3. Lateral Diffusion
3.1 Normal Diffusion
The movement of particles down a concentration gradient is coined diffusion, and it is traditionally described by the laws of Fick derived more than 150 years ago [86].
His second equation,
@
@t =Dr2 , (3.1)
shows that the change in particle concentration in timet is directly proportional to the second spatial derivative of the concentration, and the coefficient of propor-tionality, D, is called the (collective) diffusion coefficient. Here, it is assumed that D is independent of concentration, which does not always hold true. In a more general case, the right-hand side of Eq. (3.1) gets replaced by r·(Dr ).
Interestingly, diffusion does not only describe motion down a concentration gradient.
In 1827 botanist Robert Brown observed the constant jittery motion of particles ejected by pollen granules under a microscope [87]. By witnessing similar behavior of inorganic particles, he could discard the possibility that this movement stems from pollen’s origin as a part of a living organism. However, it took decades until the efforts of Albert Einstein [88] and Marian Smoluchowski [89] revealed the true nature of this “Brownian” motion. They explained the seemingly random movements of the particles by their repeated collisions with small water molecules driven by thermal motion. Einstein arrived in an equation similar to Eq. (3.1) with concentrations
18 3. Lateral Diffusion replaced by probabilities P(r, t) of finding a particle at positionr at time t. For a released particle in two-dimensional space relevant for membranes, this equation has a solution
P(r, t) = 1
4⇡Dt ⇥exp
✓ r2 4Dt
◆
, (3.2)
i.e. the probability distribution function (PDF) of particle distances from their origin r is Gaussian and spreads in time. The rate of this spreading is defined by D. The maximum of the PDF remains at r = 0 whereas its variance, the mean squared displacement (MSD), grows linearly in time as
MSD =hr2(t)i= Z 1
0
r2P(r, t)2⇡rdr= 4Dt. (3.3) Notably, the constant in front of D scales as 2d, where d is the dimension of motion.
Considering the average diffusion length lD2 = hr2(t)i, a very intuitive picture of diffusion is provided by noting that a particle undergoing normal diffusion travels on average a lateral distance of lD =p
2dDtover a time t.
It is worth noting that to obtain Eq. (3.2), no concentration gradient is required in terms of Fick’s second law ((Eq. (3.1)). Instead, it describes the self-diffusion or tracer diffusion of a single particle. The MSD in Eq. (3.3) is called the ensemble-averaged MSD (EA-MSD), as the PDF generally describes the motion of a set of identical self-diffusing particles. Fig. 3.1C demonstrates the spreading of the Gaussian distributions over time and the corresponding linear growth of the EA-MSD.
In 1908, Paul Langevin considered that when undergoing Brownian motion, a diffusing pollen grain feels a stochastic force by collisions to the solvent particles. He applied this white Gaussian noise⇠ with the autocorrelation
h⇠f(t)⇠f(t0)i= 2D (t t0) (3.4) (with Dirac delta function ) to the equation of motion and considered the overdamped case where inertial effects are insignificant. This resulted in the Langevin equation
dr
dt =⇠, (3.5)
from which he was able to extract Eq. (3.3). At this overdamped limit, Eq. (3.5) no longer necessarily describes the motion of a physical object, yet converges to the
3.1. Normal Diffusion 19
0 100 200 300 400 500
≠10 0
10
A
Time (a.u.)
Position(a.u.)
Random walk (–= 1)
0 10 20 30 40 50
0 5 10
Normal diffusion –= 1
C
Time (a.u.)
Displacement(a.u.)
0 100 200 300 400 500
B
CTRW (–= 0.8) FBM (–= 0.8)
0 10 20 30 40 50
Anomalous diffusion, –= 0.8
D
ProbabilityMSD
Figure 3.1 Visualization of some key concepts related to diffusion. A) Sample trajectory of a 1-dimensional (1D) random walker modeling normal Brownian motion. B) Sample trajectories of 1D subdiffusive motion following the continuous time random walk (CTRW) and fractional Brownian motion (FBM) mechanisms. In both cases↵ is equal to 0.8. With such an exponent, FBM is essentially indistinguishable from regular Brownian motion in panel A. CTRW is recognized from its long waiting times. C) The probability distribution P(r, t) is shown as a function of displacementr (only positive half plane is shown) and time t. For normal diffusion, this follows Eq.(3.2). The red line shows the EA-MSD, Eq (3.3).
D) Same as panel C but for anomalous diffusion with ↵= 0.8. Here, the distributionP(r, t) follows Eq. (3.17). This shape of P(r, t) is characteristic for FBM, but not CTRW. The red line shows the EA-MSD that follows Eq. (3.14), and applies to both FBM and CTRW.
mathematical concept of a random walk.
In his seminal work, Einstein also derived an expression for the diffusion coefficient
D=µkBT, (3.6)
where T is temperature, kB the Boltzmann constant, and µ the mobility of the particle, i.e. the constant of proportionality between the particle’s terminal velocity and a force applied to it. Experimental verification followed in 1909, when Jean Baptiste Perrin tracked the motion of colloidal particles, calculated their MSD, and extracted Avogadro’s constant (NA =R/kB with R the back-then known universal gas constant) from Eq. (3.6) [90].
20 3. Lateral Diffusion
3.1.1 Some Key Measurables
Next, a few fundamental concepts related to diffusion that are regularly encountered in the literature, including the remainder of this Thesis, are introduced.
