Université de Lyon, Laboratoire d'InfoRmatique en Image et Systèmes d'information, CNRS UMR5205, F-69621, France

Université de Lyon, Cardiovasculaire Métabolisme et Nutrition, Inserm UMR1060, F-69621 Villeurbanne Cédex, France

EPI BEAGLE, INRIA Rhône-Alpes, 69603 Villeurbanne, France

Abstract

Background

Cellular response to changes in the concentration of different chemical species in the extracellular medium is induced by ligand binding to dedicated transmembrane receptors. Receptor density, distribution, and clustering may be key spatial features that influence effective and proper physical and biochemical cellular responses to many regulatory signals. Classical equations describing this kind of binding kinetics assume the distributions of interacting species to be homogeneous, neglecting by doing so the impact of clustering. As there is experimental evidence that receptors tend to group in clusters inside membrane domains, we investigated the effects of receptor clustering on cellular receptor ligand binding.

Results

We implemented a model of receptor binding using a Monte-Carlo algorithm to simulate ligand diffusion and binding. In some simple cases, analytic solutions for binding equilibrium of ligand on clusters of receptors are provided, and supported by simulation results. Our simulations show that the so-called "apparent" affinity of the ligand for the receptor decreases with clustering although the microscopic affinity remains constant.

Conclusions

Changing membrane receptors clustering could be a simple mechanism that allows cells to change and adapt its affinity/sensitivity toward a given stimulus.

Background

The binding kinetics between cell surface receptors and extracellular biomolecules are critical to all intracellular and intercellular activity. Modelling and predicting of receptor-mediated cell functions are facilitated by measurement of the binding properties on whole cells. Therefore, these measurements, however elaborate, have been based on the ground of chemical enzyme/substrate formalism

These assumptions may fail in real biological systems, in particular considering membrane receptors which are restricted to only 2 of the 3 spatial dimensions

This localization and clustering may have a dramatic influence on signalling. This influence remains, however, unclear as literature reports contradictory effects of clustering/declustering on signalling (see e.g.

In any case, the impact of an inhomogeneous receptor density

In addition, several more detailed studies illustrate the possible effect of receptor clustering on receptor binding by inducing enhanced rebinding or ligand receptor switching

Notably

Considering ligand-receptor binding as a diffusion-limited reaction

In order to investigate the effects of receptor clustering on ligand binding, we present two joint approaches of ligand receptor binding at equilibrium when receptors are organized in clusters at cell surface. We consider three membrane receptor layouts illustrating three degrees of spatial correlation. These layouts, for two of which a simple ODE description is available, are studied in the context of ligand-receptor reversible binding. The three layouts are investigated following computer based simulations conjointly with an ODE formalism, the latter adapted to include spatial characteristics of receptor organization.

Ligands are assumed to diffuse freely above the membrane without interaction except when they can bind stochastically to receptors. Receptors are modelled as still positions on the membrane. Ligand-receptor complex formations are stochastic events occurring whenever a ligand is near enough a free receptor. More precisely, it occurs whenever the ligand lies in a defined area above the receptor position. This area is called the

Methods

We describe below the three possibilities of spatial correlation we have chosen to investigate. For each, we present the assumptions made in order to model them properly, the simple analytical formulation we derived whenever it was possible, and the corresponding individual-based model used in simulation. As mentioned in introduction, we consider monovalent ligands reversibly binding to monovalent receptors which are independent from each other.

No spatial correlation

The first layout consists of receptors homogeneously set on the membrane, which stands as a reference configuration of homogeneously spread receptors on the cell membrane. The classical approach to model ligand-receptor interaction is through reaction mechanism akin to enzymatic reactions. In the case of monovalent receptors, the most simple model remains the classical Ligand-Receptor Binding Equilibrium equation:

where _{1 }and backward rate constant _{-1}.

