Model and theory

In our primary model, we consider two proteins P₁ and P₂, that bind in solution with equilibrium constant , where total concentrations of the proteins are [P₁]₀ = [P₁P₂] + [P₁] and same for [P₂]₀. If these proteins can also reversibly bind to membranes via targeting a specific lipid M, and continue to bind one another, their binding equilibrium will shift as a total of nine distinct species can form (Fig 1, Methods). The bound protein-protein complexes can either be in solution or on the membrane, [P₁P₂]^sol+mem = [P₁P₂]^sol + [P₁P₂]^mem = [P₁P₂]+[P₁P₂M]+[MP₁P₂] [MP₁P₂M], and unbound species are similarly defined [P₁]^sol + [P₁]^mem = [P₁]+[MP₁], and [P₂]^sol + [P₂]^mem = [P₂]+[P₂M], where M indicates a copy of a target membrane lipid bound to P₁ or P₂. The model thus assumes each protein binds membrane via targeting a single copy of a specific lipid type. Proteins on the membrane must be able to diffuse to bind one another, which is consistent with experimental observations [23] even of RNA-protein complexes (>6000kDa) that are anchored via multiple lipid binding sites along with myristoyl groups [24]. Each of the nine distinct species will be constrained to preserve detailed balance at equilibrium, as defined by the 10 pairwise binding interactions of Fig 1 (see Methods), and the total concentrations of proteins is fixed at the same values as above, but now [P₁]₀ = [P₁P₂]^sol+mem+[P₁]^sol+mem, and the same for [P₂]₀. Similarly, [M]₀ = [M]+[MP₁]+[P₂M]+[P₁P₂M]+[MP₁P₂]+2[MP₁P₂M]. We note that species on the membrane have concentrations normally of μm^-2, matching the units of equilibrium constants in 2D (K_a^2D)^-1. All species copy numbers, whether on or off the membrane, however, can be solved for in volume units when the appropriate Solution volume/Membrane surface Area (V/A) conversion factor scales the 2D binding constants, so we always report volume units for concentrations. To quantify the change in bound protein-protein complexes as a function of membrane localization we will define an effective equilibrium constant (1) This is not a true equilibrium constant, as both bound and unbound states as defined above contain several species that do not all stepwise interconvert with one another. However, from K_a^eff and initial protein concentrations [P₁]₀ and [P₂]₀, one can immediately solve for bound complex concentration using Eq 1. If proteins cannot bind to the membrane, the value of K_a^eff will revert to the solution bound value, K_a^PP, and thus the ratio of K_a^eff / K_a^PP determines the extent to which membrane localization either enhances or diminishes protein-protein complex formation. As we discuss further below, in the extreme limits where all proteins are either in solution or on the membrane, K_a^eff reduces to a true equilibrium constant. The strength of K_a^eff is that it also quantitatively describes all the conditions in between these limits. Thus, our K_a^eff definition offers a valuable metric for quantifying the equilibrium of the model in Fig 1, which must otherwise be defined by multiple quantities.

We derive below an exact expression for K_a^eff based on the 10 individual equilibrium relations for each reversible binding reaction (Fig 1, Methods). The value of K_a^eff for any protein pair will depend on volume V, surface area A, total protein [P₁]₀, [P₂]₀, and lipid concentrations [M]₀, and all six true equilibrium constants between protein and lipid interactions in 3D and in 2D . Importantly, the K_a^2D values (with units μm²/mol, e.g.) are different from the corresponding 3D values, but they are related through K_a^2D = K_a^3D/(2σ). The variable σ, with units of length, is a thermodynamic property of each binding pair that captures changes to binding free energy as a result of surface restriction and changes in standard state units. Changes to free energy are largely entropic, due to altered rotational freedom and protein flexibility [22], although limitations on the orientation of the binding interfaces could alter the enthalpy. The bending rigidity of the membrane can also affect σ, by controlling the relative orientations that the binding pairs can adopt [25]. The variable σ thus represents an independent variable that is specific to each protein pair studied. It is possible that even if concentrations increase on the membrane surface, a decrease in K_a^2D will cause less complex formation, and we quantify this regime in the Results section. To keep track of these distinct 3D and 2D equilibrium constants, we explicitly retain the 2D superscript for 2D binding, otherwise K_a (including K_a^eff) describes a 3D constant. To derive a simple analytical expression for K_a^eff, we input the pairwise equilibria (Methods) into Eq 1 and after canceling terms, we use the equilibrium expression (2) with n = 1 or 2, to complete the derivation (Methods).

