Dual role of the nucleus in cell migration on planar substrates

Abstract

Cell migration is essential to sustain life. There have been significant advances in the understanding of the mechanisms that control cell crawling, but the role of the nucleus remains poorly understood. The nucleus exerts a tight control of cell migration in 3D environments, but its influence in 2D migration on planar substrates remains unclear. Here, we study the role of the cell nucleus in 2D cell migration using a computational model of fish keratocytes. Our results indicate that the apparently minor role played by the nucleus emerges from two antagonist effects: While the nucleus modifies the spatial distributions of actin and myosin in a way that reduces cell velocity (e.g., the nucleus displaces myosin to the sides and front of the cell), its mechanical connection with the cytoskeleton alters the intracellular stresses promoting cell migration. Overall, the favorable effect of the nucleus-cytoskeleton connection prevails, which may explain why regular cells usually move faster than enucleated cells.

Appendix

1.1 Appendix 1: Additional model details

The cell volume conservation is weakly imposed by the wave-pinning model used for actin dynamics. Since $N_A=\int _{\Omega }\psi (\rho _f+\rho _g) \mathrm {\,d}\Omega$ is kept constant, the model results show variations of cell area of at most $\sim \!15\%$. The nuclear volume conservation is imposed with the force ${{\mathbf {F}}^n_{\mathrm{vol}}}$ (see Eq. (9)), which is equivalent to the force $\Lambda ^{\star }_n[V_{n0}-V_n(t)]\varepsilon \vert \nabla \nu \vert ^2\mathbf {n}_{\Gamma _n}$. There, $\varepsilon \vert \nabla \nu \vert ^2$ localizes the force to the nuclear envelope, $\mathbf {n}_{\Gamma _n}$ is the unit normal vector to the nuclear envelope, and $\Lambda ^{\star }_n$ is a constant.

Some of the forces in Eqs. (8) and (9) can be derived from an energy functional making use of the phase-field formulation (Gomez and van der Zee 2018; Moure and Gomez 2017). Let us consider a generic functional ${\mathscr {F}}_a[\phi ,\nu ]=\int _{\Omega }\Psi _a(\phi ,\nu ) \mathrm {\,d}\Omega$. The forces applied on the cell (i.e., the cytosol) may be written as ${\mathbf {F}}^c_a=\frac{\delta {\mathscr {F}}_a}{\delta \phi }\nabla \phi$ and the forces applied on the nucleus as ${\mathbf {F}}^n_a=\frac{\delta {\mathscr {F}}_a}{\delta \nu }\nabla \nu$. In particular, ${\mathbf {F}}^c_{\mathrm{def}}$ and ${\mathbf {F}}^n_{\mathrm{def}}$ may be derived from the repulsive potential ${\mathscr {F}}_\mathrm{def}[\phi ,\nu ]=\int _{\Omega }\eta _{\mathrm{def}}(1-\phi )^2\nu ^2 \mathrm {\,d}\Omega$, such that ${\mathbf {F}}^c_{\mathrm{def}}=-\eta _{\mathrm{def}}2(1-\phi )\nu ^2 \nabla \phi$ and ${\mathbf {F}}^n_{\mathrm{def}}=\eta _{\mathrm{def}}2(1-\phi )^2\nu \nabla \nu$.

From Eq. (9), we can compute the nucleus velocity as

$$\begin{aligned} {\mathbf {u}}_n &= \frac{1}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2}\bigg \{\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2{\mathbf {u}}_c \\&+ \bigg [\gamma _n\left( \varepsilon \nabla ^2\nu -\frac{G'(\nu )}{\varepsilon }\right) \\&+\Lambda _n\left( V_n(t)-V_{n0}\right) \vert \nabla \nu \vert \\&+ \eta _{\mathrm{def}} 2(1-\phi )^2\nu + \eta _{\mathrm{poly}}(1-\phi )^2\nu \bigg ] \nabla \nu \bigg \}. \end{aligned}$$

(10)

