Oligomerization of peripheral membrane proteins provides tunable control of cell surface polarity

Abstract

Asymmetric distributions of peripheral membrane proteins define cell polarity across all kingdoms of life. Non-linear positive feedback on membrane binding is essential to amplify and stabilize these asymmetries, but how specific molecular sources of non-linearity shape polarization dynamics remains poorly understood. Here we show that the ability to oligomerize, which is common to many peripheral membrane proteins, can play a profound role in shaping polarization dynamics in simple feedback circuits. We show that size-dependent binding avidity and mobility of membrane-bound oligomers endow polarity circuits with several key properties. Size-dependent membrane binding avidity confers a form of positive feedback on the accumulation of oligomer subunits. Although insufficient by itself, this sharply reduces the amount of additional feedback required for spontaneous emergence and stable maintenance of polarized states. Size-dependent oligomer mobility makes symmetry breaking and stable polarity more robust with respect to variation in subunit diffusivities and cell sizes, and slows the approach to a final stable spatial distribution, allowing cells to “remember” polarity boundaries imposed by transient external cues. Together, these findings reveal how oligomerization of peripheral membrane proteins can provide powerful and highly tunable sources of non-linear feedback in biochemical circuits that govern cell surface polarity. Given its prevalence and widespread involvement in cell polarity, we speculate that self-oligomerization may have provided an accessible path to evolving simple polarity circuits.

Significance

Cell polarity is often defined by the asymmetric enrichment of specific polarity proteins at cell membranes. Many polarity proteins form small oligomers at the cell membrane, but the general consequences of this behavior for cell polarity have not been explored. Here, we show that that size-dependent oligomer dissociation and mobility can greatly enhance the ability of simple feedback circuits to form and dynamically stabilize asymmetric distributions of polarity proteins. Given the widespread tendency for peripheral membrane proteins to oligomerize, and the many ways in which positive feedback can arise in biochemical systems, the combination of oligomerization with a small amount of additional positive feedback may provide a general mechanism for polarizing a wide variety of unrelated cell types.

Introduction

The ability of cells to polarize is essential for a wide range of biological functions, including asymmetric cell division, polarized growth and secretion, cell migration, and multicellular tissue morphogenesis. For most cells, functional polarity is defined by asymmetric distributions of specific molecules or molecular activities. In many cells, these asymmetries are formed by molecules that exchange dynamically between a rapidly diffusing cytoplasmic pool and binding sites at the plasma membrane where they undergo slower diffusion that can be further hindered by interactions with a submembrane cytoskeleton (1,2,3), and where they interact to promote or inhibit one another’s binding or activity (3,4,5,6,7,8). These mutual interactions define biochemical feedback circuits, which form and stabilize asymmetric distributions of their component molecules. In the past several decades, a relatively small number of such circuits have been shown to underlie the formation and stabilization of polarity in a wide range of cellular and organismal contexts. Examples include circuits formed by small GTPases such as RhoA and Cdc42, their activators, inhibitors and effectors (9,10), the conserved PAR polarity circuit (11), the Min proteins in bacteria (12,13), and a collection of proteins in fission yeast, including Pom1p and Mid1p, that differentiate the pole from the mid-cell (6,7,14).

Cell surface asymmetries can emerge spontaneously or they can be induced by transient local cues that bias local recruitment (15,16,17,18) or active transport (5,18,19). However, positive feedback is required to amplify the effects of these symmetry-breaking cues and to stabilize asymmetries once the cues are gone against dissipation by dissociation and lateral diffusion. Thus, a key challenge is to understand how asymmetric distributions of membrane proteins are amplified and stabilized by the dynamic interplay of local exchange, lateral mobility, and feedback.

Building on the classical work of Turing, Meinhardt, and others (20,21,22), recent theoretical studies addressing this challenge have focused on simple abstractions of known polarity circuits expressed as mass-conserved reaction-diffusion models (23,24,25,26,27,28,29). A key conclusion from these studies is that, although linear positive feedback can support transient noisy asymmetries when the numbers of molecules are sufficiently low (30,31), stable polarity requires non-linear positive feedback on protein abundance and/or activity (26,28,32). Such studies also highlight how changes in feedback strength, protein abundance, binding rates, and mobilities can produce qualitative transitions in polarization dynamics (e.g., from spontaneous to inducible polarity (28,33)), or quantitative changes in the positions of stable domain boundaries, or the rates at which competition between multiple domains is resolved (5,26,34). In general, how specific forms of non-linear feedback, arising through specific molecular interactions, shape polarization dynamics remains poorly understood.

One potentially important source of non-linear feedback in polarity circuits is the oligomerization of peripheral membrane proteins. In recent years, an increasing number of polarity proteins have been observed to form discrete clusters on the cell membrane (3,7,35,36,37,38,39), suggesting a general role for oligomerization in cell polarity. Oligomerization confers size-dependent membrane binding avidity (39,40,41), and can also lead to size-dependent restriction of mobility through the interactions of oligomers with the submembrane cytoskeleton (3,42,43,44,45). A few previous modeling studies have considered specific examples of oligomerization reactions (3,6,38,46), but these have focused on special cases (3,46) or on how oligomerization shapes gradients formed by a local source of protein recruitment (6,38,47). To date, there has been no systematic analysis of how oligomerization of membrane proteins, and its effects on lateral mobility and dissociation, shapes their abilities to form and stabilize asymmetric distributions on the cell membrane.

Here we study a class of simple polarity models in which monomers bind reversibly to the membrane and oligomerize in the presence of positive feedback on membrane binding governed by first-order (linear) mass action kinetics. We find that non-linear dissociation kinetics emerge generically from size dependence of membrane binding avidity, that they confer a form of positive feedback on protein abundance, and that this, combined with weak linear positive feedback on membrane binding, is sufficient for polarization. We show that the strengths of oligomerization and feedback define phase boundaries separating regimes in which stable polarity is impossible to achieve, is inducible, or occurs spontaneously. Furthermore, modulating oligomerization strength provides a simple way to tune the speed of polarization, allowing the same system to rapidly approach a uniquely stable polarized state or to preserve polarized domains of arbitrary sizes as quasi-stable states over biologically relevant timescales. These basic findings extend to multiple circuit architectures and different modes of positive feedback. Our results provide general insights into how oligomerization shapes polarization dynamics. They provide a mechanistic framework for investigating how oligomerization of specific polarity proteins could explain polarization dynamics observed in living cells. Finally, given its widespread occurrence, our results suggest that oligomerization of peripheral membrane proteins may play a key role in facilitating and tuning polarization dynamics across a wide range of evolutionarily distinct polarity circuits and cell types.

Methods

A kinetic model for membrane binding and oligomerization

We consider a simple but general scenario (Fig. 1 A) in which monomers bind to a 1D cell membrane of length $L$ from a well-mixed 2D cytoplasmic pool with area $A=hL$. On the membrane, monomers undergo reversible assembly into oligomers. Cytoplasmic monomers bind directly and reversibly to membrane-bound oligomers, and membrane-bound oligomers undergo size-dependent mobility (3,42,43,44,45) and dissociation (40,41). Finally, we assume that there is positive feedback on monomer binding to membranes, proportional to the density of membrane-bound subunits.

Figure 1.

A kinetic model for membrane binding and oligomerization. (A) Schematic overview of the full kinetic model. Cytoplasmic monomers bind reversibly to the plasma membrane with rate constants $(k_{on}^m,k_{off}^m)$, where they self-associate to form linear oligomers with rate constants $(k_{on}^p,k_{off}^p)$. Cytoplasmic monomers also bind directly and reversibly to membrane-associated subunits with rate constants $(k_{on}^c,k_{off}^c)$. We assume that oligomer dissociation rates $k_{off}\left(n\right)$ and diffusivities $D\left(n\right)$ are decreasing functions of oligomer size. Curved line indicates positive feedback on membrane binding at a rate $k_f\left(M\right)$, which depends on the local density of membrane-bound subunits $M$. (B) A simpler one-species model for the total density of subunits $M$, which is valid when oligomerization kinetics are sufficiently fast (see supporting analysis for details). $F\left(M\right)(M_{tot}-M)$ and $G\left(M\right)$ represent total binding and unbinding rates. (C) Oligomerization strength $(\alpha =K_P{m}_1\left(M\right))$ plotted as function of $M$ for three different values of $K_P$, the equilibrium constant for binding of membrane-associated monomers to oligomers. (D) Mean oligomer size $(s=\frac{1}{1-\alpha })$ plotted as function of $M$ for three different values of $K_P$. (E) Effective unbinding rate $\left(\frac{G\left(M\right)}{M}\right)$ plotted as function of $M$ for three different values of $K_P$.