Our main result is then a surprisingly simple and exact analytical relationship that quantifies the equilibrium solution of our model (Fig 1) via K_a^eff and the enhancement relative to K_a^PP. (3) where γ is a dimensionless constant V/(2Aσ), σ = K_a^PP/2K_a^2D,PP, [M]_eq is the unbound lipid concentration at equilibrium, and all equilibrium constants (including K_a^eff) and concentrations are in volume units (Fig 2). [M]_eq is a function of all the model parameters, and can only be solved exactly using numerical methods (Methods); we therefore derive an additional approximate analytical equation for [M]_eq described below (Fig 2B). However, in the regime where lipids are in excess, the result of Eq 3 is particularly simple because [M]_eq~[M]₀, the total concentration of lipids (Fig 2A). Critically, this means that the initial experimental conditions then directly determine the enhancement. In this regime only two factors control enhancement, the ratio V/(2Aσ), and the dimensionless strengths of membrane localization, , which report the ratio of membrane bound versus solution proteins (Eq 2) and which we term the membrane stickiness. Hence the volume-to-surface-area ratio, K_a^PP/K_a^2D,PP, and membrane stickiness play a primary role in triggering (or preventing) constitutive assembly on membranes. The right-hand side of Eq 3 is also constant for all K_a^PP values (Fig 2A). In this regime, our Eq (3) can also be applied to extract binding affinities on membranes (K_a^2D) from experiments where binding occurs both on membranes and in solution. This practical application of our result should help simplify the relatively rare experimental characterization of protein-protein affinities on surfaces, as the proteins need not be restricted to the surface for it to work. The condition of excess lipids can be satisfied even with a lipid recruiter such as PI(4,5)P₂, present at 2.5x10⁴μm^-2 in the plasma membrane [15], or ~1% of lipids [13] (Fig 2A), as we explore further below for proteins involved in CME.

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Fig 2. The equilibrium theory developed here accurately predicts how and when membrane localization will enhance protein interaction strength.

(a) Our theory (Eq 3) shown in solid black lines in all panels, is compared with ODE simulation results shown in red circles. For reference, the blue line shows the trend for pure solution binding, i.e. K_a^eff = K_a^PP_. The green line shows the maximum achievable enhancement, occurring if all proteins were on the membrane, K_a^eff = γK_a^PP. The gray dashed lines are included to contrast the K_a^eff calculated using a simple approximation that lacks cooperativity (S1 Text section 2D). From a₁ to a₂, the K_a^PM is decreased, producing lower but still constant enhancements, as the lipids are in excess relative to the proteins. (b) The number of unbound lipids is plotted as a function of K_a^PP with all other parameters fixed (S1 Text section 3A), showing how lipid binding is a function of the protein interaction strength due to the cooperative effect (Fig 1D and 1E). The theoretical prediction for [M]_eq is shown in solid black. (c) With fewer lipid recruiters relative to total cytosolic proteins ([P]_tot/[M]_tot >1), the enhancement is less pronounced, although for weak binders (low K_a^PP) even limited membrane localization causes significant increases in complex formation. From c₁ to c₂ the protein concentrations are increased. All results use σ = 1nm, see S1 Text section 3A for all other parameters. (d) If only one partner binds the membrane, the protein interaction remains fully 3D and no enhancement occurs.

https://doi.org/10.1371/journal.pcbi.1006031.g002

We derive an additional approximate equation for [M]_eq to provide a complete equilibrium theory of complex formation applicable to all experimental regimes, and we validate this equilibrium theory through extensive simulations of ordinary differential equations (ODEs) (Fig 2 and Fig 3, Methods). To briefly outline the derivation, we consider two limiting conditions for localization to the membrane: either there are no protein-protein interactions (K_a^PP = 0), giving , or complete protein-protein complex formation (K_a^PP = ∞), giving (Fig 1D and 1E). These two bounds are indicated by dashed lines in Fig 2B and both limits are independent of K_a^PP. We can continuously interpolate between them (Fig 2B) using the definition (4) where is the root of a quadratic equation, is the root of a cubic equation, and λ is the function of K_a^PP (also the root of a quadratic) that smoothly interpolates between them (see Methods for detailed derivations). This theory then provides a complete description of the equilibrium concentrations of all species, as from K_a^eff, one can directly calculate the total complexes formed. The partitioning of complexes between solution and the membrane can subsequently be derived utilizing the equilibrium relations of Fig 1 (S1 Text section 2A). The larger the enhancement, the more complexes must be on the membrane (S1E and S1F Fig).

