Local Turgor Pressure Reduction via Channel Clustering

Assumptions

Model 1 assumes a spherical cell with the membrane permeability increased in a narrow circular patch (see Fig. 1). Glycerol, the main osmolyte, is produced at a rate, constant in both time and space within the cell. The turgor pressure is determined by a balance between production and leakage. We assume azimuthal symmetry about the center of the permeable patch, and define the angle θ relative to this center. To optimize numerical convergence, we assume a smoothly varying permeability profile of the form $P\left(\theta \right)=P_a+P_b\text{exp}\left[-{\theta }^2/{\left(\Delta \theta \right)}^2\right]$. For ease of comparison with our analytic Model 2 below, we also fit this to a model with a discontinuous permeability, having the form:

The coefficients $P_{a,b}$ of the continuous permeability are chosen so that $P\left(0\right)=P_1,P\left(\pi \right)=P_2$. We then choose $\Delta \theta$ to optimize the fit to the discontinuous permeability. For conceptual clarity, we present our results below in terms of ${\theta }_0$, $P_1$, and $P_2$.

Model 2 assumes an infinite, impermeable plane, with two long-distance concentrations $C_{\text{in}},C_{\text{out}}$, and a perfectly permeable hole of radius a (we take $C_{\text{out}}=0$ in our results presented below). These approximations allow for an analytic solution. Model 2 is a good approximation to Model 1 in certain parameter ranges, because the size of the permeable patch is small relative to the size of the cell, and the permeability in the middle of the patch is much larger than that far from the patch. In relating Models 1 and 2, we find that a good match is obtained by taking $a=1.3{\theta }_{0}R$, where R is the cell radius.

In both models, the varying concentrations of glycerol cause a spatially varying hydrostatic pressure because of the constraint of spatially constant chemical potential. The effects of the pressure on the water and glycerol diffusion coefficients, and possible pressure gradient terms in the glycerol current, are assumed small and are neglected.

Equations

In Model 1, the steady-state distribution of glycerol $C\left(\mathbf{r}\right)$ is governed by the reaction-diffusion equations:

where r is distance from the center of the cell, D is the diffusion coefficient, and α is the rate of glycerol production. The boundary conditions, taking into account the definition of the permeability as the ratio of current density to concentration discontinuity, are

where $\Delta C={\text{lim}}_{\mathit{ϵ}\to 0}\lfloor C\left(R-\mathit{ϵ},\theta \right)-C\left(R+\mathit{ϵ},\theta \right)\rfloor$ is the concentration discontinuity.

Because Model 2 assumes asymptotic concentrations rather than a production rate, there is no α term, and we have

The boundary conditions are

Here z is the coordinate perpendicular to the plane (at z = 0), and ρ is the radial coordinate, measured from the z axis going through the center of the hole.

Solution of Model 1

Model 1 has the general solution

Here we have expanded $P\left(\theta \right)$ as $P\left(\theta \right)={\sum }_{l=0}^{\infty }{A}_l{P}_l\left(\text{cos}\theta \right)$, where $P_l\left(\text{cos}\theta \right)$ is the Legendre polynomial of order l and $A_l=\left[\left(2l+1\right)/2\right]\int d\theta \text{cos}\left(\theta \right)P\left(\theta \right)P_l\left(\text{cos}\theta \right)$; the relationship between the coefficients of $P_l$ for $r>R$ and $r<R$ results from the continuity of $\partial C/\partial r$ at R. Integrating the first of expressions in Eq. 7 with $P_k\left(\text{cos}\theta \right)$ and employing the orthogonality properties of the Legendre polynomials, we see that the $C_l$ in Eqs. 10 and 11 satisfy

where (in terms of the Wigner-3j symbol) (26):

The 3j-symbols are nonzero only when $J=l+m+k$ is even, and were calculated from the identity

where $g=J/2$.