To simplify analysis, we initially ignore some the complexity associated with biological oligomers: We assume that oligomers undergo simple linear growth with no branching and no nucleation threshold, and we ignore the potential contributions of oligomer breakage and annealing. Thus, we assume that the rate constants for binding and dissociation of cytoplasmic or membrane monomers to membrane-bound oligomers are independent of oligomer size. We assume a simple form of size-dependent dissociation in which oligomers of size >1 do not dissociate from the membrane. Finally, we assume that, in the absence of positive feedback, the core membrane binding and oligomerization reactions satisfy detailed balance; that is, they do not consume energy through enzymatic reactions, active transport, etc. We will explore the consequences of relaxing each of these assumptions below.

With these assumptions in mind, we let $m_n\left(x\right)$ be the local density (molecules/unit length) of membrane-bound oligomers of size n, where n ranges from 1 to a maximal oligomer size $N$. $M\left(x\right)$ the local density of all membrane-bound subunits, $P\left(x\right)$ is the local density of all membrane-bound oligomers, C is the concentration (molecules/unit area) of cytoplasmic subunits, and $X_{tot}$ is the total number of subunits in the system. Assuming that monomer binding follows simple mass action kinetics, we write:

$$\begin{array}{l}\frac{\partial m_1}{\partial t}=D\left(1\right)\frac{{\partial }^2{m}_1}{\partial x^2}+(k_{on}^m+k_{f}M)C-k_{off}^m{m}_1-2\left(k_{on}^p{m}_1{m}_1-k_{off}^p{m}_2\right)\hfill \\ +\left(k_{on}^{c}C{m}_1-k_{off}^c{m}_2\right)-\sum _{n=3}^N\left(k_{on}^p{m}_1{m}_{n-1}-k_{off}^p{m}_n\right)\hfill \\ \frac{\partial m_n}{\partial t}=D\left(n\right)\frac{{\partial }^2{m}_n}{\partial x^2}+k_{ass}{m}_{n-1}-\left(k_{ass}+k_{diss}\right)m_n+k_{diss}{m}_{n+1}\hfill \\ \frac{\partial m_N}{\partial t}=D\left(N\right)\frac{{\partial }^2{m}_N}{\partial x^2}+k_{ass}{m}_{N-1}-k_{diss}{m}_N\hfill \\ X_{tot}=C·A+{\int }_0^{L}M\left(x\right)dx\hfill \end{array}$$ (1)

where $k_{ass}=k_{on}^p{m}_1+k_{on}^{c}C,k_{diss}=k_{off}^p+k_{off}^c$, $D\left(n\right)$ is the effective mobility of n-mers. and the final equation is an algebraic condition that enforces conservation of total subunits.

Steady-state analysis

At spatially uniform steady states, the diffusive fluxes vanish and we have $m_n^{ss}=\frac{k_{ass}}{k_{ass}}{m}_{n-1}^{ss}$ for $n=2,\dots N$. Thus, there is an exponential distribution of oligomer sizes, given by:

$$m_n^{ss}={\alpha }_{ss}^{n-1}{m}_1^{ss}$$ (2)

where

$${\alpha }_{ss}=\frac{k_{ass}}{k_{diss}}=\frac{k_{on}^p{m}_1^{ss}+k_{on}^c{C}_{ss}}{k_{off}^p+k_{off}^c}$$ (3)

represents oligomerization strength at the spatially uniform steady state. Summing series for $P_{ss}$ and $M_{ss}$, we have:

$$P_{ss}=\sum _{n=1}^N{\alpha }_{ss}^{n-1}{m}_1^{ss}=\left(\frac{1-{\alpha }_{ss}^N}{1-\alpha }\right)m_1^{ss},\text{and}{M}_{ss}=\sum _{n=1}^{N}n{\alpha }^{n-1}{m}_1^{ss}={\left(\frac{1-{\alpha }^N}{1-\alpha }\right)}^2{m}_1^{ss}$$ (4)

Since the maximal oligomer size is limited only by the size of the total pool, it is always possible to choose $M_{tot}$ (and $N$) sufficiently large that ${\alpha }_{ss}<1$ (48). Moreover, if $\alpha <1$, and $N$ is sufficiently large, then ${\alpha }_{ss}^N\ll 1$, and:

$$P_{ss}\approx \frac{m_1^{ss}}{\left(1-{\alpha }_{ss}\right)},\text{and}{M}_{ss}\approx \frac{m_1^{ss}}{{\left(1-{\alpha }_{ss}\right)}^2}$$ (5)

Alternatively, if $S=\frac{M_{ss}}{P_{ss}}=\frac{1}{1-{\alpha }_{ss}}$ is the mean oligomer size at the spatially uniform steady state, then:

$$P_{ss}\approx m_1^{ss}S,\text{and}{M}_{ss}\approx m_1^{ss}{S}^2$$ (6)

Enforcing detailed balance for basal binding kinetics

If membrane binding and oligomerization reactions satisfy detailed balance in the absence of positive feedback $(k_f=0)$, then the free energy change associated with incorporating a cytoplasmic monomer into a membrane-bound oligomer is path independent:

$$-\log \left(K_M\right)RT-\log \left(K_P\right)RT=-\log \left(K_C\right)RT$$

where

$$K_m=\frac{k_{on}^m}{k_{off}^m},K_c=\frac{k_{on}^c}{k_{off}^c},\text{and}{K}_p=\frac{k_{on}^p}{k_{off}^p}$$ (7)

and therefore:

$$\gamma =\frac{K_M{K}_P}{K_C}=1$$ (8)

Derivation of a one-species model

To focus on key factors that govern polarization in this system, we first consider a limiting case in which oligomerization kinetics are fast relative to diffusion and relative to the exchange of subunits between the cytoplasm and membrane. If oligomerization kinetics are sufficiently fast that they satisfy:

$$k_{off}^p\gg \max \left(k_{off}^m,k_{off}^c,\frac{K_{C}C}{K_P{m}_1}{k}_{off}^c\right)$$ (9)

then:

$$\alpha =\frac{k_{on}^p{m}_1+k_{on}^{c}C}{k_{off}^p+k_{off}^c}=\frac{K_p{m}_1\left(k_{off}^p+\frac{K_{C}C}{K_p{m}_1}{k}_{off}^c\right)}{k_{off}^p+k_{off}^c}\approx K_p{m}_1$$ (10)

and we can invoke a quasi-steady-state assumption to write a single equation for the total density of membrane-bound subunits $M$:

$$\begin{array}{l}\frac{\partial M}{\partial t}=D_M\frac{{\partial }^{2}M}{\partial x^2}+(k_{on}^m+k_{on}^{c}P\left(M\right)+k_{f}M)C-k_{off}^m{m}_1\left(M\right)-k_{off}^c(P\left(M\right)-m_1\left(M\right))\\ M_{tot}=C·h+\frac{1}{L}{\int }_0^{L}M\left(x\right)dx\end{array}$$ (11)

where

$$P\left(M\right)=M(1-K_P{m}_1),m_1\left(M\right)=\frac{1+2M{K}_P-\sqrt{1+4M{K}_P}}{2M{K}_P^2}$$ (12)

$D_M$ is the average subunit diffusivity, reflecting the dependence of oligomer mobility on oligomer size, and $M_{tot}=\frac{X_{tot}}{L}$ is the density of membrane-bound subunits when all subunits are membrane bound.

