### Continuous model

In this first model, membrane particles are independent, punctual and subjected to Brownian motion. The membrane is either a one or two dimensional manifold with periodic boundary conditions. This imposes periodic boundary conditions on the equations of motion.

For simplicity’s sake, we first derive the model and equations for the 1D case (equations for the 2D case are provided afterward using another method). Therefore, we consider the membrane as a segment of length

*2 L* centered on zero ([–

*L*;

*L*]); when a particle crosses one of the boundary it will appear at the other side). We assume that these particles undergo a classical Brownian motion in the position space (i.e. overdamped regime). To describe membrane heterogeneity, the diffusion coefficient depends on the position of the particle on the membrane. This so-called multiplicative noise can be efficiently modeled, in the 1D case, with the Stratonovich formalism yielding the following stochastic differential equation

Here *X*
_{
t
} describes the position of the molecule at time *t* in the segment [–*L*; *L*], *dZ/dt* is the classical Brownian noise (with zero mean and unit variance) and
a nonnegative *2L*-periodic function. Here
is thespace-dependent standard deviation of the noise. Then, the actual diffusion (denoted as
) is equal to
. We will suppose that
is continuously differentiable and > 0 everywhere.

By setting

*ρ*(

*x,t*) the probability density function of a particle, we obtain the associated Fokker-Planck equation [

40] derived from Eq.

1:

Looking for continuous and differentiable solutions, boundary conditions emerge naturally from the *2 L* periodicity. Namely, *ρ*(*L,t*) = *ρ*(–*L,t*) and
for all *t ≥* 0. Moreover, fluxes of particles are opposed on each side of the border. Writing this condition leads to
.

The solution of Eq.

2 can be found at equilibrium, yielding the constant flux

*J*:

The boundary conditions impose

*J*(–

*L*) = –

*J*(

*L*) =

*J* = 0 and consequently

with
So, up to a normalization constant, equilibrium density is the inverse of the square root of the diffusion coefficient (or equal to the square root of viscosity).

This result remains the same for 2D (or any higher dimension): Assuming a non-homogeneous brownian motion in higher dimension via

in the square [–

*L*;

*L*]

^{2} and

*dZ*
_{
i
}
*/dt* (

*i* = 1,2) being the classical Brownian noise (with zero mean and unit variance). The Fokker-Planck equation for the density function

*ρ*(

*x,y*) will be

with the boundary conditions

for all

*t ≥* 0. To solve Eq.

6, we multiply by

and integrate with respect to both

*x* and

*y* to obtain

since

An integration by parts yields

. The first integral on the right hand side of the previous equation is taken over the square [–

*L*;

*L*]

^{2} and vanishes due to toric boundary conditions (Eq.

7), yielding

Since
is > 0 everywhere the left hand side of Eq. 8 is a decreasing function and its derivative must reach zero at equilibrium.

This equilibrium therefore verifies

meaning the gradient of

is zero everywhere. That is

The same result holds for any dimension: any diffusion over a non-constant diffusion profile yields, at equilibrium, to a non constant concentration namely its inverse. Note that as it should be, in the case of homogeneous diffusion (
) the equilibrium density (*ρ*(*x*) ≡ (2 *L*)^{–1}) is independent of the diffusion coefficient *D*
_{0}.

As a consequence zones of slow diffusion (or high viscosity) will tend to gather more particles at equilibrium than any faster zones. Note that we used the Stratonovich formalism to describe non-homogeneous brownian motion of particles. This formalism make assumptions on the diffusion at the microscopic scale [41]. With other assumptions, we could have derived the same equations using the Ito formalism to obtain a slightly different result.

Briefly: the associated Fokker-Planck would have been:

yielding an equilibrium distribution

The main results hold in a stronger way (due to the square) in any dimensions. In the continuous model, all simulations described below have also been done using the Ito formalism for the random motion scheme. All the results at equilibrium described below hold qualitatively in the Ito formalism but in a stronger way quantitatively. Thus, to stress our point, we will focus on the ‘weaker’ form of our results and remain in Stratonovich formalism throughout the rem`aining of the paper.

The previous results hold for equilibrium distribution only. General transient solutions can be obtained for the one dimensional case by solving Eq.

2 entirely. Let

and

Eq.

