Université de Lyon Inserm UMR1060, F-69621, Villeurbanne Cédex, France

Université de Lyon, CNRS INSA-Lyon, LIRIS, UMR5205, F-69621, France

EPI BEAGLE, INRIA, F-69603, France

Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD, 20892, USA

Abstract

Background

In the classical view, cell membrane proteins undergo isotropic random motion, that is a 2D Brownian diffusion that should result in an homogeneous distribution of concentration. It is, however, far from the reality: Membrane proteins can assemble into so-called microdomains (sometimes called lipid rafts) which also display a specific lipid composition. We propose a simple mechanism that is able to explain the colocalization of protein and lipid rafts.

Results

Using very simple mathematical models and particle simulations, we show that a variation of membrane viscosity directly leads to variation of the local concentration of diffusive particles. Since specific lipid phases in the membrane can account for diffusion variation, we show that, in such a situation, the freely diffusing proteins (or any other component) still undergo a Brownian motion but concentrate in areas of lower diffusion. The amount of this so-called overconcentration at equilibrium issimply related to the ratio of diffusion coefficients between zones of high and low diffusion. Expanding the model to include particle interaction, we show that inhomogeneous diffusion can impact particles clusterization as well. The clusters of particles were more numerous and appear for a lower value of interaction strength in the zones of low diffusion compared to zones of high diffusion.

Conclusion

Provided we assume stable viscosity heterogeneity in the membrane, our model propose a simple mechanism to explain particle concentration heterogeneity. It has also a non-trivial impact on density of particles when interaction is added. This could potentially have an impact on membrane chemical reactions and oligomerization.

Background

In the classical fluid-mosaic model of membrane, membrane components undergo isotropic random motion akin to Brownian motion

Recently, this picture has considerably evolved towards a non-homogeneous distribution of cell-membranes components

The organization of membranes into microdomains can biologically be interesting because microdomains could strongly affect membrane functions as interacting species are likely to be in higher concentrations in the domains. Indeed a wide collection ofmembrane proteins involved in such processes have been shown to colocalize with rafts. They are thought to play an important role in various cellular processes such as trafficking and signaling to cite but a few

It is not a surprise that various models have emerged to explain the existence and stability of membrane heterogeneity

The latter model therefore rests on the assumption that rafts are well defined stable structure with mixed effects on diffusion. Indeed constrained diffusion within bounds imply an increase in concentration only temporarily since outbound protein have low probabilities to come back. In addition several other works suggest a more complicated picture

Being bound to a domain affects the protein diffusivity

We propose in this article a simple mechanism which leads to the formation of protein/lipid enriched microdomains. This mechanism is based on non-homogeneous diffusion (NHD). Indeed, many structural constituents of the membrane can alter its viscous (or diffusion) properties. As such, a membrane cannot be approximated anymore by a simple 2D manifold with a constant diffusion but rather should be described by a diffusion profile: a space (position) dependent diffusion function. The membrane composition is therefore by itself a source of heterogeneity in the displacement of trafficking proteins.

We will assume throughout this paper that the cell membrane has a non-spatially constant diffusion tensor that is temporally stable within the timeframe of the diffusion. These assumptions allow a greater generality since one never needs to assume any structurally stable component to define a domain. However we will not address the underlying mechanism behind such diffusion variation and simply assume it exists. Recently, several works have provided plausible mechanisms for viscosity alterations

By solving the corresponding equation of motion, we show that the diffusion profile relates simply to the equilibrium concentration profile. In the case of punctual particles without interaction we show that this relation is extremely simple and we give a closed form of the equilibrium. In some cases we are able to provide a closed form for the whole solution and derive a FRAP (Fluorescence Recovery After Photobleaching) estimation of the diffusion in such conditions.

We expand the model by performing simulations with non-punctual interacting particles. We show that the classical phase transition observed for this kind of system is altered by non-homogeneous diffusion – namely resulting in a shift in the transition diagram. This shift allows for extremely high concentration in the slow zones – much higher than for non-homogeneous diffusion or interaction separately – as well as higher clustering properties.

Methods

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

Here _{
t
} describes the position of the molecule at time

By setting

Looking for continuous and differentiable solutions, boundary conditions emerge naturally from the

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

The boundary conditions impose

with

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

in the square [–^{2} and _{
i
}

with the boundary conditions

for all ^{2} and vanishes due to toric boundary conditions (Eq. 7), yielding

Since

This equilibrium therefore verifies

meaning the gradient of

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 (^{–1}) is independent of the diffusion coefficient _{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

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

which is exactly the classical diffusion equation. One can immediately see that the same boundary conditions apply to

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

where

Equation 14 indicates that the time constant of the recovery is proportional to the square of the spatial average of the

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)

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)

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

where _{
ij
} is the unitary vector of the semi-line starting from particle _{
ij
} being the distance between the two particles. The interaction force

the force

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.

Results and discussion

Continuous model

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

Where _{
k
} = (

Equilibrium distribution behaved as expected. The first set of experiments is for the one dimensional case where we test various diffusion profiles (continuously differentiable _{0} = 1,

Results for 1 d experiment Simulation (dots) versus theoretical (plain line) distribution of particles that undergo non-homogeneous diffusion

**Results for 1 d experiment.** Simulation (dots) versus theoretical (plain line) distribution of particles that undergo non-homogeneous diffusion. The diffusion profile is a simple inverse gaussian function (in inset) with equation:

Transients for this simulation are displayed on Figure

Transient distribution. Simulation of

**Transient distribution.** Simulation of

In this case, due to the slow diffusivity the concentration at the peak of viscosity

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

FRAP Experiment Results for FRAP experiments compared to theoretical prediction

**FRAP experiment.** Results for FRAP experiments compared to theoretical prediction. Three experiments with different diffusion profiles are depicted.

