# Modeling of Cell Aggregation Dynamics Governed by Receptor–Ligand Binding Under Shear Flow

## Abstract

Shear-induced cell aggregation and disaggregation, governed by specific receptor–ligand binding, play important roles in many biological and biophysical processes. While a lot of studies have focused on elucidating the shear rate and shear stress dependence of cell aggregation, the majority of existing models based on population balance equation (PBE) has rarely dealt with cell aggregation dynamics upon intrinsic molecular kinetics. Here, a kinetic model was developed for further understanding cell aggregation and disaggregation in a linear shear flow. The novelty of the model is that a set of simple equations was constructed by coupling two-body collision theory with receptor–ligand binding kinetics. Two cases of study were employed to validate the model: one is for the homotypic aggregation dynamics of latex beads cross-linked by protein G-IgG binding, and the other is for the heterotypic aggregation dynamics of neutrophils-tumor cells governed by β_{2}-integrin–ligand interactions. It was found that the model fits the data well and the obtained kinetic parameters are consistent with the previous predictions and experimental measurements. Moreover, the decay factor defined biophysically to account for the chemokine- and shear-induced regulation of receptor and/or ligand expression and conformation was compared at molecular and cellular levels. Our results provided a universal framework to quantify the molecular kinetics of receptor–ligand binding in shear-induced cell aggregation dynamics.

## Keywords

Two-dimensional kinetics Cone-plate viscometer Homotypic aggregation Heterotypic aggregation Bell model Protein G-IgG bond β_{2}-Integrin and ICAM-1 bond

## List of Symbols

*a*Bond interaction range (nm)

*A*_{c}Contact area between two contact spheres (

*μ*m^{2})*A*_{c}*m*_{r}*m*_{l}*k*_{f}, (*A*_{c}*m*_{r}*m*_{l}*k*_{f})^{0}Effective forward rate, value at the moment immediately after PMN stimulation (s

^{−1})*C*;*C*_{1},*C*_{10};*C*_{2},*C*_{20}Concentration of sphere; value of sphere 1, initial value; value of sphere 2, initial value (m

^{−3})*C*_{f}, 〈*C*_{f}〉Angle factor (=(sin

^{2}*θ*_{1}sin 2*ϕ*_{1})_{max}), mean value*C*_{O}Orbit constant

*E*,*E*_{0}Adhesion efficiency, value at the moment immediately after PMN stimulation

*f*_{c},*f*_{c0}Two-body collision frequency per unit volume per sphere 2, initial value (s

^{−1})*F*;*F*_{N},*F*_{N,max};*F*_{S},*F*_{S,max}Applied force; normal force, maximum value; shear force, maximum value (pN)

*G*Shear rate (s

^{−1})*k*_{B}Boltzmann constant (=1.38 × 10

^{−23}N m K^{−1})*k*_{f},*k*_{f}^{L},*k*_{f}^{H}Forward rate, values from low and high shear rate, respectively (

*μ*m^{2}s^{−1})*k*_{r},*k*_{r}^{(n)},*k*_{r}^{0}Reverse rate, value for dissociation of

*n*-th bond, value at zero force (s^{−1})*M*Number of data points

*n*, 〈*n*〉Number of bonds, mean value

*N*Maximum number of bonds possibly to link the doublet

*p*_{n},*p*_{cn}Probability of having

*n*bonds, probability of having*n*bonds at the end moment of two-body collision (*n*= 0, 1, 2…)*P*_{a},*P*_{a}^{30}Probability of adhesion, equilibrium aggregation percentage at 30 min for latex bead homotypic aggregation

*P*_{b}Fraction of doublet break-up

*r*,*r*_{1},*r*_{2}Radius of sphere, value of sphere 1, value of sphere 2 (

*μ*m)*r*_{e}Equivalent axis ratio of doublet

*t*Arbitrary time (s)

*T*Period of doublet rotation (s)

*T*_{K}Absolute temperature (K)

*u*_{1},*u*_{2},*u*_{3}Fluid velocity,

*u*_{1}=*u*_{2}= 0 and*u*_{3}=*GX*_{2}(*μ*m s^{−1})*X*_{1},*X*_{2},*X*_{3}Cartesian coordinates (

*μ*m)*y*_{i},*y*(*x*_{i})Measurement and prediction values at

*x*_{ i }*α*_{c},*α*_{m}Decay factors at cellular and molecular level, respectively (s

^{−1})*α*_{N},*α*_{S}Normal and shear force coefficients, respectively

*ε*Two-body collision capture efficiency

*η*Medium viscosity (cP, = mPa s = 10

^{−3}N s m^{−2})*θ*_{1},*ϕ*_{1}Polar and azimuthal angles of doublet major axis with respect to

*X*_{1}*θ*_{2}Polar angle of doublet axis respect to

*X*_{2}*ϕ*_{1}^{0}Contact angle of two colliding spheres

*σ*_{i}Standard deviation

*τ*, \( \bar{\tau } \)Two-body collision duration, mean value (s)

*χ*^{2}Chi-square statistic

## Notes

### Acknowledgments

This work was supported by National Natural Science Foundation of China grants 30730032, 11072251, 10902117, and 10702075, Chinese Academy of Sciences grants KJCX2-YW-L08 and Y2010030, and National Key Basic Research Foundation of China grant 2011CB710904.

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