Transient active force generation and stress fibre remodelling in cells under cyclic loading

Abstract

The active cytoskeleton is known to play an important mechanistic role in cellular structure, spreading, and contractility. Contractility is actively generated by stress fibres (SF), which continuously remodel in response to physiological dynamic loading conditions. The influence of actin-myosin cross-bridge cycling on SF remodelling under dynamic loading conditions has not previously been uncovered. In this study, a novel SF cross-bridge cycling model is developed to predict transient active force generation in cells subjected to dynamic loading. Rates of formation of cross-bridges within SFs are governed by the chemical potentials of attached and unattached myosin heads. This transient cross-bridge cycling model is coupled with a thermodynamically motivated framework for SF remodelling to analyse the influence of transient force generation on cytoskeletal evolution. A 1D implementation of the model is shown to correctly predict complex patterns of active cell force generation under a range of dynamic loading conditions, as reported in previous experimental studies.

Appendices

Appendix A: Phenomenological remodelling of transient cytoskeletal contractility

Active force generation by cells under dynamic loading conditions has previously been investigated using phenomenological models for cross-bridge cycling and SF remodelling (McGarry et al. 2009; Reynolds and McGarry 2015). In summary, the cross-bridge tension–strain rate relationship is described by the phenomenological Hill-type relationship:

$$\begin{array}{*{20}c} {\frac{T}{{T_{0} }} = \left\{ {\begin{array}{*{20}l} {\frac{{1 - k_{\text{v}} \dot{\bar{\varepsilon }}/\eta_{\text{al}} }}{{1 + k_{k} \dot{\bar{\varepsilon }}/\eta_{\text{al}} }},} \hfill & {h < 0} \hfill \\ {1.8 - 0.8\left( {\frac{{1 - k_{\text{v}} \dot{\bar{\varepsilon }}/\eta_{\text{al}} }}{{1 + k_{k} \dot{\bar{\varepsilon }}/\eta_{\text{al}} }}} \right),} \hfill & { h \ge 0} \hfill \\ \end{array} } \right.} \\ \end{array}$$

(A1)

where $\dot{\bar{\varepsilon }}$ is the effective SF strain rate and $k_{v}$ and $k_{k}$ are model parameters. $T$ and $T_{0}$ are the SF tension and isometric tension, respectively. $\eta_{\text{al}}$ is a non-dimensional SF activation level, and its tension-dependent evolution is given as

$$\begin{array}{*{20}c} {\frac{{{\text{d}}\eta_{al} }}{{{\text{d}}t}} = \left[ {1 - \eta_{al} } \right]Ck_{f} - \left[ {1 - \frac{T}{{T_{0} }}} \right]\eta_{al} k_{b} } \\ \end{array}$$

(A2)

where C is again a non-dimensional signal and $k_{f}$ and $k_{b}$ are formation and dissociation rate constants (Deshpande et al. 2006). We now consider three forms of $\dot{\bar{\varepsilon }}$:

• (i) A Hill-type model where $\dot{\bar{\varepsilon }}$ is taken to be the instantaneous SF strain,$\dot{\varepsilon }_{\text{sf}}$, i.e. $\dot{\bar{\varepsilon }} = \dot{\varepsilon }_{sf}$. Results are shown in Fig. 11a.

  Fig. 11

  

  Steady-state force–strain loops for a Hill-type model; b fading memory model; c modified fading memory model

• (ii) A fading memory type model based on the formulation of Hunter et al. (1998) where $\dot{\bar{\varepsilon }}$ is determined from the SF strain rate history, such that

  $$\begin{array}{*{20}c} {\dot{\bar{\varepsilon }} = \alpha \mathop \int \limits_{ - \infty }^{t} {\text{e}}^{{ - \alpha \left( {t - \tau } \right)}} \dot{\varepsilon }_{\text{sf}} {\text{d}}\tau .} \\ \end{array}$$

  (A3)

  For a constant loading and unloading strain rate magnitude, $\left| {\dot{\varepsilon }_{\text{sf}} } \right|$, $\dot{\bar{\varepsilon }}$ is given as

  $$\begin{array}{*{20}c} {\dot{\bar{\varepsilon }} = \left| {\dot{\varepsilon }_{\text{sf}} } \right|\left\{ { - {\text{e}}^{ - \alpha t} + 2{\text{e}}^{{ - \alpha \left( {t - P/2} \right)}} - 2{\text{e}}^{{ - \alpha \left( {t - P} \right)}} + 2{\text{e}}^{{ - \alpha \left( {t - 3P/2} \right)}} - 2{\text{e}}^{{ - \alpha \left( {t - 2P} \right)}} + 1} \right\} ,} \\ \end{array}$$

  (A4)

  with results shown in Fig. 11b.