By substituting ${\mathbf {u}}_n$ in Eq. (2), we obtain

$$\begin{aligned}&\frac{\partial \nu }{\partial t} + \frac{\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2}{\mathbf {u}}_c \cdot \nabla \nu \\&\quad = \Gamma _{n}\left( \varepsilon \nabla ^2\nu -\frac{G'(\nu )}{\varepsilon } + c_{\nu } \varepsilon |\nabla \nu |\right) \\&\qquad -\frac{\vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2} \bigg [\gamma _n\left( \varepsilon \nabla ^2\nu -\frac{G'(\nu )}{\varepsilon }\right) \\&\qquad +\Lambda _n\left( V_n(t)-V_{n0}\right) \vert \nabla \nu \vert \\&\qquad + \eta _{\mathrm{def}} 2(1-\phi )^2\nu + \eta _{\mathrm{poly}} (1-\phi )^2\nu \bigg ]. \end{aligned}$$

(11)

In our numerical simulations, we use Eq. (11) instead of Eqs. (2) and (9), to speed up the calculations. The velocity ${\mathbf {u}}_n$, required to compute ${\mathbf {F}}^c_{\mathrm{AC}}$ in Eq. (8), is taken from Eq. (10).

1.2 Appendix 2: Numerical formulation

We consider a fixed computational domain $\Omega$. We solve Eqs. (1), (3), (4), (7), (8), and (11) on $\Omega$ using isogeometric analysis (Hughes et al. 2005), a finite element method that employs B-splines as basis functions. Note that we do not need to discretize Eq. (7). The numerical formulation is analogous to that explained in Moure and Gomez (2017). We derive a weak form of Eqs. (1), (3), (4), (8), and (11) by multiplying them with weighting functions, integrating over $\Omega$, and integrating by parts under the assumption of periodic boundary conditions. The weak form can be written as:

$$\begin{aligned}&\int _{\Omega }w_1\frac{\partial \phi }{\partial t}\mathrm{d}\Omega + \int _{\Omega }w_1\,{\mathbf {u}}_c\cdot \nabla \phi \mathrm{d}\Omega + \int _{\Omega }\Gamma _{\phi }\varepsilon \nabla w_1\cdot \nabla \phi \mathrm{d}\Omega \\&\quad +\int _{\Omega }w_1\Gamma _{\phi }\frac{G'(\phi )}{\varepsilon }\mathrm{d}\Omega - \int _{\Omega }\Gamma _{\phi }\varepsilon \nabla w_1\cdot \nabla \phi \mathrm{d}\Omega \\&\quad -\int _{\Omega }w_1\frac{\Gamma _{\phi }\varepsilon }{\vert \nabla \phi \vert }\nabla \phi \cdot \nabla (\vert \nabla \phi \vert )\mathrm{d}\Omega = 0, \end{aligned}$$

(12)

$$\begin{aligned}&\int _{\Omega }w_2\frac{\partial (\psi \rho _m)}{\partial t}\mathrm{d}\Omega - \int _{\Omega }\nabla w_2\cdot {\mathbf {u}}_c\psi \rho _m\mathrm{d}\Omega \\&\quad + \int _{\Omega }{D}_{m}\psi \nabla w_2 \cdot \nabla \rho _m\mathrm{d}\Omega = 0, \end{aligned}$$

(13)

$$\begin{aligned}&\int _{\Omega }w_3\frac{\partial (\psi \rho _f)}{\partial t}\mathrm{d}\Omega - \int _{\Omega }\nabla w_3\cdot {\mathbf {u}}_c\psi \rho _f\mathrm{d}\Omega \\&\quad + \int _{\Omega }{D}_{f}\psi \nabla w_3 \cdot \nabla \rho _f\mathrm{d}\Omega -\int _{\Omega }w_3 \psi f_\rho (\rho _f,\rho _g)\mathrm{d}\Omega = 0, \end{aligned}$$

(14)

$$\begin{aligned}&-\int _{\Omega }\mu \psi \nabla \mathbf {w}_4 :(\nabla {\mathbf {u}}_c +\nabla {\mathbf {u}}_c^T)\mathrm{d}\Omega - \int _{\Omega }{\eta }_{m}\psi {\rho }_{m}\nabla \mathbf {w}_4 :{\mathbf {I}}\,\mathrm{d}\Omega \\&\quad + \int _{\Omega }{\eta }_{f}\psi {\rho }_{f}\nabla \mathbf {w}_4 :\nabla \phi \otimes \nabla \phi \mathrm{d}\Omega - \int _{\Omega }\varsigma _c\psi \mathbf {w}_4\cdot {\mathbf {u}}_c\mathrm{d}\Omega \\&\quad -\int _{\Omega }\gamma _\phi \,\mathbf {w}_4\cdot \nabla \phi \left( \varepsilon \nabla ^2\phi -\frac{G'(\phi )}{\varepsilon }\right) \mathrm{d}\Omega \\&\quad -\int _{\Omega } \tau _{\mathrm{AC}}\varepsilon \vert \nabla \nu \vert ^2 \mathbf {w}_4\cdot {\mathbf {u}}_c\mathrm{d}\Omega + \int _{\Omega } \tau _{\mathrm{AC}}\varepsilon \vert \nabla \nu \vert ^2 \mathbf {w}_4\cdot {\mathbf {u}}_n\mathrm{d}\Omega \\&\quad -\int _{\Omega } \eta _{\mathrm{def}} 2(1-\phi )\nu ^2 \mathbf {w}_4\cdot \nabla \phi \mathrm{d}\Omega = 0, \end{aligned}$$

