# Receptor # Threshold for Cell-Cell Adhesion

How do two cells that can adhere, decide whether or not they should adhere? Typically, potentially adherent cells become adherent by increasing their adherence-receptor expression levels (R1/R2) past a certain “threshold” or “EC50” (see figure above). A classic 1984 paper defined this EC50 as a function of R1/R2‘s binding constant Ksoln as illustrated above for two average eukaryotic cells.1,2 This equilibrium model for cellular adhesion is described in more detail below.

EQUILLIBRIUM MODEL OF CELL ADHESION

This model is formulated on the idea that cell-cell adhesion is determined by a competition between two opposing forces:

• specific binding interactions (KD) between complementary receptors R1 and R2 which favors adhesion (like the hooks and loops in velcro)
• non-specific repulsion (Γ) between the cell surfaces created by: (1) “electrostatic repulsion” between negatively charged cell surfaces (due to the glycocalyx) (2) “steric stabilization effect” which is an osmostic pressure created by the exclusion of water from the cell surfaces as they come together.

These opposing forces can be defined using the equations below which are defined and discussed in detail in reference #1.

PARAMETER DEFINITIONS (≈ TYPICAL VALUES)

• Ksoln: molar dissociation constant measured in free solution
• 6 × 1026: conversion factor converting moles/Liter to molecules/m3
• d: distance between adherent cells (≈ 10-20×10-9 m)
• A: contact area between adherent cells (≈ 10-10 m2)
• bond stress: spring model of mechanical “pulling” stress between R1 and R21,2
• L: distance between cells that results in an “unstressed” bond between R1 and R2 (≈ 10×10-9 m)
• k: Boltzmanns Constant (≈ 1.38×10-23 m2s-2K-1kg )
• T: Temperature (≈ 300 K)
• κ: force constant for stretching of R1/R2 bond (≈ 10-4 N/m)3,4
• γ: compressability of glycocalyx (≈ 10-11 N)5
• τ: thickness coefficient of glycocalyx (≈ 5-20×10-9 m)5

QUADRATIC MODEL FOR BRIDGE NUMBER
Combining these equations with conservation of mass and the law of mass action the authors of reference #1 were able to model the number of bridges between cells at equilibrium as a quadratic function below:

Solving for the EC50 and then plugging in the physically reasonable values of each parameter listed above (for typical eukaryotic cells) we were able to obtain the below expression which enables one to estimate the Receptor threshold from a dissociation constant determined in free solution (Ksoln):

REFERENCES:

1. Bell, G.I.; Dembo, M.; Bongrand, P. Cell Adhesion: Competition between Nonspecific Repulsion and Specific Bonding. Biophys. J. 1984, 45, 1051-1064.
2. Lauffenburger, D.A. Receptors: Models for Binding, Trafficking and Signalling, Oxford University Press 1993.
3. Levy, R.M.; Karplus, M. Vibrational approach to the dynamics of the alpha-helix. Biopolymers, 1979 18, 2465-2495.
4. Suezaki, Y.; Go, N. Fluctuations and mechanic strength of alpha-helixes of ployglycine and poly (L-alanine). Biopolymers, 1976 15, 2137-2153
5. Napper, D.H. Steric Stabilization. J. Colloid Interface Sci. 1977 58, 390-407