To calculate the coefficients numerically, we approximate the infinite-dimensional system with one of finite dimension by truncating the summation at L coefficients, resulting in an L × L system (600 by 600, unless otherwise mentioned) to be solved for the coefficients $C_l$. The $A_l$ were obtained as numerical integrals. We find that the solution to the truncated system converges as L increases. As a measure of the convergence of the solution we used the variation of the concentration difference across the membrane. The choice of L = 600 is justified by Fig. 2, which indicates that the solution converges for ∼L > 150 for the baseline parameters; for the slowest converging cases, the error is <1%. Thus we see that the approximate solution of Eqs. 4 and 5 provides an accurate solution to the problem.

Solution of Model 2

The solution to Model 2 uses the solution for the case $C_{\text{in}}=-C_{\text{out}}$ given in Crank (27). We summarize the method used to obtain the solution. The approach uses the correction to the solution that would hold if no hole were present. The sign of the correction is opposite on each side. We thus have

where $\tilde{C}\left(z,\rho \right)$ is the correction for $z>0$.

At z = 0, the boundary conditions (Eq. 9) require that $C\left(\mathbf{r}\right)$ be continuous inside the hole, while outside the hole $\partial C/\partial z=0$. In terms of the correction, this becomes $\tilde{C}\left(0^{+},\rho \right)=\left(1/2\right)\left(C_{\text{in}}-C_{\text{out}}\right)$ for $\rho <a,{\left(\frac{\partial \tilde{C}\left(z,\rho \right)}{\partial z}\right)}_{z=0}=0$ for $\rho >a$.

Equation 8 for $\tilde{C}$ is solved in the Supporting Material by an expansion in Bessel functions. The result, for $C_{\text{out}}=0$, is

where $\lambda =\left(1/a^2\right)\left({\rho }^2+z^2-a^2\right)$ and $\zeta ={\left({\lambda }^2+\left(4z^2/a^2\right)\right)}^{1/2}$.

This model is expected to reproduce the spherical-model results in certain limits. In the spherical model there is a finite discontinuity across the hole, while in the planar model, the concentration is continuous. In the spherical model, there is some membrane current outside the patch, whereas in the planar model there is none. Finally, the spherical model has a curved geometry, while that in the planar model is flat. We thus expect the planar model to approximate the spherical one well only if $P_{1}a/D$ is large, $P_{2}R/D$ is small, and $a/R$ is small. The second and third conditions hold for all our parameter sets, but the first holds only at the upper end of the values of $P_1$ that we have treated.

Model 1—spherical cell

The parameter values are given in Table 1. We vary ${\theta }_0$ and $P_1$ because their values are poorly known. In choosing our range of values for ${\theta }_0$, we note that the radius of the invagination is ∼30 nm, corresponding to $\theta =0.01$, while the radius of the ribosome-free zone around the invagination indicating the presence of F-actin is ∼100 nm (12), corresponding to $\theta =0.033$. The value of ${\theta }_0$ is likely not much smaller than the radius of the invagination region, because the turgor pressure needs to be reduced over the whole invagination region. However, portions of the permeable region outside the invagination region can contribute to the turgor pressure reduction in the invagination region. Thus we used a range of ${\theta }_0$ values between 0.006 and 0.03. As a baseline value, we use ${\theta }_0=0.02$.

We base our range of values of $P_1$ on $P_c$, the single-channel permeability of a channel protein such as Fps1 in S. cerevisiae, which can transport glycerol to the outside of the cell. We are not aware of experimental estimates of the single-channel permeability of Fps1 or other glyceroporins. The value of $P_c$ was thus determined as being in the middle of the range of water permeability values quoted in Yang and Verkman (28), noting that glycerol permeability values are similar to water permeability values (29). To estimate $P_1$ from $P_c$, we assume a constant density of channels. The permeability is taken to be

where $n_c=N_c/A_p$ is the channel number density per unit area and $A_c$ is the area per channel. The term with the minus sign is the total area taken up by the channels, and is included to avoid double-counting of the permeability. The second relation holds approximately because $P_1\gg P_2$. Our range of $P_1$ values corresponds to values of the center-to-center channel distance ranging down to 10 nm; the baseline value of $4\times {10}^{-2}$ cm/s corresponds to a spacing of 16 nm. Because the measured global permeability values (30) include the effects of channels such as Fps1, and $P_2$ corresponds to regions of the membrane where channels are depleted by clustering into patches in other regions, we choose our value of $P_2$ at the lower end of the values quoted in Smith (31).