Rearranging terms and using $P-m_1=K_P{m}_{1}P$, we have:

$$\frac{\partial M}{\partial t}=k_{off}^m\left[\frac{D_M}{k_{off}^m}\frac{{\partial }^{2}M}{\partial x^2}+\frac{k_{on}^m}{k_{off}^m}\left(1+\frac{k_{on}^c}{k_{on}^m}P+\frac{k_f}{k_{on}^m}M\right)C-m_1\left(1+\frac{k_{off}^c}{k_{off}^m}{K}_{P}P\right)\right]$$

Introducing dimensionless variables:

$$\hat{t}=k_{off}^{m}t,\hat{x}=\frac{x}{L},\hat{M}=\frac{M}{M_{tot}},\hat{P}=\frac{P}{M_{tot}}\text{and}\hat{C}=\frac{Ch}{M_{tot}}$$ (13)

and dropping the hats, we obtain:

$$\begin{array}{c}\frac{\partial M}{\partial t}=D\frac{{\partial }^{2}M}{\partial x^2}+K\left(1+BP+fM\right)C-m_1(1+\gamma BP)\\ C=\left(1-{\int }_0^{1}M\left(x\right)dx\right)\end{array}$$ (14)

where:

$$D=\frac{D_M}{k_{off}^m{L}^2},K=\frac{K_M}{h},B=\frac{k_{on}^c{M}_{tot}}{k_{on}^m},f=\frac{k_f{M}_{tot}}{k_{on}^m}\text{and}\gamma =\frac{K_p{K}_m}{K_c}$$

are dimensionless parameters.

Computing an effective dissociation rate for the one-species model

For the one-species model, we define an effective dissociation rate constant for membrane-bound subunits to be the ratio of the subunit dissociation rate $m_1(1+\gamma BP)$ divided by the total density of membrane-bound subunits $M$:

$$k_{diss}^{eff}=\frac{m_1(1+\gamma B)}{M}$$ (15)

Using Eqs. 5 and 6 above, this becomes:

$$k_{diss}^{eff}=(1+\gamma B){(1-\alpha )}^2=\frac{1+\gamma B}{S^2}$$ (16)

From Eq. 12:

$$\frac{\partial m_1}{\partial M}=\frac{m_1}{M}\left(\frac{1-\alpha }{1+\alpha }\right)$$ (17)

Since $0<\alpha <1$, it follows that $m1\left(M\right)$ is a monotonically increasing function of $M$. Therefore, $\alpha =K_P{m}_1$ and $S=\frac{1}{1-\alpha }$ are also monotonically increasing functions of $M$ (Fig. 1 C and D), and the effective dissociation rate $k_{diss}^{eff}$ is a non-linear decreasing function of $M$ (Fig. 1 E).

General conditions for spontaneous polarization

Equation 14 has the general form:

$$\frac{\partial M}{\partial t}=D\frac{{\partial }^{2}M}{\partial x^2}+F\left(M\right)\left(1-{\int }_0^{1}M\left(x\right)dx\right)-G\left(M\right)$$ (18)

where $F\left(M\right)$ and $G\left(M\right)$ define membrane binding and unbinding kinetics, respectively, as a function of $M$. For spatially uniform solutions, the diffusion term vanishes, ${\int }_0^{1}M\left(x\right)dx=M$, and Eq. 18 becomes:

$$\frac{dM}{dt}=F\left(M\right)\left(1-M\right)-G\left(M\right)$$ (19)

Graphical flux balance analysis (Fig. S1) reveals that there can be either one or two stable solutions of Eq. 19, and these correspond to spatially uniform solutions of Eq. 18 that are stable with respect to spatially uniform perturbations. We seek conditions in which these solutions become unstable with respect to small perturbations of the form:

$$M\left(x\right)=M_{ss}+\Delta \left(x\right)$$ (20)

where $\Delta \left(x\right)$ is a spatially varying perturbation satisfying

$${\int }_0^1\Delta \left(x\right)dx=0$$ (21)

Inserting Eq. 20 into Eq. 18 and linearizing, we obtain:

$$\frac{\partial \Delta }{\partial t}=D\frac{{\partial }^2\Delta }{\partial x^2}+\beta \Delta$$ (22)

where:

$$\beta ={\left(\frac{\partial F}{\partial m}\frac{G}{F}-\frac{\partial G}{\partial m}\right)}_{M=M_{ss}}$$ (23)

The general solution to Eq. 22 can be written:

$$\Delta (x,t)=\sum _{n=1}^{\infty }{\alpha }_n{e}^{\beta -k^{2}D}\cos \left(kx\right),k=2\pi n$$ (24)

For any small initial perturbation $\Delta (x,0)=\sum _{n=1}^{\infty }{a}_n\cos \left(kx\right)$, spatial differences defined by Eq. 24 can grow if, and only if, $\beta >k^{2}D$ for some $n\ge 1$. Therefore, the conditions for spontaneous polarization are:

$${\left(\frac{\partial F}{\partial m}\frac{G}{F}-\frac{\partial G}{\partial m}\right)}_{M=M_{ss}}>4{\pi }^{2}D$$ (25)

Note that, for a 2D membrane, the term on the right-hand side of Eq. 25 becomes $8{\pi }^{2}D$, which produces a small shift in the phase boundary.

Conditions for inducible polarization

We used local perturbation analysis (LPA (33)) to determine conditions for coexistence of stable spatially uniform and polarized steady states and to determine the size of a local perturbation required to induce a transition from the spatially uniform to the polarized state. Briefly, we consider the dynamic response to a finite-amplitude perturbation from a stable spatially uniform steady state $M=M_{ss}$ within an infinitesimally small spatial domain. We assume that diffusion is sufficiently slow $(D_M\approx 0)$ that local changes within this domain do not affect the bulk pools of membrane-bound or cytoplasmic protein. The equation for the local response $M_l$ then writes:

$$\frac{d{M}_l}{dt}=F\left(M_l\right)\ast C_{ss}-G\left(M_l\right)$$ (26)

where

$$C_{ss}=\frac{G\left(M_{ss}\right)}{F\left(M_{ss}\right)}$$ (27)

Equation 26 has a fixed point at $M_l=M_{ss}$ corresponding to the spatially uniform steady state. When $D_M=0$, spontaneous polarization occurs when this fixed point is unstable, i.e., when

$${\left(\frac{\partial F}{\partial M_l}\frac{G}{F}-\frac{\partial G}{\partial M_l}\right)}_{M_l=M_{ss}}>0$$ (28)

consistent with the result from linear stability analysis when $D_M=0$ (see supporting mathematical appendix).

If the fixed point at $M_l=M_{ss}$ is stable, then a stably polarized steady state exists when Eq. 26 has an unstable steady state at $M_l=M_{th}$, for $M_{th}>M_{ss}$. $M_{th}$ then defines the threshold size of a perturbation required to induce a transition to the polarized steady state (neglecting diffusion).

In practice, we used MATLAB to identify a point $M_l=M_{th}>M_{ss}$ where:

$$\frac{d{M}_l}{dt}=F\left(M_{th}\right)\ast C_{ss}-G\left(M_{th}\right)=0$$ (29)

within the range:

$$M_{ss}\le M_l\le M_{ss}\ast 1000$$ (30)

and we assessed its stability by determining the sign of:

$${\left(\frac{\partial F}{\partial M_l}\ast C_{ss}-\frac{\partial G}{\partial M_l}\right)}_{M_l=M_{th}}$$ (31)

Mapping phase diagrams for spontaneous and inducible polarization

We defined the oligomerization strength to be the value of $\left(\alpha \right)$ evaluated at the spatially uniform steady state:

$$\alpha =\frac{k_{on}^p{m}_1^{ss}+k_{on}^c{C}_{ss}}{k_{off}^p+k_{off}^c}$$ (32)

For the one-species model and/or when basal exchange reactions obey detailed balance, this reduces to:

$$\alpha =K_P{m}_1^{ss}$$ (33)

We defined the strength of feedback on membrane binding $\left(F_{mb}\right)$ to be the ratio of subunit flux onto the membrane due to feedback over the basal flux, evaluated at the spatially uniform steady state. For the simple one-species model with indirect positive feedback:

$$F_{mb}=\frac{k_f{M}_{ss}}{k_{on}^m}=f{M}_{ss}$$ (34)

For different variants of the one-species model, in the Supporting material, we derive specific algebraic forms of the conditions for spontaneous polarization in terms of $F_{mb}$ and $\alpha$ from Eq. 25, and these are given in the main text below. To map the complete phase diagrams for spontaneous and/or inducible polarization numerically, we fixed $M_{ss}=1$, systematically sampled values for $\alpha$ and $F_{mb}$, assigned values to a subset of model parameters, used the algebraic formulae for $\alpha$ and $F_{mb}$ to compute values for the remaining parameters, and then used simulations or LPA to assess outcomes (see Supporting material for details).