• (iii) A modified fading memory type model where only shortening (negative) strain rates contribute to $\dot{\bar{\varepsilon }}$, such that

  $$\begin{array}{*{20}c} {\dot{\bar{\varepsilon }} = \alpha \mathop \int \limits_{ - \infty }^{t} {\text{e}}^{{ - \alpha \left( {t - \tau } \right)}} \dot{\varepsilon }_{\text{m}} {\text{d}}\tau ,} \\ \end{array}$$

  (A5)
  $$\begin{array}{*{20}c} {\dot{\varepsilon }_{\text{m}} = \left\{ {\begin{array}{*{20}c} {\dot{\varepsilon }_{\text{sf}} } & {\dot{\varepsilon }_{\text{sf}} \le 0} \\ 0 & {\dot{\varepsilon }_{\text{sf}} > 0} \\ \end{array} } \right. .} \\ \end{array}$$

  (A6)

  For a constant loading and unloading strain rate magnitude, $\left| {\dot{\varepsilon }_{\text{sf}} } \right|$, $\dot{\bar{\varepsilon }}$ is given as

  $$\begin{array}{*{20}c} {\dot{\bar{\varepsilon }} = \left| {\dot{\varepsilon }_{\text{sf}} } \right|\left\{ {{\text{e}}^{{ - \alpha \left( {t - P/2} \right)}} - {\text{e}}^{{ - \alpha \left( {t - P} \right)}} + {\text{e}}^{{ - \alpha \left( {t - 3P/2} \right)}} - {\text{e}}^{{ - \alpha \left( {t - 2P} \right)}} } \right\} ,} \\ \end{array}$$

  (A7)

  with results shown in Fig. 11c.

Predicted results are shown in Fig. 11a. All models approximate the following features of the experimental measurements of Wille et al.: (i) a reduction in peak tension with increasing $\varepsilon_{\max }$ and (ii) a nonzero tension at the end of each cycle for $\varepsilon_{\max }$ = 5% and 10%. However, the Hill-type model incorrectly predicts a nearly constant cell force during loading and unloading. The fading memory model predicts a transient change in force during each loading cycle, but the concave–convex curve shapes do not resemble experimental curves. The modified fading memory predicts near-linear behaviour during loading and a convex unloading curve. This is approximately similar to the experimental measurements of Wille et al.

Analyses presented in this appendix using a phenomenological modelling approach demonstrate that transient cell contractility during dynamic loading cannot be replicated using a classical Hill-type contractility mode. Fading memory approaches improve the predictions of transient force generation, and fading memory effective strain rates appear require an asymmetric between loading and unloading contributions.

In summary, the thermodynamically consistent mechanistically based cross-bridge cycling model and SF formulation presented in the main paper provides a physical rationalisation of the apparent fading memory behaviour in the experimental results of Wille et al.

Appendix B: Nonlinear viscoelastic model

Here, we provide a more detailed derivation for the nonlinear model summarised in Sect. 2.3. We consider a nonlinear Maxwell unit (nonlinear spring in series with a linear dashpot) in parallel with a second nonlinear spring. The following assumptions may be made regarding the system shown in Fig. 3:

• (i) The total stress $\sigma_{\text{cell}}^{\text{pass}} = \sigma_{nle1} + \sigma_{mw} \quad \left( {\text{B1}} \right)$,

• (ii) The total strain $\varepsilon = \varepsilon_{mw} = \varepsilon_{nle1} \quad \left( {\text{B2}} \right)$,

• (iii) $\sigma_{mw} = \sigma_{nle2} = \sigma_{\text{v}} \quad \left( {\text{B3}} \right)$,

• (iv) and $\varepsilon_{mw} = \varepsilon_{nle2} + \varepsilon_{\text{v}} \quad \left( {\text{B4}} \right)$.

Initially, let us describe the spring stress–strain relations as

$$\begin{array}{*{20}c} {\sigma_{i} = f\left( {\varepsilon_{i} } \right) = f_{i} ,\quad i = nle1, nle2} \\ \end{array}$$

(B5)

Differentiating a nonlinear elastic spring with respect to time gives

$$\begin{array}{*{20}c} {\dot{\sigma }_{nle} = \frac{\text{d}}{{{\text{d}}t}}\left( {\sigma_{nle} } \right) = \frac{{{\text{d}}f_{nle} }}{{{\text{d}}\varepsilon_{nle} }}\frac{{{\text{d}}\varepsilon_{nle} }}{{{\text{d}}t}} = \frac{{{\text{d}}f_{nle} }}{{{\text{d}}\varepsilon_{nle} }}\dot{\varepsilon }_{nle} ,} \\ \end{array}$$

(B6)

Combining with the strain (rate) constraint, we find:

$$\begin{array}{*{20}c} {\dot{\varepsilon }_{mw} = \dot{\varepsilon }_{nle2} + \dot{\varepsilon }_{\text{v}} = \frac{{\dot{\sigma }_{nle2} }}{{\frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}}} + \dot{\varepsilon }_{\text{v}} ,} \\ \end{array}$$

(B7)

so that

$$\begin{array}{*{20}c} {\frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}\dot{\varepsilon } = \dot{\sigma } + \frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}\dot{\varepsilon }_{\text{v}} .} \\ \end{array}$$

(B8)

Also, considering the stress rate:

$$\begin{array}{*{20}c} {\dot{\sigma }_{\text{cell}}^{\text{pass}} = \dot{\sigma }_{nle2} + \dot{\sigma }_{nle1} ,} \\ \end{array}$$

