A thermodynamically motivated model for stress-fiber reorganization

Abstract

We present a model for stress-fiber reorganization and the associated contractility that includes both the kinetics of stress-fiber formation and dissociation as well as the kinetics of stress-fiber remodeling. These kinetics are motivated by considering the enthalpies of the actin/myosin functional units that constitute the stress fibers. The stress, strain and strain rate dependence of the stress-fiber dynamics are natural outcomes of the approach. The model is presented in a general 3D framework and includes the transport of the unbound stress-fiber proteins. Predictions of the model for a range of cyclic loadings are illustrated to rationalize hitherto apparently contrasting observations. These observations include: (1) For strain amplitudes around 10 % and cyclic frequencies of about 1 Hz, stress fibers align perpendicular to the straining direction in cells subjected to cyclic straining on a 2D substrate while the stress fibers align parallel with the straining direction in cells constrained in a 3D tissue. (2) At lower applied cyclic frequencies, stress fibers in cells on 2D substrates display no sensitivity to symmetric applied strain versus time waveforms but realign in response to applied loadings with a fast lengthening rate and slow shortening. (3) At very low applied cyclic frequencies (on the order of mHz) with symmetric strain versus time waveforms, cells on 2D substrates orient perpendicular to the direction of cyclic straining above a critical strain amplitude.

Appendix: Chemical potentials of the stress-fiber proteins

Here, we derive expressions for the chemical potential $\chi _{u}$ of the aggregate of unbound molecules that form a single functional unit and the chemical potential $\chi _{b}$ of a functional unit within a stress-fiber comprising n units. These chemical potentials are derived using the enthalpies and the constraints imposed on the formation of stress fibers in deriving the kinetic Eq. (2.16). We shall employ classical statistical mechanics using the assumption of non-interacting particles, i.e., an ideal system.

Consider a segment subtending a unit solid angle within the RVE. Within this segment, there are $\eta$ stress fibers each comprising n functional units and aggregates of unbound molecules that form $N_{u}/(2\pi )$ functional units. For notational simplicity, we denote $\bar{N}_{u} \equiv N_{u}/(2\pi )$, $\bar{N}_{b}\equiv \eta n$ and $\bar{N}_{T}\equiv \bar{N}_{u}+\bar{N}_{b}$. First consider the mixing between the $\bar{N}_{L}$ lattice sites and the $\bar{N}_{u}$ unbound aggregates of molecules. The $\bar{N}_{u}$ identical aggregates of molecules and the $(\bar{N}_{L}-\bar{N}_{u})$ identical empty lattice sites in the mixture can be arranged in

$$\begin{aligned} W=\frac{\bar{N}_{L}!}{\bar{N}_{u}!(\bar{N}_{L}-\bar{N}_{u})!}, \end{aligned}$$

(8.1)

ways and Boltzmann’s entropy formula then gives the entropy of mixing as

$$\begin{aligned} \varDelta S_{u} =k_{B} \hbox {ln}W. \end{aligned}$$

(8.2)

Using Stirling’s approximation ($\ln M!\approx M\ln M-M$ for large M) we have

$$\begin{aligned} \varDelta S_{u} =-k_{B}\left[ \bar{N}_{u} \hbox {ln}\bar{N}_{u} \!+\!(\bar{N}_{L}\!-\!\bar{N}_{u})\hbox {ln} (\bar{N}_{L}\!-\!\bar{N}_{u}) \!-\!\bar{N}_{L} \hbox {ln}\bar{N}_{L}\right] .\nonumber \\ \end{aligned}$$

(8.3)

Upon assuming that the entropy of the unbound molecules prior to mixing with the lattice is zero, the chemical potential of the unbound molecules is given by

$$\begin{aligned} \chi _{u} \equiv \frac{\mu _{u}}{n^{R}}\!-\!T\frac{\partial \varDelta S_{u} }{\partial \hat{N}_{u}}=\frac{\mu _{u}}{n^{R}}+k_{B} T\hbox {ln}\left[ \!\frac{{\hat{N}}_{u}}{2\pi ~{\hat{N}}_L \left( 1-\frac{\hat{N}_{u}}{\hat{N}_{L}} \right) }\!\right] ,\nonumber \\ \end{aligned}$$

