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

d X t = D ( X t ) ∘ d Z

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 D a nonnegative 2L-periodic function. Here D is thespace-dependent standard deviation of the noise. Then, the actual diffusion (denoted as D ) is equal to D x = D 2 x . We will suppose that D 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:

∂ ρ ( x , t ) ∂ t = 1 2 ∂ ∂ x D ( x ) ∂ ∂ x D ( x ) ρ ( x , t )

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

The solution of Eq. 2 can be found at equilibrium, yielding the constant flux J:

J = D ( x ) ∂ ∂ x D ( x ) ρ ( x )

The boundary conditions impose J(–L) = –J(L) = J = 0 and consequently

ρ ( x ) = Ω D ( x )

with Ω = ∫ − L L d x D ( x ) − 1 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

d X t d Y t = D ( x , y ) 0 0 D ( x , y ) ∘ d Z 1 d Z 2

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

∂ ρ ( x , y , t ) ∂ t = d i v ( D ∇ ( D ρ ) )

with the boundary conditions

∂ ∂ x D ( x , y ) ρ ( x , y , t ) | ± L , y       = ∂ ∂ x D ( x , y ) ρ ( x , y , t ) | x , ± L = 0

for all t ≥ 0. To solve Eq. 6, we multiply by D ρ and integrate with respect to both x and y to obtain 1 2 ∂ ∂ t ∫ ∫ D ρ 2 d x d y = ∫ ∫ D ρ d i v ( D ∇ ( D ρ ) ) d x d y since ∂ ∂ t ρ 2 = 1 2 ρ ∂ ρ ∂ t An integration by parts yields ∫ ∫ D ρ d i v ( D ∇ ( D ρ ) ) d x d y = ∫ [ − L ; L ] 2 D 2 ρ ∇ ( D ρ ) − ∫ ∫ ∇ ( D ρ ) D ∇ ( D ρ ) d x d y . 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

1 2 ∂ ∂ t ∫ ∫ D ( x , y ) | ρ | 2 d x d y = − ∫ ∫ D | ∇ ( D ρ ) | 2 d x d y

Since D 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

∫ ∫ D | ∇ ( D ρ ) | 2 d x d y = 0

meaning the gradient of D ρ is zero everywhere. That is

ρ ( x , y ) = Ω D ( x , y )

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 ( D ( x ) ≡ D 0 ) 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:

∂ ρ ( x , y , t ) ∂ t = d i v ( ∇ ( D 2 ρ ) )

yielding an equilibrium distribution

ρ ( x , y ) = Ω D 2 ( x , y )

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 x ˜ ( x ) = ∫ − L x d u D ( u ) and p ( x ˜ , t ) = D ( x ) ρ ( x , t ) Eq. 2 becomes

∂ p ∂ t = 1 2 ∂ 2 p ∂ x ˜ 2

which is exactly the classical diffusion equation. One can immediately see that the same boundary conditions apply to p by replacing x by x ˜ . Equilibrium solution for classical diffusion on a circle is a constant density. So D ( x ) ρ ( x , t ) = p ( x ˜ , t ) = Ω . In addition, we obtain the transient solutions (on a circle) by replacing x by ∫ − L x d u D ( u ) 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 | x | < r (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

τ r = γ r 2 D

where γ is a constant. Replacing r by ∫ − r r d u D ( u ) and using D ( r ) = D 2 ( r ) , we can account for an heterogeneous diffusion dependent half-time recovery.

τ r = γ ∫ − r r d u D ( u ) 2 = γ ∫ − r r d u D ( u ) 2

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.

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:

d x j d y j = D 2 ( x j , y j ) ∑ i = 1 , i ≠ j N f ( r ij ) u ij + D ( x j , y j ) d Z x d Z y

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:

P ( r ) = ∈ σ r 12 − 2 σ r 6

the force f itself being the opposite of the gradient of the potential f ( r ) = − d P ( r ) d r . 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.

We present in this section the results obtained via simulation of particles that undergo equation of motion such as Eq. 1. We used the Milstein numerical schema for the Stratonovich formalism 52 ,

X i ( t k + 1 ) = Δ t D ( X i ( t k ) ) Z + Δ t 2 D ( X i ( t k ) ) D ′ ( X i ( t k ) ) Z 2

Where Xi (t) is the position of particles i at time t (t _k  = (k + 1)Δt) and D and D ′ are the diffusion profile and its gradient respectively. Various diffusion profiles were tested as well as different number of particles. For the 1 dimensional experiment 100,000 particles were used whereas 1,000,000 were used for the 2 dimensional case. We tested various time step and took the highest value (Δt = 0.01)) that yielded no change in equilibrium.

Equilibrium distribution behaved as expected. The first set of experiments is for the one dimensional case where we test various diffusion profiles (continuously differentiable 2 L-periodic). As an example, we present two of them: In one case, Figure 1, the diffusion profile is a Gaussian function D ( x ) = D 0 ( 1 − d exp ( − x 2 / ω ) with D ₀ = 1, d = 0.9 and ω = 0.1 (displayed in inset). The theoretical distribution Ω / D ( x ) and the simulated particle distribution are depicted by a plain line and dots respectively.

Transients for this simulation are displayed on Figure 2 as ρ(t,x) for 0 ≤ t ≤ 10 and x∈[–1:1]. The initial condition is ρ(0,x) = δ (x – 0.5) a Dirac function centered on 0.5.

In this case, due to the slow diffusivity the concentration at the peak of viscosity ρ(t,0.0) is the lowest value for a long period: transiently the slow zone is under populated.