In choosing D, we assumed a cytoplasmic value 10 times smaller than the water value given in Lide (32). This value is somewhat uncertain, and we treat the effects of varying D in the Discussion. The production rate α was obtained by imposing the constraint $k_{\text{B}}T{N}_A\Delta C\left(\pi \right)=\text{Π}$. We note that because all concentrations in the model are proportional to α, changes in α will not affect fractional reductions in turgor pressure.

Fig. 3 a shows the typical behavior of the concentration as a function of radius, going from inside the cell, through the cell membrane $r=3\mu \text{m}$, and continuing outside the cell. Each of the curves shows a dropoff in C approaching the membrane from the inside, with a corresponding feature outside the membrane. This leads to a reduction in the discontinuity $\Delta C$, which causes a corresponding reduction in the turgor pressure $\text{Π}$ according to Eq. 2. As seen in the figure, the effect is smaller when the patch width ${\theta }_0$ is reduced and larger when $P_1$ is increased. Fig. 3 b shows $\Delta C$ as function of angle relative to the patch center. The maximum reduction in $\Delta C$, 31%, occurs at the center $\theta =0$ of the patch, but the effect is also pronounced for $\theta >{\theta }_0$; at $\theta ={\theta }_0$ the reduction is 22%. Thus significant local reductions in turgor pressure occur for our baseline parameter values.

Fig. 4 a shows the concentration difference reduction as a function of $P_1$ for a range of values of ${\theta }_0$. All the curves have a similar shape, with the reduction becoming larger as ${\theta }_0$ increases.

Scaling form for concentration difference reduction

The model has two small dimensionless parameters: $P_{2}R/D=1.1\times {10}^{-3}$, and $a/R=1.3{\theta }_0\approx 0.008-0.04$. The smallness of these parameters suggests that the ratio of $\Delta C$ at the center of the patch to $\Delta C$ at the other end of the cell, $\delta c=\left|\Delta C\left(0\right)/\Delta C\left(\pi \right)\right|$, might depend only weakly on $P_2$ and R, provided that a is kept constant. As mentioned above, there is no dependence on α, because increasing α by a certain factor changes both $\Delta C\left(0\right)$ and $\Delta C\left(\pi \right)$ by the same factor. If $P_2$ and R have only weak effects on $\delta c$, then $\delta c$ will be determined almost entirely by $P_1$, a, and D. There is only one dimensionless combination of these parameters that could enter $\delta c$, so a relation of the form

should hold to high accuracy, where f is an unknown function. This relation makes two predictions: (1) That $\delta c$ is independent of $P_2$; we have tested this prediction by increasing and decreasing $P_2$ by a factor of two, and as expected, found changes of only 0.5%. (2) If $P_1^{50\text{\%}}$ is the value of $P_1$ required to achieve a 50% reduction in $\Delta C$, so that $\delta c=0.5$, then $P_1^{50\text{\%}}\propto 1/a$. Our test of this prediction is given in Fig. 4 b, which shows that the relationship

holds to an accuracy of 2%.

Model 2—planar geometry

Fig. 5 a shows results for Model 2. It is seen that the concentration drops over a range of values of z comparable to the hole width a. We compared these results to those for the spherical model by identifying a with $1.3{\theta }_{0}R$, z with $r-R$, $C_{\text{in}}$ with the concentration at the center of the cell, and taking $C_{\text{out}}=0$; the prefactor in a optimized the agreement between the models. We also used a large value of $P_1$, to optimize agreement with the planar case. The agreement is quite good, with the main difference being that the discontinuity $\Delta C$ vanishes in the planar model while it has a small nonzero value in the spherical model.

