Boundary Flow-Induced Membrane Tubulation Under Turgor Pressures

Abstract

During clathrin-mediated endocytosis in yeast cells, a small patch of flat membrane is deformed into a tubular shape. It is generally believed that the tubulation is powered by actin polymerization. However, studies based on quantitative measurement of the actin molecules suggest that they are not sufficient to produce the forces to overcome the high turgor pressure inside of the cell. In this paper, we model the membrane as a viscous 2D fluid with elasticity and study the dynamic membrane deformation powered by a boundary lipid flow under osmotic pressure. We find that in the absence pressure, the lipid flow drives the membrane into a spherical shape or a parachute shape. The shapes over time exhibit self-similarity. The presence of pressure transforms the membrane into a tubular shape that elongates almost linearly with time and the self-similarity between shapes at different times is lost. Furthermore, the width of the tube is found to scale inversely to the cubic root of the pressure, and the tension across the membrane is negative and scales to the cubic root squared of the pressure. Our results demonstrate that boundary flow powered by myosin motors, as a new way to deform the membrane, could be a supplementary mechanism to actin polymerization to drive endocytosis in yeast cells.

1. Introduction

Clathrin-mediated endocytosis (CME) is an essential process in eukaryotic cells for transporting materials, uptaking nutrients, and regulating signal transduction [1,2,3,4,5,6]. During CME, a patch of flat plasma membrane is bent inward into the cell and severed to release a vesicle. This process requires overcoming the membrane bending energy and membrane tension, which resist membrane deformation [7,8]. In walled cells, the inward deformation is additionally resisted by turgor pressure, which pushes the plasma membrane against the cell wall. The pressure strongly opposes endocytic membrane invagination [9,10]. In yeast cells, the osmotic pressure can reach 1–1.5 MPa [11,12]. High turgor pressure is speculated to shape vesicles into narrow tubular structures (∼30 nm) in yeast [1,7], while in mammalian cells with lower pressure (∼1 kPa), membrane invagination forms spherical shapes ∼100 $\mathrm{nm}$ in diameter [13].

In recent years, several theoretical and computational studies have been proposed to explain the deformation mechanisms involved in CME [14,15,16,17,18,19,20,21,22]. These studies are mostly based on assumptions related to mammalian cells, particularly under low turgor pressure conditions, and primarily focus on the effect of membrane tension on CME. Only a few studies have explored CME under high osmotic pressure conditions [17,19,22]. These studies indicate that when the osmotic pressure reaches 1 MPa, the resistance to inward deformation required for forming a clathrin-coated pit can reach up to 3000 pN [17,19]. With the assistance of clathrin, this resistance is reduced to approximately 2000 pN [22]. It is generally believed that the polymerization of actin filaments generates forces at the invagination tip through push-and-pull mechanisms, overcoming the resistance and facilitating membrane invagination [19,23,24,25,26]. However, experimental estimates suggest that no more than 200 actin filaments are involved in CME [27]. Even if all filaments contribute, each filament would need to generate a force of 10 pN, which far exceeds the experimentally observed polymerization force of approximately 1 pN per filament [28]. In other words, under high osmotic pressure, the pulling force provided solely by actin filaments is unlikely to be sufficient for the membrane to form vesicles.

Myosin is a motor protein that converts chemical energy into mechanical work, playing a crucial role in muscle contraction, cell division, and endocytosis [29,30,31,32,33]. In CME, myosin motors assemble at endocytic sites, forming a ring-like structure that connects the actin meshwork to the plasma membrane [26]. Studies have suggested that myosin provides additional mechanical force in clathrin-mediated endocytosis (CME) by anchoring to actin [23,32,33]. To address the puzzle that during CME in yeast cells, actin polymerization alone is not able to produce enough force to overcome the huge turgor pressure and bend the membrane into a tubular shape, we hypothesize that during this process, the interaction between myosin and the plasma membrane may also induce lipid flow, driving phospholipid molecules toward the endocytic site. In our study, we incorporate the factor of boundary lipid flow into our model to explore its potential role in the endocytic process.