Some experimental techniques track the motion of a single particle. Therefore, to obtain reasonable statistics, a single long measurement is performed over a time ⇥, and the MSD is averaged over time and given as a function of lag time . An MSD value at lag time corresponds to the average MSD over all time intervals that have a length present in the trajectory. This time-averaged (TA) MSD reads
2( ) = 1
⇥
Z ⇥ 0
[r(t+ ) r(t)]2dt, (3.7) and the integral is often discretized. In cases where many trajectories are measured, the TA-MSD can also be averaged over N different trajectories as
h 2( )i= 1 N
XN i=1
i2( ) (3.8)
to further improve the quality of the TA-MSD as an ensemble- and time-averaged MSD (EA-TA-MSD). Curiously, sometimes such a time average is not well-defined but depends on the duration of the measurement. This aging phenomenon is crucial for processes where waiting times are not bound, such as continuous time random walks discussed in Section 3.2.2.
Processes such as Brownian motion that fulfill hr2( )i= lim⇥!1 2( ) are called ergodic. However, it is relatively often observed in biological systems that the time and ensemble averages are not equal, and the system shows ergodicity breaking.
This behavior can be characterized by the spread of TA-MSD curves, yet it is best captured in the ergodicity breaking parameter [91]
EB( ) =h 2( )i h ( )i2 =h 2( )i 1, (3.9) where
( ) =
2( )
h 2( )i. (3.10)
For ergodic processes, EB eventually converges to zero.
3.1. Normal Diffusion 21
3.1.2 Free Area Model for Lipid Diffusion
Plenty of effort has been invested in developing models that predict diffusion co-efficients based on other measurable quantities, as this would remove the need for carrying out often tedious and expensive measurements. The free volume and free area models describe diffusion as a process, where a particle diffuses by repeatedly jumping to an opening volume or area in its vicinity. The origin of such models lies in the free volume concept developed by Cohen and Turnbull to describe the three-dimensional diffusion in hard-sphere solutions [92]. Their purely geometrical reasoning was combined with the ideas of Eyring [93] by Macedo and Litovitz, who extended the model to include an energetic term [94]. Their equation reads
D⇠exp
✓ v⇤ vf
+ Ev⇤ RT
◆
, (3.11)
where v⇤ is called critical volume,vf is the available free volume, and accounts for the overlap in this free volume. The activation energy in constant volume, i.e. the energy required for a diffusing particle to break free from its surroundings prior to jumping to another vacant site, is given by Ev⇤, whereas R andT are the universal gas constant and temperature, respectively.
The free volume model evolved into a free area model as it was applied to planar lipid bilayers by several teams [95, 96, 97, 98]. However, the complete description was provided by Almeidaet al. [10]. Their result replaces the critical area parameter
a⇤ by the cross-sectional close-packed area of a diffusing lipid a0 [95], and reads
D= 3.224⇥10 5
rT a(T) M exp
✓ a0
a(T) a0 + Ea
RT
◆
. (3.12)
Here, a(T) is the average area of a lipid at temperature T in units of Å2, Ea is the activation energy, and M is the molar mass.
This model has been successfully employed to describe the temperature-dependence of lipid diffusion in protein-free model membranes [10]. Moreover, it links the decrease in diffusion coefficients due to the addition of cholesterol to the reduction in the free volume in the membrane [10] and fits the data measured for lipid monolayers [99].
The fundamental assumptions of the free area model are that 1) lipids are solid cylinders with a well-defined cross-sectional area, 2) the lipids move with discrete
22 3. Lateral Diffusion and rapid jumps to a nearby pocket of approximately their own size, 3) these pockets open at a rate faster than the jumps occur, and 4) a jump only takes place if a lipid can break free from the interactions with its neighbors, i.e. overcome the activation energy barrier.
Some of these assumptions are clearly oversimplifications. The lipids are obviously not cylinders. Instead, their area varies drastically across the lipid leaflet [100]. Moreover, the link between the activation energy Ea and the similar Arrhenius activation energy concept remains unclear [101]. Computer simulations do not support the picture of discrete jumps, but instead suggest that lipids move as concerted flows of loosely-defined clusters along the membrane [11, 102]. Such flows have also been detected by experiments [103, 104].
3.1.3 The Saffman–Delbrück Model for Protein Diffusion
The well-known Stokes–Einstein relation is the application of Eq. (3.6) to spherical objects diffusing in three dimensions with a mobility ofµ= (6⇡⌘R) 1. Here,⌘ is the dynamic viscosity of the solvent andR the radius of the diffusing object. It would be tempting to derive a similar formulation for a two-dimensional system in which a disk, presenting a membrane protein, diffuses along a liquid sheet, presenting the membrane. Unfortunately, the infamous Stokes’ paradox states that this problem does not have a steady-state solution.
Fortunately, this limitation can be overcome by applying some additional boundary conditions to the problem. In their seminal work, Philip Saffman and Max Delbrück studied three such cases, one of which considered a membrane being surrounded by a solvent with a non-zero viscosity [12, 13]. Also, they considered the cylindrical mem-brane proteins and the memmem-brane to share a non-zero thickness. These assumptions led to the Saffman–Delbrück (SD) model [12, 13]
DSD= kBT 4⇡µmh ⇥
ln
✓hµm
µfR
◆
, (3.13)
where h is the thickness of the membrane, µm and µf are the viscosities of the membrane and the surrounding solvent, respectively, and is the Euler–Mascheroni constant equal to ⇠0.577. The central difference to the Stokes–Einstein relation is the weak logarithmic size-dependence of the diffusion coefficients.
3.2. Anomalous Diffusion 23