(15)

$$\begin{aligned}&\int _{\Omega }w_5\frac{\partial \nu }{\partial t}\mathrm{d}\Omega + \int _{\Omega }w_5\frac{\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2}{\mathbf {u}}_c\cdot \nabla \nu \mathrm{d}\Omega \\&\quad + \int _{\Omega }\Gamma _{n}\varepsilon \nabla w_5\cdot \nabla \nu \mathrm{d}\Omega \\&\quad +\int _{\Omega }w_5\Gamma _{n}\frac{G'(\nu )}{\varepsilon }\mathrm{d}\Omega - \int _{\Omega }\Gamma _{n}\varepsilon \nabla w_5\cdot \nabla \nu \mathrm{d}\Omega \\&\quad -\int _{\Omega }w_5\frac{\Gamma _{n}\varepsilon }{\vert \nabla \nu \vert }\nabla \nu \cdot \nabla (\vert \nabla \nu \vert )\mathrm{d}\Omega \\&\quad + \int _{\Omega }w_5\frac{\vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2}\gamma _{n}\left( \varepsilon \nabla ^2\nu -\frac{G'(\nu )}{\varepsilon } \right) \mathrm{d}\Omega \\&\quad +\int _{\Omega }w_5\frac{\vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2} \Lambda _n\left( V_n(t)-V_{n0}\right) \vert \nabla \nu \vert \mathrm{d}\Omega \\&\quad +\int _{\Omega }w_5\frac{\vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2} \eta _{\mathrm{def}} 2(1-\phi )^2\nu \mathrm{d}\Omega \\&\quad +\int _{\Omega }w_5\frac{\vert \nabla \nu \vert ^2}{\varsigma _n\nu +\tau _\mathrm{AC}\varepsilon \vert \nabla \nu \vert ^2} \eta _{\mathrm{poly}} (1-\phi )^2\nu \mathrm{d}\Omega = 0, \end{aligned}$$

(16)

where $\{w_i({\mathbf {x}})\}_{i=1,\ldots ,5}$ are the weighting functions. In Eqs. (13)–(15), $\psi$ and ${\mathbf {u}}_n$ must be replaced by $\phi -\nu$ and the expression given in Eq. (10), respectively. We have not substituted $\psi$ and ${\mathbf {u}}_n$ into Eqs. (13)–(15) to keep the notation clear. Note that Eqs. (12) and (16) become singular in regions far from the interface due to the curvature term (Biben et al. 2005). To address this issue, we propose a simple method similar to that used in Biben et al. (2005). We multiply the curvature $c_{\iota }$ by $\alpha _{\iota } = 0.5+0.5\mathrm {\,tanh}[\mathrm {\,log}_{10}(\vert \nabla \iota \vert +\Upsilon _{\iota })]$, with $\iota =\phi ,\nu$; see Eqs. (1) and (11). In our computations, we took $\Upsilon _{\phi }=7$ and $\Upsilon _{\nu }=3.5$. This regularizes the equations without altering the results. For similar reasons, in Eqs. (15) and (16) we replace $\varsigma _c\psi$ and $\varsigma _n\nu$ by $\varsigma _c\psi +\varsigma _{\epsilon }$ and $\varsigma _n\nu +\varsigma _{\epsilon }$, respectively, to avoid singularities and improve the condition number of the linear system. Here, $\varsigma _{\epsilon }$ is a small constant (we took $\varsigma _{\epsilon }={0.2}\,\hbox {pN s}\,\upmu \hbox {m}^{-3}$) such that $0<\varsigma _{\epsilon }<<\varsigma _c$. The Galerkin form is obtained by replacing the unknowns and weighting functions with discrete approximations. Let us denote the basis functions as $N^A$. We replace $\phi ({\mathbf {x}},t)$ and $w_1({\mathbf {x}})$ with

$$\begin{aligned} \phi ^h({\mathbf {x}},t)=\sum _A \phi ^A(t)N^A({\mathbf {x}}), \quad \quad w^h_1({\mathbf {x}})=\sum _A w_1^A N^A({\mathbf {x}}), \end{aligned}$$