(8.4)

where ${\hat{N}}_L \equiv \bar{N}_{L}/N_{0}$. Since $\bar{N}_{u} \ll \bar{N}_L$ (i.e., dilute assumption), $\chi _{u}$ simplifies to

$$\begin{aligned} \chi _{u} =\frac{\mu _{u}}{n^{R}}+k_{B} T\hbox {ln}\left( {\frac{{\hat{N}}_{u}}{2\pi {\hat{N}}_L }} \right) . \end{aligned}$$

(8.5)

Now consider the stress fibers. In deriving the kinetic Eq. (2.16), we have assumed that there exists an intermediate stage where the unbound molecules first cluster into packets comprising unbound molecules that can form n functional units and some of these packets react to form stress fibers also comprising n functional units. To calculate the chemical potentials of the bound molecules within stress fibers and the unbound molecules in the intermediate stage, consider the following two mixing processes. First consider the mixing between $\bar{N}_{u} /n$ identical packets of unbound proteins and $\bar{N}_{b} /n$ identical packets bound proteins where $\bar{N}_T \equiv \bar{N}_{u} +\bar{N}_{b}$. Using Boltzmann’s entropy formula, the entropy of mixing in this process is

$$\begin{aligned} \varDelta S_{b} =k_{B} \hbox {ln}\left[ {\frac{\left( {\bar{{N}}_{b} /n+\bar{{N}}_{u} /n} \right) !}{\left( {\bar{{N}}_{b} /n} \right) !\left( {\bar{{N}}_{u} /n} \right) !}} \right] , \end{aligned}$$

(8.6)

which simplifies using Stirling’s approximation to

$$\begin{aligned} \varDelta S_{b}= & {} -k_{B} \left[ \left( {\frac{\bar{{N}}_{b} }{n}} \right) \hbox {ln}\left( {\frac{\bar{{N}}_{b} }{n}} \right) +\left( {\frac{\bar{{N}}_{u}}{n}}\right) \hbox {ln}\left( {\frac{\bar{{N}}_{u}}{n}}\right) \right. \nonumber \\&\left. -\left( {\frac{\bar{{N}}_T }{n}} \right) \hbox {ln}\left( {\frac{\bar{{N}}_T }{n}}\right) \right] . \end{aligned}$$

(8.7)

The chemical potentials of the bound proteins and unbound proteins after this first step are

$$\begin{aligned} \chi _{b1} \equiv \frac{\mu _{b} }{n^{R}}-T\frac{\partial \varDelta S_{b} }{\partial \bar{N}_{b}}=\frac{\mu _{b} }{n^{R}}+k_{B} T\hbox {ln}\left( {\frac{2\pi {\hat{\eta }} ~\hat{n}}{{\hat{N}}_{u}}}\right) ^{\frac{1}{n}} \end{aligned}$$

(8.8)

and

$$\begin{aligned} \chi _{I1} \equiv \frac{\mu _{b} }{n^{R}}-T\frac{\partial \varDelta S_{b} }{\partial \hat{N}_{u}}=\frac{\mu _{u} }{n^{R}}+k_{B} T\hbox {ln}\left( {\frac{{\hat{N}}_{u} }{2\pi {\hat{\eta }} ~\hat{n}}} \right) ^{\frac{1}{n}}, \end{aligned}$$