Analytic theory of patch current and reduction in concentration difference

For the purposes of finding a simple formula for the reduction of $\Delta C\left(0\right)$ in Model 1, we first develop an expression for the total current I through the patch. We calculate I in Model 2 as a starting point. Inspection of Eq. 16 shows that if $C\left(\overrightarrow{r},\rho \right)$ is the solution for a hole of radius a, then $C\left(\overrightarrow{r}/\eta ,\rho /\eta \right)$ is the solution for a hole of radius $\eta a$. In other words, the spatial scale of the solution is proportional to a. Because the current density in the hole is proportional to $D\partial C/\partial z\propto a^{-1}$, and the area of the hole is proportional to $a^2$, it follows that $I\propto a$. Inserting the additional proportionality to $C_{\text{in}}$ and D, we obtain

where ν = 1.98 is obtained numerically from the solution given in Eq. 16.

We next calculate the current I through the permeable patch in Model 1 in two limiting cases. We assume that the permeability has the constant value $P_1$ over a disk of radius a, and the value $P_2$ outside that disk, and use $\Delta C\left(\pi \right)$ as a measure of the bulk concentration discontinuity. In the limit of low permeability $P_{1}a/D\ll 1$, $\Delta C$ should be constant over the interior surface of the cell, because diffusion will eliminate concentration gradients. Then $I=P_1\pi a^2\Delta C\left(\pi \right)$. In the high-permeability limit $P_{1}a/D\gg 1$, the planar model is applicable, and we use Eq. 20. We identify $C_{\text{in}}$ in Eq. 20 with $\Delta C\left(\pi \right)$, because the concentration outside the cell at $\theta =\pi$ is very small, so that $C\simeq \Delta C$. Thus $I=\nu Da\Delta C\left(\pi \right)$.

To construct a formula for intermediate values of $P_{1}a/D$, we view osmolytes flowing through the patch as experiencing two resistances in series, corresponding to diffusion to the patch and subsequent permeation of the patch. Each process is defined by a resistance $\Delta C\left(\pi \right)/I$. The resistance to diffusion is the total resistance for very large $P_1$, which from the discussion above is $1/\nu Da$. Similarly the resistance to permeation is obtained by taking $P_{1}a/D$ very small, and equals $1/P_1\pi a^2$. Adding the resistances, we obtain $\Delta C\left(\pi \right)/I=1/\nu Da+1/P_1\pi a^2$, or

Finally, I is related to $\Delta \overline{C}$, the average of $\Delta C$ over the permeable patch. By the definition of the permeability, $I=\pi a^2{P}_1\Delta \overline{C}$. Putting this into Eq. 21, we have

As seen in Fig. 5 b, this formula gives an excellent description of the simulation results, provided that we use a fitted value of $\nu =2.9$. It emphasizes the role of the key dimensionless parameter $P_{1}a/D$.

Extent of turgor pressure reduction in yeast

To evaluate the potential relevance of local turgor pressure reduction to endocytosis in yeast, we estimate a plausible value of $P_1$. Superresolution imaging of aquaporin clusters in brain cells (33) found an area of 36 nm² per aquaporin, corresponding to a center-to-center spacing of 6 nm. Because endocytic protein patches contain a number of other membrane proteins, we use a conservative estimate of a 10 nm center-to-center spacing, corresponding to a channel density $n_c$ of 10¹² cm⁻². Taking the single-channel permeability to be 10 × 10⁻¹³ cm⁻³ (Table 1), we find $P_1=0.1\text{cm}/\text{s}$. Above in Fig. 4 b, we found that for this value of $P_1$, a 50% reduction in the turgor pressure will occur as long as the patch radius is 45 nm or greater. Thus patches only slightly larger than the invagination size can reduce the turgor pressure significantly. A patch of 45 nm would contain ∼60 channels. This number is comparable to the counts of the coat proteins known to be important for endocytosis in budding yeast. For example, the maximum count of the linker protein Sla2 is ∼50 (34). In addition, this channel cluster size is similar in magnitude to the 35-channel aquaporin clusters previously observed in Smith and Verkman (33).

