Cell protrusions and tethers: a unified approach

Abstract

Low pulling forces applied locally to cell surface membranes produce viscoelastic cell surface protrusions. As the force increases, the membrane can locally separate from the cytoskeleton and a tether forms. Tethers can grow to great lengths exceeding the cell diameter. The protrusion-to-tether transition is known as the crossover. Here we propose a unified approach to protrusions and tethers providing, to our knowledge, new insights into their biomechanics. We derive a necessary and sufficient condition for a crossover to occur, a formula for predicting the crossover time, conditions for a tether to establish a dynamic equilibrium (characterized by constant nonzero pulling force and tether extension rate), a general formula for the tether material after crossover, and a general modeling method for tether pulling experiments. We introduce two general protrusion parameters, the spring constant and effective viscosity, valid before and after crossover. Their first estimates for neutrophils are 50 pN μm(-1) and 9 pN s μm(-1), respectively. The tether elongation after crossover is described as elongation of a viscoelastic-like material with a nonlinearly decaying spring (NLDs-viscoelastic material). Our model correctly describes the results of the published protrusion and tether pulling experiments, suggesting that it is universally applicable to such experiments.

Figure 1

Tether development under a constant pulling force F, before (black) and after (gray in print/red online) crossover, and tether crossover extensions. (a) Illustration of a cellular tether forming under F = 45 pN. Based on modeling an average neutrophil tether, the crossover occurs at t = 0.33 s initiating the tether's dynamic equilibrium state. The equilibrium tether radius/cell radius ratio of ∼0.007 (21) is not captured by the drawing. (b) Constant force tether extension network, showing the modeled tether extension L for different constant pulling forces F (for F = 5 pN to F = 115 pN in increments of 10 pN). The curve (green online) corresponding to the threshold force, F_th, defines the upper boundary for the region where a crossover cannot occur. (c) The tether crossover extension L_cr|F for the pulling process under constant force of F, shown as a function of F. The quantity F_c is the critical force at which L_cr|F reaches zero. The figure is based on the neutrophil parameters listed in Table 1.

Figure 2

Modeled tether extension rate $\dot{L}$ as a function of corresponding (i.e., present at the same time) pulling force F for a constant F, or constant $\dot{L}$, or constant $\dot{F}$, as indicated, compared with the set W (force-extension rate points satisfying Eq. 2). The number near W indicates the tether's crossover time. The circle indicates the tether's dynamic equilibrium state. In the case of a constant $\dot{F}$, a dynamic equilibrium state cannot be established. The parameters are as in Table 1.

Figure 3

Modeled tether extension and elastic stiffness, before (black) and after (gray in print/red online) crossover, for the constant loading rate pulling process. (a) Constant loading rate tether extension network, showing the tether extension L for different constant loading rates $\dot{F}$ (for $\dot{F}=10\text{pN}{\text{s}}^{-1}$ to $\dot{F}=110\text{pN}{\text{s}}^{-1}$ in increments of 10 pN s⁻¹). (b) The effective spring function σ_eff(t, σ) representing the instantaneous stiffness of the tether elastic component for three constant loading rates $\dot{F}$, as indicated. The parameters are as in Table 1.

Figure 4

Schematic representation of the tether material. The unit is composed of a viscous component (of effective viscosity η_eff) represented by a dashpot, and an elastic/nonlinearly decaying (NLD)-elastic component (of effective spring function σ_eff(t, σ), where σ is the tether spring constant) represented by a spring/NLD spring (an initially linear spring, which becomes nonlinear at crossover and decays nonlinearly as increasing F approaches the critical force F_c). F(t) is the pulling force, L(t) is the tether extension, and $\widehat{L}\left(t\right)$ is the crossover extension for the pulling process under constant force of F(t), i.e., $\widehat{L}\left(t\right)=L_{\text{cr}|F\left(t\right)}$. The parameters are as in Table 1.

Figure 5

Modeled tether extension L, before (black) and after (gray in print/red online) crossover, for two cases (as indicated) of constant pulling force for the system, F_sy, in the experiment of Shao et al. (4). The extension functions are overlaid on the experimental data of Shao et al. (4) for two individual tethers. The functions are based on r_c = 4.5 μm and the other parameters as in Table 1.

Figure 6

Pulling and plateau force data for the experiment of Evans et al. (9) and Heinrich et al. (10). (a) Modeled pulling force F, before (black) and after (gray in print/red online) crossover, for three cases (as indicated) of constant extension rate for the system, ${\dot{L}}_{\text{sy}}$. With time, each curve approaches a plateau force F_plateau. (b) Modeled $\left(F_{\text{plateau}},{\dot{L}}_{\text{sy}}\right)$, points which make the set W, compared to the experimental $\left(F_{\text{plateau}},{\dot{L}}_{\text{sy}}\right)$ data of Heinrich et al. (10) shown as solid diamonds. The gray curve represents the ${\dot{L}}_{\text{sy}}$-versus-F_plateau function proposed by Heinrich et al. (10) according to the indicated formula (where F_plateau is given in pN and ${\dot{L}}_{\text{sy}}$ is given in μm s⁻¹). The diagrams in the figure are based on r_c = 4.3 μm, κ_com = κ₁κ₂/(κ₁ + κ₂), where κ₁ = 500 pN μm⁻¹ and κ₂ = 800 pN μm⁻¹, and the other parameters are as in Table 1.