Mechanosensitivity of Membrane Budding and Trafficking

Abstract

Intracellular compartments continually exchange material transported by small vesicles or tubules, which are formed in the membrane of the donor compartments and eventually fuse with the membrane of the receptor compartments. The formation and fission of a membrane bud giving rise to a new object and the fusion are controlled to some extent by the mechanical properties of the membranes, in particular their tension. In this chapter, we review the different mechanisms of vesicle and tubule budding and analyze the influence of the membrane tension on these processes using basic considerations of thermodynamics and mechanics. In any case, vesicle and tubule production can be impaired at high enough tension. Next, we discuss the influence of tension on membrane fusion, which is a less understood problem. Finally, since the release/absorption of vesicles or tubules should affect the tension of the donor/receptor, we speculate about the possible regulatory role of the membrane tension on intracellular trafficking and compartments stability.

Appendices

Appendix 1: Shape Equations for Axisymmetric Membrane

The shape of a membrane with cylindrical symmetry can be characterized by the functions r(s), z(s), and ψ(s), where s is the arc length along the shape contour in a plane at a fixed azimuthal angle. r and z are the usual cylindrical coordinates and ψ is the angle between the radial and the tangent vectors, see Fig. 2.

In the most general case in which the membrane undergoes a pressure difference between each side and a force pulling along the z-axis at the contour boundaries, the shape of the membrane minimizes the free energy,

$$\displaystyle \begin{aligned} {\mathcal{G}}={\mathcal{F}}_{\mathrm{m}} - pV - fL \ , {} \end{aligned} $$

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The second term is the energy cost associated with the volume change with $V=\pi \int _0^{s_1}r^2\sin \psi ds$ the volume enclosed by the membrane and p the pressure difference across the membrane. The last term is included when $L=z(0)-z(s_1)=\int _0^{s_1}\sin \psi ds$ is fixed; f is then the force exerted by the membrane at s = 0 and s = s ₁ in the z direction.

In order to minimize ${\mathcal {G}}$ with respect to r(s) and ψ(s) accounting for the constrain (2), one has to introduce a Lagrange multiplier γ(s) and minimize the functional,

$$\displaystyle \begin{aligned} \begin{array}{rcl} S[r(s),\psi(s)]=\frac{\mathcal{G}}{2\pi\kappa}+\int_0^{s_1} \gamma(s)(\dot r-\cos\psi)ds=\int_0^{s_1}{\mathcal{L}}(\psi,\dot\psi,r,\dot r)ds \end{array} \end{aligned} $$

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with,

$$\displaystyle \begin{aligned} {\mathcal{L}}=\frac{1}{2}\left(\dot\psi +\frac{\sin\psi}{r}-C_0\right)^2r+\frac{\sigma}{\kappa} r- \frac{p}{2\kappa}r^2\sin\psi-\frac{f}{2\pi\kappa}\sin\psi+\gamma (\dot r-\cos\psi) \end{aligned} $$

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The condition δS = 0 leads to the Euler–Lagrange equations $\frac {\partial {\mathcal {L}}}{\partial r}-\frac {d}{ds}\frac {\partial {\mathcal {L}}}{\partial \dot r}=0$ and $\frac {\partial {\mathcal {L}}}{\partial \psi }-\frac {d}{ds}\frac {\partial {\mathcal {L}}}{\partial \dot \psi }=0$,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \ddot\psi &\displaystyle =&\displaystyle - \frac{\dot \psi\cos\psi}{r} +\frac{\cos\psi\sin\psi}{r^2}-\frac{p}{2\kappa}r\cos\psi + \frac{\gamma\sin\psi}{r}-\frac{f}{2\pi\kappa}\frac{\cos\psi}{r}\ , \\ \dot\gamma&\displaystyle =&\displaystyle \frac{1}{2}\left(\dot\psi - C_0\right)^2-\frac{1}{2}\frac{\sin^2\psi}{r^2}+\frac{\sigma}{\kappa}-\frac{p}{\kappa}r\sin\psi\ , {} \end{array} \end{aligned} $$