Equation 22 shows that the turgor pressure reduction is enhanced if D is smaller than the value given in Table 1. It will also be enhanced if diffusion in the cell wall outside the membrane is slower than in the cytoplasm. Because treating a cell wall with finite thickness would be significantly more complicated than the calculations we have done, we instead have considered a model in which the diffusion coefficient is reduced uniformly outside the membrane. This corresponds to changing the boundary conditions in Eqs. 7. We find that if the outside value of the diffusion coefficient is reduced by 50%, $P_{50\text{\%}}$ is reduced by 55%.

For the turgor pressure reduction to facilitate endocytosis, it must persist when the membrane is bent inward, up to the point where a tubule forms and late-arriving curvature-generating proteins assist the process. We see no mathematically practical way of performing the calculations for an inward-bent membrane. However, we note that diffusion currents in general are much more sensitive to the maximum size of their target than to its shape. For example, the diffusion current to a perfectly absorbing cylindrically symmetric ellipsoid of semimajor axis a and semiminor axis b, with $a\gg b$, differs from the current to a perfect absorbing sphere of radius a by only a logarithmic factor of $\left(1/\text{ln}\left(2a/b\right)\right)$ (35); the current to a disk of radius a differs from that to a sphere of radius a by only a factor of $\left(2/\pi \right)$ (27). Therefore the current I from the inside of the cell through the permeable patch should depend mainly on the maximum dimension of the patch rather than its deformation. Further, $I=N_c{P}_c\Delta \overline{C}$, where $N_c$ is the number of channels, $P_c$ is the single-channel permeability, and $\Delta \overline{C}$ is the average concentration difference across the membrane over the area of the patch. If $N_c$ and $P_c$ are constant, then $\Delta \overline{C}$ will also be relatively independent of patch deformation. Thus the turgor pressure reduction will persist even when the membrane is bent inwards, until it forms a tubule.

The presence of permeable patches can also cause a substantial decrease in the bulk turgor pressure if α is unaffected by the patches. Using the maximum estimate of $P_1=0.1\text{cm}/\text{s}$, together with the corresponding patch radius of 45 nm, we find a global turgor pressure reduction of 52%. This does not affect our key results because we focus on the ratio of the turgor pressure at the permeable patch to that in other parts of the cell. Because the permeable patch is small, the measured turgor pressure corresponds to the calculated value away from the patch. The effect on the bulk turgor pressure will be reduced if $P_2$ is larger than is assumed in Table 1, because a lower fraction of the total osmolyte current will flow through the patch. It will also be reduced if D is smaller, because this reduces the required values of $P_1$ and a, according to Eq. 19.

Relation to experiments

We are not aware of experiments currently in the literature that can test the hypothesis of local turgor pressure reduction in yeast. However, the efficiency of invagination is greatly reduced in deletion mutants of the channel Fps1 (9), an effect that could increase turgor pressure either locally or globally. Fps1 localizes in membrane patches (25). To establish the extent of channel clustering at endocytic sites, one could use two-color fluorescence imaging to measure colocalization of Fps1 and with endocytosis markers such as the early scaffold protein Ede1 or the polymerized-actin proxy Abp1. Colocalization of channel patches with endocytic proteins would indicate that the turgor pressure is locally reduced around the endocytic site. Appearance of the channels before Ede1 would suggest that the channels have a function in marking the endocytic site. On the other hand, appearance of the channels after Abp1 could mean that polymerized actin is important for recruiting the channels. Such a possibility is suggested by the documented effects of polymerized actin on channel dynamics in other types of cells (36, 37, 38). If the channel patches do colocalize with endocytic sites, superresolution measurements of the distribution of channels could provide a concrete test of our hypothesis.