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Fig 3. Protein interactions aided by strong protein-lipid interactions, abundant lipid recruiters, and low protein concentrations benefit most widely from membrane localization.

a,b) Enhancement ratios from ODE simulation (colored lines) and theory (black solid lines in all panels). The dashed black line is the upper limit for the enhancement ratio given by K_a^eff/K_a^PP = γ. σ = 1nm. Between the limiting cases where the membrane surface area is either too large to enhance binding (V/A⟶0) or is too small to effect binding (V/A⟶∞), a broad region of enhanced binding occurs. The vertical gray line is the V/A ratio for the yeast plasma membrane (0.5μm), for reference, and most physiologic V/A values fall in the range ~0.05–20μm (S3 Table). Within this physiologic range of V/A values is where we generally observe the largest enhancements. The cell geometry pictures provide examples of how one can produce different V/A ratios along the x-axis relative to the central sphere. A maximum enhancement for each parameter set is reached at a value of V/A where lipids still outnumber proteins (shown on upper x axis). Increasing (a) protein-lipid affinities and (b) lipid concentrations produces greater possibilities for enhancement. (c) Increases in the membrane stickiness (K_a^PM[M]_eq) produces monotonic increases in enhancement for all values of γ>1. (d) For lower expression levels relative to binding strength (K_a^PP = 10⁶M^-1), membrane localization can act as a switch to turn on assembly from <50% to >50% (shaded areas). (e) Timescales to equilibrate were calculated from simulations of both ODEs (solid lines) and reaction-diffusion (RD) (green points) (Methods). Weak lipid binding can reduce speeds (blue) relative to pure solution binding (dashed). The approximate theoretical bounds shown here for time-scales of binding either purely in solution (dashed) or on the membrane (gray) derive from the kinetics of irreversible association (S1 Text section 4C).

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Bounds on binding enhancement are determined by V/A and K_a^PP/K_a^2D,PP

Using our main result, Eq 3, one can predict when and by how much membrane binding will enhance complex formation of binding pairs without performing any simulation or experiment. Further, we can assess whether this enhanced complex formation is robust to perturbations in binding affinities or concentrations. To establish possible values for K_a^eff, we first ask: are there any cases where membrane binding will reduce protein complexation, i.e. K_a^eff<K_a^PP? To answer this, we consider the case where all proteins are on the membrane (), such that we have pure 2D binding and K_a^eff = γK_a^PP. Reduced protein complexation will occur only if γ<1, or V/A<2σ. The size of σ controls the relative strength of K_a^2D,PP vs. K_a^PP and varies from one binding pair to another, but experiment and theory indicate it is of the nanometer length-scale [21, 22, 26]. We therefore collected the V/A ratios for a wide range of cell types and organelles to illustrate that in nearly all cells, V/A>2σ (~20nm) and thus membrane localization will enhance binding (S2 Fig). Indeed, targeting the plasma membrane in most cells results in γ values much greater than 1, in the range 10–1000 (S2 Fig). Ultimately, the V/(Aσ) ratio is absolutely central in controlling observed enhancement, as it sets the maximum achievable K_a^eff. In most cases proteins will end up mixed between solution and membrane, and from Eq 3 this gives us K_a^PP≤K_a^eff<γK_a^PP (Fig 2). In the cases where only one protein binds to lipids, all 2D localization benefits are lost and no enhancement occurs: K_a^eff = K_a^PP (Fig 2E).

Cooperativity emerges in protein-lipid binding

An important feature that our analysis captures is the coupling that emerges between protein-protein affinity and protein-lipid binding due to membrane localization. If a bound protein-protein complex localizes to the membrane by binding one lipid, binding to a second lipid then becomes a 2D rather than a 3D search [12] (Fig 1D and 1E). Thus, stabilization of proteins on the membrane is achieved not only through strong protein-lipid interactions, but by feedback from strong protein-protein interactions. This cooperative effect for lipid binding (binding of one lipid changes the affinity for the second lipid) produces the unexpected result that the number of proteins bound to the membrane is dependent on the protein-protein interaction strength (Fig 2B). To contrast, one could instead consider lipid binding as simply partitioning proteins between solution and membrane, after which they form complexes as in Fig 1A and 1C. This simplified interpretation, shown in gray dashed lines in Fig 2, does not capture the cooperative effect (Fig 1D and 1E) and is clearly wrong for strong binding proteins that rely on cooperativity to stabilize complex formation (see S1 Text section 2D for details).

How binding enhancement responds to perturbations and factors external to the model

With our theory, it is possible to directly probe how changes in cell geometry, binding affinities, or concentrations will regulate enhancement. Although the model is too simple to fully describe multi-component assembly, by varying the input parameters to mimic competing cytosolic factors, one can evaluate the relative importance of concentration, affinities, and geometry. This is particularly true of equilibrium in vitro experiments. For changes to geometry, we first note that the equilibrium results depend only on the ratio and not the absolute size of V or A. The membrane does not have to surround the solution volume like the plasma membrane but can reflect binding to the outsides of liposomes, for example, which allows for studying any V/A ratio. Although it may seem that increasing V/A (through γ in Eq 3) always increases K_a^eff, this is not the case when [M]₀ is kept constant in its natural units of μm^-2 (it is then converted into Volume units by multiplication by A/V). V and A thus also control the initial copy numbers of proteins and lipids separately, such that large values of V/A have a great excess of proteins over lipids. This drives K_a^eff⟶K_a^PP (Fig 3A and 3B). For most physiologic V/A values (~0.05–20μm S3 Table) however, and physiologic concentrations of proteins (1nM-10μM) or lipids (10³−10⁵μm^-2 S4 Table), proteins are not in great excess, meaning significant enhancement is achievable depending on the membrane localization strength (Fig 3A and 3B).