2. Model and Methods

2.1. Geometry of the Membrane Surface

We assume the membrane surface $\mathrm{\Gamma }$ is axisymmetric around the z-axis, and is described by the vector

$$\mathbf{X}(u,\varphi ,t)=\left(r\right(u,t)cos\varphi ,r(u,t)sin\varphi ,z(u\left)\right).$$

(1)

Here, u is a parameter defined on a fixed interval $u\in [0,1]$ with $u=0$ corresponding the membrane tip and $u=1$ corresponding to the membrane base (Figure 1). We introduce two axillary variables $\psi (u,t)$ and $h(u,t)$, which are related with $r(u,t)$ and $z(u,t)$ via the equations

$$r^{\prime }=hcos\psi ,z^{\prime }=-hsin\psi .$$

(2)

Here, the prime indicates the partial derivative with respect to u. With these variables, the two principal curvatures $c_1$ and $c_2$ of the membrane surface $\mathrm{\Gamma }$ can be simplified as

$$c_1={\displaystyle \frac{sin\psi }{r}},c_2={\displaystyle \frac{{\psi }^{\prime }}{h}}.$$

(3)

We stress that our choice of the parameter u, which is defined on a fixed interval $[0,1]$, as the spatial coordinate to parameterize the membrane shape is different from the typical choice of using the arclength s as the coordinate [34,35]. As a matter of fact, the arclength s is related with u via the equation

$$s^{\prime }=h.$$

(4)

With u as the coordinate, we can formulate the membrane shape dynamics as solving a number of ordinary differential equations defined on a fixed interval with boundary conditions, i.e., a boundary value problem (BVP) (see Appendix A). There are many available numerical tools to solve the BVP efficiently. However, if using the arclength s as the coordinate, the total arclength S, i.e., the interval of the BVP, changes over time with the membrane shape. A special shooting method has to be used to solve the BVP with varied interval [34]. We therefore prefer using u as the coordinate to s.

2.2. Flow Field on the Membrane Surface

To describe the lipid flow on the membrane surface $\mathrm{\Gamma }$, we introduce the velocity field

$$\mathbf{v}=v_u{\mathbf{e}}_u+v_{\varphi }{\mathbf{e}}_{\varphi }+v_n\mathbf{n},$$

(5)

where $v_u(u,t)$ and $v_{\varphi }(u,t)$ denote the in-plane lipid flow along the meridional direction and the azimuthal direction, respectively, and $v_n(u,t)$ denotes the out-of-plane deformation of the surface. The two unit vectors ${\mathbf{e}}_u={\partial }_u\mathbf{X}/h$ and ${\mathbf{e}}_{\varphi }={\partial }_{\varphi }\mathbf{X}/r$ constitute the basis vectors of the tangential space on $\mathrm{\Gamma }$. The unit normal vector reads $\mathbf{n}={\mathbf{e}}_u\times {\mathbf{e}}_{\varphi }$. In case the membrane has axisymmetry and there is no imposed azimuthal flow at the boundary, i.e., $v_{\varphi }(u=1)=0$, the azimuthal flow $v_{\varphi }$ has been proven to be zero [36]. Therefore, for the rest of the paper, we set $v_{\varphi }=0$ and only consider the meridional flow $v_u$ and deformation velocity $v_n$.

The lipid flow field $\mathbf{v}$ is related with the membrane shape $\mathbf{X}$ via the equation

$$\mathbf{v}={\displaystyle \frac{d\mathbf{X}}{dt}},$$

(6)

where $df/dt\equiv {\partial }_{t}f+(q/h){\partial }_{u}f$ denotes the total derivative of an arbitrary (scalar or vectorial) surface function f. The different choices of q lead to distinct representations of the surface flow, with $q=0$ corresponding to the Lagrangian representation and $q=v_u$ to the Eulerian representation. Other choices of q are termed Arbitrary Lagrangian–Eulerian representation [37]. Once q is specified, the dynamics of the geometric variables can be determined from the flow field $v_u$ and $v_n$ via the equations

$$\begin{array}{ccc}\hfill {\partial }_{t}r& =& (v_u-q)cos\psi +v_{n}sin\psi ,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\partial }_{t}z& =& (q-v_u)sin\psi +v_{n}cos\psi ,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\partial }_t\psi & =& -{\displaystyle \frac{v_n^{\prime }}{h}}+(v_u-q){\displaystyle \frac{{\psi }^{\prime }}{h}},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\partial }_{t}h& =& -q^{\prime }+v_u^{\prime }+v_n{\psi }^{\prime }.\hfill \end{array}$$