Next, we present the results of NHD on a two-dimensional torus. The diffusion profile consists of 4 randomly positioned Gaussian patches centered on (_{
i
},_{
i
}) ∈ [–^{2} with 1 ≤ _{0} = 1,

Results for 2 d experiment A) The diffusion profile in light spectrum (toward red is slower) for 4 randomly “cholesterol” islands

**Results for 2 d experiment. A**) The diffusion profile in light spectrum (toward red is slower) for 4 randomly “cholesterol” islands. **B**) 2 d histogram of simulated particles positions at equilibrium. **C**) Comparison with theoretical prediction for a cross section (

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.

Force field interactions

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 Δ_{
i
}(

Collision detection was included to allow a simple crowding experiment (i.e

Assuming cholesterol diffusion is reduced by around one order of magnitude

Particles Interactions Results Concentration of particles in the “cholesterol” patch for two diffusion ratios

**Particles interactions results.** Concentration of particles in the “cholesterol” patch for two diffusion ratios

Particles Maps Position map of particles for A) **= 0 and****B)****500 and****= 2, C)****= 4000 and D)****= 10000.**

**Particles maps.** Position map of particles for **A**) ** ε** = 0 and r = 2,

As expected, the overconcentration effect due to non-homogeneous diffusion is dramatically increased via particles interaction. As Figure

To numerically assess this situation, we compute the clustering coefficient as a function of the interaction strength: _{
c
}
_{
i
} where _{
c
} is the number of clusters and _{
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 _{0} (

Effects on Clustering Clustering coefficient (see text for definition) as a function of the interaction strength

**Effects on clustering.** Clustering coefficient (see text for definition) as a function of the interaction strength ** ε** for control

To mimic cholesterol effect, we deliberately chose diffusion ratios ^{
2
} ranges from 2 to 4 instead of 10) in almost all experiments

**Particle evolution for r = 2 and no interactions
(Є = 0).**Evolution under non-homogeneous diffusion of 1,000 particles
(depicted in green) for 1000 time step. A square slow patch on the upper
left corner has a diffusion ratio of 2 and particles interaction is limited to
collision. This is the typical NHD simulations when, at equilibrium, there
will be twice more particles in the slow patch.

Click here for file

**Particle evolution for r = 2 and no interactions
(Є = 10000).**Evolution under non-homogeneous diffusion of 1,000
particles (depicted in green) for 1000 time step. A square slow patch on
the upper left corner has a diffusion ratio of 2 and particles interaction is
set to an intermediate level Є = 10000. For this important level of
interactions, clusters appear everywhere but they are always more
pronounced in the slow patch.

Click here for file

**Particle evolution for r = 10 and no interactions
(Є = 0).**Evolution under non-homogeneous diffusion of 1,000 particles
(depicted in green) for 500 time step. A square slow patch on the upper left
corner has a diffusion ratio of 10 and particles interaction is limited to
collision. The important amount of slowing allows for an important overconcentration
(that will ultimately be 10 fold).

Click here for file

**Particle evolution for r = 10 and no interactions
(Є = 10000).**Evolution under non-homogeneous diffusion of 1,000
particles (depicted in green) for 500 time step. A square slow patch on
the upper left corner has a diffusion ratio of 10 and particles interaction
strength is set to 10000. In this last experiment,the over-concentration
appears first on the border of the patch. The difference in the phase
transition is extremely more pronounced in the slow patch.

Click here for file

Diffusion Value Dependence Concentration ratio as a function of the ratio of diffusion for three interactions strength

**Diffusion value dependence.** Concentration ratio as a function of the ratio of diffusion for three interactions strength ** ε** = 0 (triangles),

These results first confirm the predictions for punctual particles. With the parameters tested and as shown on Figure

Conclusion

Cell membrane can display a wide variety of heterogeneity in its physical properties, viscosity and constituents composition. Our model describes the possible emergence of what we call a minimal membrane domain based on only one feature of the membrane, namely its local viscosity. We show that, everything else being equal, domain with high viscosity tends to gather statistically a larger proportion of the diffusive particles. Particles are not trapped within these domains but merely tend to spend more time in them. In addition, we show that the ratio of concentration of particles in and out domain is proportional to the square root of the inverse ratio of their local diffusion. As such, protein free membrane zones are seen as a product of high diffusivity and not from forbidding constraints – contrary to the hypothesis developed in

Moreover, while this model is fairly general, we emphasize the particular role of cholesterol. Cholesterol is ubiquitous in cell membranes and its influence on membrane diffusion has been well documented. Therefore, it is not surprising, from our point of view, that cholesterol has been found in higher quantity in virtually all microdomains exhibited so far. For many authors, its presence even characteries membrane domains

Looking for experimental results, authors have reported the impact of temperature and cholesterol depletion on membrane heterogeneities. Cholesterol removal should decrease the colocalization of membrane components as in

Important diffusion variation leads not only to a quantitative particles concentration variation but also to a qualitative variation. Protein interactions alter significantly the particles concentration landscape when combined with NHD. We show that cluster of particles are more stable in slower zones and appear for lower values of interactions. Our model predicts that domains of slow diffusion will alter affinity interactions. That is, for example oligomerization will happen preferentially inside these domains. This latter prediction has been observed on model membrane where modifications of the cholesterol content trigger oligomerization

Competing interest

The authors declare that they have no competing interests.

Authors’ contributions

HAS designed the study, performed the simulations, analyzed the simulation data and drafted the manuscript. AC and GB conceived the study, analyzed the data and drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgment

This work has been partly supported by the Institut of Complex System of Rhones-Alphes (IXXI).