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The authors have declared that no competing interests exist.

Conceived and designed the experiments: TRH JE RG DGM. Performed the experiments: TRH JE. Analyzed the data: TRH JE. Contributed reagents/materials/analysis tools: TRH JE. Wrote the paper: TRH JE RG DGM. Designed the software used in analysis: TRH JE.

Cell-surface receptors are the most common target for therapeutic drugs. The design and optimization of next generation synthetic drugs require a detailed understanding of the interaction with their corresponding receptors. Mathematical approximations to study ligand-receptor systems based on reaction kinetics strongly simplify the spatial constraints of the interaction, while full atomistic ligand-receptor models do not allow for a statistical many-particle analysis, due to their high computational requirements. Here we present a generic coarse-grained model for ligand-receptor systems that accounts for the essential spatial characteristics of the interaction, while allowing statistical analysis. The model captures the main features of ligand-receptor kinetics, such as diffusion dependence of affinity and dissociation rates. Our model is used to characterize chimeric compounds, designed to take advantage of the receptor over-expression phenotype of certain diseases to selectively target unhealthy cells. Molecular dynamics simulations of chimeric ligands are used to study how selectivity can be optimized based on receptor abundance, ligand-receptor affinity and length of the linker between both ligand subunits. Overall, this coarse-grained model is a useful approximation in the study of systems with complex ligand-receptor interactions or spatial constraints.

The current importance of cell surface receptors as primary targets for drug treatment explains the increasing interest in a mathematical and quantitative description of the process of ligand-receptor interaction. Recently, a new generation of synthetic chimeric ligands has been developed to selectively target unhealthy cells, without harming healthy tissue. To understand these and other types of complex ligand-receptor systems, conventional chemical interaction models often rely on simplifications and assumptions about the spatial characteristics of the system, while full atomistic molecular dynamics simulations are too computationally demanding to perform many particle statistical analysis. In this paper, we describe a novel approach to model the interaction between ligands and receptors based on a coarse grained approximation that includes explicitly both spatial and kinetic details of the interaction, while allowing many-particle simulations and therefore, statistical analysis at biologically relevant time scales. The model is used to study the binding properties of generic chimeric ligands to understand how cell specificity is achieved, its dependence on receptor concentration and the influence of the distance between subunits of the chimera. Overall, this approach proves optimal to study other ligand-receptor systems with complex spatial regulation, such as receptor clustering, multimerization and multivalent asymmetric ligand binding.

This is a

Extracellular signals, such as morphogens and hormones, bind to specific receptors on the cell surface to activate signaling cascades that ultimately regulate key cell decisions, such as proliferation, migration or apoptosis. In multicellular organisms, dysregulation of this receptor-initiated signaling can lead to uncontrolled cell proliferation and cancer. Nowadays, around 60% of all commercial drugs are designed to target specific receptors on the cell surface. Due to this role in stimulus recognition and upstream regulation of cell signaling, mathematical modeling of ligand-receptor interaction constitutes a major effort in the development and rational design of novel therapeutic strategies. The majority of these models are based on a chemical kinetics description of ligands in a three-dimensional environment that bind to receptors diffusing in a two-dimensional surface

Although these assumptions may be valid in simple scenarios, they neglect several important regulatory mechanisms induced by the structural details of the interaction, such as diffusion inhomogeneities or conformational changes after multimerization, or sequential binding

In this work we present a coarse-grained approximation to ligand-receptor interactions, which allows for a computationally feasible study of the dynamics of systems with different sets of ligands and receptors at biologically relevant temporal and spatial scales. The model takes into account both spatial and kinetic features of the interaction, while allowing many particle simulations, and statistical analysis. The interaction between ligand and receptor is described by two basic parameters (angular specificity and strength) which can be tuned to fit a broad range of affinities and dissociation rates to model different ligand-receptor pairs. First, we show that the kinetics of ligand binding and unbinding behave as predicted by chemical kinetics theory, in terms of diffusion and receptor abundance. Then we correlate the parameters defining the interaction (angular specificity and strength) with the experimentally relevant affinity and dissociation rate parameters. As a case study, the model is applied to the analysis of the binding properties of generic chimeric ligands, where we show how cell specificity is achieved and how it depends on receptor diffusion and length of the linker between ligand subunits of the chimera.

