Pattern formation by curvature-inducing proteins on spherical membranes

We will adopt a continuum elastic model of a closed membrane, which might represent a model vesicle or a biological cell, and study the stability of spherical shapes to perturbations in the presence of curvature-inducing proteins that decorate the membrane. The shape of a quasi-spherical membrane can be written in spherical coordinates as ${\bf{R}}(\theta ,\phi )=R[1+u(\theta ,\phi )]\hat{{\bf{r}}}$, where R is the radius of the unperturbed sphere, $u(\theta ,\phi )$ is a scalar function that describes the deviations from the sphere, and $\hat{{\bf{r}}}$ is the radial unit vector, see figure 2. The distribution of proteins on the membrane can be described in a similar way, with the surface number density $\rho (\theta ,\phi )={\rho }_{0}[1+\psi (\theta ,\phi )]$. Here, ${\rho }_{0}$ is the average protein number density, i.e. ${\rho }_{0}=N/4\pi {R}^{2}$ if N is the total number of proteins on the membrane, and the function $\psi (\theta ,\phi )$ represents the deviations from a homogeneous distribution of proteins.

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We will assume that each protein covers a patch of membrane of area a₀, and imposes a spontaneous curvature C_p on the membrane, see figure 1. The bending free energy of the membrane can then be written within the spontaneous curvature model [20–23] as

$$\begin{eqnarray}&&{F}_{{\rm{b}}}=\displaystyle \frac{\kappa }{2}\int {\rm{d}}A\ [{C}^{2}-2{C}_{{\rm{p}}}\rho {a}_{0}C],\end{eqnarray} \tag{ 1 }$$

where κ is the bending rigidity of the membrane, and C is the local membrane curvature, with $C={C}_{1}+{C}_{2}$, where C₁ and C₂ are the two principal curvatures. The second term inside the integral represents the simplest possible coupling between protein density and local curvature. It can also be interpreted as a position-dependent spontaneous curvature ${C}_{0}(\theta ,\phi )\equiv {C}_{{\rm{p}}}\rho (\theta ,\phi ){a}_{0}$, which varies from ${C}_{0}=0$ in the absence of proteins, with $\rho =0$, to ${C}_{0}={C}_{{\rm{p}}}$ for full coverage of proteins, with $\rho =1/{a}_{0}$. The local membrane curvature $C(\theta ,\phi )$ can be written explicitly as a function of $u(\theta ,\phi )$, as described in [24].

Besides the bending contributions to the free energy, we need to take into account the entropic contributions due to the mixing and density fluctuations of the proteins. To lowest order, this contribution to the free energy can be incorporated as

$$\begin{eqnarray}&&{F}_{{\rm{d}}}=\displaystyle \frac{1}{2\chi }\int {\rm{d}}A\ [{\xi }^{2}{({\rm{\nabla }}\psi )}^{2}+{\psi }^{2}].\end{eqnarray} \tag{ 2 }$$

Here, χ and ξ are the compressibility and the correlation length of the protein density fluctuations, respectively. The first term in the integral penalises the creation of interfaces between high protein density and low protein density regions, whereas the second term penalises deviations from a homogeneous protein distribution.

We will also consider the effect of the tethering of the membrane to a cell wall or actomyosin cortex, by including a harmonic confinement potential of the form

$$\begin{eqnarray}&&{F}_{{\rm{h}}}=\displaystyle \frac{{k}_{\mathrm{te}}{R}^{2}}{2}\int {\rm{d}}A\ {u}^{2},\end{eqnarray} \tag{ 3 }$$

where k_te is an effective spring constant per unit area, which in general may include contributions from specific interactions (i.e. proteins that directly link the membrane to the wall/cortex) as well as non-specific interactions such as steric repulsion, van der Waals attraction or electrostatic attraction/repulsion. Within this effective description, the cell wall/cortex is taken to be spherical and rigid (i.e. much more rigid than the membrane), and k_te penalises deviations of the membrane position from the (optimal) equilibrium membrane-wall distance.