Previous experiments have demonstrated that the effect described here, variations of osmotic pressure difference across the membrane caused by variations in channel density, does function on larger spatial scales. For example, segregation of aquaporins and Na⁺/H⁺ channels to the leading edge of cancer cells led to osmolyte concentration gradients over the length of the cell, which in turn created internal pressure gradients that drove migration through artificial channels (39). Here we predict that gradients of osmolytes over much smaller distances, caused by higher densities of channels, will lead to osmotic pressure variations over these smaller distances.

Application to plant cells

The high turgor pressure in plants also poses conceptual difficulties in rationalizing the mechanobiology of endocytosis, and some early articles actually doubted the existence of the process for this reason (40, 41). Endocytosis in plants has some parallels to the process in yeast, in particular its polymerized-actin dependence (42, 43). The difficulties of overcoming high turgor pressure in plants by an actin-dependent process might thus be eased by a local turgor pressure reduction mechanism like that we discuss here for yeast.

Alternative mechanisms

Several other possible mechanisms might deal with the difficulty of generating sufficient force via actin polymerization and curvature-generating proteins. A calculation based on the total number of type-I myosins in fission yeast (10), assuming that each one can generate 2 pN of force, suggested that they could overcome the turgor pressure, but only over a smaller region of radius 10 nm. The geometric arrangement of myosins that would supply forces with the correct orientation and distribution is also not known. One possibility is that the myosins turn actin filaments into “superpolymerizers” by 1) binding the filaments to the membrane and 2) moving to the growing tip of the actin filament after each polymerization event, keeping it far enough from the membrane that new subunits can be added freely (44). In this hypothesis, myosins act somewhat like formins, which drive processive elongation of actin filaments while bound to the membrane. An alternate possibility is one analogous to tubulation of giant unilamellar vesicles driven by type-1 myosin (45), where the driving force for tubulation came from myosin motion along a preexisting actin scaffold. We are not aware of measurements of the force-generating capability of yeast type-1 myosins, but mammalian type-1 myosins are able to generate up to 5 pN of force per molecule (46). It is also possible that there are growing actin filaments that were not detected in the previous filament counts based on Arp2/3 complex-mediated branching (20, 23). These could, for example, be generated by domains on the actin nucleator Las-17 that act independently of the Arp2/3 complex (47).

It has also been suggested that polymerization of actin filaments along the membrane could generate much larger forces in a wedgelike geometry (48). The transmission of such forces to the membrane would require a very stiff actin gel (4). There is no proof that a gel with such properties exists, but the high cross-linking fraction in endocytic actin patches could make them much stiffer than other forms of polymerized actin.

Another possibility is that curvature-generating proteins could be providing a very large fraction, say 90%, of the force overcoming turgor pressure. However, if this were the case, one would expect that the large heterogeneity between cells, and between endocytic events in the same cell, would often lead to the curvature-generating proteins exerting sufficient force to drive the process by themselves, in the absence of actin polymerization. Correlative electron-microscopy and light-microscopy experiments (12) have shown that this is not the case: substantial membrane deformation seldom or never occurs without polymerized actin.

Finally, the global turgor pressure could be overestimated. The turgor pressure is typically measured indirectly (15), by exposing the cell to increasing solute concentrations and measuring the volume change. The turgor pressure is proportional to the minimal external solute concentration increase that causes the cell wall to collapse, on the assumption that the membrane collapses at zero turgor pressure. These methods could overestimate the turgor pressure because cells react to increasing external solute concentration by increasing the internal solute concentration, thus requiring higher external concentrations for collapse (25). Osmoregulation could also confound mechanically based turgor pressure measurements (16), if internal osmolytes are upregulated in response to force opposing cell growth. Direct methods using an indentor that penetrates the cell wall are impractical in yeast because of the small system size. But experiments in plant cells (14) have shown good consistency between indirect and direct methods. Therefore, for the global turgor pressure in yeast to be strongly overestimated, osmoregulation mechanisms in yeast would have to be more efficient than in plants.