Timescales of assembly vary strongly with binding affinity

Our theory (Eq 3) only describes the equilibrium state of the model. However, we can determine speeds of assembly via simulation. Now, the binding rates and the absolute values of V and A (not just the ratio) will influence the kinetics (all simulation inputs in S2, S3, and S4 Datasets). For these time-scales, we find that protein-lipid affinities K_a^PM are most often shown to be critical in controlling the overall time-scales of complexation, even driving slow-downs in speeds relative to solution binding (Fig 3E, S4 Fig). Changes in diffusion from solution to the membrane (about 100 times slower) affect the magnitude of association and dissociation rates and are captured implicitly in our ODE simulations (Methods), and explicitly in our spatially resolved reaction-diffusion simulations [19, 20]. However, the influence of diffusion on the reaction rates is rarely a dominant factor in physiological rate regimes (S4 Fig), indicating it is the binding strengths rather than slow 2D diffusion that determine assembly speeds. However, we note that our comparison of ODE and RD kinetics was performed in relatively small RD systems due to simulation costs, and it is true that as spatial dimensions increase, times to diffuse to reach the membrane will influence the overall equilibration times. The timescales we calculated for protein pairs and scaffold mediated systems (S4B and S4C Fig) were performed using ODEs at their corresponding cellular dimensions (S3 Table): V = 1200 μm³ (human) and V = 37.2 μm³ (yeast). Performing RD simulations at these dimensions would produce slower relaxation times, particularly for human cells, due to the time required to reach the surface. Crowding would also lower effective diffusion constants of proteins, although the decrease in time-scales to equilibrate would be negligible unless binding rates were strongly diffusion-influenced (Methods).

Biological relevance for proteins in CME

To test the biological relevance of membrane localization for driving complex formation and assembly, we collected biochemical (Table 1, S1 Table, S2 Table), concentration (Table 1, S4 Table), and cellular geometry data (S3 Table) for interactions among 37 membrane targeting proteins in yeast and human cells, including 22 proteins involved in clathrin-mediated endocytosis (CME). We first study only individual protein pairs that can bind according to our model of Fig 1, (S1 Table) shown in Fig 4A: the membrane binding proteins AP-2, DAB2, ARH, FCHo1, FCHo2, HIP1, HIP1R, PICALM, SH3GL2, EPN, AP180, SLA2, and SYP1. In Fig 4B and 4C we show results of binding between specific pairs. We used cytosplasmic concentrations of the proteins (Table 1) and the targeted lipids (S4 Table), and the relative Volume and Area from their respective cell types (S3 Table). Binding constants are collected from previous experimental studies (Table 1, S1 Table, S3 Dataset), and for 2D binding constants we test values of σ = 1nm (Fig 4) and 10nm (S5 Fig). Our results thus provide quantitative insight into how these pairs in isolation would use membrane localization at physiologic conditions to drive their protein-protein interactions. For some proteins, such as AP-2, solution values have been measured with partners (Fig 4A, S1 Table), but further experiments indicate that the proteins undergo minimal binding in solution due to conformational regulation [1]. Despite this additional regulation, membrane localization will still increase complex formation relative to what is observed in solution (γ>1), so the effect is quantified here using the measured value (S1 Table). Using our theory along with simulations for verification and time-scales, we find that affinities of these CME binding pairs can be enhanced 10–1000 fold by binding to membranes (Fig 4B, S5 Fig for results with σ = 10nm). With limited binding in solution for most pairs, membrane localization then triggers a dramatic increase in complex formation (Fig 4C, S5 Fig). The central adaptor protein AP-2 is responsible for many of these interactions, showing the capacity to trigger assembly with nearly all of its binding partners (Fig 4C, green bars). Even though we assume binding is possible for AP-2 in solution, it is still quite limited prior to localization. Not surprisingly, knockdown of AP-2 in mammalian cells causes severe disruption of endocytosis [28], underlining its secondary importance only to the irreplaceable clathrin [28] and PI(4,5)P₂ [29]. We note that AP-2 can potentially bind up to three PI(4,5)P₂ copies [30, 31], meaning that there will be less free lipids available for each AP-2. With fewer lipids, enhancement and complexation will be reduced, but is still quite large (S5 Fig). For some proteins such as FCHo1 (SYP1 in yeast), binding affinities (K_a^PP or K_a^PM) are not available, and this F-BAR protein does not target a single lipid specifically. However, by considering ranges of membrane stickiness values, we can use our method to identify which combinations (S6 Fig) best describe the experimental observation that these proteins only localize effectively to membranes when they can bind other proteins [18, 32]. We find for this protein, membrane stickiness values of ~0.5 produce membrane targeting that is sensitive to protein-protein interactions, whereas once values exceed ~1, no partners are needed to target the membrane effectively (S6 Fig).