A two-species model with cross-inhibition

We also considered a scenario in which two proteins $A$ and $B$, with cytoplasmic concentrations $C_A,C_B$, and membrane densities $M_A,M_B$, bind reversibly to the membrane and promote one another’s dissociation. We assume that $A$, but not $B$, oligomerizes at the membrane, and we ignore direct binding of cytoplasmic $A$ monomers to membrane-bound oligomers. Assuming that oligomerization of $A$ is fast relative to monomer exchange of $A$ monomers with the membrane. Invoking a quasi-steady-state approximation as above, and choosing appropriate units for length and density (see supporting mathematical appendix), we write a pair of partial differential equations for $M_A$ and $M_B$:

$$\begin{array}{ll}\frac{\partial M_A}{\partial t}\hfill & =D_A\frac{{\partial }^2{M}_A}{\partial x^2}+k_{on}^A{C}_A-(k_{off}^A+k_{BA}{M}_B)m_1^A\left(M_A\right)\hfill \\ \frac{\partial M_B}{\partial t}\hfill & =D_B\frac{{\partial }^2{M}_B}{\partial x^2}+k_{on}^B{C}_B-(k_{off}^B+k_{AB}{M}_A)M_B\hfill \end{array}$$ (35)

with conservation conditions:

$$\begin{array}{ll}{C}_A\hfill & =\left(1-{\int }_0^1{M}_A\left(x\right)dx\right)\hfill \\ C_B\hfill & =\left(1-{\int }_0^1{M}_B\left(x\right)dx\right)\hfill \end{array}$$

where $m_1^A$ is the density of A monomers:

$$m_1^A\left(M_A\right)=\frac{1+2M_A{K}_P^A-\sqrt{1+4M_A{K}_P^A}}{2M_A{\left(K_P^A\right)}^2}$$ (36)

$M_A$, $M_B$, $C_A$, and $C_B$ are scaled variables, and $D_A,D_B,k_{on}^A,k_{on}^B,k_{off}^A,k_{off}^B,k_{AB},k_{BA}\text{and}{K}_P^A$ are scaled constants with units of $1/t$.

In the supporting mathematical appendix, we show that this system has a single spatially uniform steady state, which is always stable with respect to spatially uniform perturbations, and we derive general conditions for spontaneous polarization:

$$\left(\frac{J_A}{1+J_A}\right)\left(\frac{J_B}{1+J_B}\right)>\left(\frac{1-\alpha }{1+\alpha }\right)\left(1+\left(\frac{D_A^{\ast }}{1+J_A}\right)\left(\frac{{\left(1-\alpha \right)}^3}{1+\alpha }\right)\right)\left(1+\frac{D_B^{\ast }}{1+J_B}\right)$$ (37)

where $\alpha =K_P^A{m}_1^A$ is the oligomerization strength, $J_A=\frac{k_{BA}{M}_B}{k_{off}^A}$ and $J_B=\frac{k_{AB}{M}_A}{k_{off}^B}$ are the cross-inhibition strengths, and $D_A^{\ast }=\frac{4{\pi }^2{D}_A}{k_{off}^A}$ and $D_B^{\ast }=\frac{4{\pi }^2{D}_B}{k_{off}^B}$ are scaled diffusivities. Note that, when diffusivity is negligible, this reduces to:

$$F_{ma}>\left(\frac{1-\alpha }{1+\alpha }\right)$$ (38)

where:

$$F_{ma}=\left(\frac{J_A}{1+J_A}\right)\left(\frac{J_B}{1+J_B}\right)$$ (39)

Numerical simulations

We performed simulations using custom scripts written in MATLAB. We discretized the 1D membrane domain into 100 equally sized compartments and used a standard center-difference method to approximate Eqs. 1 and 11 as systems of ordinary different equations, which we solve using MATLAB’s built-in ode45 function. We used periodic boundary conditions in simulations to test for symmetry breaking, and no-flux boundary conditions in simulations to determine dynamics. For simulations of the full kinetic model, we set the maximal oligomer size to $N=50$.

Results

Oligomerization and positive feedback on membrane binding determine the potential for polarization

We began by considering a limiting case in which oligomerization dynamics are fast relative to monomer exchange between cytoplasm and membrane. Under these conditions, the full kinetic Eq. 1 simplifies to a single equation for the total abundance of membrane-bound subunits $M$ (see section “derivation of a one-species model”; Eq. 14; Fig. 1 B), which we refer to as the one-species model and reproduce here for clarity:

$$\begin{array}{c}\frac{\partial M}{\partial t}=D\frac{{\partial }^{2}M}{\partial x^2}+F\left(M\right)C-G\left(M\right)\\ C=\left(1-{\int }_0^{1}M\left(x\right)dx\right)\end{array}$$ (40)

where

$$F\left(M\right)=K\left(1+BP+fM\right),G\left(M\right)=m_1(1+\gamma BP)$$ (41)

$m_1=m_1(M,K_P)$ and $P=P(M,K_P)$ are monotonically increasing functions of $M$ (see section “methods”; Eq. 12), and $K_P$, $D$, $K$, $B$, $f$, and $\gamma$ are dimensionless parameters that determine $\left(K_P\right)$ oligomerization strength, $\left(D\right)$ the mean diffusivity of membrane-bound subunits, $\left(K\right)$ basal membrane binding, $\left(B\right)$ subunit binding to membrane-bound oligomers, $\left(f\right)$ feedback on membrane binding, and $\left(\gamma \right)$ whether the basal binding kinetics consume energy (i.e., whether they obey detailed balance).

For the range of model parameters considered here, the one-species model (and thus the full kinetic model) has a single spatially uniform steady state (Fig. S1), and one of three qualitatively distinct behaviors are possible (Fig. 2 A–C): Spontaneous polarization, in which the uniform steady state is unstable to all perturbations; inducible polarization, in which the uniform steady state is stable, but stable polarity can be triggered by a sufficiently large transient perturbation; and no polarization, in which the uniform steady state is globally stable. Thus, we sought to map how these possible outcomes depend on oligomerization, positive feedback on membrane binding, and oligomer mobility.

Figure 2.

Strength of oligomerization and positive feedback define the potential for polarization. (A–C) Examples of three qualitatively distinct polarization regimes: (A) no polarization, (B) inducible polarization, and (C) spontaneous polarization. (D) Spontaneous symmetry breaking as a function of $\alpha$ (or mean oligomer size $S$) and feedback strength $F_{mb}$. The dotted line indicates the boundary between regimes in which the uniform steady state is stable (respectively unstable) to small perturbations. (E) The minimal size of a local perturbation (measured as local fold increase over steady-state concentration) required to induce polarization under conditions where symmetry breaking is not spontaneous, as determined by LPA (see Supporting material). (F and G) Dependence of spontaneous and inducible polarization on scaled monomer diffusivity $D=\frac{D_1}{k_{off}^m{L}^2}$. For a typical length scale $L=40\mu m$, and monomer dissociation rate $k_{off}^m=1$, the scaled values represent (left to right) $D_1=\left(0.1,0.16,0.4,1.6\right)\frac{\mu m^2}{s}$. For a typical monomer diffusivity $D_1=0.1\frac{\mu m^2}{s}$, and dissociation rate $k_{off}^m=1$, the scaled values represent (left to right) $L=\left(\mathrm{40,30,20,10}\right)\mu m$. The dotted line indicates the predicted boundary for spontaneous symmetry breaking from linear stability analysis. (F) A scenario in which diffusion is size independent $(D_M=D_1)$, and (G) represents a scenario in which oligomers of size greater than one do not diffuse. $(D_M=D_1{(1-\alpha )}^2)$. See Supporting material for detailed description of simulations.