The further steps involve some generally implicit assumptions: the complex concentration variation will be the sum of two parts. The negative rate of complex dissociation will be _{-1 }times the complex number. The statistical process underneath this assumption relies basically upon a time independent (exponential) undocking probability

On the other hand, the complex formation equation is based on what is called

where lower case indicates quantities of corresponding species. The total number of receptors will be denoted as _{0 }and _{0}. Note for later that we have two ways to retrieve the dissociation constants: first, using the _{50 }(efficient concentration 50) that is the amount of ligand needed to generate occupation of half the receptors at equilibrium. In this case, this amount is

Over stacked receptors

Spatial correlation of receptors should in itself modify Eq. 2, as the joint probability to find both reactants in the same vicinity is no longer independent for close receptors. Thus, we first propose an extreme case that has an analytical derivation. Let us assume we have _{0 }receptors which are divided among clusters of size _{0}/_{i }
_{0 }= _{i}
_{i}
_{
i-1 }and _{
i
}(

At this point we simply partitioned the number of clusters _{0}/_{i}

From this we can derive a set of ODE's that describe the evolution of concentrations of these components, where we can assume a homogeneous medium. At equilibrium, we obtain a very general formula

where we can relate simply the different association/dissociation constants. We assume that a receptor with _{i }
_{1 }but _{-i
}= _{-1}, so _{i }
_{1}. Due to the shared affinity zone, we will assume in this model that the potential to bind a free site will be independent of the number of free sites. Therefore the on rate _{i }
_{1 }because it defines the transition from _{
i-1 }to _{
i
}through binding of 1 ligand to 1 site. This event happens with the same probability as the transition _{1}. Then getting rid of the 1 subscript (_{1})

and

with

Several theoretical dose-response (for dimensionless ligand dose

Model validation and clusters of over stacked receptors

**Model validation and clusters of over stacked receptors**. A) Dose response for reference size _{50 }≈ _{50 }=

In the dimensionless case (_{50}, we can note that when

The real _{50 }obtained by numerical computation is compared to Eq. 7 on Figure _{50}. The local conclusion of this simple analysis is that we can expect modification of the receptor occupation at equilibrium whenever the spatial configuration of the receptors is changed. Introducing correlations in the probabilities of encounter by spatial organization modifies the receptor occupation. In addition, the apparent affinity seems to decrease with the clustering of receptors.

By overstacking affinity zones, even partially, this configuration creates a "strong" spatial correlation which influences dramatically the complex formation rate: within a cluster of receptors, the occupation of a receptor affinity zone is directly dependent of the occupation of affinity zones of the other receptors, since they are totally or partially the same. In order to address the issues stated above, we now propose to investigate what may happen if affinity zones remain distinct from each other inside a cluster of receptors, but "weak" spatial correlation is still induced by placing receptors contiguously. We propose to examine this case using a simulation framework, as no simple mathematical derivation could be obtained.

Contiguous receptors

We introduce in this section a particle simulation framework that was used to detect the effect of clustering, by modelling clusters of receptors with contiguous but non-overlapping affinity zones. This configuration is taken to be the opposite extreme of over stacked receptors in terms of spatial configuration. That is, within a cluster, receptors are still close to each other, but the presence of ligand in the vicinity of one receptor does not influence the binding of a ligand with receptors of the same cluster: their affinity zones are contiguous.

The simulation is restricted to a 2D environment, and a 1D membrane. Ligands are particles in a 2D environment (see Figure

Simulation environment

**Simulation environment**. Top panel: On the left is a cartoon view of the 2D membrane of area _{t}_{r}

where _{i}

Receptors are punctual but localized only on the bottom line of the environment area. Their diffusion is neglected and they will therefore remain at their initial position throughout the simulations. To simulate docking, we chose a very simple formalism: each receptor has an affinity zone - a square above its position - where there is a constant probability _{1 }for a ligand to bind whenever it is found itself in. Of course, a ligand can only bind to a free receptor. No binding event can occur for an already bound receptor. In addition, the bound ligand cannot diffuse as long as it stays bound. Finally, when formed, the complex has a constant probability to dissociate _{-1}. Upon dissociation, the ligand molecule resumes its Brownian approximated motion, starting from the center of upper edge of the affinity zone it just left. This is to avoid bias in rebinding events; the probability at the next time step for the ligand to return into the affinity zone or to move away will be equal.