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Fig 4. Membrane localization triggers strong complex formation for pairs of protein binding partners involved in clathrin-mediated endocytosis.

(a) Interactions between lipid binding human (green) and yeast (blue) CME proteins are shown along with K_d^PP values (μM) measured (red text), inferred through structural and functional homology (green), or estimated (blue) (Table 1, S1 Table, S3 Dataset). Sizes indicate concentrations (Table 1). (b) Enhancements for each of these CME binding pairs, following the model of Fig 1, were calculated using Eq 3 and verified through numerical simulation of ODEs. Pairs involving human AP-2 are in dark green, light green points involve AP-2 with cargo binding-adjusted K_a^PM, with red and blue points showing other human and yeast proteins, respectively. The x-axis is the average membrane stickiness for each pair, and here we assume σ = 1nm (see S5 Fig for σ = 10nm). For poorly characterized lipid binding affinities, we considered ranges of values (error bars in x), producing ranges of enhancements (error bars in y). Gray lines are guides for fixed V/A ratios. Protein names in parentheses are homo-dimers. (c) The percent of proteins in complexes increases from solution binding levels (gray bars) as a result of membrane localization (colored bars). All results and parameters used for all data points in S3 Dataset.

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We further interrogate two additional mechanisms for stabilization at the membrane by lipid binding proteins such as AP-2, epsin, and Dab2 [3]. First, they each bind transmembrane cargo after membrane localization, which acts to effectively increase the K_a^PM by increasing their residence time on the membrane. K_a^PM is a factor of ~40 higher for AP-2 binding to PI(4,5)P₂ when cargo is available [30]. Interestingly, these cargo stabilized interactions (Fig 4B, light green) do not make a significant impact on complexation when we assume the full 1% PI(4,5)P₂ concentration is free to bind, as the numerous lipids outweigh a need for stronger binding (Fig 4C, light green). However, when we evaluate complexation with PI(4,5)P₂ pools diminished by a factor of 10 due to assumed competition from other PI(4,5)P₂ binders, now cargo stabilization via higher K_a^PM does help recover strong complexation on the membrane (S5 Fig). This suggests that cargo binding, which is a main functional goal of CME, becomes a significant regulator of adaptor stabilization when competition from multiple adaptors limits PI(4,5)P₂ binding. Second, when these adaptors can bind multiple partners with distinct appendage domains [33], we see more proteins on the membrane due to the increased difficulty of un-tethering from the membrane domain (S6B Fig).

In the cellular environment, CME proteins are of course not in isolation and can both compete and cooperate with one another to form higher order assemblies, induce conformational changes, and occupy lipid binding sites on the membrane. Thus we can only speculate about the role of localization in nucleating clathrin-coated pits in vivo. However, based on the above analysis showing that, physiologically, γ is greater than 1, membrane localization will drive clathrin towards complex formation. The initial nucleation of clathrin-coated pit sites is difficult to resolve experimentally because of the challenges in tracking the many participatory proteins simultaneously, and because prior to cage formation, the density of molecules is, by definition, low. Experiments have tracked the role of AP-2 and clathrin in nucleating sites [34], which we discuss below.

Scaffold mediated interactions of CME proteins also exploit membrane localization

To go beyond our Fig 1 model of pairwise protein binding and thus characterize how scaffold proteins (Table 1: ITSN1, EPS15, EDE1 and SLA2) stabilize complex formation at the membrane despite not directly interacting with the lipids (model in S7 Fig, list of interactions in S2 Table), we simulate systems of ODEs, as Eq 3 no longer applies (Methods). We thus simulate interactions involving three proteins, two of which can bind lipids but not each other, and the third that binds both peripheral membrane proteins but not the membrane (Fig 5A). Our results in Fig 5B and 5C show that while scaffold mediated complexes can still capitalize on 2D localization for binding (Fig 5B), because localization is now mediated by peripheral membrane proteins that are at much lower concentrations than the lipid recruiters, we find that the increase in complex formation is less robust (Fig 5C), and is limited by concentration of the scaffold protein (S8A Fig).

