Effects of domain unfolding and catch-like dissociation on the collective behavior of integrin–fibronectin bond clusters

Abstract

To gain more understanding of cell–matrix adhesion, we consider an idealized theoretical model of a cluster of integrin–fibronectin bonds at the cell–matrix interface subjected to a dynamic ramping. The distributions of bond traction and interfacial deformation are assumed to obey classical elastic equations, whereas the dissociation/association of individual bonds as well as unfolding/refolding of fibronectin domains are described by stochastic equations. Through stochastic-elasticity coupling, we perform Monte Carlo simulations to investigate how the collective behavior and adhesion performance of the integrin–fibronectin-mediated interface are influenced by two characteristics newly incorporated in the modeling, i.e., catch-like dissociation between integrin and fibronectin, and unfolding of repeated domains in fibronectin. The probable unfolding of fibronectin domains is found to have profound effects on the resultant adhesion energy of the integrin–fibronectin-mediated interface, and governs the failure model transiting between uniform decay and catastrophic crack-like rupture.

Graphic abstract

Focal adhesions containing integrin–fibronectin bond clusters play a vital role in a wide variety of cellular processes or functions. We develop a stochastic-elasticity modeling framework for such specific interaction at cell–matrix interface, by unifying the elastic descriptions of force distribution and deformation, and the stochastic descriptions of bond dissociation/association and domain unfolding/folding. The present modeling reveals a rich array of out-of-equilibrium behaviors of the integrin–fibronectin-mediated interface, and may provide clues for the development of artificial domain-containing structures and functional surfaces to modulate the collective behavior of bond-mediated cell–matrix adhesion.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11672268 and 91748209) and the Fundamental Research Funds for Central Universities of China (Grant 2020XZZX005-02).

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Correspondence to Jin Qian.

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Executive Editor: Xi-Qiao Feng

Appendix

Appendix

Two-state model for catch bond between integrin and fibronectin

The two-state model assumes that there are two binding sites for \({\alpha_{5} }\beta_{1}\)-fibronectin bond, with the following ordinary kinetic equations that govern the evolution of states in these two allosteric sites:

$${{\frac{{{\text{d}}P_{{1}} \left( {t,f} \right)}}{{{\text{d}}t}} = - k_{{{12}}} \left( f \right)P_{{1}} \left( {t,f} \right) + k_{{{21}}} \left( f \right)P_{{2}} \left( {t,f} \right)},\,\frac{{{\text{d}}P_{{2}} \left( {t,f} \right)}}{{{\text{d}}t}} = - k_{{{21}}} \left( f \right)P_{{2}} \left( {t,f} \right) + k_{{{12}}} \left( f \right)P_{{1}} \left( {t,f} \right)},$$
(A1)

where \({P_{{1}} \left( {t,f} \right)}\), \({P_{{2}} \left( {t,f} \right)}\) denote the probability for the allosteric sites subjected to force \(f\) to exist in states 1 and 2 at time \(t\), respectively. \({k_{{{12}}} \left( f \right) = k_{{{12}}}^{{0}} {\text{exp}}\left[ {{{x_{{{12}}} f} \mathord{\left/ {\vphantom {{x_{{{12}}} f} {(k_{{\text{B}}} T}}} \right. \kern-\nulldelimiterspace} {(k_{{\text{B}}} T}})} \right]}\) and \({k_{{{21}}} \left( f \right) = k_{{{21}}}^{{0}} {\text{exp}}\left[ { - {{x_{{{21}}} f} \mathord{\left/ {\vphantom {{x_{{{21}}} f} {(k_{{\text{B}}} T)}}} \right. \kern-\nulldelimiterspace} {(k_{{\text{B}}} T)}}} \right]}\) are the rates for the reversible transition between the two states, with \({k_{{{12}}}^{{0}} }\), \({k_{{{21}}}^{{0}} }\) being the rate constants in the absence of force and \({x_{{{12}}} }\), \({x_{{{21}}} }\) being the widths of the activation barrier for the corresponding transition to occur. It is natural to assume that when \({f = {0}}\) at the beginning of evolution, the two probabilities are in thermodynamic equilibrium, thus the right-sides in Eq. (A1) can be set to zero. Therefore, the initial values of probability at two sites should be \({P_{{1}} \left( {0,0} \right) = {{k_{{{21}}}^{{0}} } \mathord{\left/ {\vphantom {{k_{{{21}}}^{{0}} } {\left( {k_{{{12}}}^{{0}} + k_{{{21}}}^{{0}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {k_{{{12}}}^{{0}} + k_{{{21}}}^{{0}} } \right)}}}\) and \({P_{{2}} \left( {0,0} \right) = {{k_{{{12}}}^{{0}} } \mathord{\left/ {\vphantom {{k_{{{12}}}^{{0}} } {\left( {k_{{{12}}}^{{0}} + k_{{{21}}}^{{0}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {k_{{{12}}}^{{0}} + k_{{{21}}}^{{0}} } \right)}}}\), respectively. Using these two initial conditions, the solution to Eq. (A1) for \(f \ne 0\) is given by