Lastly, we consider the possibility that the membrane is connected to a membrane area reservoir at constant membrane tension. A constant membrane tension is typical of biological cells, [25, 26] and can be mimicked in model vesicle systems by the use of micropipette aspiration. The contribution of a membrane tension σ to the free energy is

$$\begin{eqnarray}&&{F}_{{\rm{t}}}=\sigma \int {\rm{d}}A.\end{eqnarray} \tag{ 4 }$$

The total free energy can finally be written as the sum of these four contributions, with

$$\begin{eqnarray}&&F={F}_{{\rm{b}}}+{F}_{{\rm{d}}}+{F}_{{\rm{h}}}+{F}_{{\rm{t}}}=\int {\rm{d}}A\ { \mathcal F }\end{eqnarray} \tag{ 5 }$$

with the free energy density

$$\begin{eqnarray}&&{ \mathcal F }\equiv \displaystyle \frac{\kappa }{2}[{C}^{2}-2{C}_{{\rm{p}}}\rho {a}_{0}C]+\displaystyle \frac{1}{2\chi }[{\xi }^{2}{({\rm{\nabla }}\psi )}^{2}+{\psi }^{2}]+\displaystyle \frac{{{kR}}^{2}}{2}{u}^{2}+\sigma .\end{eqnarray} \tag{ 6 }$$

In addition, we will explicitly impose constraints on the volume enclosed by the membrane (representing osmotic balance), so that

$$\begin{eqnarray}&&\displaystyle \frac{4\pi {R}^{3}}{3}=\int {\rm{d}}V\end{eqnarray} \tag{ 7 }$$

as well as on the total number of proteins N on the membrane, so that

$$\begin{eqnarray}&&N={\rho }_{0}4\pi {R}^{2}=\int {\rm{d}}A\ \rho \end{eqnarray} \tag{ 8 }$$

at all times.

The effective force exerted on the membrane in the radial direction will be balanced by a frictional force, leading to a dynamical equation for the shape of the membrane as a function of time t

$$\begin{eqnarray}&&{\partial }_{t}u(\theta ,\phi ,t)=-{L}_{u}\displaystyle \frac{\delta F}{\delta u(\theta ,\phi )},\end{eqnarray} \tag{ 9 }$$

where L_u is a transport coefficient corresponding to the membrane mobility.

On the other hand, the dynamical equation describing the diffusion of the proteins on the membrane can be written in the form of a continuity equation

$$\begin{eqnarray}&&{\partial }_{t}\psi (\theta ,\phi ,t)+{\rm{\nabla }}\cdot {\bf{J}}=0\end{eqnarray} \tag{ 10 }$$

with a current density ${\bf{J}}=-{L}_{\psi }{\rm{\nabla }}\mu$, where L_ψ is another transport coefficient and $\mu =\delta { \mathcal F }/\delta \psi (\theta ,\phi )$ is the chemical potential. Putting all together, the dynamical equation for the protein density becomes

$$\begin{eqnarray}&&{\partial }_{t}\psi (\theta ,\phi ,t)={L}_{\psi }{{\rm{\nabla }}}^{2}\left(\displaystyle \frac{\delta F}{\delta \psi (\theta ,\phi )}\right).\end{eqnarray} \tag{ 11 }$$

The Laplacian operator on a sphere can be written as ${{\rm{\nabla }}}^{2}\equiv -\tfrac{1}{{R}^{2}}{\hat{L}}^{2}$, with the operator

$$\begin{eqnarray}&&-{\hat{L}}^{2}\equiv \displaystyle \frac{1}{{\rm{\sin }}\theta }{\partial }_{\theta }({\rm{\sin }}\theta {\partial }_{\theta })+\displaystyle \frac{1}{{{\rm{\sin }}}^{2}\theta }{\partial }_{\phi }^{2}.\end{eqnarray} \tag{ 12 }$$

This operator is diagonal in the basis of spherical harmonics ${Y}_{{\ell }m}(\theta ,\phi )$. In particular, it satisfies

$$\begin{eqnarray}&&{\hat{L}}^{2}{Y}_{{\ell }m}(\theta ,\phi )={\ell }({\ell }+1){Y}_{{\ell }m}(\theta ,\phi ).\end{eqnarray} \tag{ 13 }$$