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Fig 5. Scaffold-mediated interactions can also exploit membrane localization to drive complex formation in clathrin-mediated endocytosis.

a) When sets of three proteins can form a complex and two of them also bind to lipids, localization is again capable of driving stronger complex formation. Yeast proteins in blue and human proteins in green, where the uppermost four (SLA1, EDE1, EPS15, ITSN1) are scaffold proteins that do not bind lipids. Affinities follow the same color scheme as Fig 4, and we have included only binding interactions between these proteins that are shown in parts b-c. (b) Because our primary model (Fig 1) no longer applies, we cannot use Eq 3 to predict enhancements, and instead use simulations of ODEs. The scaffold model is defined in S1 Text section 1C and S7 Fig, along with the definitions of K_a^eff,SP and K_a^sol,SP, where neither is a true equilibrium constant and the equilibrium results of the simulations are used to calculate their defined values. Without membrane localization, K_a^eff,SP→K_a^sol,SP. Scaffold proteins (s in the labels) are eps15/ede1 (orange), or itsn1/sla1 (pink). (c) Complexation involves all three proteins (see S1 Text section 1D) and here again, the percent of proteins in complexes increases from solution binding levels (gray bars) as a result of membrane localization (colored bars). All results and parameters in S4 Dataset.

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Clathrin cage nucleation and BAR domain oligomerization can exploit membrane localization

Thus far we have not discussed clathrin, the central component of the CME vesicles that does not actually bind to lipids itself. In vitro experiments find that clathrin polymerization on the membrane (via adaptor binding) is more robust than occurs in solution (with adaptors still present), supporting a role for membrane localization in its nucleation and assembly [2]. Clathrin is a trimeric protein with three binding sites to target peripheral membrane proteins. It polymerizes with itself into hexagonal lattices without competition from the peripheral membrane proteins. Thus, its interactions with peripheral membrane proteins not only increase the quantity of protein bound to the membrane, it can help drive 2D polymerization between clathrin trimers. Through (non-spatial) stochastic simulations (Methods) set-up to mimic recent in vitro experiments [1], we explored a range of clathrin-clathrin interaction strengths to show how membrane recruitment by the AP-2 adaptor [1] can enhance clathrin polymerization yield (S8 Fig). Although these simulations lack molecular structure or spatial resolution, they can track formation of multi-protein complexes and the important role of affinity and concentration in controlling these complexes. We find that clathrin localizes to the membrane first via AP-2 binding before assembling into cages in 2D for the most reasonable K_ds of 10–100μM [35]. This result is supported by evidence from in vivo experiments that probe the early stages in the nucleation of clathrin coated pit sites through tracking of AP-2 and clathrin [34]. They found that clathrin arrives at the membrane most frequently (75%) as a single trimer, and bound to at least one but most often two AP-2 molecules [34]. Nucleation can then initiate in two ways: (A) another clathrin trimer localizes to the membrane via AP-2 and these trimers dimerize in 2D or (B) another clathrin trimer is directly recruited by the clathrin on the surface. Although the subsequent clathrin dimerization events were not resolved experimentally, preventing definitive evidence of membrane localized clathrin-clathrin assembly, the fact that each clathrin is bound to AP-2 suggests that AP-2 binding of clathrin is a prerequisite for initial clathrin dimerization. From our simulations, the (A) nucleation process is markedly dominant. There is a strong driving force both from affinity and from concentration for AP-2 to bind any of the 25000 PI(4,5)P₂/um², and correspondingly minimal drive for a solution clathrin to bind a small number of clathrin trimers localized to the surface. We note that because clathrin also arrives at the membrane as dimers or higher order complexes 25% of the time, solution binding of clathrin also contributes to nucleation of pit sites, but to a much lower extent [34]. Interestingly, once pit sites have formed, assembled clathrin cages exchange with solution clathrin with the aid of ATP-consuming proteins that facilitate remodeling of the clathrin cage [36]. Thus clathrin-clathrin interactions from solution certainly play an important role in the cell in maturing the pit sites [36].

CME proteins with BAR domains that dimerize, appear to oligomerize only on the membrane, and are functionally important for driving membrane deformation [17, 18] can also exploit localization to drive their binding interactions. We study isolated FCHO1/2 oligomerization and endophilin (SH3GL2) oligomerization, again using non-spatial stochastic simulations (S9 Fig). Here again we consider a range of K_a^PM values to capture uncertainty in the membrane stickiness of these domains. We find the stoichiometry of the dimerization pair (homo or hetero) is central in determining whether large oligomers form. With matched pairs, homodimers such as endophilin form larger oligomers that feedback into higher stabilization at the membrane, whereas the disparity in FCHo1 and FCHo2 concentration (S1 Table) produces more isolated dimers. Experiments have shown that BAR domains exhibit stronger binding to curved membranes [17]. Because we lack this cooperative feedback in our model between oligomers tubulating membranes and thus potentially increasing affinity for subsequent proteins, our result can be interpreted as a lower bound on observed oligomerization.

In all cases, an important outcome of these strong binding interactions on the membrane is that they are difficult to disassemble, consistent with findings that unproductive assembly events observed in vivo [37] require the ATP-driven uncoating machinery for disassembly [2, 38]. Our results demonstrate that establishing the physiologic significance of these polymerization observations depends not only on protein concentrations and solution conditions, but also the V/A ratio. Thus, this ratio should be regarded as a critical factor in designing in vitro experiments to better reflect in vivo behavior.