In addition, we simulated FRAP experiments and compared the diffusion coefficients computed through simulation to the theoretical ones as a function of the FRAP radius r (the FRAP experiment was centered on the slowest diffusion point). Figure 3 shows the results for three experiments, one classical homogeneous diffusion with D ( x ) = 1 and two NHD with a centered gaussian D ( x ) = 1 − d e − x 2 / ω with d = 0.9 with ω = 0.1 and ω = 0.01 (shown in inset). Triangles, squares and circles are diffusion coefficient computed from the particle simulation for the cases D x = 1 , D = D ( x ) with ω = 0.1 and ω = 0.01 respectively. The lines are their theoretical counterparts computed using Eq. (14).

Next, we present the results of NHD on a two-dimensional torus. The diffusion profile consists of 4 randomly positioned Gaussian patches centered on (x _i ,y _i ) ∈ [–L; L]² with 1 ≤ i ≤ 4 and D i ( x , y ) = D 0 ( 1 − d e x p ( − ( ( ( x − x i ) 2 + ( y − y i ) 2 ) / ω ) with D ₀ = 1, d = 0.9 and ω = 0.1. The overall diffusion profile is the average D ( x , y ) = 1 4 ∑ i = 1 4 D i ( x , y ) . Figure 4 shows the diffusion profile (light spectrum heightmap, Figure 4A) and the 2D histogram of the resulting equilibrium distribution of particles (Figure 4B). The last panel (4C) compares a cross-section (x = 0.2 is constant) of theoretical and simulated distributions. This cross-section can be visualized as a black line on the heightmap of Figure 4A and as a white line on Figure 4B.

In both one and two dimensional cases, the equilibrium distribution of particles that undergo non-homogeneous diffusion matches the theoretical prediction. Slower zones concentrate more particles than the others. In addition this over-concentration isproportional to the amount of slowing.

Simulations were performed for the particles with interaction. To take into account steric hindrance, the numerical schema (Eq. 17) was modified by adding an interaction strength vector Δtf(X _i (t)).

Collision detection was included to allow a simple crowding experiment (i.e ∈ = 0 in Eq. 16). We present the following experiment: On a 2D square (with toric boundary conditions) we insert a squared slow patch (on the upper leftcorner) where diffusion D is r = D 0 / D times smaller than the diffusion ( D 0 ) in the remaining of the space: a “cholesterol” patch. Another square of the same surface away from the cholesterol patch is used as control patch. Numbers of particles are counted in both patches at equilibrium and normalized by the total number of particles and the size of the patch.

Assuming cholesterol diffusion is reduced by around one order of magnitude 53 54 55 56 , those numbers are displayed on Figure 5 for two conservative (i.e higher) values of the diffusion ratio r = D 0 / D = 2 (black square) and r = 1.4 (black circles). The dashed lines are the theoretical predictions for the concentration in the patch in case of punctual particles (that is y = r). For low interaction strength, the punctual particles prediction remains a good approximation. However when the interaction increases, concentration of particles in the cholesterol patch increases dramatically while there is no such increase in the control patch. On Figure 6 screenshots of particles positions are displayed (r = 0.5) for ∈ = 0 (A), ∈ = 500(B), ∈ = 4000(C) and ∈ = 10000(D) – see also supporting videos.

As expected, the overconcentration effect due to non-homogeneous diffusion is dramatically increased via particles interaction. As Figure 5 shows it, a great portion of the particles is concentrated in the cholesterol patch. In this extreme case, the patch is virtually composed of one big cluster of particles in interaction whereas there are no clusters in the remaining of the space.

To numerically assess this situation, we compute the clustering coefficient as a function of the interaction strength: c(f) = N _c /N _i where N + N _c is the number of clusters and N _i is the number of particles involved. Clusters were computed by creating a graph of particles whose nodes are the particles themselves and link between two nodes exists if the distance between their centers is below ar ₀ (a = 2.7 was chosen a bit higher than σ = 2.5). A value close to 1 indicates no clustering while a more pronounced clustering yields smaller values. Results are displayed in Figure 7. The values are computed in three situations: in the first two, diffusion is homogeneous with diffusion D = 1 (squares) and D = 0 . 5 (circles). The remaining case (triangles) is the one with a cholesterol patch with ratio r = 2 ( D 0 = 1 as in the previous experiments). In all cases, the clustering coefficient decreases with the interaction strength and thereforeclusters appear (one needs higher interaction strength for the case D = 1 ). Moreover, the lower the viscosity the deeper the clustering. However, for high enough interaction strength, the cholesterol patch does not behave as an homogeneous slow zone: whereas cluster sizes seem to reach a plateau in the slow diffusion experiment, clusters were bigger in the cholesterol patch.

To mimic cholesterol effect, we deliberately chose diffusion ratios r 2 = D 0 2 / D 2 = D 0 / D to be conservative i.e lower than one order magnitude of the diffusion ratio (r ² ranges from 2 to 4 instead of 10) in almost all experiments 53 54 55 56 . However, for completeness’ sake, we finally test several values of diffusion ratios. Results are summed up in Figure 8 for three values of the interaction strength (∈) and for ratios going from r = 1 (no slowing) to r = 10. Typical extremal dynamics are available on the supporting Additional file 1: Video S1, Additional file 2: Video S2, Additional file 3: Video S3 and Additional file 4: Video S4.

These results first confirm the predictions for punctual particles. With the parameters tested and as shown on Figure 8, the crowding only experiment did not differ much from the predictions. In addition, we show that particle interactionsin the form of Lennard-Jones potential can strongly modify both the magnitude and the nature of the resulting overconcentration. Around conservative values for diffusion, the so-called cholesterol patch can gather up to 9 times more particles than expected for an homogeneous medium. In addition, these particles tend to be more in clusters than any other situation. Our results can only be obtained and explained by combining both NHD and particle interactions. The cholesterol patch – the domain – spontaneously gathers particles and increases the stability of their interactions.