Random blebbing motion: A simple model linking cell structural properties to migration characteristics

If the plasma membrane of a cell is able to delaminate locally from its actin cortex, a cellular bleb can be produced. Blebs are pressure-driven protrusions, which are noteworthy for their ability to produce cellular motion. Starting from a general continuum mechanics description, we restrict ourselves to considering cell and bleb shapes that maintain approximately spherical forms. From this assumption, we obtain a tractable algebraic system for bleb formation. By including cell-substrate adhesions, we can model blebbing cell motility. Further, by considering mechanically isolated blebbing events, which are randomly distributed over the cell, we can derive equations linking the macroscopic migration characteristics to the microscopic structural parameters of the cell. This multiscale modeling framework is then used to provide parameter estimates, which are in agreement with current experimental data. In summary, the construction of the mathematical model provides testable relationships between the bleb size and cell motility.

Figure 1

A blebbing muscle stem cell with fluorescent actin skeleton at two different time points, illustrating the rounded form of the cells and blebs. Used with permission from the Skeletal Muscle Development Group, University of Reading. Scale bars, 5 $\mu \mathrm{m}$.

Figure 2

Schematic diagram of stresses acting on the membrane of the cell. (a) Definition of adhesion forces and geometric variables. (b) Adhesion forces, surface tensions, and cellular pressures defined on a small section of the membrane. Variable definitions: $(z_{rc},y_{rc})$, reference configuration of the membrane; $(z,y)$, solution position of the membrane; $\sigma$, reference configuration arc length; $s$, solution configuration arc length; $\theta$, angle normal to membrane solution configuration; $\mathit{F}$, Force vector produced by adhesions; $\delta$, angle of adhesion action; $t_s$ and $t_{\psi }$, surface tensions; $P_e$ and $P_i$, external and internal cell pressures, respectively. For further details, see text.

Figure 3

Schematic diagram illustrating the bleb and cell geometry. For the $i\text{th}$ bleb we define the variables: $r_{bi}$, bleb radius; ${\theta }_{bi}$, neck opening angle of bleb; $({\varphi }_i,{\psi }_i)$, polar angles denoting the bleb's position; ${\theta }_{ci}$, neck opening angle of cell.

Figure 4

Schematic diagram of adhesions acting between a substrate and (a) a cell or (b) a cell and a bleb. Variable definitions: $(r_c,{\theta }_c)$, cell radius and neck angle; $(r_b,{\theta }_b)$, bleb radius and neck angle; $L$, width of resting adhesion layer; $\alpha L$, maximum width of adhesion layer; ${\mathrm{\Theta }}_c$, half angle subtended by the cell's adhesion pad; ${\mathrm{\Theta }}_b$, half angle subtended by the bleb's adhesion pad; $h_c$, separation distance of bleb and substrate; $h_b$, separation distance of cell and substrate; $d$, distance between cell and bleb centers.

Figure 5

Roots of the force and torque balance equations. The points 1–4 are the values of ${\mathrm{\Theta }}_b$ and ${\mathrm{\Theta }}_c$ that simultaneously solve both equilibrium functions. Although we are specifically interested in the nonnegative quarter plane, the whole plane is shown to illustrate the rotational symmetry that is present in the solutions. Parameters are $r_c=5\mu \mathrm{m}, r_b=1.5\mu \mathrm{m}, {\theta }_c=1/5,L={10}^{-3}\mu \mathrm{m},\alpha =2$, and ${\theta }_b$ is found through Eq. (26).