Biological relevance for membrane remodeling pathways in yeast

Lastly, our analysis motivates why diverse proteins that target membranes in yeast can follow pathways to assembly both similar and distinct from the CME proteins. In particular, the CME protein pairs produce limited protein-protein complexes when isolated in solution, but experience large enhancements due to membrane localization, triggering widespread protein-protein interactions only after binding to the membrane (Fig 4). For 15 yeast proteins involved in daughter cell budding, lipid regulation, and morphogenesis, we studied their pairwise binding in using cytoplasmic (yeast) concentrations (Table 1), lipid concentrations (S4 Table), cytoplasmic V/A ratios (S3 Table) and experimentally measured protein-lipid affinities (Table 1). We find binding enhancements are high (100–1000), similar to the CME proteins, indicating that binding will be promoted once proteins are on the membrane (S10 Fig, S2 Dataset). Although enhancements were readily measured for these yeast binding pairs because they were independent of K_a^PP values (S2 Dataset), we could not directly compare complexation for these interactions as we did for the CME interactions because they lacked any K_a^PP data. For binding enhancements, we found an exception in the coat forming proteins targeting endosomes (VPS5, VPS17, SNX4, SNX41), which only exhibit enhancements <20. These proteins target the PI(3)P lipid but most bind only weakly (K_d^PM>100μM) [39], limiting their enhancements despite a favorable V/A ratio at the endosome (S10 Fig, S3 Table). Unlike in CME, however, these coat proteins form stable interactions in solution [40]. Thus, rather than membrane binding triggering protein interactions, we would first expect the reverse: strong protein interactions in solution function to target and stabilize protein at the membrane through the cooperative effect (Fig 1D and 1E, Fig 2B). We test how the binding of the retromer components VPS5 and VPS17 to the endosome will be significantly enhanced by forming a higher order assembly in solution with the strong lipid binding cargo adaptor, SNX3 (S9 Fig). SNX3 targets PI(3)P with stronger affinity (~2μM) than either VPS5 or VPS17 [39], and is known to improve recruitment of the retromer to endosomes [40]. Once these small pre-assembled coat subunits are on the membrane, they can then continue to exploit localization to form larger protein coats.

Conclusions

We conclude by noting that assembly on membranes is regulated to occur at specific times or sub-cellular locales, and our theory provides a useful aid in predicting the changes in local protein, lipid concentrations, and affinities that are necessary to trigger (or prevent) such assembly. Ultimately, our theory is most powerfully applied to interpreting in vitro results, due to the simplifying assumptions of the model, and can improve the design and quantitative interpretation of assays probing multi-protein complexation at membrane surfaces. Also, given known protein-protein and protein-lipid binding affinities, our theory can quantitatively predict the results of in vitro experiments that mimic Fig 1, thus avoiding the need for such measurements. Our results indicate that even relatively low lipid concentrations (i.e. PI(4,5)P₂ at ~1% of plasma membrane lipids) can be sufficient in many cases to stabilize proteins to membranes and drive protein-protein interactions. We found that additional factors, such as cargo binding by adaptor proteins in CME, are only strong regulators of membrane localization or protein interactions under specific conditions. Since cargo-binding is known to influence the success of vesicle formation in vivo[41, 42], this suggests that the condition where total PI(4,5)P₂ concentration is reduced to mimic competition from other proteins is more physiologically relevant. A fruitful means of exploring in more detail the role of cytoplasmic factors, as well as spatial heterogeneity, crowding, and non-equilibrium dynamics, is through reaction-diffusion simulations, although we note the results will then be dependent on many additional parameters. Overall, the theory we provide here offers a general and useful quantitative guide for predicting when or if membrane localization plays a role in the cellular control of self-assembly.

Theoretical derivations

Definition of microscopic and macroscopic rates for simulation

To simulate the systems of ordinary differential equations (ODEs) for Fig 1 (S1E Fig, S1A Text section), we need macroscopic rates, and to simulate the single-particle reaction-diffusion system (RD), we need microscopic rates (also known as intrinsic rates in the Smoluchowski theory [43]) in both 3D and 2D. The macroscopic rates emerge based on the dynamics of the more detailed microscopic system, and can therefore be constructed to optimally match the kinetics of the ODE simulations to the RD simulations. We note that these definitions are specific to the kinetics, as the equilibrium of both simulation approaches will be identical due to their matching equilibrium constants.