(8.9)

respectively, where $\partial \bar{N}_{u} /\partial \bar{N}_{b} =-1$ as the mixing process occurs at constant $\bar{N}_T$. Second, recall that the unbound aggregate of proteins occupy lattice sites and thus we mix the $\bar{N}_{u}$ unbound protein aggregates with the $\bar{N}_L$ lattice sites lattice while not mixing the $\bar{N}_{u} /n$ and $\bar{N}_{b} /n$ packets. The entropy of mixing of this process is given by Eqs. (8.1) and (8.2). Again using the dilute assumption $(\bar{N}_{u} \ll \bar{N}_L)$, the chemical potentials of the bound and unbound molecules in their intermediate stage follow as

$$\begin{aligned}&\chi _{b} \equiv \chi _{b1} \!-T\frac{\partial \varDelta S_{u} }{\partial \hat{N}_{b}}\!=\frac{\mu _{b} }{n^{R}}\nonumber \\&\quad +\,k_{B} T\hbox {ln}\left[ \!{\left( {\frac{2\pi {\hat{\eta }} ~\hat{n}}{{\hat{N}}_{u} }} \right) ^{\frac{1}{n}}\left( {\frac{{\hat{N}}_{u} }{2\pi {\hat{N}}_L }} \right) }\!\!\right] , \end{aligned}$$

(8.10)

and

$$\begin{aligned}&\chi _I \equiv \chi _{I1} -T\frac{\partial \varDelta S_{u} }{\partial \hat{N}_{u} }=\frac{\mu _{u}}{n^{R}}\nonumber \\&\quad +\, k_{B} T\hbox {ln}\left[ {\left( {\frac{{\hat{N}}_{u} }{2\pi {\hat{\eta }}~\hat{n}}} \right) ^{\frac{1}{n}}\left( {\frac{{\hat{N}}_{u} }{2\pi {\hat{N}}_L }} \right) } \right] , \end{aligned}$$

(8.11)

respectively. In deriving Eqs. (8.10) and (8.11), we have used the fact that now $\partial \bar{N}_{u} /\partial \bar{N}_{b} =1$ as in this step we do not change the number of bound and unbound molecules with respect to each but rather calculate the variation in the entropy while changing the number of stress-fiber protein molecules with respect to the fixed number of lattice sites.

Comparing the chemical potentials $\chi _I$ and $\chi _{u}$, we see that the clustering reaction is endergonic and the unclustering reaction is exergonic when ${\hat{N}}_{u}/({2\pi {\hat{\eta }} \hat{n}})>1$ and vice-versa when ${\hat{N}}_{u}/(2\pi {\hat{\eta }}\hat{n})<1$. This is rationalized by the fact that when the stress-fiber concentration is small compared to the unbound protein concentration, the geometrical constraints imposed by the stress fibers are small and clustering requires an entropy reduction but the situation is reversed at high stress-fiber concentrations. We emphasize that the unbound proteins in their intermediate clustered state are unstable (due to their low entropy compared to their unclustered counterparts) and not physically present in the system at any given time. Rather this intermediate state is a transient state in the reaction for the formation/dissociation of the stress fibers.

Equilibrium between the bound and unbound proteins occurs when their chemical potentials equalize, i.e., $\chi _{u}=\chi _{b}$. Setting ${\dot{\hat{\eta }}}=0$ in Eq. (2.16) and simplifying reduces Eq. (2.16) to the condition $\chi _{u} =\chi _{b}$ consistent with the chemical potentials derived here from statistical mechanics considerations.

Finally, we note that the free energy of the stress-fiber proteins within a RVE is given as

$$\begin{aligned} g=N_{u}\chi _{u} +\int \limits _{-\pi /2}^{\pi /2}\int \limits _{0}^{\pi } ~(\eta n\chi _{b})\hbox {sin}\theta \mathrm{d}\theta \mathrm{d}\varphi , \end{aligned}$$

(8.12)

and this expression can be integrated over the entire cell volume $V_{0}$ to give the free energy of the stress-fiber proteins in the cell as

$$\begin{aligned} G=\frac{1}{\frac{4\pi }{3}(n^{R}\ell _{0}/2)^{3}}\int \limits _{V_{0}} g~\mathrm{d}V. \end{aligned}$$

(8.13)