$${P_{{1}} \left( {t,f} \right) = q\left( f \right) + \left[ {q\left( {0} \right) - q\left( f \right)} \right]{\text{exp}}\left[ { - {t \mathord{\left/ {\vphantom {t {\tau_{r} \left( f \right)}}} \right. \kern-\nulldelimiterspace} {\tau_{r} \left( f \right)}}} \right],\quad P_{{2}} \left( {t,f} \right) = {1} - P_{{1}} \left( {t,f} \right),}$$
(A2)

with

$${\tau_{{\text{r}}} \left( f \right) = {{1} \mathord{\left/ {\vphantom {{1} {\left[ {k_{{{12}}} \left( f \right) + k_{{{21}}} \left( f \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {k_{{{12}}} \left( f \right) + k_{{{21}}} \left( f \right)} \right]}}}$$
(A3)

being a characteristic relaxation time of the two-state allosteric site and \({q\left( f \right) = k_{{{21}}} \left( f \right)\tau_{{\text{r}}} \left( f \right)}\) being a normalized quantity that represents the transition rate \({k_{{{21}}} }\), subjected to external force \(f\). Using Bell’s model, the escape rates of integrin–fibronectin bond from the two binding sites are determined by \({k_{{1}} = k_{{1}}^{{0}} {\text{exp}}\left[ {{{x_{{1}} f} \mathord{\left/ {\vphantom {{x_{{1}} f} {(k_{{\text{B}}} T)}}} \right. \kern-\nulldelimiterspace} {(k_{{\text{B}}} T)}}} \right]}\), \({k_{{2}} = k_{{2}}^{{0}} {\text{exp}}\left[ {{{x_{{2}} f} \mathord{\left/ {\vphantom {{x_{{2}} f} {(k_{{\text{B}}} T)}}} \right. \kern-\nulldelimiterspace} {(k_{{\text{B}}} T)}}} \right]}\), respectively. Here, \({k_{{1}}^{{0}} }\), \({k_{{2}}^{{0}} }\) are the rate constants in the absence of force, and \({x_{{1}} }\), \({x_{{2}} }\) are the widths of the corresponding activation barrier. Combining two dissociation rates with the probabilities at two allosteric sites in Eq. (A2), we obtain the dissociation rate of the catch bond as

$${k_{{{\text{off}}}}^{{{\text{catch}}}} \left( {t,f} \right) = k_{{1}} \left( f \right)P_{{1}} \left( {t,f} \right) + k_{{2}} \left( f \right)P_{{2}} \left( {t,f} \right)}.$$
(A4)

Extracting \({{\varvec{\upalpha}}}_{5}{{\varvec{\upbeta}}}_{1}\)-fibronectin dissociation kinetics by excluding the effect of anchoring protein

We should notice that in the original atomic force microscopy (AFM) experiments on \({\alpha_{{5}} \beta_{{1}} }\)-fibronectin dissociation [20], the integrin molecules were immobilized on the substrate surface through anchoring protein GG-7, adding an extra slip bond with a dissociation coefficient \({k_{{{\text{off}}}}^{{{\text{slip}}}} = k_{{\text{s}}}^{{0}} {\text{exp}}\left[ {{{x_{{\text{s}}} f} \mathord{\left/ {\vphantom {{x_{{\text{s}}} f} {(k_{{\text{B}}} T)}}} \right. \kern-\nulldelimiterspace} {(k_{{\text{B}}} T)}}} \right]}\). Here, \({k_{{\text{s}}}^{{0}} }\) is the rate constant at zero force, and \({x_{{\text{s}}} }\) is the width of the energy barrier. The values of slip bond parameters (for GG-7) are found to be \({k_{{\text{s}}}^{{0}} = {0}{\text{.016 s}}^{{ - 1}} }\) and \({x_{{\text{s}}} = {0}{\text{.27 nm}}}\) [39], which means a quite strong linking between integrin and GG-7. In the presence of sequential chain of the allosteric and slip contacts, the survival probability of GG-7-\({\alpha_{{5}} \beta_{{1}} }\)-fibronectin chain, \({P_{{{\text{all}}}} \left( {t,f} \right)}\), should obey the following kinetic equation:

$${\frac{{{\text{d}}P_{{{\text{all}}}} \left( {t,f} \right)}}{{{\text{d}}t}} + \left[ {k_{{{\text{off}}}}^{{{\text{catch}}}} \left( {t,f} \right) + k_{{{\text{off}}}}^{{{\text{slip}}}} \left( f \right)} \right]P_{{{\text{all}}}} \left( {t,f} \right) = {0}}.$$
(A5)

