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Biology Articles » Biophysics » Kinetics of Receptor-Ligand Interactions in Immune Responses » Theoretical framework for receptor-ligand kinetics

Consider a second-order forward and first-order reverse reaction,

(1)

where R, L, and B denote respectively the receptor, ligand, and bond. In 3D binding, kinetics of a soluble ligand binding to a receptor follows a simple, deterministic kinetic equation,

dB/dt = k_{f }[ R][ L]− k_{r }[ B] (2)

where [R], [L], and [B] denote the concentrations of receptor, ligand, and bond respectively (in unit of M), and kf (in unit of M^{-1}·s^{-1}) and kr (in unit of s^{-1}) are the forward and reverse rates respectively. Ka (= kf/kr) is the binding affinity (in unit of M^{-1}) when the reaction reaches an equilibrium state.

2D binding of receptor-ligand interactions in cell-cell or cell-substrate adhesions is a stochastic process regulated by applied forces. On one hand, the stochastic nature of such a binding can be described using a probabilistic model. The basic idea is to define the probability of bonds, instead of concentration of bonds, since the adhesion is no longer a deterministic process. Upon a small system kinetics first proposed by McQuarrie (23), a probabilistic modeling has been developed and the adhesion probability, Pa, at contact time t follows (24-30),

(3)_{ }

where Ka ^{0} (in unit of μ m^{2}) and kr ^{0} are respectively the zero-force binding affinity and reverse rate, mr and ml are respectively the site densities of receptor and ligand (in unit of μ m^{-2}), and Ac is the contact area (in unit of μ m^{2}). 2D kinetic parameters of AcmlKa ^{0} (if mr is known) or AcKa ^{0} (if both mr and ml are known) and kr ^{0 }can be predicted by fitting the experimental measurements of binding curves (Pa ~ t curves) to the model (Eq. 3), and 2D forward rate kf (in unit of μ m·s^{-1}) can be obtained by the definition (= Ka kr ^{0}).

On the other hand, force regulates the formation and dissociation of bonds in blood flow. Two parameters are used to quantify the effect: one is the bond rupture force or bond strength, and the other is bond lifetime. Bond rupture force depends on the rate of force application, or force loading rate (20, 31-40), and other extrinsic physical parameters (41). Bond lifetime is governed by external forces, as proposed by Bell (17) and Dembo (18),

(4)

B where kr is the reverse rate at force f, a is the interaction range, and kB and T are respectively the Boltzmann constant and absolute temperature. Noting that the bond lifetime, τ , is the reciprocal of reverse rate (τ = 1/kr) at any given force, f, Eq. 4 gives out two mechanisms of forced dissociation of bonds: bond lifetime τ decreases with f (if positive symbol is given in Eq. 4) which is termed as slip bond (17, 42-45), or increases with f (if negative symbol is given in Eq. 4) which is termed as catch bond (18, 46-49). Experimental measurements of bond lifetime at systematically varied forces can be used to determine the force dependence of bond lifetime, which is used to test the theoretical predictions (Eq. 4).

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