For the one-species model, the effective dissociation rate constant for membrane-bound subunits is the total subunit dissociation rate divided by the density of membrane-bound subunits $k_{diss}^{eff}=\frac{G\left(M\right)}{M}$. As shown in Fig. 1 C–E (and see section “methods”), $k_{diss}^{eff}$ is a non-linear decreasing function of $M$, whose steepness increases with $K_P$, reflecting the positive dependence of membrane binding avidity on oligomerization strength $(\alpha =K_P{m}_1)$ and mean oligomer size $\left(S\right)$, and the positive dependence of $\alpha$ and $S$ on $M$. Thus, the negative dependence of subunit dissociation on density represents a form of oligomerization-dependent positive feedback on the accumulation of membrane-bound subunits, whose strength increases with $K_P$. Below we first consider how oligomerization and positive feedback on membrane binding shape the potential for spontaneous or inducible symmetry breaking when diffusion is slow and direct binding of cytoplasmic monomers to membrane-bound oligomers can be neglected ($D\approx 0$ and $B\approx 0$ in Eq. 40). Then we consider how diffusion and/or direct binding change the outcome.

Under these conditions (see Supporting material), linear stability analysis shows that the spatially uniform steady state is unstable, i.e. spontaneous symmetry breaking occurs, when

$$F_{mb}>\frac{(1-\alpha )}{2\alpha }$$ (42)

where $\alpha =K_P{m}_1^{ss}$ quantifies the strength of feedback due to oligomerization, and $F_{mb}=f{M}_{ss}$ quantifies the strength of feedback on membrane binding, both measured at the spatially uniform steady state (Fig. 2 D). For values of $F_{mb}$ and $\alpha$ that make the spatially uniform steady state stable, we used LPA (33) to determine the minimal size of a local perturbation that can induce a stably polarized state. Plotting the complete phase diagram for both spontaneous and inducible polarization (Fig. 2 E) shows that both oligomerization and linear positive feedback are required for spontaneous symmetry breaking. That is, both $\alpha$ and $F_{mb}$ must be positive. This is consistent with basic physical principles: a system that does not consume energy cannot form a spatial pattern at equilibrium. This is consistent with theoretical studies showing that symmetry breaking requires non-linear positive feedback of density on the net rate of accumulation (26,28,32). In particular, if the binding rate $F\left(M\right)C\propto M^k$ and the unbinding rate $G\left(M\right)\propto M^l$, then we must have $k>l$ for spontaneous symmetry breaking (28). For linear feedback on membrane recruitment and no oligomerization, $k=l=1$. For oligomerization alone, $k=0$ and $0<l<1$. Thus, only the combination of linear feedback on recruitment ($k=1$) and oligomerization ($0<l<1$) can satisfy $k>l$.

Importantly, although some positive feedback on membrane recruitment is required, the amount decreases rapidly with increasing mean oligomer size, such that, for mean oligomer sizes greater than a few subunits, positive feedback on recruitment must only deliver a fractional increase over the basal on-rate to induce spontaneous polarization. As a simple comparison, for a mean oligomer size of 2.5, positive feedback on monomer binding must only deliver an approximately $35\%$ increase over basal rates to drive symmetry breaking. By contrast, to achieve the same result (with the same feedback strength at steady state) with cooperative positive feedback on monomer recruitment would require approximately fourth-order dependence (Fig. S2). Finally, in the absence of diffusion, when the spatially uniform steady state is stable, a polarized state can always be induced by a sufficiently large local input. The threshold for induction decreases with increasing $F_{mb}$ or $\alpha$, reaching zero at the spontaneous polarization boundary.

We used numerical simulations to examine the consequences of relaxing the simplifying assumptions made above or introducing more realistic representations of oligomerization. We confirmed that the position of the spontaneous polarization boundary is largely unaffected by these changes, including relaxing the assumption that oligomerization kinetics are fast relative to exchange with the cytoplasmic pool (Fig. S3A), relaxing the sharp size dependence of oligomer dissociation (Fig. S3B), allowing oligomerization to occur in the cytoplasm as well as on the membrane (Fig. S3C), introducing a nucleation threshold (Fig. S3D), allowing oligomers to break and re-anneal (Fig. S3E), or allowing branched oligomer growth (Fig. S3F). Some of these perturbations produced deviations from an exponential distribution of cluster sizes at steady state (Fig. S3G). Thus, additional complexities associated with oligomerization appear to have little or no effect on the overall dynamics of polarization in this system.

We also assessed how increasing diffusivity affects the potential for both spontaneous and inducible polarization. Because interactions with the submembrane cytoskeleton can induce sharp dependence of oligomer mobility on size (3,38), we considered two limiting scenarios: size-independent diffusivity, and an effective size-dependent diffusivity in which oligomers of size $\ge 2$ are immobile $(D_{eff}=D_1{(1-\alpha )}^2)$. When diffusivity is non-negligible, the conditions for spontaneous polarization are given by (see the supporting mathematical appendix)

$$\begin{array}{c}{F}_{mb}>\frac{4{\pi }^{2}D\frac{1+\alpha }{{(1-\alpha )}^2}+(1-\alpha )}{2\alpha -4{\pi }^{2}D\frac{1+\alpha }{{(1-\alpha )}^2}}\\ 2\alpha -4{\pi }^{2}D\frac{1+\alpha }{{(1-\alpha )}^2}>0\end{array}$$ (43)

Consistent with intuition, for size-independent diffusion, increasing monomer diffusivity reduced the potential for both spontaneous and induced polarization to an extent that increased with monomer diffusivity (Fig. 2 F). However, these effects were relatively mild for typical diffusivities of membrane proteins $\left(D=0.1\frac{\mu m^2}{\sec }\right)$, cell lengths $(L=10-40\mu m)$, and monomer dissociation rates $\left(k_{off}^m\approx \frac{1}{\sec }\right)$ (see Fig. 2 legend for scaled diffusivities corresponding to these typical values). Moreover, for the size-dependent diffusion scenario, these effects are sharply reduced (Fig. 2 G). In summary, we find that oligomerization plus linear positive feedback on monomer binding provides a robust mechanism for spontaneous polarization of membrane-bound proteins. Although diffusion can degrade this mechanism, the effect is mild and is reduced by size-dependent decrease in oligomer mobility.

Direct binding of cytoplasmic monomers to membrane-bound oligomers can promote or antagonize polarization under different conditions

We next asked how the ability to polarize is shaped by direct binding of cytoplasmic monomers to membrane-bound oligomers. We considered two general scenarios: one in which there is positive feedback on membrane binding and the basal oligomerization and exchange reactions obey detailed balance ($k_f>0$, $\gamma =1$), and one in which there is no positive feedback on membrane binding and the basal reactions do not obey detailed balance ($k_f=0$, $\gamma \ne 1$).

When positive feedback on membrane binding is active and basal oligomerization and exchange reactions obey detailed balance (Fig. 3 A; see section “methods”), neglecting the effects of diffusion, the conditions for spontaneous polarization are given by (see supporting mathematical appendix, section “specific conditions for spontaneous polarization,” case 1):

$$F_{mb}>\left(1+B_{dir}\right)\frac{(1-\alpha )\left(1+B_{dir}\right)}{2\alpha \left(1+B_{dir}\right)-B_{dir}}$$ (44)

where $B_{dir}=B{P}_{ss}$ quantifies the rate of binding to membrane-bound oligomers, relative to the rate of binding the membrane itself.