(17)

respectively, where the $\phi ^A$s are referred to as control variables. The spatial discretization of the rest of the unknowns and weighting functions is analogous.

Our semi-implicit time-stepping scheme is based on the generalized-$\alpha$ method (Chung and Hulbert 1993; Jansen et al. 2000). The algorithm presents two particular features: first, the explicit, rather than implicit, treatment of the term $\rho _g$; see Eq. (7) and second, the redefinition of the tangent matrix and residual vector to improve the condition number of the tangent matrix. The procedure consists of replacing certain rows of the tangent matrix and the associated elements of the residual vector; see Moure and Gomez (2017) for more details. The tangent matrix rows and the residual vector elements to be replaced are associated with the control variables of $\rho _m$ and $\rho _f$ and are determined according to the value of $\psi$ (unlike the procedure explained in Moure and Gomez (2017), which employs the value of $\phi$). We use the Newton–Raphson method to solve the resulting nonlinear system. In our simulations, we employ a periodic domain of $50\times {50}\upmu \hbox {m}^2$, which is meshed with $200\times 200$${\mathscr {C}}^1$-continuous quadratic B-spline elements. We use a time step of ${0.05}\hbox {s}$.

1.3 Appendix 3: Parameter estimation

The parameters used in our simulations are listed in Table 3. We keep constant all the parameters except those related to the nucleus dynamics. Unless otherwise stated, the parameters $\tau _{\mathrm{AC}}$, $\eta _{\mathrm{def}}$, $\eta _{\mathrm{poly}}$, $\gamma _n$, and $\varsigma _n$ take the values indicated in Table 3.

The parameter $\Lambda _n$ is estimated from the value of the nucleus bulk modulus $K_n$. In our 2D problem, we assume plane stress, which leads to the relation $K_n=\frac{E_n}{2(1-2\nu _n)}$, where $E_n$ is the nucleus Young’s modulus and $\nu _n$ is the nucleus Poisson’s ratio. The force ${\mathbf {F}}^n_{\mathrm{vol}}$ can be understood as a pressure acting on the nucleus, induced by nuclear volume changes. Therefore, we can compare ${\mathbf {F}}^n_\mathrm{vol}=\Lambda ^{\star }_n[V_{n0}-V_n(t)]\varepsilon \vert \nabla \nu \vert ^2\mathbf {n}_{\Gamma _n}$ (see “Appendix 1”) with the definition of the bulk modulus: $dP=-K_n\frac{dV}{V}$, where P is pressure and V is volume. If we neglect the term $\varepsilon \vert \nabla \nu \vert ^2$, which localizes the force ${\mathbf {F}}^n_{\mathrm{vol}}$ to the nuclear envelope, we identify $dV=-[V_{n0}-V_n(t)]$ and $\frac{K_n}{V}=\Lambda ^{\star }_n$. If we take $V=V_{n0}$, we obtain $\Lambda ^{\star }_n=\frac{E_n}{2(1-2\nu _n)V_{n0}}$. Finally, the parameter $\Lambda _n$ is defined as $\Lambda _n=\varepsilon \Lambda ^{\star }_n=\frac{\varepsilon E_n}{2(1-2\nu _n)V_{n0}}$. Note that the latter expression is not dimensionally consistent because we omitted the term $\varepsilon \vert \nabla \nu \vert ^2$. If we assume that $\varepsilon \vert \nabla \nu \vert ^2={1}{\upmu \hbox {m}^{-1}}$ and incorporate it into the previous derivation, the expression for $\Lambda _n$ becomes dimensionally consistent.