To leading order in u and ψ, and taking into account the constraints (7) and (8) on the enclosed volume and total number of proteins on the membrane, we can write equations (9) and (11) as

$$\begin{eqnarray}&&{\partial }_{t}u(\theta ,\phi ,t)=-{L}_{u}\left[\left(\displaystyle \frac{\kappa }{{R}^{2}}{\hat{L}}^{2}+\sigma \right)({\hat{L}}^{2}-2)u+{k}_{\mathrm{te}}{R}^{2}u+\displaystyle \frac{\kappa {C}_{{\rm{p}}}{a}_{0}{\rho }_{0}}{R}({\hat{L}}^{2}-2)\psi \right]\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}&&{\partial }_{t}\psi (\theta ,\phi ,t)=-\displaystyle \frac{{L}_{\psi }}{{R}^{2}}\left[\displaystyle \frac{1}{\chi }{\hat{L}}^{2}\psi +\displaystyle \frac{1}{\chi }\left(\displaystyle \frac{{\xi }^{2}}{{R}^{2}}\right){\hat{L}}^{4}\psi +\displaystyle \frac{\kappa {C}_{{\rm{p}}}{a}_{0}{\rho }_{0}}{R}{\hat{L}}^{2}({\hat{L}}^{2}-2)u\right].\end{eqnarray} \tag{ 15 }$$

We can write the solutions $u(\theta ,\phi ,t)$ and $\psi (\theta ,\phi ,t)$ as a sum of spherical harmonics, which provide a complete set of orthogonal functions on the sphere, so that

$$\begin{eqnarray}&&u(\theta ,\phi ,t)=\displaystyle \sum _{{\ell },m}{u}_{{\ell }m}(t){Y}_{{\ell }m}(\theta ,\phi )\,\,\mathrm{and}\,\,\psi (\theta ,\phi ,t)=\displaystyle \sum _{{\ell },m}{\psi }_{{\ell }m}(t){Y}_{{\ell }m}(\theta ,\phi ),\end{eqnarray} \tag{ 16 }$$

where ${u}_{{\ell }m}$ and ${\psi }_{{\ell }m}$ are the amplitudes of the corresponding modes, and we have ${\ell }=0,1,2...$ and $| m| \leqslant {\ell }$. However, the constraints (7) and (8) on the enclosed volume and total number of proteins on the membrane imply that the zero-amplitudes u₀₀ and ${\psi }_{00}$ cannot be varied independently. Explicitly imposing these constraints results in expressions for u₀₀ and ${\psi }_{00}$ as a function of the squared amplitudes of all modes

$$\begin{eqnarray}&&\sqrt{4\pi }{u}_{00}=-\displaystyle \sum _{{\ell },m}{u}_{{\ell }m}^{2},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\sqrt{4\pi }{\psi }_{00}=\displaystyle \frac{1}{2}\displaystyle \sum _{{\ell },m}[2-{\ell }({\ell }+1)]{u}_{{\ell }m}^{2}-2\displaystyle \sum _{{\ell },m}{u}_{{\ell }m}{\psi }_{{\ell }m}.\end{eqnarray} \tag{ 18 }$$

Equations (17) and (18) imply that u₀₀ and ${\psi }_{00}$ are a function of the higher-order amplitudes, and furthermore, that they are of quadratic order (they are equal to a sum of ${u}_{{\ell }m}^{2}$ and ${u}_{{\ell }m}{\psi }_{{\ell }m}$ terms). For this reason, the u₀₀ and ${\psi }_{00}$ terms are negligible to linear order, and we can rewrite (16) as

$$\begin{eqnarray}&&u(\theta ,\phi ,t)\simeq \displaystyle \sum _{{\ell }\geqslant 1,m}{u}_{{\ell }m}(t){Y}_{{\ell }m}(\theta ,\phi )\,\,\mathrm{and}\,\,\psi (\theta ,\phi ,t)\simeq \displaystyle \sum _{{\ell }\geqslant 1,m}{\psi }_{{\ell }m}(t){Y}_{{\ell }m}(\theta ,\phi ).\end{eqnarray} \tag{ 19 }$$

