On Some Aspects of the Thermodynamic of Membrane Recycling Mediated by Fluid Phase Endocytosis: Evaluation of Published Data and Perspectives

Abstract

The theoretical and experimental description of fluid phase endocytosis (FPE) requires an asymmetry in phospholipid number between the two leaflets of the cell membrane, which provides the biomechanical torque needed to generate membrane budding. Although the motor force behind FPE is defined, its kinetic has yet to be determined. Based on a body of evidences suggesting that the mean surface tension is unlikely to be involved in endocytosis we decided to determine whether the cytosolic hydrostatic pressure could be involved, by considering a constant energy exchanged between the cytosol and the cell membrane. The theory is compared to existing experimental data obtained from FPE kinetic studies in living cells where altered phospholipid asymmetry or changes in the extracellular osmotic pressure have been investigated. The model demonstrates that FPE is dependent on the influx and efflux of vesicular volumes (i.e. vesicular volumes recycling) rather than the membrane tension of cells. We conclude that: (i) a relationship exists between membrane lipid number asymmetry and resting cytosolic pressure and (ii) the validity of Laplace’s law is limited to cells incubated in a definite hypotonic regime. Finally, we discuss how the model could help clarifying elusive observations obtained from different fields and including: (a) the non-canonical shuttling of aquaporin in cells, (b) the relationship between high blood pressure and inflammation and (c) the mechanosensitivity of the sodium/proton exchanger.

Appendices

Appendix 1

Recapitulation of Membrane Budding Linked to Phospholipid Number Asymmetry at the Vesicular Scale

In this model, and as stated earlier, the mean surface tension will be neglected given that there is no need for a vesicle to pull adjacent membrane to be formed if membrane recycling (i.e. endocytosis and exocytosis) is considered. As demonstrated previously [17], the energy of a membrane patch budding of radius R _V, of thickness h, and of neutral surface area S can be described by the sum of two terms. The first term describes the force driving membrane budding, which is associated with the endogenous difference in surface tensions between the leaflets of the plasma membrane, and linked to the phospholipid number asymmetry:

$$ \delta \Upphi_{\Updelta \sigma } = {\frac{{h\Updelta \sigma_{0} }}{{2R_{\text{V}} }}}\delta S, $$

(22)

where Δσ₀ = −2KδN ₀/N ₀ is the difference in surface tensions, K the elastic modulus of leaflets, δN ₀ the number of phospholipids in excess in the inner leaflet (compared to the outer one) and N ₀ the average phospholipid number in each leaflet.

The second term is the bending energy, which corresponds to the resistance to membrane curvature. However, as the local biomechanical moment related to the membrane tension as a magnitude smaller than the bending modulus ($\sigma_{0} h^{2} /k_{\text{c}} \sim 0.1$, h ∼ 5 nm) and accordingly, the surface tension can be ignored in the expression of the bending energy:

$$ \delta \Upphi_{\text{c}} = 2k_{\text{c}} {\frac{\delta S}{{R_{\text{V}}^{{^{2} }} }}}, $$

(23)

where k _c is the membrane bending modulus. As a result, the competition between these two energies (Eqs. 22, 23), provides an optimal budding radius:

$$ \left( {R_{\text{V}} } \right)_{0} = - {\frac{{8k_{\text{c}} }}{{h\Updelta \sigma_{0} }}}\sim \left( {{\frac{{\delta N_{0} }}{{N_{0} }}}} \right)^{ - 1} . $$

(24)

Equation 24 demonstrates that the vesicle radius is inversely proportional to the phospholipid number asymmetry. Finally, inserting Eq. 24 into the sum of Eqs. 23 and 22 allows the classical determination of the energy released by the membrane after completion of the formation of a vesicle:

$$ \left( {\Upphi_{\text{V}} } \right)_{0} = \left. {\Upphi_{{\Updelta \sigma_{0} }} } \right|_{{R_{\text{V}} }} + \left. {\Upphi_{\text{c}} } \right|_{{R_{\text{V}} }} = - 8\pi k_{\text{c}} . $$

(25)

Equation 25 is independent of the vesicle’s size. In conclusion, this model assumes that the membrane will release membrane vesicles to cancel the difference in surface tensions stored at the cell membrane level.

