Poisson distribution, in which the probability for kfluorescence molecules (or photons) to be found in the confocal volume is

PFt=k=eλλkk!E65

where λ=Ftis the average number of fluorescence molecules (or photons) in the confocal volume. This assumption is reasonable for a diffusion process since the arrival process of infinitely many identical independent diffusion processes was shown to be a Poisson process [27]. Importantly, the mean (or expectation) and variance of a Poisson random variable are known to be equal

EFt=k=0kPFt=k=Ftσ2=EFtFt2=Ft.E66

Since we assumed that Ftfollows the Poisson statistics that has equal variance and mean

Ft2G0=ΔFtΔFt+0tE67
=ΔFt2t
=σ2
=Ft

by Eq. (66). On the other hand, by Eq. (64)

G0=Q2σ2ωπE68

which indicates that

1Ft=Q2σ2ωπE69

By replacing the bulk parameters in Eq. (47) with 1/Ft

Gτ=1Ft11+τ/τDE70

As we saw, with a Poisson distribution assumption on Ft, we can readily determine the average density of fluorescence molecules as well as the average number of fluorescence molecules in the confocal volume. Similar to FRAP equations, FCS equations in higher spatial dimensions can be found by similar calculations

Gτ=1Ft11+τ/τDxy2DE71
Gτ=1Ft11+τ/τDxy1+τ/τDz3D

where τDxy=ωxy2/4Dand τDz=ωz2/4Dwith ωxy= the half-width of the beam at e2in x/ydirection and ωz= the half-width of the beam at e2in zdirection.

5. Conclusion

Diffusion plays a crucial role within biological systems in many different temporal and spatial scales from various perspectives. It is a dominant way for biological organisms to transport multiple molecules to desirable locations for cell signaling. However, to quantify the molecular diffusion, especially in live cells, is still challenging although a couple of tools are available, including fluorescence recovery after photobleaching and fluorescence correlation spectroscopy. Although FRAP and FCS were originally developed to study biological diffusion processes, they are now being applied not only to a diffusion process but also to a broad range of biochemical processes, including binding kinetics and anomalous diffusion. Since the derivation of FRAP and FCS equations for many biochemical processes shares many common steps with the diffusion FRAP and FCS equations, it is essential to understand the mathematical theory behind the diffusion FRAP /FCS equation [18, 22, 25, 28, 29, 30, 31, 32]. In this study, we provide a simple and straightforward derivation of FRAP/FCS equation for free diffusion based on calculus-level mathematics, so that FRAP/FCS equations and its applications are accessible to a broad audience. Although the applications of these FRAP and FCS equations to cell membrane biophysics from experimental perspectives can be a very important topic, it is beyond the scope of this chapter and therefore will not be covered here. These topics are well documented in various references, and interested readers are referred to [20, 31, 33], and references therein. We hope that this tutorial is understandable as well as gives readers a solid theoretical foundation for FRAP and FCS, bridging the gap between experimental and theoretical aspects of FRAP and FCS.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited.

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Minchul Kang (September 30th 2020). Diffusion Theory for Cell Membrane Fluorescence Microscopy, Fluorescence Methods for Investigation of Living Cells and Microorganisms, Natalia Grigoryeva, IntechOpen, DOI: 10.5772/intechopen.91845. Available from:

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