(B9)

so that

$$\begin{array}{*{20}c} {\dot{\sigma }_{\text{cell}}^{\text{pass}} = \frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}\dot{\varepsilon } + \frac{{{\text{d}}f_{nle1} }}{{{\text{d}}\varepsilon_{nle1} }}\dot{\varepsilon } - \frac{{{\text{d}}f_{nle1} }}{{{\text{d}}\varepsilon_{nle1} }}\dot{\varepsilon }_{\text{v}} } \\ \end{array}$$

(B10.1)

$$\begin{array}{*{20}c} { = \left( {\frac{{{\text{d}}f_{nle1} }}{{{\text{d}}\varepsilon_{nle1} }} + \frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}} \right)\dot{\varepsilon } - \frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}\dot{\varepsilon }_{\text{v}} } \\ \end{array}$$

(B10.2)

$$\begin{array}{*{20}c} { = \frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }}\left( {\dot{\varepsilon } - \dot{\varepsilon }_{\text{v}} } \right) + \frac{{{\text{d}}f_{nle1} }}{{{\text{d}}\varepsilon_{nle1} }}\dot{\varepsilon }.} \\ \end{array}$$

(B10.3)

The specific form for the nonlinear spring function is given by:

$$\begin{array}{*{20}c} {\sigma_{nle} = \beta \left( {\exp \left( {\alpha \varepsilon_{nle} } \right) - 1} \right) = f_{nle} , } \\ \end{array}$$

(B11)

where $\varepsilon_{nle}$ is the strain in the spring, and $\alpha_{nle}$ and $\beta_{nle}$ (${\text{kPa}}$) are material constants. The derivative with respect to strain is then:

$$\begin{array}{*{20}c} {\frac{{{\text{d}}f_{nle} }}{{{\text{d}}\varepsilon_{nle} }} = \alpha \beta \exp \left( {\alpha \varepsilon_{nle} } \right).} \\ \end{array}$$

(B12)

The dashpot stress is:

$$\begin{array}{*{20}c} {\sigma_{\text{v}} = \xi \dot{\varepsilon }_{\text{v}} ,} \\ \end{array}$$

(B13)

where $\xi$ is the viscosity of the dashpot (${\text{kPa}}\;{\text{s}}$). Rearranging:

$$\begin{array}{*{20}c} {\dot{\varepsilon }_{\text{v}} = \frac{{\sigma_{\text{v}} }}{\xi } = \frac{1}{\xi }\left( {\sigma_{\text{vis}} - \sigma_{nle1} } \right)} \\ \end{array}$$

(B14.1)

$$\begin{array}{*{20}c} { = \frac{1}{\xi }\left( {\sigma_{\text{vis}} - \beta_{1} \left( {\exp \left( {\alpha_{1} \varepsilon } \right) - 1} \right)} \right).} \\ \end{array}$$

(B14.2)

${\text{d}}f_{nle2} /{\text{d}}\varepsilon_{nle2}$ may be arranged as:

$$\begin{array}{*{20}c} {\frac{{{\text{d}}f_{nle2} }}{{{\text{d}}\varepsilon_{nle2} }} = \alpha_{2} \beta_{2} \exp \left( {\alpha_{2} \varepsilon_{nle2} } \right)} \\ \end{array}$$

(B15.1)

$$\begin{array}{*{20}c} { = \alpha_{2} \left( {\beta_{2} + \beta_{2} \left( {\exp \left( {\alpha_{2} \varepsilon_{nle2} } \right) - 1} \right)} \right)} \\ \end{array}$$

(B15.2)

$$\begin{array}{*{20}c} { = \alpha_{2} \left( {\beta_{2} + \sigma_{nle2} } \right) = \alpha_{2} \left( {\beta_{2} + \sigma_{\text{vis}} - \sigma_{nle1} } \right)} \\ \end{array}$$

(B15.3)

$$\begin{array}{*{20}c} { = \alpha_{2} \left( {\beta_{2} + \sigma_{\text{vis}} - \beta_{1} \left( {\exp \left( {\alpha_{1} \varepsilon } \right) - 1} \right)} \right).} \\ \end{array}$$

(B15.4)

From equation B10.3, the constitutive equation for the complete viscoelastic model follows as:

$$\begin{array}{*{20}c} {\dot{\sigma }_{\text{cell}}^{\text{pass}} = \alpha_{1} \beta_{1} \exp \left( {\alpha_{1} \varepsilon } \right)\dot{\varepsilon } + \alpha_{2} \left\{ {\beta_{2} + \sigma_{\text{cell}}^{\text{pass}} - \beta_{1} \left( {\exp \left( {\alpha_{1} \varepsilon } \right) - 1} \right)} \right\}(\dot{\varepsilon } - \frac{1}{\xi }\left( {\sigma_{\text{cell}}^{\text{pass}} - \beta_{1} \left( {\exp \left( {\alpha_{1} \varepsilon } \right) - 1} \right)} \right).} \\ \end{array}$$

(B16)