Appendix 2

Second-Order Determination of the Effect of the Difference in Pressures on the Vesicles Radius

Using the assumptions described in the legend of Fig. 1a including Eq. 6b, the membrane bending (Eq. 23) and difference in surface tension (Eq. 22) energies; the expression relating any changes in the vesicle radius to the difference in pressures applied, is given by the following optimization when a vesicle is created:

$$ \delta_{{R_{\text{V}} }} \Upphi_{\text{V}} = S_{\text{V}} \delta_{{R_{\text{V}} }} \left[ {2k_{\text{c}} /R_{\text{V}}^{2} + h\Updelta \sigma_{0} /2R_{\text{V}} + \Updelta PR_{\text{V}} /3} \right] = 0. $$

(26)

In Eq. 26, $\Updelta P = P_{\text{ex}} - P_{\text{cell}}^{ 0} .$ It follows that any changes in the vesicle radius and the difference of pressure applied are related by the following equation:

$$ \Updelta P/3 - 4k_{\text{c}} /R_{\text{V}}^{3} - h\Updelta \sigma_{0} /2R_{\text{V}}^{2} = 0. $$

(27)

Using Eq. 24 and noting that $P_{\text{cell}}^{ 0} = P_{\text{ex}}^{ 0}$ for the initial isotonic pressure, Eq. 27 can be re-expressed as:

$$ \Updelta \bar{P}\bar{R}_{\text{V}}^{3} + 2\bar{R}_{\text{V}} - 2 = 0, $$

(28)

where $\bar{R}_{\text{V}} = R_{\text{V}} /(R_{\text{V}} )_{0}$ is the ratio between the vesicle radius altered by the difference in osmotic pressure and the vesicle radius in isotonic condition, and $\Updelta \bar{P} = \Updelta P/P_{\text{cell}}^{0}$. In addition, using second derivative in R _V of Eq. 26 in conjunction with Eq. 28, it follows that the regime of membrane vesiculation in hypotonic medium is permitted (i.e. $\delta_{{R_{\text{V}} }}^{2} \Upphi_{\text{V}} > 0$) so long that:

$$ - \Updelta \bar{P} = (P_{\text{cell}}^{0} - P_{\text{ex}} )/P_{\text{cell}}^{0} \le (2/3)^{3} . $$

(29)

Finally, to determine how the vesicle radius is affected when cells are incubated in hypotonic medium, a second-order development in ΔP of R _V has been determined. Replacing $\bar{R}_{\text{V}} = 1 + a\Updelta \bar{P} + b\Updelta \bar{P}^{2}$ into Eq. 28 and equating each pre-factor of $\Updelta \bar{P}$ and ΔP ² to be equal to zero lead to:

$$ \bar{R}_{\text{V}} = R_{\text{V}} /(\bar{R}_{\text{V}} )_{0} = 1 - (P_{\text{ex}} /P_{\text{cell}}^{0} - 1)/2 + 3(P_{\text{ex}} /P_{\text{cell}}^{0} - 1)^{2} /4. $$

(30)

Appendix 3

Further Justifications for the Use of Classical Hydrodynamic in the Case of Membrane Vesiculation

Using $\text{Re} = R_{\text{V}}^{2} /t_{\text{V}} \nu ,$ where R _V ∼ 50 nm [17], t _V ∼ 5 × 10⁻² s [79] and ν ∼ 10⁻⁶ m²/s the kinematic viscosity of the cytoplasm, approximated to that of water in the first instance, one finds $\text{Re} \sim 10^{ - 12} < < 1.$

In addition, the spatial and temporal scales involved during vesiculation imply that visco-elastic laws do not apply. Indeed, the visco-elastic properties of the cytosol have been investigated in living cells using endosomes (~600-nm diameter) enclosing nanomagnetic probes under a rotational magnetic field [78]. From these results, it has been demonstrated that at this scale (i.e. ~100 nm diameter), the cytosol has a fluid-like behaviour if the characteristic time scale of a given biological event, such as membrane vesiculation, is ≥10⁻² s. Thus, the use of classical hydrodynamic laws at low Re is justified for the study of FPE as t _V ∼ 5 × 10⁻² s.

This latter point is further justified as, in this regime of membrane vesiculation, the viscosity contrast between the cytosol and the membrane is low and, therefore, the membrane viscosity does not impact on FPE. This can be demonstrated as follows. The viscosity contrast modelled by Saffman and Delbruck [80] allows one to determine which viscosity parameter, i.e. water (μ _w) or membrane (μ _m), viscosity limits lipids movement. For example, considering a single lipid assumed to have a cylinder like shape (i.e. cross-section radius, a, and length, h) the viscosity contrast is given by: hμ _m/aμ _w. At the lipid scale, as μ_m ∼ 1 − 10 P [81] and μ_w ∼ 1 cP it follows $h\mu_{\text{m}} /a\mu_{\text{w}} > > 1,$ and the intramembrane viscosity effects lipids movement. Considering now the vesicular scale, the viscosity contrast becomes: $h\mu_{\text{m}} /R_{\text{V}} \xi ,$ where, R _V, is the size of the vesicle and, ξ, the viscosity encountered by the patch of membrane budding. As ξ ∼ 10 P [78] and R _V ∼ 50 nm (i.e. R _V/h ∼ 10), it follows $h\mu_{\text{m}} /R_{\text{V}} \xi < < 1.$ Therefore, the membrane budding is not dependent on the membrane viscosity but on the cytosolic viscosity.