Table 3 List of model parameters

1.4 Appendix 4: Initial conditions

The initial conditions are defined as

$$\begin{aligned}&\phi ({\mathbf {x}},0)=\phi _0({\mathbf {x}})=0.5-0.5\mathrm {\,tanh}\left[ \frac{2\sqrt{2}}{\varepsilon }\left( d_c({\mathbf {x}})-R_c\right) \right] , \end{aligned}$$

(18)

$$\begin{aligned}&\nu ({\mathbf {x}},0)=\nu _0({\mathbf {x}})=0.5-0.5\mathrm {\,tanh}\left[ \frac{2\sqrt{2}}{\varepsilon }\left( d_c({\mathbf {x}})-R_n\right) \right] , \end{aligned}$$

(19)

$$\begin{aligned}&\rho _m({\mathbf {x}},0)=\rho _{m0}\left[ \phi _0({\mathbf {x}})-\nu _0({\mathbf {x}})\right] , \end{aligned}$$

(20)

$$\begin{aligned}&\rho _f({\mathbf {x}},0)= \left\{ \begin{array}{ll} \rho _{f0}^M \left[ \phi _0({\mathbf {x}})-\nu _0({\mathbf {x}})\right] &\;\text {if }y\ge y_c, \\ \rho _{f0}^m \left[ \phi _0({\mathbf {x}})-\nu _0({\mathbf {x}})\right] &\;\text {if }y<y_c, \end{array} \right. \end{aligned}$$

(21)

$$\begin{aligned}&\rho _g({\mathbf {x}},0)=\rho _{g0}\left[ \phi _0({\mathbf {x}})-\nu _0({\mathbf {x}})\right] , \end{aligned}$$

(22)

$$\begin{aligned}&{\mathbf {u}}_c({\mathbf {x}},0)=0, \end{aligned}$$

(23)

where $d_c({\mathbf {x}})$ is the distance to the center of the cell, denoted here by $\mathbf {C}=(x_c,y_c)$. $R_c$ and $R_n$ are the initial radius of the cell and the nucleus, respectively. The initial myosin and actin densities are defined by the parameters $\rho _{m0}$, $\rho _{f0}^M$, $\rho _{f0}^m$, and $\rho _{g0}$. Unless otherwise stated, we use the values $R_c={10.5}\upmu \hbox {m}$, $R_n={4.3}\,\upmu \hbox {m}$, $\rho _{m0}={0.3}\,\upmu \hbox {m}^{-2}$, $\rho _{f0}^M={1.5}\,\upmu \hbox {m}^{-2}$, $\rho _{f0}^m={0.17}\,\upmu \hbox {m}^{-2}$, and $\rho _{g0}={1.76}\,\upmu \hbox {m}^{-2}$.

In the simulations corresponding to a cell spreading on a $75\times {5}\,\upmu \hbox {m}^2$ adhesive micropattern, we take $R_c={9.5}\,\upmu \hbox {m}$, $R_n={5.5}\,\upmu \hbox {m}$, $\rho _{m0}={0.165}\,\upmu \hbox {m}^{-2}$, $\rho _{f0}^M=\rho _{f0}^m={2.4}\,\upmu \hbox {m}^{-2}$, and $\rho _{g0}={5}\,\upmu \hbox {m}^{-2}$.

1.5 Appendix 5: Simulations of cells with fixed nucleus geometry

We fixed the nucleus shape by imposing a uniform nucleus velocity in Eq. (2), i.e., ${\mathbf {u}}_n({\mathbf {x}},t)={\mathbf {u}}_n^{\star }(t)$. Thus, we solve Eq. (2) instead of Eq. (11) in our system of equations. The velocity ${\mathbf {u}}^{\star }_n(t)$ is defined as a function of the distance between the nucleus and the cell’s rear, $d^{\star }\!(t)$, such that

$${\mathbf{u}}_{n}^{{ \star}} (t) = \left\{ {\begin{array}{*{20}l} {1.1{\mathbf{u}}_{c}^{R} (t)\left[ {2\left( {\frac{{d^{{ \star }} (t)}}{{d_{M} }}} \right)^{3} - 3\left( {\frac{{d^{{ \star }} (t)}}{{d_{M} }}} \right)^{2} + 1} \right]} & {{\text{if}}\;0 < d^{{ \star }} (t) < d_{M},} \\ 0 & {{\text{if}}\;d^{{ \star }} (t) \ge d_{M},} \\ \end{array} } \right.$$

(24)

where $d_M={2}\,\upmu \hbox {m}$ and ${\mathbf {u}}_c^{R}(t)$ is the y-velocity of the membrane at the cell’s rear.