# The Effect of Thermal Fluctuation on the Receptor-Mediated Adhesion of a Cell Membrane to an Elastic Substrate

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

_{RL}. The unbinding rate constant k

_{off}takes Bell’s formula [23] as ${k}_{off}={k}_{off}^{0}\mathrm{exp}\left(f{x}_{b}/{k}_{B}T\right)$, where f is the force acting on a ligand–receptor bond by the membrane pulling, x

_{b}describes how strongly the reaction rate change with force, ${k}_{off}^{0}$ is the rate at zero force, k

_{B}is Boltzmann constant, and T is the temperature. As showed by Bell [23], for the case of classical particle diffusion, rates of the diffusion step are simple functions of particle diffusivities and encounter distance. In contrast, the two-step reaction in this model is complicated by the membrane fluctuation and we resort to computer simulations. Throughout this paper, we use the closed bond ratio ϕ, which is simply defined as the number of closed ligand–receptor bonds divided by the total number of receptors in the membrane, to characterize the adhesion strength. The objective of this paper is to carry out numerical simulations of the chemical reaction in Equation (1) to calculate the closed bond ratio ϕ, and to study how it depends on substrate rigidity and receptor density by involving the membrane fluctuation step. It shall be mentioned that previous cell adhesion models [23,24,25,26,27] mainly focused on the rupture strength of a cluster of bonds under an external pulling force, while in our present work, we are concerned with the equilibrium closed bond ratio in the case of no external pulling force.

^{2}, L is the membrane side length, ${\Lambda}_{q}=1/\left(4\eta q\right)$, η is the viscosity of the surrounding fluid, and $q=\left|q\right|$, $q=\left(2\pi \alpha /L,2\pi \beta /L\right)$, α and β are integers from 0 to α

_{max}. Note that, in the numerical simulations, two wavelength cutoffs are defined: λ

_{min}= L/α

_{max}and λ

_{max}= L. The stochastic force ${\zeta}_{q}\left(t\right)$ has a Gaussian distribution with $\langle \left|{\zeta}_{q}\left(t\right)\right|\rangle =0$ and $\langle \left|{\zeta}_{q}\left(t\right){\zeta}_{{q}^{\prime}}\left({t}^{\prime}\right)\right|\rangle =2{k}_{\mathrm{B}}T{L}^{2}{\Lambda}_{q}^{-1}{\delta}_{q{q}^{\prime}}\delta \left(t-{t}^{\prime}\right)$, and ${F}_{q}\left(t\right)$ is the deterministic force derived from potential energies which will be described in detail later. Numerical integration is used to update ${h}_{q}$ in Equation (2) from time t to t + Δt, and $h\left(x\right)$ is obtained by inverse Fourier transform from ${h}_{q}$. At the end of each time step, ${k}_{on}^{0}$ is calculated for each open bond if ligand–receptor distance is within the encounter distance R

_{RL}, and ${k}_{off}={k}_{off}^{0}\mathrm{exp}\left(f{x}_{b}/{k}_{\mathrm{B}}T\right)$ is computed for each closed bond with given bond force f calculated based on the membrane position. Then, ${k}_{on}^{0}\Delta t$ and ${k}_{off}\Delta t$ are compared with a uniformly distributed random number in [0,1] to determine whether binding or unbinding occurs.

_{L}denotes the vertical position of rest unbound ligands, ${x}_{mn}=\left(m\Delta ,n\Delta \right)$ are uniform locations of ligand–receptor pairs where Δ denotes receptor spacing as shown in Figure 1. The receptor density is then characterized by 1/Δ

^{2}. The total number of receptors in the domain of interest is denoted by ${\left({N}_{b}\right)}^{2}$. The binary function φ

_{mn}describes ligand–receptor binding status with φ

_{mn}= 1 for closed bonds and φ

_{mn}= 0 for open bonds. The force derived from ${E}_{L}$ is ${F}_{q}^{L}=-{\displaystyle \sum _{m=0}^{{N}_{b}-1}{\displaystyle \sum _{n=0}^{{N}_{b}-1}k\left[h\left({x}_{mn}\right)-{h}_{L}\right]\mathrm{exp}\left(-iq\cdot {x}_{mn}\right)}}$. Since the substrate is a solid bulk material, the thermal fluctuation of the substrate shape is neglected in this model. Therefore, unbound ligands embedded in the substrate are considered to be at rest with the substrate. The elevated ligands elastically restore to their rest positions when unbinding from receptors and such retracting process takes no time by assuming the time scale of elastic restoring is small compared to membrane fluctuation time scale. The substrate is modeled using a soft-wall repulsive interaction ${E}_{w}$ = ${\epsilon}_{w}{\displaystyle {\int}_{A}{\left({\sigma}_{w}/(h+{\sigma}_{w}-{h}_{sub})\right)}^{8}dx}$, and E

_{w}is truncated so that ${E}_{w}\equiv 0$ when h > h

_{sub}where h

_{sub}is the upper bound of the repulsive interaction range, h

_{sub}= 0 is assumed in this paper. Here, σ

_{w}and ε

_{w}are parameters determining the repulsive interaction strength. Note that other types of enthalpic repulsive interactions [35], if considered, can also be absorbed into E

_{w}. The force acting on the membrane derived from E

_{w}is

## 3. Results

_{min}= 20 nm, B = 20 pN·nm, and η = 0.06 Poise, the relaxation time is estimated as [37] $\tau =4\eta {\lambda}^{3}/B{\left(2\pi \right)}^{3}$~40 ns.