Figure 3.

Direct binding of cytoplasmic monomers to membrane-bound oligomers can promote or antagonize polarization. (A) Schematic view of the scenario in which there is indirect feedback on monomer recruitment and detailed balance is enforced for membrane binding and oligomerization $\gamma =1$. (B) Spontaneous symmetry breaking as a function of $\alpha$ or mean oligomer size $S$ and feedback strength $F_{mb}$ for different values of $B_{dir}$, measured at the uniform steady state. The grayscale lines indicate the boundary across which the uniform steady state goes from stable (bottom left) to unstable (top right) for different values of $B_{dir}$. (C) Schematic illustrating the case where there is no indirect positive feedback and the basal kinetics obey $\gamma <1$. (D) Spontaneous symmetry breaking as a function of $\alpha$ or mean oligomer size $D$ and feedback strength $F_{dir}$ for different values of $\gamma B_{dir}$. The grayscale lines in (C) and (D) indicate phase boundaries separating regimes in which the uniform steady state is stable (bottom left) and unstable (top right).

Examining Eq. 44, or plotting the phase boundary for spontaneous polarization (Fig. 3 B), shows that, for this scenario, increasing direct binding $\left(B_{dir}\right)$ increases the strength of positive feedback required for symmetry breaking. This effect is largest when oligomerization is weak (i.e., when $\alpha$ is small). The origins of this effect can be understood intuitively as follows. When detailed balance is satisfied, a dynamic balance of binding and unbinding fluxes stabilizes the spatially uniform steady state. Positive feedback must overcome this effect to induce symmetry breaking. Increasing direct binding increases the magnitude of these stabilizing fluxes and thus increases the strength of positive feedback required to induce spontaneous symmetry breaking. Numerical simulations of the full kinetic model generalize these results to the case where oligomerization is faster than exchange. As oligomerization kinetics become slower, the phase boundary shifts upward in the $F_{mb}$ versus $\alpha$ plane (Fig. S4). However, the effect of increasing direct binding on polarization persists. Thus, direct binding to oligomers opposes polarization driven by indirect positive feedback when the basal oligomerization and exchange kinetics obey detailed balance.

We then considered the alternative scenario involving direct binding of cytoplasmic subunits to membrane-bound oligomers in which there is no positive feedback on membrane binding ($f=0$ in Eq. 11), but one or more of the basal exchange and oligomerization reactions are driven in a way that breaks detailed balance $\left(\gamma \ne 1\right)$. For example, this could arise if phosphorylation of subunits within oligomers increases their affinity for the membrane. For this scenario, linear stability analysis reveals that spontaneous polarization can occur in the absence of positive feedback on membrane binding, but only if $\gamma <1$ (see supporting mathematical appendix, section “specific conditions for spontaneous polarization,” case 2). When $\gamma <1$, there is a net flux of subunits from the cytoplasm into oligomers at steady state, given by $J_{net}=B{P}_{ss}\left(KC-\gamma m_1^{ss}\right)$, which is analogous to the flux delivered by positive feedback on membrane binding (compare Fig. 3 A and C). Importantly, $J_{net}$ increases with the abundance of membrane-bound oligomers $P_{ss}$. Thus, when detailed balance is broken, the net flux of subunits into membrane-bound oligomers constitutes a form of positive feedback. We defined the strength of this feedback $F_{dir}$ to be the ratio of the net flux $J_{net}$ to the basal rate of membrane binding, evaluated at the spatially uniform steady state:

$$F_{dir}=\frac{J_{net}}{K_{M}C}$$ (45)

Again, neglecting diffusion, the conditions for spontaneous polarization are (see supporting mathematical appendix, section “specific conditions for spontaneous polarization,” case 2):

$$F_{dir}>\frac{\left(1+\gamma B_{dir}\right)(1-\alpha )}{1-(1+\gamma B_{dir})(1-\alpha )}$$ (46)

Plotting the spontaneous polarization boundary for different values of $\gamma B_{dir}$ reveals that, for mean oligomer sizes greater than a few subunits, the net flux delivered by breaking detailed balance must only be a small fraction of the basal monomer binding rate to support spontaneous symmetry breaking (Fig. 3 D). The spontaneous polarization boundary shifts up and to the right with increasing values of $\gamma B_{dir}$. However, the strength of feedback also depends on $B_{dir}$ and $\gamma$ through (see supporting mathematical appendix, section “specific conditions for spontaneous polarization,” case 2):

$$F_{dir}=\frac{(1-\gamma )B_{dir}}{1+\gamma B_{dir}}$$ (47)

Reducing $\gamma$ promotes symmetry breaking both by reducing the amount of feedback required to break symmetry and by increasing the actual feedback strength. In contrast, increasing $B_{dir}$ increases the amount of feedback required but also increases the feedback strength itself. For most values of $\alpha$, $\gamma$, and $B_{dir}$, the net effect of increasing $B_{dir}$ is to drive the system toward spontaneous symmetry breaking. Numerical simulations show that these results hold approximately for the full kinetic model when oligomerization kinetics are sufficiently fast (Fig. S4). In summary, these results show that direct oligomer binding opposes polarization driven by indirect positive feedback, when basal binding reactions satisfy detailed balance. However, when basal binding reactions break detailed balance in the right direction $\left(\gamma <1\right)$, direct binding supports a form of feedback that can drive polarization.

Mean oligomer size tunes the rate at which asymmetries grow and the stability of polarity boundaries

We next considered how oligomerization affects the dynamic evolution of polarized states. Simulations confirmed that, for both the one-species and full-kinetic models, stably polarized states are characterized by single-peaked distributions (Fig. 4 A and D and not shown). We focused on two scenarios that correspond to different routes to polarity observed in living cells: one in which asymmetries grow from the spatially uniform steady state during spontaneous symmetry breaking or in response to a transient localized cue (Fig. 4 A, Fig. S5), and one in which an initially broad distribution that would be stable in the absence of diffusion evolves toward the stable peaked steady state through diffusion and exchange (Fig. 4 D).

Figure 4.

Mean oligomer size tunes polarization dynamics. (A) Example dynamics of polarization when symmetry breaking is induced by a local threefold increase in membrane protein concentration $M$ at a single point in space. Each line represents the spatial distribution of protein at a different time point. (B) The logarithm of maximal local value of $M$ over time, showing a region in which $M_{\max }$ grows exponentially. (C) Asymmetry growth rate during the exponential phase plotted as a function of mean oligomer size $S$, given different relationships between oligomer size, effective diffusivity, and effective subunit dissociation rates. The bold lines indicate the average of values measured while sampling feedback strength $F_{mb}$ between 2 and 10, and the error bars represent the maximal and minimal values. The orange line represents the case where only diffusion is size dependent, the magenta line represents the case where only the dissociation rate is size dependent, and the dark blue line represents the case where both are size dependent. (D) Example of the temporal evolution of a step-change distribution that is stable in the absence of diffusion. (E) The boundary position, measured as the inflection point in the spatial profile of the distribution, as a function of time. The position moves at a constant rate (boundary speed). (F) Boundary speed plotted as a function of mean oligomer size $S$, given different relationships between oligomer size and effective diffusivity and effective dissociation. The bold lines indicate the average of values measured sampling $F_{mb}$ between 2 and 10 and the error bars represent the maximal and minimal values. Orange line, size-dependent diffusion; magenta line, size-dependent dissociation; dark blue line, size-dependent diffusion and dissociation; light blue line, no size dependence. See Supporting material: Determining boundary speed for detailed description of simulations. (G) Plot showing the predicted time for the boundary to shift 1 $\mu m$ as a function of mean oligomer size $S$, given $F_{mb}=1$, $D=0.1\mu m^2{s}^{-1}$, and $\frac{k_{off}^p}{k_{off}^m}=0.1$. See Supporting material for detailed description of simulations in (A)–(G). (H) Schematic illustrating the trade-offs that emerge from tuning the mean oligomer size.