Inserting (19) into (14) and (15), we can rewrite the dynamical equations as separate equations for each of the ${\ell }\geqslant 1$ modes. Introducing a rescaled time variable

$$\begin{eqnarray}&&\tau \equiv \left(\displaystyle \frac{\kappa {L}_{u}}{{R}^{2}}\right)t\end{eqnarray} \tag{ 20 }$$

as well as dimensionless parameters

$$\begin{eqnarray}&&K\equiv \displaystyle \frac{{k}_{\mathrm{te}}{R}^{4}}{\kappa },\,T\equiv \displaystyle \frac{\sigma {R}^{2}}{\kappa },\,P\equiv \displaystyle \frac{{\xi }^{2}}{{R}^{2}},\,M\equiv \displaystyle \frac{{L}_{\psi }}{\kappa {L}_{u}\chi },\,S\equiv {\rho }_{0}{a}_{0}{C}_{{\rm{p}}}R,\,\mathrm{and}\,B\equiv \displaystyle \frac{\kappa \chi }{{R}^{2}}\end{eqnarray} \tag{ 21 }$$

the equations become

$$\begin{eqnarray}&&-{\partial }_{\tau }{u}_{{\ell }m}=\{[{\ell }({\ell }+1)+T]({\ell }+2)({\ell }-1)+K\}{u}_{{\ell }m}+S({\ell }+2)({\ell }-1){\psi }_{{\ell }m}\end{eqnarray} \tag{ 22 }$$

and

$$\begin{eqnarray}&&-(1/M){\partial }_{\tau }{\psi }_{{\ell }m}={BS}{\ell }({\ell }+1)({\ell }+2)({\ell }-1){u}_{{\ell }m}+{\ell }({\ell }+1)[1+P{\ell }({\ell }+1)]{\psi }_{{\ell }m}.\end{eqnarray} \tag{ 23 }$$

The solutions to (22) and (23) will have the form

$$\begin{eqnarray}&&{u}_{{\ell }m}(\tau )={u}_{{\ell }m}(0){{\rm{e}}}^{\lambda \tau },\,\,{\psi }_{{\ell }m}(\tau )={\psi }_{{\ell }m}(0){{\rm{e}}}^{\lambda \tau }.\end{eqnarray} \tag{ 24 }$$

Inserting these solutions back into (22) and (23), and setting the determinant of the coefficients to zero, we can obtain an equation for the growth rates λ of the characteristic modes of the system, which reads

$$\begin{eqnarray}&&{\lambda }^{2}+b\lambda +c=0\end{eqnarray} \tag{ 25 }$$

with coefficients

$$\begin{eqnarray}&&b\equiv K-2T+{\ell }({\ell }+1)[M+T-2+{\ell }({\ell }+1)({MP}+1)]\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&c\equiv M{\ell }({\ell }+1)\{[K+({\ell }({\ell }+1)+T)({\ell }+2)({\ell }-1)][1+P{\ell }({\ell }+1)]-W{({\ell }+2)}^{2}{({\ell }-1)}^{2}\}\end{eqnarray} \tag{ 27 }$$

where we have defined the parameter

$$\begin{eqnarray}&&W\equiv {{BS}}^{2}=\kappa \chi {\rho }_{0}^{2}{a}_{0}^{2}{C}_{{\rm{p}}}^{2}.\end{eqnarray} \tag{ 28 }$$

The two characteristic modes of the system given by the solutions to (25) can finally be written as

$$\begin{eqnarray}&&{\lambda }_{\pm }=\displaystyle \frac{1}{2}[-K+2T-{\ell }({\ell }+1)[M+T-2+{\ell }({\ell }+1)({MP}+1)]]\\ &&\ \ \pm \displaystyle \frac{1}{2}\sqrt{{[K-2T-{\ell }({\ell }+1)[M-T+2+{\ell }({\ell }+1)({MP}-1)]]}^{2}+4{MW}{\ell }({\ell }+1){({\ell }+2)}^{2}{({\ell }-1)}^{2}}.\end{eqnarray} \tag{ 29 }$$