_{min}, and σ

_{w}from the soft wall interaction.

_{min}is varied. This is rather counterintuitive, because small wavelength modes with relatively small fluctuation seem to be less affected by the hard wall confinement. Varying membrane periodic size from 800 nm to 240 nm only results in negligible changes in the p – d relation as indicated by the overlapping three curves, as shown in Figure 2b. In addition, increasing the length scale parameter σ

_{w}renders the wall potential softer and thus causes the p – d relation to deviate from Equation (4). By fitting the simulation data that is closest to Equation (4), we obtain c ~ 0.078, which is in good agreement with other simulation results (see the references cited in [40]). Note that, due to the entropic repulsive force, a weak pressure p

_{0}pushing the membrane towards the substrate is necessary in our FSBD-MC simulations below to prevent the membrane from diffusing far away from the substrate in dynamic processes.

_{b}is the ligand–receptor binding energy. Assuming ε

_{b}to be 10 k

_{B}T yields ${k}_{on}^{0}/{k}_{off}^{0}$~10

^{4}. In the literature, ${k}_{on}^{0}$ was estimated to be from 10

^{−3}to 1 ns

^{−1}for hapten-antibody [23], and ${k}_{off}^{0}$ was measured to be from 10

^{−11}to 10

^{−6}ns

^{−1}for integrins [42,43]. Since varying the substrate Young’s modulus is equivalent to changing the MSL spring constant k if other segments of the linkage remain unchanged, the lumped spring stiffness k is used to represent the change of substrate stiffness. Typical morphologies of the fluctuating membrane in our FSBD-MC simulations are shown in Figure 3.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A schematic of a fluctuating membrane adhering to an elastic substrate via ligand–receptor binding: (

**a**) side view; (

**b**) top view. Note that out-of-plane fluctuation amplitude is exaggerated for better visualization. Structurally, the membrane drawn here includes the glycocalyx layer.

**Figure 2.**(

**a**) Free membrane fluctuation spectrum. Simulation parameters are as follows L = 800 nm, k

_{B}T = 4.3 pN·nm, B = 20 pN·nm ~ 5 k

_{B}T, η = 0.06 Poise, time step Δt = 0.5 ns, total simulation time t

_{total}= 1 ms, and λ

_{min}= 20 nm. A simulation snapshot showing fluctuation magnitude and morphology in physics space is plotted as an inset. (

**b**) Entropic repulsive interaction between the fluctuating membrane and the substrate. Simulation parameters are as follows: k

_{B}T = 4.3 pN·nm, B = 20 pN·nm, η = 0.06 Poise, ε

_{w}= 0.043 pN/nm, time step Δt = 0.5 ns, and total simulation time t

_{total}= 0.6 ms. The triad number denotes $\left({\sigma}_{w},{\lambda}_{\mathrm{min}},L\right)$ in units of nm.

**Figure 3.**Snapshots of fluctuating membranes in the FSBD-MC simulations. (

**a**) k = 1 pN/nm; (

**b**) k = 100 pN/nm.

**Figure 4.**The closed bond ratio as functions of (

**a**) the spring constant k and (

**b**) the receptor density. Simulation parameters are as follows: p = 4 × 10

^{−5}pN/nm

^{2}, h

_{L}= −1.2, σ

_{w}= 4 nm, and ε

_{w}= 0.01 k

_{B}T/nm

^{2}.

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**MDPI and ACS Style**

Marzban, B.; Yuan, H.
The Effect of Thermal Fluctuation on the Receptor-Mediated Adhesion of a Cell Membrane to an Elastic Substrate. *Membranes* **2017**, *7*, 24.
https://doi.org/10.3390/membranes7020024

**AMA Style**

Marzban B, Yuan H.
The Effect of Thermal Fluctuation on the Receptor-Mediated Adhesion of a Cell Membrane to an Elastic Substrate. *Membranes*. 2017; 7(2):24.
https://doi.org/10.3390/membranes7020024

**Chicago/Turabian Style**

Marzban, Bahador, and Hongyan Yuan.
2017. "The Effect of Thermal Fluctuation on the Receptor-Mediated Adhesion of a Cell Membrane to an Elastic Substrate" *Membranes* 7, no. 2: 24.
https://doi.org/10.3390/membranes7020024