(51)

and to the boundary conditions at s = 0 and s = s ₁,

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \psi~{\mathrm{fixed}}~~{\mathrm{or}}~~\frac{\partial {\mathcal{L}}}{\partial \dot \psi}=\dot\psi+\frac{\sin\psi}{r}-C_0=0\ ,\\ &\displaystyle &\displaystyle r~{\mathrm{fixed}}~~{\mathrm{or}}~~\frac{\partial {\mathcal{L}}}{\partial \dot r}=\gamma=0 . {} \end{array} \end{aligned} $$

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Equation (51) together with Eq. (2) forms a close set of differential equations of 4th order complemented by four boundary conditions (52).

In the usual case where the contour length s ₁ is not fixed, then $H=\dot r\partial _ {\dot r}{\mathcal {L}}+\dot \psi \partial _ {\dot \psi }{\mathcal {L}}-{\mathcal {L}}=0$, which gives,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{r\dot\psi^2}{2}- \frac{r}{2}\left(\frac{\sin\psi}{r} - C_0\right)^2-\frac{\sigma}{\kappa} r +\frac{p}{2\kappa}r^2\sin\psi+\gamma\cos\psi+\frac{f}{2\pi\kappa}\sin\psi=0\ .\qquad \end{array} \end{aligned} $$

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This equation can be combined with (51) to eliminate γ and obtain a lowest order equation in ψ and r, Eq. (4).

Appendix 2: Model for Dynamical Cluster of Kinesin at Tubule Tip

The force f required to pull a tubule is usually larger than the stall force f _s ∼ 10 pN of a single motor. Several motors, localized at the tip of the tubule, then work cooperatively to extract a tube [53, 58]. Tubule formation then relies on two conditions: (1) the formation of a stable cluster of N motors at the tip, and (2) the load on each motor (f∕N assuming that the force created by the membrane is equally distributed among the motors) should be smaller than the stall force. Let’s consider the first condition. A cluster of motors at the tubule tip is sustained by an influx J _b of motors moving along the tube, and looses motors that unbind the microtubule at a rate dependent of their load, $k_{\mathrm {u}}\exp \left (\frac {fa}{Nk_{\mathrm {B}}T}\right )$, where k _u is the unbinding rate at zero load and a is the typical distance of the motor–microtubule interaction. The influx J _b depends on the density of motors on the membrane, and kinetic parameters such as the motor velocity, the binding and unbinding rates [53, 58]. The flux balance,

$$\displaystyle \begin{aligned} k_{\mathrm{u}}N\exp\left(\frac{fa}{Nk_{\mathrm{B}}T}\right)=J_{\mathrm{b}}\ , \end{aligned} $$

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determines N, the number of motors in the cluster. A stable cluster can exist only if $\frac {J_{\mathrm {b}}k_{\mathrm {B}}T}{k_{\mathrm {u}}}fa>e$ where e ≃ 2.71 is the base of natural logarithm. In this case, N is in the range $\frac {J_{\mathrm {b}}}{k_{\mathrm {u}}}<N<\frac {J_{\mathrm {b}}}{k_{\mathrm {u}}}e$. Then accounting for the second condition, f∕N < f _s, tubule extraction by the collective action of molecular motors is possible if the force f (28) exerted by the membrane is lower than a critical value,

$$\displaystyle \begin{aligned} f<f_c~~~~{\mathrm{with}}~,~~~~f_c=\left\{ \begin{array}{lll} \frac{k_{\mathrm{B}}T}{a}\frac{J_{\mathrm{b}}}{k_{\mathrm{u}}}\frac{1}{e} &{\mathrm{if}} & f_{\mathrm{s}}a>k_{\mathrm{B}}T\\ f_{\mathrm{s}}\frac{J_{\mathrm{b}}}{k_{\mathrm{u}}}\exp\left(-\frac{f_{\mathrm{s}}a}{k_{\mathrm{B}}T}\right) &{\mathrm{if }}& f_{\mathrm{s}}a<k_{\mathrm{B}}T \end{array} \right. \end{aligned} $$

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