During both spontaneous symmetry breaking and after a transient local perturbation to the uniform steady state, the growth of asymmetries proceeds through an exponential phase (Fig. 4 B, Fig. S5). Although the rate of exponential growth depends on initial conditions, its variation with model parameters is tightly correlated across different initial conditions (Fig. S5). Thus, we considered how the rate of asymmetry growth during the exponential phase depends on oligomerization and feedback strengths.

The asymmetry growth rate showed weak dependence on feedback strength but decreased sharply with mean oligomer size (Fig. 4 C, blue curve). The observed decrease closely matched the predicted decrease in effective dissociation rate with oligomer size $(\approx \frac{1}{S^2})$. Indeed, scaling the monomer off rate to enforce size-independent dissociation at steady state abolished the dependence of polarization speeds on mean oligomer size (Fig. 4 C, orange curve), although nullifying size dependence of diffusivity had no effect on polarization speed (Fig. 4 C, magenta curve). Thus size-dependent kinetics of oligomer release promotes the ability to polarize, while slowing the rate at which asymmetries grow.

In many polarizing cells, (e.g., (18,19,49)), a transient cue induces the rapid enrichment of polarity proteins within a broad spatial domain, and this distribution can then evolve after the cue has gone through further diffusion and exchange. To determine how oligomerization and feedback shape the timescale on which this occurs, we initialized simulations with broad plateau-shaped distributions that are stable in the absence of diffusion, and then tracked the position of the domain boundary over time in the presence of diffusion. In all such simulations, the domain boundary position moves at an approximately constant speed toward the stable peaked steady state (Fig. 4 E). Like the asymmetry growth rate, boundary speed showed weak dependence on positive feedback but decreased sharply with increases in mean oligomer size (Fig. 4 F). However, unlike for the asymmetry growth rate, the size dependence of boundary speed depends on multiplicative contributions from size-dependent dissociation and mobility (Fig. 4 F). Slowing depolymerization kinetics further decreases both polarization and boundary speed (Fig. S6). Given typical diffusivities of membrane proteins and oligomerization kinetics (see Fig. 4 legend), the time for the boundary to shift by $1\mu m$ increased sharply with mean oligomer size, reaching >1,000 times the monomer binding lifetime for a mean oligomer size of 10. Thus, oligomerization of membrane proteins can allow cells to “remember” spatial asymmetries imposed by external inputs as quasi-stable states over times much longer than the residence times of individual proteins.

Analysis of a simpler two-compartment model for the case in which diffusion is negligible (see supporting mathematical appendix) reveals that, when the system is tuned for spontaneous symmetry breaking, there is no limit to the size of a polarity domain that a cell can stabilize. When the system is tuned for inducible polarization, the maximal size of the quasi-stable polarity domain depends on both oligomerization and feedback strength and becomes smaller with increasing distance from the phase boundary (Fig. S7).

In summary, we find that increasing mean oligomer size enhances the ability to polarize and to maintain quasi-stable domain boundaries at different positions, but at the expense of reducing the rate at which asymmetries grow from a spatially uniform state (Fig. 4 H). We note, however, that polarity cues can readily overcome this tradeoff by promoting rapid asymmetry growth through directed oligomer transport (5,19,41,50) or by promoting local oligomer dissociation (51).

Saturating feedback slows symmetry breaking but has weak effects on boundary speed relative to oligomerization

Previous studies have shown that introducing saturating feedback kinetics into simple mass-balanced reaction-diffusion models can lead to slower polarization and slower resolution of multipolar states by competition (26). To ask how the effects of saturation compare with those that arise through oligomerization, we introduced a simple form of saturating feedback $F_{sat}=f\frac{M}{M+K_{sat}}$ (Fig. 5 A). The conditions for spontaneous polarization then become (see supporting mathematical appendix):

$$F_{sat}>\frac{D^{\ast }\frac{1+\alpha }{{\left(1-\alpha \right)}^2}+1-\alpha }{\alpha \left(\frac{2+R}{1+R}\right)-\left(\frac{R}{1+R}\right)-D^{\ast }\frac{1+\alpha }{{\left(1-\alpha \right)}^2}}$$ (48)

where $R=\frac{M_{ss}}{K_{sat}}$ is a simple measure of saturation at the uniform steady state.

Figure 5.

Saturating feedback reduces the potential for polarization and the rate at which asymmetries grow but has weak effects on boundary speed. (A) Plot of simple saturating feedback. (B) Phase diagrams produced using LPA and showing how regimes with spontaneous, inducible, or no polarity shift with increasing saturation at uniform steady state, measured by $\frac{M_{ss}}{K_{sat}}$. (C and D) The effect of increasing $\frac{M_{ss}}{K_{sat}}$ on the dynamics of polarization and boundary stability, assessed by numerical simulations. (C) Asymmetry growth rate plotted as a function of mean oligomer size $S$, for different values of $\frac{M_{ss}}{K_{sat}}$. (D) Boundary speed plotted as a function of mean oligomer size $S$, for different values of $\frac{M_{ss}}{K_{sat}}$. See Supporting material for detailed description of simulations.

Combining LPA and simulations, we find that introducing saturation increases the strength of feedback and/or oligomerization required for spontaneous polarization, and reduces the range of parameter values for which polarity can be induced through local perturbations (Fig. 5 B). Saturation reduces both asymmetry growth rate and boundary speeds (Fig. 5 C and D). However, although the effects on asymmetry growth rate are significant relative to the effects of oligomerization (Fig. 5 C), the effects on boundary speed are much weaker (Fig. 5 D).

The role of oligomerization in shaping polarization dynamics extends to different feedback topologies

Thus far, we considered forms of feedback in which a protein acts locally to promote its own accumulation on the cell membrane. To explore the generality of these results, we considered an alternative model in which two peripheral membrane proteins $A$ and $B$ bind the membrane at constant rates and act locally to promote one another’s dissociation at rates that are linear functions of their local densities (Fig. 6 A). We assume that $A$ forms oligomers but $B$ does not, and that only monomers dissociate. Assuming that oligomerization of A is fast relative to exchange, we derived a simple two-species model describing the mean abundance of A and B at the membrane (see section “methods”). Considering a limiting case in which diffusion is slow, we find that spontaneous polarization occurs when:

$$F_{ma}>\frac{1-\alpha }{1+\alpha }$$ (49)

where

$$F_{ma}=\left(\frac{J_A}{1+J_A}\right)\left(\frac{J_B}{1+J_B}\right)$$ (50)

is a composite measure of feedback through mutual inhibition, and $J_A$ and $J_B$ are the ratios of feedback-dependent to basal dissociation rates for $A$ and $B$ respectively. Plotting $F_{ma}$ versus $\alpha$ (Fig. 6 B) shows that, as for one-species models with positive feedback on local recruitment, combinations of relatively weak oligomerization and feedback are sufficient for spontaneous polarization. LPA reveals that, even when polarization does not occur spontaneously, it can be induced by sufficiently large local perturbations for a broader range of $F_{ma}$ and $\alpha$ values (Fig. 6 C). Finally, as for models with positive feedback on recruitment, the asymmetry growth rate and boundary speeds both decrease sharply as a function of mean oligomer size (Fig. 6 D). Thus, oligomerization enables polarization and shapes polarization dynamics in very similar ways for networks with very different feedback topologies.

Figure 6.