Figure 6

A blebbing cell, adhering to a flat substrate. In all cases the full nonlinear system, Eqs. (23, 24, 25, 26, 27), (36), and (40), were solved numerically using a Newton-Raphson iteration scheme. The center shows a fully rendered three-dimensional cell adhered to a two-dimensional flat substrate, with a bleb extended to its maximum distance. The black line along the three-dimensional body illustrates the plane of symmetry normal to the substrate. The surrounding images are cross sections of a blebbing cell. (a) Initial spherical cell adhered to the substrate, which also illustrates the concentric shells of the adhesions at their resting length and full extended length. The inset illustrates a magnified section of the adhesion pad. The black solid lines illustrate size of the adhesion pad, which depends on ${\mathrm{\Theta }}_c$. (b) A cell with a bleb that was initiated at an angle $\varphi >0$, which will never touch the substrate and, thus, extends to its maximum size. The variables $\varphi ,{\rho }_c,{\theta }_c,R_b$, and ${\theta }_b$ are also presented in the image. (c) A bleb is extended at an angle $\varphi =\mathrm{\Phi }$, hence, the bleb is just touching the substrate and, so, the bleb is able once again to extend to its maximum size. (d) A bleb for which $\varphi \in [-\pi /2+{\mathrm{\Theta }}_c+{\theta }_c,\mathrm{\Phi }]$ and, thus, it is unable to grow to its maximum size, before it interacts with the substrate. Parameters are ${\rho }_c=5\mu \mathrm{m}, {\theta }_c=1/5,L={10}^{-3}\mu \mathrm{m}, \alpha =2,\kappa C=1000\mathrm{pN}/\mu {\mathrm{m}}^3,A=400\mathrm{pN}/\mu \mathrm{m}, \mu =0.5$ and, initially, $\mathrm{\Delta }P=40\mathrm{pN}/\mu {\mathrm{m}}^2$.

Figure 7

Comparing numerical and linearized approximate solutions for (a) $r_c$ and (b) $r_b$ within the region $\varphi \in [-\pi /2,\mathrm{\Phi }]$. Parameters are the same as described in the caption of Fig. 6.

Figure 8

Comparison of the probability density function for the random variable $W$ calculated approximately from ${10}^5$ observations of a simulated blebbing cell (shown as a histogram) and directly, through Eq. (57). The parameter values are $r_c=5\mu \mathrm{m}, {\theta }_c=1/5$, (a) $\mathrm{\Phi }=-\pi /4$ and (b) $\mathrm{\Phi }=0$. Note that the first column of the histogram contains the Dirac $\delta$ function contribution.

Figure 9

Comparison of the mean and standard deviation of $W$ as estimated from ${10}^5$ stochastic simulations of a simulated blebbing cell or calculated from Eqs. (58) and (59). The black solid and dashed lines represent analytically derived quantities, while the (blue) points represent the sample mean and the thin (blue) vertical lines represent one standard deviation about the mean. The parameter values are $r_c=5\mu \mathrm{m}$ and ${\theta }_c=1/5$.

Figure 10

Visualizing the relationship between the mean and maximum values of $W$. (a) Comparison of the mean and maximum of $W$, as $\mathrm{\Phi }$ is varied, using Eqs. (58) and (48). (b) Divergence rate of the mean and maximum of $W$, as $\mathrm{\Phi }$ is varied. The parameter values are the same as in Fig. 9.

Figure 11

Comparisons of the probability density functions of a cell being at a position $(x,y)$ over time, where time is measured in terms of the number of blebbing events, (a) $t=3$ blebbing events; (b) $t=6$ blebbing events; (c) $t=9$ blebbing events; (d) $t=18$ blebbing events. The filled circles illustrate a two-dimensional histogram derived from ${10}^5$ simulations. Each histogram column has a base area of $1\mu {\mathrm{m}}^2$. The shaded smooth surface is the Gaussian approximation, Eq. (63). To aid visualization, the simulations and approximation have been projected along the $x$ and $y$ directions illustrating the fit. The parameter values are $r_c=5\mu \mathrm{m}, {\theta }_c=1/5$, and $\mathrm{\Phi }=-\pi /4$. Note that the $z$ axis has been truncated at 0.01. The coloration of the surface and points is to aid visualization only.

Figure 12

Illustrating the influence of ${\theta }_c$ on (a) $R_b$ and $\widehat{w}$, produced through Eq. (29) and combining Eqs. (41) and (48), respectively; and (b) ${\theta }_b^{\text{max}}$ and $\mathrm{\Phi }$, produced through Eq. (30) and Eq. (41), respectively. (c), (d), and (e) help demonstrate the trends seen in (a) and (b) as they are cell profiles that illustrate the maximum bleb extension, hence, $r_b=R_b,w=\widehat{w},{\theta }_b={\theta }_b^{\text{max}}$, and $\varphi =\mathrm{\Phi }$. The parameter values are the same as described in the caption of Fig. 6 and in (c), (d), and (e), ${\theta }_c=\pi /16,\pi /8$, and $\pi /4$, respectively.

Figure 13

Illustrating the relationship between ${\theta }_c$ and $\langle W^2\rangle$, produced by combining Eqs. (29), (30), (41), (48), and (59). The parameter values are the same as in Fig. 6.