Using this formalism, it is very simple to relate the parameters of the simulation with the association constant of the ligand/receptor binding. Indeed, at equilibrium, the number of receptor-ligand complexes that are dissociating per time step is equal to _{-1}

Assuming the classical framework _{r}
_{t }
_{r }
_{t }
_{1}.

This produces the relation (since what comes out must be equal to what comes in at equilibrium), and using _{0 }-

to obtain the classical equation:

with

Eq. 9 allows a direct comparison with the dissociation constant. It relates simply with docking and undocking probability plus what we called before the affinity zone: the surface available for binding.

Results

Unless otherwise specified, the parameters are identical for all simulations. The simulations were performed for a sufficient number of time steps to ensure equilibrium was reached, which is around 10^{3 }for the selected parameters. The number of receptor is fixed and is _{0 }= 500. Similar runs were performed with _{0 }∈ {1000, 2000, 5000, 10000}, showing no qualitative or quantitative differences with _{0 }= 500. Thus, the latter value for _{0 }was chosen to limit finite-sized effects and computational time. The time step ^{-2 }and ^{5 }(using a space ratio _{T }
^{5}
_{R}
_{-1}/_{1 }= 1 with _{1 }= _{-1 }= 0.1. The results obtained would have to be considered within the correct regime of reaction, that is reaction-limited or diffusion-limited. As the simulated reaction is either one or the other possibility, results cannot be interpreted in the same way. Our concern being the effect of the spatial organization of receptors on binding at equilibrium, we would like to make sure that we simulated ligand-receptor binding in the diffusion-limited regime, so the observation of an effect of clustering can specifically be related to diffusion and geometrical aspects. In order to check whether the simulations were reaction-limited or diffusion-limited, we compared the average mean first passage time (MFPT) of a ligand molecule in a receptor affinity zone to the reaction time-scale.

A diffusion time scale several orders of magnitude larger than the reaction one characterizes diffusion-limited reactions. An estimation of the average MFPT can be obtained using the asymptotic formula from _{0 }traps of surface area _{r }

Finally, the number of occupied sites at equilibrium is computed throughout all simulations, and displayed normalized with respect to _{0 }= 500.

No spatial correlation: homogeneous receptor distribution

In the case of evenly distributed receptors (see Figure ^{5}, _{1 }= _{-1 }= 0.01). The two others values for _{-1 }= 0.1 = 10_{1}) and _{-1 }= 0.001 = _{1}/10). The results for the several runs are displayed on Figure _{r}
_{t }

To obtain a good approximation of the slope at origin and the _{50}, more runs were necessary for low concentrations and for values near expected the _{50 }(i.e 1, 0.1). But, all in all, the minimal number of runs is 10 for any given concentration and parameters set. Due to their smallness, error bars are actually negligible - the radius of data points is larger.

As the figures show it and for each parameter set tested, the particles simulation framework is consistent with the predicted behavior: a curvilinear Michaelian-type curve with the correct affinity _{r}
_{t}
_{1 }and _{-1}).

Over stacked receptors

Spatial correlation in the case of receptors with stacked affinity zones - Figure

Three degrees of spatial correlation implied by over stacked receptors (

Simulations were in perfect agreement with the mathematical derivations presented in the Models section for both type of layouts (as in Figure

Contiguous receptors

We present in Figure

Effect of clustering for contiguous receptors

**Effect of clustering for contiguous receptors**. A) Dose response for _{50 }to control _{50 }(i.e. for

The dose response curves are compared, all other parameters being equal, to the control case where receptors are homogeneously spread. In Figure _{0 }= 500. So _{50 }has increased and the response always lies below the control one, in a weaker but similar way than in the over stacked case seen previously.

Figure _{50 }can be estimated respectively by linear regression and non-linear least square fitting. For the slopes at origin, simple linear regressions of occupation rate against dose were performed, using values between 0 and 0.05_{50 }were estimated by fitting data using Hill functions - a widely used model for non-Michaelian kinetics _{50}.

