Curvature sensing as an emergent property of multiscale assembly of septins

Significance

Cells use their membrane curvature to coordinate the activation and spatiotemporal compartmentalization of molecules during key cellular processes. Recent works have identified different proteins that can sense or induce membrane curvature from nanometer to micrometer scale. Septins are nanoscopic cytoskeletal proteins that preferentially bind to membranes with a narrow range of micrometer-scale curvatures. Yet, the sensing mechanism remains ambiguous. Using a combination of microscopy and kinetic modeling, we show that, unlike most proteins that sense curvature in a single protein scale, curvature sensing in septins emerges from their multistep hierarchical assembly on the membrane. This introduces a protein localization scheme where the same protein can be deployed for differential binding sensitivities in different cellular contexts.

Abstract

The ability of cells to sense and communicate their shape is central to many of their functions. Much is known about how cells generate complex shapes, yet how they sense and respond to geometric cues remains poorly understood. Septins are GTP-binding proteins that localize to sites of micrometer-scale membrane curvature. Assembly of septins is a multistep and multiscale process, but it is unknown how these discrete steps lead to curvature sensing. Here, we experimentally examine the time-dependent binding of septins at different curvatures and septin bulk concentrations. These experiments unexpectedly indicated that septins’ curvature preference is not absolute but rather is sensitive to the combinations of membrane curvatures present in a reaction, suggesting that there is competition between different curvatures for septin binding. To understand the physical underpinning of this result, we developed a kinetic model that connects septins’ self-assembly and curvature-sensing properties. Our experimental and modeling results are consistent with curvature-sensitive assembly being driven by cooperative associations of septin oligomers in solution with the bound septins. When combined, the work indicates that septin curvature sensing is an emergent property of the multistep, multiscale assembly of membrane-bound septins. As a result, curvature preference is not absolute and can be modulated by changing the physicochemical and geometric parameters involved in septin assembly, including bulk concentration, and the available membrane curvatures. While much geometry-sensitive assembly in biology is thought to be guided by intrinsic material properties of molecules, this is an important example of how curvature sensing can arise from multiscale assembly of polymers.

Cells and their internal compartments take on a spectacular array of shapes. How cells generate and utilize their shape to best suit their functions has been a central question in biology since the advent of the light microscope. Since then, much has been learned about how cells generate complex shapes, though little is known about how they sense their shape. Most of the work done on cell shape sensing has been done in the context of endocytosis and vesicle trafficking, where nanometer-scaled proteins assemble onto nanometer-scaled membrane curvatures. For example, proteins containing BAR domains and/or amphipathic helices are well known for their roles in both sensing extreme curvature and deforming the membrane to aid in vesicle formation (1–4). A combination of electrostatic and hydrophobic interactions allows for high-affinity binding of these proteins to curved membranes (1, 4). Convexly (or positively) curved membranes are characterized by an increase in the size and number of lipid-packing defects or exposed hydrophobic regions between lipid species, which act as binding sites for amphipathic helices (5, 6). Cells can use the complex physicochemical landscape of membrane curvature, which includes the shape and size of lipid head groups, the length of fatty acyl chains, and the electrostatic content as important features to drive preferential assembly of proteins onto curved surfaces (7–9).

Changes to cell shape, like those that occur during cytokinesis or polarized growth, require changes on a micrometer scale. This raises a fundamental question: How can cells use nanometer-scaled assemblies to detect micrometer-scale changes in membrane curvature? This disconnect in scale between molecules and geometry is striking, as a micrometer-scale curvature is essentially a flat surface for the typical peripheral membrane protein. However, these shallow curvatures are found throughout the cell at both organelles and the plasma membrane, suggesting that cells must have the means to detect structures of this magnitude. SpoVM, a bacterial protein involved in sporulation, is one of the few documented micrometer-scale curvature sensors and uses a unique amphipathic helix to embed into the membrane similarly to nanometer-scale curvature sensors (10). Another documented class of micrometer-scale curvature sensors is that of septins.

Septins are GTP-binding proteins that can be found throughout many branches of life, including fungi, algae, and animals but appear to have been lost in major lineages such as land plants (11–14). Septins are present in the cytosol, associated with actin and microtubules, and bound to the plasma membrane where septins act as scaffolds to coordinate cell cycle progression, polarized growth, and fungal pathogenesis (15–24). While septin organization at the plasma membrane is complex and controlled by a host of effector proteins and posttranslational modifications (25–29), septin filaments notably localize and orient along regions of positive membrane curvature such as at the mother–bud neck during bud emergence in yeast (30), along branch points in filamentous fungi (31), when engulfing intracellular bacteria (32), along cell protrusions in human fibroblasts (31), and along the base of dendrites in neurons (19, 20). In vitro reconstitution of recombinant yeast and human septins on supported lipid bilayers of varying curvatures displays a narrow preference for micrometer-scale curvature, attributed to an amphipathic helix domain in the C-terminal extension of Cdc12 in yeast and Sept6 in humans (9, 31, 33).