(10)

The Lagrangian ($q=0$) and Eulerian ($q=v_u$) representation are subject to the drawback that the dynamics of h in Equation (10) depend on the local flow $v_u$ and $v_n$. As a result, h becomes spatially inhomogeneous and might cause mathematically singular points at certain u where $h=0$. A re-meshing technique is always needed to overcome the mathematical but not physical singularity [36]. In this paper, instead of specifying the explicit form of q, we drop the dynamic Equation (10) and impose $h^{\prime }=0$ over time, i.e., h is a constant with respect to u. This means that h is the total contour length of the membrane profile that could change over time (see Figure 1). To see that this choice of q is mathematically plausible, we take a derivative of Equation (10) with respect to u and impose $h^{\prime }=0$ to solve for q. The solution reads

$$q(u,t)=v_u(u,t)+{\int }_0^{u}d\overline{u}{v}_n(\overline{u},t){\psi }^{\prime }(\overline{u},t)+c_1\left(t\right)+c_2\left(t\right)u,$$

(11)

where $c_1\left(t\right)$ and $c_2\left(t\right)$ are integration constants to be determined from suitable boundary conditions. In practice, the choice of q in Equation (11) imposes $h^{\prime }=0$, but leads to impractical integro-differential equations. To circumvent the issue, we do not specify the expression of q but set ${\partial }_{t}r$, ${\partial }_{t}z$, and $v_u$ as independent variables and express q in the following form

$$\begin{array}{c}\hfill q=v_u-cos\psi {\partial }_{t}r+sin\psi {\partial }_{t}z,\end{array}$$

(12)

which is derived from Equations (7) and (8). We derive equations for $v_u,{\partial }_{t}r,{\partial }_{t}z$ via variational methods and use $h^{\prime }=0$ as a constraint in the variation. For a more detailed description of the method, we refer the readers to Ref. [38], where we systematically describe the mathematical basis of the method. Note that $h^{\prime }=0$ implies h is a constant with respect to u but not t. In fact, $h=h\left(t\right)$ varies over time and is treated as an unknown parameter of the boundary value problem in each iteration of the time step $\Delta t$ described in Appendix A.5.

Inspired by the experimental observation that myosin motors form a ring-like structure at the base of the endocytic pit [26], we hypothesize that myosin motors produce stresses at the boundary that pump lipid molecules into the endocytic pit, thus creating a boundary flow $v_b$. Mathematically, it is expressed as the boundary condition for the flow field

$$v_u(u=1)=v_b.$$

(13)

2.3. Variational Formulation of the Dynamic Membrane Shape Equations

We treat the membrane as an elastic 2D surface with a bending rigidity of $\kappa$, and the lipid flow as a 2D incompressible viscous fluid with a viscosity of $\eta$. The dynamic equations that govern the shape evolution of the membrane surface and the flow field on the surface are determined from the force balance equations. In case the dynamics is overdamped, the dynamic equations can be derived from a variational formulation [39]. In this paper, we adopt the variational formulation by constructing the Rayleigh functional

$$R[v_u,{\partial }_{t}r,{\partial }_{t}z,{\partial }_t\psi ,{\partial }_{t}h,{\partial }_t\alpha ,{\partial }_t\beta ,\sigma ]=D+{\displaystyle \frac{dF}{dt}}-{\displaystyle \frac{d{W}_{ext}}{dt}}+L,$$

(14)

where D denotes half of the energy dissipation rate due to internal membrane viscosity and external friction from the surroundings, $dF/dt$ denotes the elastic energy change rate of the membrane, $d{W}_{ext}/dt$ denotes the work per unit time exerted by the external forces ${\mathbf{f}}^{ext}$, and L denotes the incompressibility constraint imposed by the Lagrangian multiplier $\sigma$ and the geometric constraint (2) imposed by the Lagrangian multiplier $\alpha$ and $\beta$. The detailed expression for D, $dF/dt$, $d{W}_{ext}/dt$, and L are provided in Appendix A.