Because b in (25) always satisfies $b\gt 0$ for all modes with ${\ell }\geqslant 1$, we know that the amplitude with the smaller value, ${\lambda }_{-}$, is always negative for all ℓ-modes. On the other hand, the larger one, ${\lambda }_{+}$, might be positive or negative depending on the ℓ-mode and on the values of the parameters W, K, T, P, and M. It is also worth noting that the stability analysis is independent of the value of m of the spherical harmonics. This ultimately arises from the fact that the eigenvalues of the Laplacian of a spherical harmonic are independent of its m-value.

The physical significance of the five dimensionless parameters is the following. The parameter W represents the protein-induced spontaneous curvature, and increases both with the average density ${\rho }_{0}$ of proteins on the membrane and with the characteristic spontaneous curvature C_p of these proteins. The parameter K represents the strength of the confinement of the membrane by its interaction with the rigid cell wall/cortex. The parameter T represents the magnitude of the membrane tension. The parameter P compares the correlation length of the protein density fluctuations to the size of the cell or vesicle. Given that correlation lengths are typically of the order of nanometers whereas cell or vesicle sizes are of the order of micrometers, P will generally be small, and will decrease or increase with increasing or decreasing cell/vesicle size, respectively. Finally, the parameter M compares the typical timescale of the changes in membrane shape (${R}^{2}/{L}_{u}\kappa$) to that of changes in protein distribution ($\chi {R}^{2}/{L}_{\psi }$). Importantly, we note that all five dimensionless parameters are always positive.

We have shown above that pattern formation in a spherical membrane containing curvature-inducing proteins is controlled by the four dimensionless parameters W, K, T and P, which represent the number of curvature-inducing proteins on the membrane, the strength of the membrane tethering to the cell wall/cortex, the membrane tension, and the correlation length of protein fluctuations, respectively. An important question is then: what are the typical values of these parameters in real systems, and to what extent can they be controlled by a biological cell, or tuned in experiments with model vesicles?

The parameter to which the system is most sensitive is W, see figures 3–5. Even if K, T, and P vary across many orders of magnitude, the critical value W above which pattern formation occurs always stays in the proximity of ${W}^{* }\approx 1$, except in the extreme case of very high K (or T) and $P\approx 1$ simultaneously, see figure 4. Going back to the definition of W in terms of the dimensionful system parameters in (28), we see that the requirement $W\geqslant {W}^{* }\approx 1$ implies that $\kappa \chi {\rho }_{0}^{2}{a}_{0}^{2}{C}_{{\rm{p}}}^{2}\gtrsim 1$. Here, κ is the bending rigidity of the membrane, χ is the compressibility of the protein density fluctuations, ${\rho }_{0}$ is the average density of curvature-inducing proteins on the membrane, a₀ is the lateral area of a single protein, and C_p is the protein spontaneous curvature. Furthermore, in the limit of low protein density ${\rho }_{0}{a}_{0}\ll 1$, the compressibility χ can be approximated by $\chi \approx 1/({k}_{{\rm{B}}}T{\rho }_{0})$, [27] where k_B is Boltzmann's constant and T is the temperature. In this limit, the requirement for pattern formation thus becomes $(\kappa /{k}_{{\rm{B}}}T){\rho }_{0}{a}_{0}^{2}{C}_{{\rm{p}}}^{2}\gtrsim 1$. In general, the protein spontaneous curvature C_p will be of the order of the (inverse) characteristic length of the protein $\sqrt{{a}_{0}}$, so that we can take ${C}_{{\rm{p}}}^{2}{a}_{0}\approx 1$. We finally conclude that pattern formation typically occurs for average protein densities satisfying

$$\begin{eqnarray}&&{\rho }_{0}{a}_{0}\gtrsim {k}_{{\rm{B}}}T/\kappa .\end{eqnarray} \tag{ 35 }$$

Here, ${\rho }_{0}{a}_{0}$ is simply the dimensionless area fraction of membrane covered by the protein. Typical values of the bending rigidity of membranes range from 10 to $100\,{k}_{{\rm{B}}}T$, leading to a critical protein coverage of the order of ${\rho }_{0}{a}_{0}\sim 0.01$–0.1. Importantly, the range obtained self-consistently validates the low protein density assumption made above. Furthermore, such coverages are within the range achievable both in biological cells as well as in model vesicles. In this picture, a biological cell could up- or down-regulate the expression of the curvature-inducing protein in order to switch between patterned and non-patterned conformations, see figure 4. Furthermore, the concentration of curvature-inducing proteins on the membrane could be directly controlled in experiments with model vesicles.