Oligomerization promotes polarization and slows polarization dynamics when combined with mutual antagonism. (A) Reaction diagram showing mutually antagonistic effects between two membrane-binding proteins. (B) Analytical solution for the boundary between unstable and stable uniform states in phase space determined by $\alpha$ and $F_{ma}=\frac{J_A}{1+J_A}\frac{J_P}{1+J_B}$. (C) The potential for polarization, as assessed by LPA, plotted in terms of $\alpha$ or $S$ and $F_{ma}$ for $J_A=1$ (top) and $J_A=5$ (bottom). Blue indicating spontaneous polarization, yellow indicating inducible polarization, and orange indicating no polarization. (D) Asymmetry growth rate plotted as a function of mean oligomer size $S$, given different relationships between oligomer size and effective diffusivity and effective dissociation. The orange line represents the case where only diffusion is size-dependent, the magenta line represents the case where only the dissociation rate is size dependent, and the dark blue line represents the case where both are size dependent. (E) Boundary speed plotted as a function of mean oligomer size $S$ for $F_{ma}=0.64$, given different relationships between oligomer size and effective diffusivity and effective dissociation. Orange line, size-dependent diffusion only; magenta line, size-dependent dissociation only; dark blue line, size-dependent diffusion and dissociation; light blue line, no size dependence. See Supporting material for detailed description of simulations.

Discussion

The ability to self-oligomerize is common to many peripheral membrane proteins that adopt asymmetric distributions in polarized cells. Here we have explored how this ability shapes the performance of simple feedback circuits that underlie cell polarization. We focused on two properties that emerge as natural consequences of oligomerization: size-dependent membrane binding avidity (40,41) and size-dependent mobility (45). We find that both properties sharply enhance the ability of simple circuits to form and stabilize cell surface polarity. Both also contribute to controlling the rate at which asymmetries grow spontaneously or in response to local cues and the rate at which they evolve toward stably polarized states. Overall, our results reveal how strength of oligomerization, which determines mean oligomer size, can act as a physiological control parameter that determines if, when, and how fast polarization can occur.

Oligomerization promotes symmetry breaking through positive feedback and by reducing the dissipative effects of diffusion

Cell surface asymmetries are formed and maintained by the local differences in the relative rates of binding and unbinding (27) working against the dissipative effects of diffusion. Theoretical studies have established that both spontaneous emergence and maintenance of stable asymmetries require non-linear positive feedback on accumulation (26,28,32). For peripheral membrane proteins, this can involve either increased binding rates, or decreased unbinding rates, with increasing density. To amplify local asymmetries around a spatially uniform steady state, the net rate of accumulation must increase with cell surface density at the steady state (see Eq. 25 (26,28)). For a typical protein that binds the membrane with zero-order (density-independent) kinetics and unbinds with first-order (linear in density) kinetics, this implies that the combination of positive feedback on binding and detachment must produce a stronger than linear dependence of net accumulation rate on density, to amplify local asymmetries.

Here, we find that size-dependent membrane binding avidity endows oligomers generically with a form of positive feedback in which the effective dissociation rate constant decreases with increasing density. This form of feedback is insufficient, by itself, to drive spontaneous symmetry breaking. However, it strongly reduces the amount of additional positive feedback required to break symmetry to an extent that increases with increasing oligomer size. Even for weak oligomerization, characterized by a mean oligomer size of 2, positive feedback on monomer binding need only deliver an approximately $50\%$ increase over basal rates to drive symmetry breaking. As a simple comparison, achieving the same result (with the same feedback strength) with cooperative positive feedback on monomer recruitment would require greater than fourth order dependence (Fig. S7).

We also find that the size-dependent decrease in oligomer mobility can sharply reduce the dissipative effects of diffusion, which would otherwise degrade the potential for symmetry breaking. This effect would be weak for the weak size dependence of oligomer diffusivities observed in pure membranes. However, interactions with a submembrane cytoskeleton can lead to a much sharper size-dependent reduction in oligomer mobility. Moreover, the magnitude of this effect will increase with the strength of oligomerization. Thus, oligomerization of peripheral membrane proteins provides a tunable form of feedback and tunable control of protein mobility that can greatly enhance the potential for symmetry breaking and polarization. Importantly, the strength of this effect can be readily controlled by modulating the equilibrium binding constant for self-association; e.g., through phosphorylation.

Direct binding of cytoplasmic monomers to oligomers can either inhibit or drive polarization

For proteins that can bind membranes and self-oligomerize, there are two paths by which a cytoplasmic monomer can join membrane-bound oligomers: either indirectly, by first binding the membrane and then diffusing into contact with an oligomer, or by direct binding to the oligomer. Classical studies (52,53) have characterized the relative contributions of these two binding modes to protein absorption on membranes (53) or to ligand binding and uptake (52).

Here we characterized the relative contributions of direct and indirect biding modes to polarization. We find that the contribution of direct binding depends on whether or not the basal membrane binding and oligomerization kinetics obey detailed balance (i.e., whether or not they consume energy). When detailed balance is satisfied and polarization is driven by positive feedback on monomer binding to the membrane, direct binding to oligomers reduces the potential for polarization. However, another unanticipated mode of feedback emerges when detailed balance is broken in a way that favors a steady-state flux of monomers from the cytoplasm into oligomers. This could occur in many ways. For example, oligomers could recruit an enzyme that locally modifies monomers or phospholipids to promote association between monomers and the cell membrane. Regardless of how it arises, net flux from cytoplasm into oligomers at steady state is a form of positive feedback because its magnitude increases with overall density of membrane-bound subunits. As with other forms of feedback, we find that the potential for polarization increases with feedback strength (measured as the ratio of the net flux from cytoplasm into oligomers $\left(J_{net}\right)$ over basal monomer binding rate), and increasing oligomerization strength reduces the amount of additional feedback required for polarization. However, in this case, increasing direct binding increases feedback strength because it increases $\left(J_{net}\right)$.

Our results therefore highlight two general modes of positive feedback on recruitment that could drive symmetry breaking and polarization: one relies on increasing the effective number of membrane binding sites for cytoplasmic monomers; the other relies on breaking detailed balance to favor a net flux into oligomers. Because binding to the cell surface is a diffusion limited process, the relative densities of membrane binding sites and oligomers will determine which of these two modes is the more effective driver of cell polarity.

Oligomerization provides tunable control over the speed and mode of polarization

Our results also reveal how oligomerization allows tunable control over the speed of polarization. Weak oligomerization favors rapid growth of asymmetries and a rapid approach to a steady state in which the position of polarity boundaries is dictated by binding/unbinding kinetics and diffusivities intrinsic to the polarity circuit through mechanisms such as wave pinning (25). Stronger oligomerization reduces the rate at which local asymmetries grow and the rate at which they approach steady state. This allows external cues that promote local binding (14,17,54,55) or unbinding (51) of polarity factors, or their rapid transport by actomyosin flows (18,19,49) to impose spatial asymmetry patterns that are then maintained as quasi-stable states over much longer timescales. Although increasing oligomerization strength to maintain asymmetries far from steady state will also necessarily decrease the rate at which asymmetries grow, this can be readily overcome through the rapid control of local binding/unbinding or transport by external inputs. Thus, modulating the strength of oligomerization can allow tunable control, across different cells or within the same cell over time, over the relative extents to which circuit-specific reaction-diffusion dynamics and external inputs determine the spatial distributions of polarity proteins.

Homo-oligomerization as a versatile source of non-linear feedback and tunable mobility for cell polarity

The ability to self-oligomerize is common to a large fraction of peripheral membrane proteins. By recent estimates, something like $50\%$ of cytoplasmic and membrane proteins form homo-oligomers (56,57), although the majority of these form homo-oligomers of fixed size. This abundance is thought to be driven at least in part by the ease with which homodimeric interfaces can arise through random mutation and through selection for the many functional advantages (unrelated to polarization) conferred by homo-oligomerization (56,58), and also through entrenchment of randomly occurring, selective neutral mutations (59). Therefore, the frequent presence of oligomeric proteins in polarity circuits may be the result from exaptation of a frequently occurring property of proteins for an essential function in polarization. Importantly, because the generic form of non-linear positive feedback and tunable control of mobility conferred by oligomerization do not depend on polarity circuit architecture, or other modes of feedback, the barrier to exaptation is low.

Author contributions

C.F.L. and E.M.M. designed the research, carried out simulations, analyzed the data, and wrote the article.

Editor: Ben O'Shaughnessy.

Supporting material

Document S1. Figures S1–S7, supporting mathematical appendix, and

Supporting material

Document S2. Article plus Supporting

material