The variation of the Rayleigh functional $\delta R$ against small perturbations leads to a number of ordinary differential equations and the associated boundary conditions, together forming a well-defined boundary value problem. We numerically solve the problem with the MATLAB solver bvp5c (MATLAB R2020b (version 9.9)) and obtain the shape evolution of the membrane $r(u,t)$ and $z(u,t)$, the lipid flow field $v_u(u,t)$, as well as the membrane tension $\sigma (u,t)$. The detailed description of the boundary value problem are provided in Appendix A.

2.4. Non-Dimensionalization and Choice of Parameters

We fix the base radius $R_b$, which is defined as the radial coordinate at the base $R_b\equiv r(u=1)$ and rescale all the length in units of $R_b$ and all the time in units of the characteristic time $\tau =\eta R_b^2/\kappa$. They together determine a characteristic velocity $v_0=R_b/\tau =\kappa /\left(\eta R_b\right)$, which is used to non-dimensionalize all the velocities. In addition, we express the membrane tension in units of ${\sigma }_0=\kappa /R_b^2$ and the turgor pressure in units of $p_0=\kappa /R_b^3$. For a typical choice of the parameters that are relevant for the clathrin coated pit during CME, $\kappa =4.1\times {10}^3\mathrm{pN}·\mathrm{nm}$ [40], $\eta ={10}^{-1}\mathrm{pN}·\mathrm{s}/\mathrm{nm}$ [41], and $R_b=30\mathrm{nm}$ [7], the corresponding time unit $\tau =22\mathrm{ms}$, and the velocity unit $v_0=1.37\mathrm{\mu }\mathrm{m}/\mathrm{s}$. The characteristic values for the membrane tension and the turgor pressure are ${\sigma }_0=4.6\mathrm{pN}/\mathrm{nm}$ and $p_0=0.15\mathrm{pN}/{\mathrm{nm}}^2$, respectively.

3. Results

3.1. Different Boundary Flow Velocities Induce Various Vesicle Morphologies

We first investigate the evolution of the membrane shape under different boundary flow velocities $v_b$ in the absence of turgor pressure (Figure 2). We start the simulation with a nearly flat membrane. The height of the membrane increases gradually over time as a result of the boundary flow, and the rate of increase slows down progressively (Figure 2a). At a certain time, the initially flat membrane becomes $\mathrm{\Omega }$-shaped and develops a neck. The width of the neck $W_n$ tends to stabilize over time, and under higher flow velocities, the stabilization occurs more rapidly (Figure 2b).

When the boundary flow is slow (small $v_b$), the membrane grows into a spherical shape, similar to the vesicles commonly found in mammalian cells during CME. Under rapid boundary flow (large $v_b$), the membrane exhibits a parachute shape (Figure 2c,d). As time evolves, the shape of the membrane scales proportionally. By normalizing the membrane shape with respect to the total arc length, the membrane shapes at different times and under different flow velocities almost coincide with each other (Figure 2c,d, Insets).

The in-plane flow $v_u$ within the membrane is non-uniform, with its maximum always occurring near $u=1$ (Figure A1). By comparing the curves of the radial coordinate $r\left(u\right)$ and the flow velocity $v_u\left(u\right)$, we find that the maximum in-plane flow is always located at the neck of the membrane with a local minimum of r, regardless of the boundary flow velocity and the time points (Figure 2e,f). The Lagrangian multiplier $\sigma \left(u\right)$ that imposes the incompressibility condition and serves as the membrane tension is close to zero for most part of the membrane, except near the base where the membrane is subject to highly compressive stress with negative $\sigma$ (Figure 2g,h). When plotting the tension $\sigma$ against the rescaled arclength u at different times, tensions at later times are almost overlapped. The effect is more pronounced for rapid boundary flows.