_{50 }and slope at origin obtain via fitting are displayed in Figure

The graph Figure _{50 }gradually increases with cluster size until a plateau is reached at around 170% of the control value. Similarly the slope at origine decreases down to 50% of the control value. Observing dose response curves from similar experiments, but with increasing cluster size, leads to observing different affinities for the ligand for receptors at a global scale, whereas the intrinsic affinity of each individual receptor remained equal. The saturation at high cluster sizes is merely due to the fact that no more clustering can be induced once extreme cluster sizes are reached, which are limited by the fixed number of receptors.

The Hill coefficient _{50 }estimation. The very slight variation of Hill coefficient can hardly support any qualitative or quantitative conclusions about clustering effect in the contiguous receptors case, as the Hill function is not pertinent here as a mechanistic model.

Clustering enhances response by increased rebinding

Intuitively, receptor clustering should induce two opposite effects that counter themselves: enhanced rebinding to close receptors, but decreased ligand-receptor encounter probability. In other words, when receptors are clustered, ligands spend on average more time diffusing before encountering a receptor. Indeed the membrane is not evenly covered and has large receptor-free zones. On the other hand, once bound a ligand will be released in a richer receptor area when receptors are clustered thereby allowing a greater rebinding probability. In order to explore the effect of this rebinding, we perform the following experiment: instead of releasing a ligand at the edge of its former cognate receptor affinity zone when it undocks, the ligand is relocated randomly within the entire medium.

By imposing this random repositioning of ligands after unbinding, the simulation bypasses the potential effect of rebinding, as ligands are on average reinjected quite far from the membrane.

Receptor occupation is then only caused by spatial and temporal independent complex formation. Comparison between dose response curves in such a case and standard simulations may then qualitatively illustrate the part of response alteration which is only due to clustering-enhanced rebinding.

Dose response from such simulations are compared with the standard simulations presented so far i.e. the simulations described in the previous section) for the same clustering (i.e. same

Effect of rebinding on receptor occupation at equilibrium

**Effect of rebinding on receptor occupation at equilibrium**. A) Comparison of dose response curves between _{50 }obtained with random reinjection normalized by _{50 }obtained with normal reinjection. White bars: ratio obtained for random reinjection using _{50 }computed for various cluster sizes normalized by _{50 }with no clusters (

As mentioned above, the effect of random reinjection strongly affects the receptor occupation even in the unclustered case. Since black bars are increasing with clustering, removing rebinding events has a stronger importance the more the receptors are clustered. It was expected since ligands have a higher probability to rebind when receptors are available in the vicinity. Moreover white bars show that the impact of clustering can be greatly increased via random reinjection when normalized by unclustered case (up to ten times the _{50 }as compared to results in Figure

Clustering through partially overlapping receptors

Between clusters of over stacked receptors and clusters of adjacent receptors, we investigate an intermediate scenario, in which clusters are composed of receptors with partially overlapped zones. Responses are computed for a single dose

Receptor occupation when affinity surfaces partially overlapped within a cluster

**Receptor occupation when affinity surfaces partially overlapped within a cluster**. A) Comparison of occupation as a function of relative overlap of affinity surfaces. On a single curve, points correspond to the same experiment, for a fixed ligand concentration, but with varying overlap. Error bars are ± standard deviation. B) Cartoon representing increasingly overlapped receptor affinity surfaces within clusters.

As the overlap increases, at fixed number of receptors set in a fixed number of clusters, the effective surface covered by receptors decreases, and so decreases the receptor occupation at equilibrium, from 0% to 100% overlap within a continuum. When in clusters, receptors can possibly share a common affinity zone with some of its neighbors. The decreases in apparent affinity is therefore more pronounced in that case. A similar behavior was observed for each cluster size tested.