The ODE simulations do not account for space or explicit diffusion. Here, we define their rates to implicitly account for changes to diffusion and thus best match the RD simulation kinetics. That way, discrepancies between kinetics of ODE and RD results can be attributed to explicit spatial heterogeneity influencing the binding interactions. Macroscopic association (on-) rates can be defined in 3D from the intrinsic rate of the Smoluchowski model via the relation [44]: (9) where k_a is the intrinsic association rate that captures the barrier to complex formation for species in contact at binding radius σ, and D_tot is the sum of both species’ diffusion constants. The macroscopic off-rate can be defined in all dimensions via (10) The intrinsic dissociation rate k_b is defined via the corresponding equation, k_b = k_a/K_a, with all off-rates having the same units in all dimensions of s^-1. In 2D, there is no single macroscopic rate constant independent of the system size or concentrations [20]. However, one can define a macroscopic 2D rate, built on theory from Szabo et al [45], that provides optimal agreement with the corresponding spatial reaction-diffusion simulations via [20]: (11) where (12) is a length scale that is defined based on the more concentrated of the reacting species P₁ or P₂ in the surface area A.

The important interpretation of Eqs 9 and 11 is that, unless k_a is large, even substantial (factor of 10 or more) changes to the diffusion constant will have a relatively small impact on the macroscopic rate. It is not until macroscopic rates reach values of ~10⁶-10⁷M^-1s^-1 that they become strongly diffusion influenced and thus sensitive to changes in diffusion.

Our 2D intrinsic rates are defined relative to our 3D rates via (13) and unbinding rates (14) which produces the equilibrium relation defined in the main text, K_a^2D = K_a^3D/(2σ). We assume here that the dissociation rates are the same from 3D to 2D. It is the association rates that capture two species finding one another in a specific spatial dimension. This definition of Eq 13 also can be shown to preserve the reactivity of the binding interaction in the Smoluchowski model from 3D to 2D, independent of changes to diffusion (S1 Text section 4A). For the macroscopic 2D rates, k_on^2D, we used Eq 13 in Eq 11, which allows us to capture effects of diffusion towards timescales of binding in k_on^2D, as D^2D is ~100 times lower than D^3D. Transitioning from solution to the membrane via binding lipid or protein involves a 3D search, and thus uses the corresponding 3D rates. See S1 Text section 3B for further discussion.

Ultimately, the results of K_a^eff are only sensitive to equilibrium constants such as K_a^2D and therefore the size of σ, rather than sizes of relative rates. This length scale σ encodes thermodynamic properties of the molecules involved in the binding reaction and is of the nanometer range [22]. In general, the value of σ therefore depends on the proteins involved, but σ (or K_a^2D), is almost never measured. We extract σ~7nm (from V/A = 6.7μm and K_a^eff/K_a^PP≈500) in the experimental measurement of 2D binding between calmodulin and a target peptide [21]. Smaller σ values have been observed [26]. For simulations, we thus used either 1 or 10nm. We used the same value for the protein-protein (σ^PP) or protein-lipid (, ) 2D binding interactions, although only σ^PP appears in Eq 3. The size of these values is constrained to ensure an equilibrium steady-state is reached, and the simplest solution has that .

Computer simulation methods

Collecting biochemical data, in vivo geometry, and concentrations

In Table 1 we list all the human and yeast proteins for which we were able to collect sufficient biochemical data on lipid and protein interactions. The 20 lipid-binding yeast proteins were retained from a larger list of 139 peripheral membrane proteins (PMP) identified from the Uniprot database as having lipid binding activity in yeast (S1 Dataset). Between this set of 139 PMPs, we found 396 interactions via BioGRID, however, only 17 pairs (S1 Table) involved partners with known K_a^PM’s. The 15 human proteins studied are all involved in CME and their biochemical data (Table 1, S1 Table, S3 and S4 Datasets) was collected via extensive literature curation. To study scaffold-mediated interactions (S2 Table), we identified all possible interactions that involved a non-membrane binding protein that could simultaneously and non-competitively bind to two of our PMPs. For the yeast proteins, these interactions could be identified from the manually curated interface interaction network for CME proteins [48]. There was a relatively small number of examples where a single scaffold protein was capable of bridging two PMPs (S2 Table). These interactions in humans/yeast involved clathrin/clathrin, eps15/ede1, or itsn1/sla1.

In S3 Table we collected volume and surface areas for cells and organelles with justifications provided. Because the cytoplasmic volume typically constitutes 50–60% of the total cell volume in mammalian cells, our V/A ratios set the solution volume as 60% of the total cell volume for all cell types. Lipid concentrations are collected in S4 Table. The concentrations of specific lipids on specific membranes have only been quantified in a few cases, such as PI(4,5)P₂ having an average concentration of 2.5x10⁴μm^-2 on the plasma membrane in mouse fibroblasts [15]. We used this concentration as a gold standard, due to its relative consistency across measurements [13, 15], and other phosphoinositide concentrations were quantified relative to this one. We curated literature to collect the necessary copy numbers of each lipid in the cell, and their distributions across organelles. Lastly, protein concentrations were defined from copy numbers measured in yeast [49] and human cells [50] (Table 1).