This equation can be solved by combining Eqs. (A2) to (A4) together with the initial condition that the bond is intact at \({t = {0}}\) (\({P_{{{\text{all}}}} \left( {{0},f} \right) = {1}}\))

$${P_{{{\text{all}}}} \left( {t,f} \right) = {\text{exp}}\left\{ { - {t \mathord{\left/ {\vphantom {t {\tau_{{{\text{all}}}} \left( f \right)}}} \right. \kern-\nulldelimiterspace} {\tau_{{{\text{all}}}} \left( f \right)}} + h\left( f \right)\left\{ {{\text{exp}}\left[ { - {t \mathord{\left/ {\vphantom {t {\tau_{{\text{r}}} \left( f \right)}}} \right. \kern-\nulldelimiterspace} {\tau_{{\text{r}}} \left( f \right)}}} \right] - {1}} \right\}} \right\}},$$
(A6)

where \({\tau_{{{\text{all}}}} \left( f \right) = k_{{2}} \left( f \right) + \frac{{k_{{{21}}} \left( f \right)\left[ {k_{{1}} \left( f \right) - k_{{2}} \left( f \right)} \right]}}{{k_{{{12}}} \left( f \right) + k_{{{21}}} \left( f \right)}} + k_{{{\text{off}}}}^{{{\text{slip}}}} \left( f \right)}\) is a combination of rate coefficients, and

$${h\left( f \right) = \left[ {q\left( {0} \right) - q\left( f \right)} \right]\left[ {k_{{1}} \left( f \right) - k_{{2}} \left( f \right)} \right]\tau_{{\text{r}}} \left( f \right)},$$
(A7)

The average lifetime \({T_{{{\text{all}}}} }\) of this sequential chain is given by integration of the survival probability over time

$$T_{{{\text{all}}}} = \int\limits_{0}^{\infty } {P_{{{\text{all}}}} \left( {t,f} \right){\text{d}}t} .$$
(A8)

Choosing appropriate parameters for the two-state model in Eq. (A4) as follow, \({x_{{1}} = x_{{2}} = {0}{\text{.15 nm}}}\)\({x_{{{12}}} = x_{{{21}}} = {0}{\text{.6 nm}}}\), \({k_{{1}}^{{0}} = {\text{3 s}}^{{ - 1}} }\), \({k_{{2}}^{{0}} = {0}{\text{.003 s}}^{{ - 1}} }\), \({k_{{{12}}}^{{0}} = {\text{2 s}}^{{ - 1}} }\), \({k_{{{21}}}^{{0}} = {\text{20 s}}^{{ - 1}} }\), we can theoretically reproduce the experimental data. Neglecting the \({k_{{{\text{off}}}}^{{{\text{slip}}}} \left( f \right)}\) term in Eq. (A5), we can derive a new force dependence of lifetime for the integrin–fibronectin bonding

$$T_{{{\text{catch}}}} = \int\limits_{0}^{\infty } {P_{{{\text{catch}}}} \left( {f,t} \right){\text{d}}t},$$
(A9)

with the survival probability of catch bond being

$${P_{{{\text{catch}}}} \left( {t,f} \right) = {\text{exp}}\left\{ { - {t \mathord{\left/ {\vphantom {t {\tau_{{{\text{catch}}}} \left( f \right)}}} \right. \kern-\nulldelimiterspace} {\tau_{{{\text{catch}}}} \left( f \right)}} + h\left( f \right)\left\{ {{\text{exp}}\left[ { - {t \mathord{\left/ {\vphantom {t {\tau_{{\text{r}}} \left( f \right)}}} \right. \kern-\nulldelimiterspace} {\tau_{{\text{r}}} \left( f \right)}}} \right] - {1}} \right\}} \right\}},$$
(A10)

where \({\tau_{{{\text{catch}}}} = k_{{2}} + \frac{{k_{{{21}}} \left( {k_{{1}} - k_{{2}} } \right)}}{{k_{{{12}}} + k_{{{21}}} }}}\) and the expressions for \({\tau_{{\text{r}}} \left( f \right)}\) and \({h\left( f \right)}\) are shown in Eqs. (A3) and (A7). Therefore, the probability of catch bond dissociation is

$${P_{{{\text{off}}}} = {1} - P_{{{\text{catch}}}}}.$$
(A11)

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Lin, J., Wang, Y. & Qian, J. Effects of domain unfolding and catch-like dissociation on the collective behavior of integrin–fibronectin bond clusters. Acta Mech. Sin. (2021). https://doi.org/10.1007/s10409-020-01039-x

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Keywords

  • Integrin–fibronectin complex
  • Cell–matrix adhesion
  • Domain unfolding
  • Catch bond
  • Rate dependence
  • Modulus effect