2 becomes

which is exactly the classical diffusion equation. One can immediately see that the same boundary conditions apply to *p* by replacing *x* by
. Equilibrium solution for classical diffusion on a circle is a constant density. So
. In addition, we obtain the transient solutions (on a circle) by replacing *x* by
in the general solutions of the classical diffusion. Note that depending on the initialconditions, particles can take a longer time to reach equilibrium in the slow diffusing zones. Therefore, transiently it may happen that zones of slow diffusion will gather less particles than in faster zones.

On biological membranes, transient solutions are impossible to measure. The classical solution is to estimate them indirectly via imaging techniques such as Fluorescence Recovery After Photobleaching (FRAP). Bleaching particles amounts to create a new initial distribution

*ρ*(

*x*,0) =

*g*(

*x*). Particles inside the bleaching are removed (actually bleached) while the others are untouched:

*g*(

*x*) = 0 for

(for a bleaching beam of radius

*r* centered on zero). In that case, the classical approximation of the half-time to recovery [

42,

43] is

where

*γ* is a constant. Replacing

*r* by

and using

, we can account for an heterogeneous diffusion dependent half-time recovery.

Equation 14 indicates that the time constant of the recovery is proportional to the square of the spatial average of the *inverse* of the square root of diffusivity. Using this equation, one can, theoretically, extracts the diffusion profile from FRAP data [44]. Unfortunately, for a general diffusion profile, no such transient solution can be found for higher dimensions.

At this point, we can already make some useful predictions on membrane particle distribution. Indeed, some membrane components are known to alter membrane viscosity. Our model suggests that, around components that increase viscosity, diffusing particles will be in higher concentration – overconcentration – compared to faster zones. A good candidate could be cholesterol: Indeed, cholesterol is found in high amount on virtually all mammal cells and is known to rigidify membranes and cholesterol-enriched membranes display lower diffusion coefficient (by around one order of magnitude) [45, 46]. In this case our model predicts that membrane cholesterol islands (patches) are *originating* membrane domains. In other words and without any more assumptions, stable cholesterol domains should imply by means of slow diffusion a protein-riched domain.

In addition, our model estimates this domains to concentrate more particles in proportion to the slowing. Indeed this overconcentration is simply proportional to the square root of the ratio of diffusion. For example, diffusion an order of magnitude slower should provide slow domains with around three times more particles.

Conversely, when cholesterol concentration is low enough or high enough for the membrane to exhibit an approximatively constant diffusion profile, our model predicts no domain.

### Force field interactions

This first model is neglecting some crucial features of membrane diffusing particles. As such, our model assumes independent particles that never collide nor hinder each other and that cannot interact. But both crowding and particle interactions are known to yield inhomogeneous particle distribution. We expand the previous model to take into account steric hindrance and possible particle interactions (e.g. protein-protein interactions) [47].

To study the combined effect of non-homogeneous diffusion and particle interaction, we used 2D particle simulations. As such, particle interactions must be modeled at the same scale as the diffusion itself: the mesoscopic scale. In order to do so, particles were subjected to non-homogeneous Brownian motion and short-distance interactions [

48,

49]. Assuming there are

*N* particles, we introduce a new equation of motion for a particle

*j* as:

where

*u*
_{
ij
} is the unitary vector of the semi-line starting from particle

*i* to particle

*j*,

*r*
_{
ij
} being the distance between the two particles. The interaction force

*f* is derived from the 6 – 12 Lennard-Jones empirical potential accounting for van der Waals interactions:

the force *f* itself being the opposite of the gradient of the potential
. The two parameters *∈* and *σ* describe the shape of the interaction. For a given *σ* increasing *∈* increases the strength of the interaction (as well as its range) and clusters of particles appear. The average size ofclusters with increasing *∈* yields a phase transition from *N* clusters of size 1 to one cluster of size *N*[50, 51].

However, in the case of non-homogeneous diffusion, the phase transition will be shifted to lower interaction strength in the slower zones. As such, fast zones will act as a particle provider (well) and slow zones as particles sink. We expect therefore to obtain in the slow zones a higher over-concentration compared to the fast zones but also a higher degree of clustering. This over-concentration should be higher than both effects (interaction and non-homogeneous diffusion) taken separately.