3.2. Pressure Makes a Dramatic Difference in Membrane Morphology and Growth Dynamics

Since walled cells, including yeast cells, often exhibit significant osmotic pressure that hinders CME, in this section we fix the boundary flow velocity $v_b$ at a relatively large value $v_b/v_0=-1$ and explore the effect of pressure on membrane morphology (Figure 3). In the presence of pressure, the height of the membrane increases linearly over time, and the rate of increase is faster with higher pressure (Figure 3a). This is different from the nonlinear growth of the membrane height in the absence of pressure (Figure 2a). As time evolves, the membrane also develops a neck and the neck width $W_n$ finally reaches a plateau that weakly depends on the magnitude of the pressure (Figure 3b).

Another difference made by the pressure is that the parachute-shaped membrane in the pressure-free condition is compressed into a tubular shape in the presence of pressure (Figure 3d). Furthermore, the lateral growth of the membrane is inhibited by the pressure such that the radius of the tube remains almost constant from top to bottom over time. Therefore, the scaling behavior of the membrane shape over time is lost. When normalizing the membrane shapes at different times with respect to the total arc length $h\left(t\right)$ for various pressures (Figure A3), it is found that the scaling behavior reserves up to $p/p_0=-0.02$ when the boundary flow $v_b/v_0=-1$. At $p/p_0=-0.2$, the rescaled membrane shapes no longer overlap with each other.

The magnitude of the in-plane flow $v_u$ in the presence of pressure still has its maximum at the neck (Figure 3e,f). However, different from the gradual increase of $v_u$ up to a position near the neck from top to bottom in the pressure-free condition, the magnitude of $v_u$ away from the tip quickly reaches a plateau and has a big drop near the membrane neck under pressure (Figure A2). The zero membrane tension $\sigma$ far from the base in the pressure-free condition becomes a negative value when pressure is present, which implies that the membrane is subject to compressive stresses (Figure 3g,h).

3.3. The Scaling Behavior Between the Width of the Tubular Membrane and the Pressure

As we have found in the previous section that the presence of pressure transforms the membrane from a parachute shape into a tubular shape (Figure 3c,d), and the radius of the tube is almost a constant from top to bottom and over time, we investigate the scaling behavior between the width of the tube W and the pressure p. Here, we define $W\equiv r_{\max }$ as the widest radial coordinate of the membrane. When looking at the temporal evolution of the tube width W, it is found that W increases at first and gradually reaches a plateau. The time to reach the plateau is reduced with increasing pressure (Figure 4a,b). Under larger pressure, the tube width W becomes narrower. If W is rescaled with the characteristic length

$$R_{\mathrm{p}}={\left({\displaystyle \frac{\kappa }{2p}}\right)}^{1/3},$$

(15)

$W/R_{\mathrm{p}}$ for different pressures p collapse to the same value when time is long enough (Figure 4a,b). Since $R_p\propto p^{-1/3}$, we conclude that the final membrane width W is inversely proportional to the cubic root of the pressure, i.e., $W\propto p^{-1/3}$. This conclusion is independent of the choice of the boundary flow $v_b$ (Figure A4a,b).

3.4. The Scaling Behavior Between the Membrane Tension and the Pressure

In the presence of pressure, the membrane tension away from the base assumes a negative value (Figure 3g,h). We now investigate how the pressure alters the tension $\sigma$ at the membrane tip, i.e., $\sigma (u=0)$. The temporal evolution of $\sigma (u=0)$ is shown in Figure 4c. When the membrane is nearly flat at the beginning of the simulation, the tension $\sigma (u=0)$ is large and negative. As times evolves, the tension finally relaxes to a less negative value and becomes stable (Figure 4c). Higher pressures lead to more negative stable tension values. Furthermore, a simple dimensional analysis of the membrane tension suggests that $\sigma \propto \kappa /W^2$. Since the tube radius $W\propto p^{-1/3}$, we conjecture that $\sigma \propto p^{2/3}$. When normalizing the tension with ${\sigma }_p={\left(4\kappa \right)}^{1/3}{p}^{2/3}$, we find that its dynamic evolution at different pressures indeed collapse to the same value over time (Figure 4d). We conclude that the stable tension value across the membrane in the presence of pressure is proportional to $p^{2/3}$. This conclusion is independent of the choice of the boundary flow $v_b$ (Figure A4c,d)