Let us now turn to the dimensionless parameters $K\equiv {k}_{\mathrm{te}}{R}^{4}/\kappa$ and $T\equiv \sigma {R}^{2}/\kappa$, which both act as geometric constraints on the membrane: K represents the confinement of the membrane due to its interaction/tethering to the cell wall or cortex, whereas T represents the membrane tension, which acts to minimise the cell membrane area. It is interesting to note that, while model membranes such as Giant Unilamellar Vesicles show clear shape fluctuations due to thermal excitation of bending modes, [28] eukaryotic cells or bacteria do not show such fluctuations. The latter is an indication that, in such systems, membrane confinement and tension must overpower bending, and consequently that in these systems $K\gg 1$ and/or $T\gg 1$, as can be confirmed by the quantitative estimates that follow.

Estimates of the confinement strength k_te of biological membranes due to the interaction with the corresponding cell wall or cortex do not abound in the literature. In [29], the density of membrane-cortex linkers in eukaryotic cells was estimated to be around ${\rho }_{\mathrm{link}}\approx 100\,\mu {{\rm{m}}}^{-2}$, whereas the spring constant of a typical linker was estimated to be ${k}_{\mathrm{link}}\approx {10}^{-4}$ N m⁻¹. The effective tethering strength should go as ${k}_{\mathrm{te}}\approx {\rho }_{\mathrm{link}}{k}_{\mathrm{link}}$, leading to the estimate ${k}_{\mathrm{te}}\approx {10}^{10}$ J m⁻⁴ $\approx \,2.5\times {10}^{6}\,{k}_{{\rm{B}}}T\,{\mu {\rm{m}}}^{-4}$. Considering a typical range of bending rigidities $\kappa =10$–$100\,{k}_{{\rm{B}}}T$, and a typical cell radius ranging from from $R=1$ to $10\,\mu {\rm{m}}$, we find values for K ranging from $2.5\times {10}^{4}$ up to $2.5\times {10}^{9}$. Cells could then actively switch between patterned and non-patterned conformations by down- or up-regulating the concentration of linker proteins between the plasma membrane and the cell wall/cortex, see figure 5(a).

The typical tension of cellular membranes, on the other hand, has been extensively measured for different cell types, and can range from $\sigma =3$ pN μm⁻¹ for epithelial cells up to about $\sigma =300$ pN μm⁻¹ for keratocytes [25, 26]. Using the range $\sigma =3$–300 pN μm⁻¹ for the membrane tension, together with the estimates $\kappa =10$–$100{k}_{{\rm{B}}}T$ for the bending rigidity of the membrane and R = 1–$10\,\mu {\rm{m}}$ for a typical cell radius, we obtain values of T ranging from 7 to $7\times {10}^{5}$. Cells can actively regulate their own tension in order to maintain homeostasis [25]. In this way, cells could switch between patterned and non-patterned conformations by actively decreasing or increasing the tension of their plasma membrane, see figure 5(b). Furthermore, triggering of pattern formation via a decrease in membrane tension could be explored in experiments using model Giant Unilamellar Vesicles aspirated by micropipettes, which allows direct experimental control over the membrane tension.