Using a coarse-grained approximation, the ligand-receptor interaction is characterized by the geometry of the reactants and by the chemical nature of their binding. The first feature sets the interaction specificity by inducing a steric contribution that limits the binding region to a specific area of the receptor molecule; the chemical contribution, in turn, sets the strength of the bond.

Within this representation, ligands and receptors are simplified as spherical particles of diameters

To account for the interaction between ligand and receptor

The functions

These functions are chosen so that they and their derivatives are continuous at

The ligand-receptor interaction potential can be identified as the effective shape of the receptor seen by a ligand molecule (

The ligand-receptor interaction is studied using molecular dynamics simulations where ligands of reduced mass

Typically, 4 to 32 independent simulations are run for each set of parameters, and the results are averaged to calculate rate constants. Standard deviations due to system fluctuations are represented by error bars.

Simulations were performed using in-house code where the model was implemented in a Verlet algorithm (source code available as

To validate our coarse-grained model, we first apply it to characterize simple ligand-receptor interactions. The chemical interaction between a freely diffusing ligand

One of the main advantages of our modeling framework is that both rate constants can be directly computed using statistical many-particle analysis: for the dissociation rate constant

We thus fit the amount of complexes as a function of time to an exponential to estimate

To calculate

To investigate how the affinity/dissociation rates depend on the two main parameters of our model, we calculate

(

Other aspects of the monovalent ligand-receptor system can be characterized within our modeling framework, such as the effect of ligand diffusion on the binding kinetics. The measured kinetic rates

Here,

Overall, we have shown that the coarse-grained model reproduces the main aspects of the ligand-receptor interaction, and provides a good description to statistically obtain the affinity and dissociation rates via dynamical simulations. We next apply our model to characterize more complex ligand-receptor scenarios.

One of the most promising strategies to improve the efficiency and selectivity of drug-based therapies is the use of synthetic chimeric ligands

To test our coarse-grained description of the chimeric ligand-receptor interaction, we implement a chimeric ligand formed by two subunits representing the

This force leads to an average end-to-end distance between

Two different types of receptors, referred as

The selective potential of chimeric ligands against specific cell types relies on the differential expression of the

Apart from the kinetic rates of both

Affinity rate as a function of the linker length for different average distances between receptors. The simulations were done for

The linker length

Results plotted in

In this work we have introduced a tunable coarse grained model for the simulation of ligand-receptor interactions. Our model represents a trade-off between a detailed description of the spatial aspects of the interaction and computational efficiency. Basic spatial features are described by a geometric factor accounting for the directional specificity of the interaction, while ligand-receptor affinity is modulated by the depth of a Lennard-Jones potential. This approach captures relevant spatial constraints while allowing for an explicit description of the diffusive dynamics. The simplicity of the model, together with the computational efficiency of the presented algorithms, facilitates the application of the model to study many-particle systems with geometrical constraints or multiple interactions that cannot be explicitly solved with theoretical considerations of molecular binding or with all-atom simulations. We note, however, that other potentially relevant geometric aspects of the ligand-receptor interaction, such as asymmetry and steric complementarity of the binding pocket and ligand can not be captured by this model.

The model applied to a monovalent ligand-receptor interaction allows to statistically determine the values for the affinity and dissociation rates and link it to the model parameters. The reduced dependence of the dissociation rates on the strength of the ligand-receptor interaction (

As a proof of concept of the capabilities of this type of coarse-grained models applied to ligand-receptor systems, we implemented within this framework a model for chimeric ligands to study the dependence of their selective potential on the concentration of both receptor subtypes, affinity of ligand subunits towards their corresponding receptor, or linker length. Our model shows that the selectivity of the chimera towards cells over-expressing

Experimentally, longer linkers are more difficult to implement due to potential cleavage of the linker (basically, due to multiple repeats of the same amino acid sequence in the linker), so the shortest linker possible is often implemented experimentally

The presented coarse grained model represents a powerful tool to understand general properties of complex ligand-receptor systems with no analytical solutions available, or where analytical approximations remain to be validated. This is the case of systems with many agents involved, multi-interactions between species or situations with special geometries. Some examples could involve receptor dimerization and its influence on binding, multivalent binding and receptor cross-linking. A careful definition of the geometric factor would also allow multiple ligands to bind simultaneously to the same receptor. In addition, the coarse-grained approximation could be implemented as a module for more complex systems including downstream signaling propagation, or even cell membrane dynamics described via generic coarse grained models, opening the possibility to study the effects of membrane fluctuations or membrane inhomogeneties on ligand-receptor dynamics.

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