4. Discussion

Yeast cells have been widely used as a model system to study CME due to its simple genetic manipulation and fast growth. Clathrin-coated pits in yeast cells exhibit a tubular shape [7], differing from the spherical-shaped pit in mammalian cells [13]. There have been extensive theoretical studies on modeling CME in yeast cells [16,17,18,22]. Most studies have predominantly focused on actin polymerization as the primary force-generating mechanism for membrane invagination. Dmitrieff and Nedelec [17] proposed that actin polymerization generates localized forces to counteract turgor pressure in yeast cells, while Walani et al. [16] emphasized the synergistic role of BAR domains and actin in promoting vesicle growth. Hassinger et al. [18] primarily considered the influence of membrane tension on vesicle formation. The deformation of membrane is driven by the spontaneous curvature induced by clathrin proteins, as well as forces exerted by actin polymerization. However, the role of turgor pressure was neglected in their model. Notably, Ma and Berro [22] highlighted the importance of clathrin coating in reducing membrane bending energy during CME in yeast cells. Their model suggests that clathrin lattices lower the force threshold for actin-driven invagination. However, the force threshold is still up to $2000\mathrm{pN}$, far beyond the force that can be generated by actin polymerization, given the estimation of the number of actin filaments during CME in yeast cells from quantitative experiments [25,27].

In this paper, we have shown the possibility of driving membrane tubulation with boundary flows of lipids alone. To sustain such a boundary lipid flow, a compressive stress is needed at the base of the membrane. Super-resolution studies of the protein organization during CME in yeast cells suggests that myosin I motors are clustered near the base of the membrane and form a ring-like structure [26]. They serve as an anchor that connects the membrane with the actin assembly [32]. The mechanical properties of myosin I motors are suitable for a power-generating machine that possibly pumps lipid flows into the endocytic membrane [33]. To sustain the growth of a tubular-shaped membrane against turgor pressures, the boundary tension ${\sigma }_b\equiv \sigma (u=1)$, which is defined as the tension at the base, is about 10 ${\sigma }_0$ (Figure 3h). Given the tension unit ${\sigma }_0=4.6\mathrm{pN}/\mathrm{nm}$, the total force needed to drive the boundary flow is about $2\pi R_b{\sigma }_b=8670\mathrm{pN}$, which is a huge cost. Therefore, boundary flow alone cannot be the main mechanism that drives membrane tubulation during CME in yeast cells and likely plays a supplementary role in addition to the actin polymerization.

We have neglected in this study the effect of clathrin molecules and BAR proteins that could generate membrane curvatures, as well as actin polymerization that could generate forces. They are the major players of CME, and how these proteins interact with the boundary flow will be an interesting topic. In particular, investigation on how the boundary flow along with clathrin and BAR proteins helps reduce the actin polymerization force needed for endocytosis will be our future work.

5. Conclusions

In this paper, we have investigated the dynamic process of membrane shape evolution driven by boundary lipid flow under different turgor pressures. The results show that even in the absence of external forces, a membrane shape with a narrow neck can form solely due to boundary lipid flow, with low flow velocities inducing a spherical shape that closely resembles those found in mammalian cells. Under turgor pressure, the membrane is compressed into tubular shapes whose height grows linearly with time. The maximum flow velocity inside the membrane always occurs at the neck region. Higher turgor pressure leads to a reduction in membrane width, and the final stable width W of the membrane is inversely proportional to the cubic root of the turgor pressure, i.e., ($W\propto p^{-1/3}$). Furthermore, the entire membrane develops strongly negative tension in the presence of pressure and is found to be proportional to $p^{2/3}$. Our results demonstrate that myosin-powered boundary flow at the base of the endocytic pit could be a supplemental mechanism to actin polymerization to help drive invagination of membrane against turgor pressure during CME in yeast cells.