Let us finally examine the dimensionless parameter P, defined as $P\equiv {\xi }^{2}/{R}^{2}$, where ξ is the correlation length of the protein density fluctuations and R is the radius of the cell or vesicle. The correlation length ξ is a measure of the distance at which proteins or protein clusters can sense each other, typically via membrane-mediated interactions in the absence of other long-ranged interactions. Previous work [15–18] has shown that the typical length scale of membrane-mediated interactions is the size of the curvature-inducing element itself, so that we can use an estimate of $\xi =10$–20 nm. On the other hand, the radius of cells or cellular compartments, as well as of model vesicles, can range between R = 100 nm and $10\mu {\rm{m}}$. With this, we find a range of $P\sim {10}^{-6}$–10⁻². In this range of values with $P\ll 1$, as described above, pattern formation is tighly controlled by the number of curvature-inducing proteins, with an instability occurring as soon as $W\gtrsim 1$. Moreover, the value of P directly controls the ℓ-order of the first unstable during pattern formation, and as a consequence controls the typical size of the protein-rich, highly-curved domains. The consequences of this fact are discussed in the following section.

4.2.1. Cell division

Cell division requires polarisation of the cell, so that the spherical symmetry of the cell is broken, leading to two identifiable poles as well as an equatorial line. Spontaneous pattern formation via an instability due to the presence of curvature-inducing proteins, as described here, provides a simple mechanism for such a symmetry breaking. This occurs for the mode ${\ell }=2$ of the instability, see figure 6(a), which can be the first unstable mode as long as $P\gt 1/18\simeq 0.056$, as determined from (34) and displayed on figures 4 and 5. For a typical cell size of $R=1\,\mu {\rm{m}}$, such values of P would correspond to a correlation length ξ for protein density fluctuations larger than 230 nm. This value appears too high for a typical protein, given that the correlation length is expected to be of the order of the protein size, i.e. a couple of tens of nanometers. It could, however, be a plausible value for protein clusters, composed of a few tens of proteins with lateral sizes of the order of 100 nm. If such clusters arose by a separate mechanism, a curvature-instability such as the one described here could lead to cell polarisation.

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Several proteins, many of them related to cell division, are known to preferentially localise at the poles of bacterial membranes [30–34]. We note, however, that the generic mechanism proposed here is distinct and unrelated to the well-studied Min system, which serves to localise the FtsZ protein ring in rod-shaped bacteria [10, 11]. The Min system involves both membrane bound as well as cytosolic components, and locates the bacterial equator via an oscillatory mechanism. The mechanism proposed here might however explain why FtsZ proteins can spontaneously self-assemble on vesicles, even in the absence of the Min system [8, 9]. In eukaryotic cells, cell division is mediated by the cytoskeleton, in particular by the mitotic spindle and the cleavage furrow. The mechanism underlying the initial positioning of this cell division apparatus is however not well understood at the molecular level [35, 36].

Even if the mechanism described here did not play a direct role in cell division, it provides a generic pathway for symmetry breaking and the initiation of division of spherical membranes into two equally sized daughters, using only a minimal number of ingredients. As such, it could serve as a plausible mechanism for the division of protocells, as well as of synthetic cells in bottom-up synthetic biology [37–39].

4.2.2. Large-scale protein organisation

4.2.3. Nano-sized membrane rafts

To summarise, we have explored in detail pattern formation in spherical membranes that contain curvature-inducing proteins. Pattern formation arises from the interplay between membrane curvature energy, protein density fluctuations, and geometric constraints such as membrane tension and confinement forces due to the tethering of the membrane to the cell wall/cortex. We have shown that pattern formation in this system is controlled by just four dimensionless parameters, W, K, T, and P, defined in (21) and (28). These parameters represent the number of curvature-induced proteins on the membrane, the confinement of the membrane due to the cell wall/cortex, the membrane tension, and the correlation length of protein density fluctuations, respectively. In most circumstances, pattern formation is expected to occur as the result of an increase in the average surface density of proteins (i.e. the total number of proteins on the membrane surface), or of a relaxation of the geometric constraints on the membrane due to membrane tension or membrane tethering to the cell wall/cortex. The patterns that arise consist of protein-rich, highly-curved domains that alternate with protein-poor, weakly-curved domains. We hypothesise that spontaneous pattern formation as described here might be exploited by biological cells as a way to regulate their geometry in situations that require spatial organisation, symmetry breaking or polarisation of the cell, using only a minimal number of ingredients.