Spreading of receptors

On the other side, we simulated situations where the affinity zone width (_{50 }ratios compared to control (for

Receptor spacing, affinity zone size and clustering

**Receptor spacing, affinity zone size and clustering**. A) Cartoon representing clusters of receptors with different receptor size width (r) on affinity zone width (b) ratio. A fixed affinity zone as it was used in simulation with a increasing receptor width leads to an increasing r/b ratio and therefore to sparser clusters of receptors. B) Ratio of fitted _{50 }to control _{50 }(i.e. for

Ligand diffusion

The simulations were so far performed with ligand diffusion coefficient ^{-4}), still comparing homogeneous receptor spacing and receptor clustering. After having checked that the equilibrium is reached, we could observe that the receptor occupation in function of the dose decreased, but still reached the same saturation value. We then compared apparent affinities in function of cluster size. Figure

Clustering effect with different ligand diffusion coefficients

**Clustering effect with different ligand diffusion coefficients**. Ratio of fitted _{50 }to control _{50 }(i.e. for

Conclusions

The presented computational model transcribes the necessity of proximity for reactants to interact and combines it with the probabilistic nature of biochemical reactions at microscopic scale. The use of approximated Brownian motion in real coordinates and binding through affinity surfaces in a continuous medium allows the investigation of ligand-receptor reactions at microscopic scale and potentially reduces latent finite size effects of discrete lattices simulations. Modelling receptor as affinity zones with probabilistic binding allows to directly relate simulation parameters with ODE formalism.

Several configurations are explored by means of simulations. First, the model was validated for homogeneous receptor repartition by checking simulation concordance with the classic Michaelian equation. Two extreme cases of clustering were then tested, inducing spatial correlation either considering two possibilities. Within a cluster, receptors could be so close to each other that they interact with ligand particles contained exactly in the same area. Or alternatively, receptor affinity zones could simply be adjacent without overlapping. For receptors with stacked affinity zones, simulations still match the mathematical description.

For contiguous receptors, as no simple mathematical formulation is available, simulations are the only way to explore the potential effect of clustering. Some additional experiments are also performed to study more specifically some local aspects of ligand-receptor interaction, such as rebinding or the effect of partial receptor overlap.

Results suggest some insights about the receptor colocalization effects on ligand-receptor binding, observed on membrane receptors occupation. The ligand-receptor encounter probability is lower when receptors are clustered, because an inhomogeneous membrane covering leads to depleted zones and highly concentrated zones which both contain the same concentration of ligand. Thus, ligand molecules roaming in such depleted zones do not encounter receptors and actual reacting quantities are decreased compared to what is assumed to interact in homogeneous configuration. But, receptor clustering also increases the rebinding probability, in accordance with previous works

Lipid rafts and other membrane structuring components could then serve as signalling modulators by adapting cell sensitivity through receptor clustering. A single kind of receptor could be declined in various apparent affinities by dynamic clustering, and thus be sufficient to give the cell some flexibility in terms of signal response, whereas producing several different types of receptor with different affinities would consume a lot more resources.

Individual-based simulations provide insights into how spatial configuration of complex systems impact the processes they generate. They produce valuable results at both spatio-temporal microscopic scale - e.g. first-time encounter probability, ligand-receptor residence time, average distance travelled between rebinding events distributions - and macroscopic scale, such as receptor occupation at equilibrium, or pharmacodynamic dose-response. Individual-based models also allow for more complete implementations of the biological reality of the studied phenomena. For example, receptor diffusion could be allowed, or receptors could be set in clusters whose size is drawn from pertinent distribution laws, such as normal, exponential or power laws. Simulations would then provide valuable results on the robustness of observed effects of clustering towards realistic and noisy spatial configurations.

Results suggest that receptor clustering has an impact on signalling by itself, without incorporating any specific receptor-receptor interactions in the model. However, it should be interesting to explore specific biological interactions with the model, such as receptor transphosphorylation, hetero/homodimeric receptors or allosteric competition between binding sites, which could be easily implemented and experimented. Simulations could be used to study more complex signalling systems such as G-Protein-based pathways and would inspire useful intuitions for biological experiments, as they provide insights on the functional impact of spatial configurations on the mechanics of signalling.

Authors' contributions

BC helped to design the study, performed the simulations, analyzed the simulation data and drafted the manuscript. HS conceived the study, analyzed the data and drafted the manuscript. All author read and approved the final manuscript.

Acknowledgements

BC holds a fellowship from la Région Rhône-Alpes. We gratefully acknowledge support from the CNRS/IN2P3 Computing Center (Lyon/Villeurbanne - France), for providing a significant amount of the computing ressources needed for this work. We thank Andrew Fowler for his critical reading of the manuscript.