While the number of septin genes and their expression varies by organism and tissue type, studies using recombinant yeast, C. elegans, and mammalian septins commonly indicate that septin subunits assemble into palindromic, rod-shaped hetero-oligomeric complexes that constitute the minimal unit (34–36). Septin assembly at sites of membrane curvature begins with oligomers binding to the membrane where they may diffuse and assemble into nonpolar filaments that can both fragment into shorter filaments as well as connect with nearby filaments to form higher-order structures, including pairs, rings, and gauzes (35, 37–39). The mechanism by which these hierarchical steps of assembly contribute to septins’ curvature-sensing function remains largely unexplored.

As in earlier studies, here, we use silica beads of varying diameters that are treated with small unilamellar vesicles to form spherical supported lipid bilayers (40). In earlier studies, we used mixtures of beads of multiple diameters (polydispersed) to study the curvature-sensitive binding of septins and found that septin adsorption is maximized on 1-μm beads, consistent with the curvature where they are found to localize in cells (9, 31). As we will show here, mono-dispersed (single curvature) and bidispersed (two curvatures) assays exhibit different adsorption patterns and apparent curvature sensitivities. These observations suggest that membranes of different curvatures can interact/compete nonlocally and that the “optimal” curvature sensed by septins is not fixed—it is instead a function of the assembly process and available membrane platforms.

We use a combination of single-molecule imaging, scanning electron microscopy, time-lapse confocal microscopy, biophysical modeling, and computer simulations to propose a unified kinetic model that connects several different time-dependent processes to the observed curvature sensing of septin assemblies. Our model correctly predicts the observed time-dependent septin bindings at different bulk concentrations to beads of different radii. It also predicts the different behaviors observed in monodispersed and bidispersed assays. From our model, the difference between the assays arises due to the competition between beads of different radii as a shared, finite pool of septins is depleted by septin assembly on the beads. We provide experimental support for this prediction by varying the total membrane surface area to alter the degree of depletion. We find that processes of diffusion and end-on annealing of bound filaments are not sufficient for curvature sensing; instead, we hypothesize that they control the rate of formation of a foundation layer of septins. The model predicts that septin binding is dominated by cooperative interactions between bound septin filaments in the foundation layer and free septin oligomers in the bulk to create curvature-dependent assemblies.

Results

Time-Dependent Adsorption of Septins.

To study the assembly process of membrane-bound septins, we used confocal microscopy and measured time-dependent septin adsorption onto membrane-coated beads with assays that were developed and used in our earlier studies (9, 41). One main difference between our current and past studies is that here, the adsorption is measured over time, whereas in previous studies, we reported only the adsorption at steady state. Second, here, we directly compare assays where all spherical supported lipid bilayers are the same radius (monodispersed) with assays of two different radii (bidispersed), also referred to as the competition case. As in previous studies, the total membrane surface area is kept constant between all conditions unless specifically stated, and the surface area of 1-μm to 5-μm beads in bidispersed cases is always 1:1.

The fluorescence intensity of septins was measured through time and as a function of bead size with diameter 1 μm or 5 μm, each at bulk concentrations 6.25, 12.5, 25, and 50 nM septins in monodispersed assays (Fig. 1 A and B). The time-dependent adsorption onto 1-μm and 5-μm beads in competition (bidispersed assays) at bulk concentration 25 nM is displayed in Fig. 1C. The main observations can be summarized as follows:

1. The time-dependent adsorption resembles a sigmoidal curve and can be divided into three sequential periods, regardless of the bead radius and bulk concentration: i) initiation period, where the adsorption remains limited and increases slowly, ii) growth period, where both the adsorption and adsorption rates increase superlinearly with time, and iii) saturation period, where the adsorption rate begins to slow down and adsorption approaches steady state (Fig. 1D); also, SI Appendix, section F.

2. In monodispersed assays, the initiation and growth timescales are reduced (adsorption rates are increased) by increasing the bulk concentration of septins and membrane curvature.

3. Most interestingly, we observe larger steady-state adsorption on 1-μm compared to 5-μm beads only when the two bead sizes are coincubated. In contrast, when only a single bead size is present in the reaction, we find larger steady-state adsorption levels on 5-μm beads than those on 1-μm beads, while controlling for surface area and concentration of septins in the bulk (Fig. 1 C and F). Similar steady-state differences were seen when comparing 1-μm and 3-μm beads in previous studies (9). Notably, higher adsorption on 1-μm beads in the competition setting is associated with its shorter lag times and faster growth rates, compared to 5-μm beads, indicating kinetic differences in the assembly process on the two curvatures when they coexist.

Fig. 1.

Time-lapse microscopy on spherical supported lipid bilayers shows context-dependent septin assembly kinetics. (A–C) First row: Schematic representations of the view of monodispersed (A and B) and bidispersed assays (C) containing silica microspheres coated in a lipid bilayer (purple) of 75% DOPC and 25% phosphatidylinositol; the total membrane surface area is kept fixed in all experiments. Second row: The time variations of the representative focal slices of septin adsorption (green) at various concentrations onto (A) monodispersed 1 μm and (B) 5 μm beads and (C) bidispersed mixture of beads at bulk concentration n_b = 12.5 nM. The lipid channel is in magenta, and the scale bars are 1 μm in (A), 2 μm in (B), and 1 μm in (C). Third row: Quantification of septin adsorption. Error bars correspond to the SE. (D) Diagram of the adsorption process, separated into three regimes: initiation, growth, and saturation. (E) Subprocesses involved in the septin adsorption process: ① binding of a single-septin oligomer in the bulk; ② the diffusion of bound septins on the membrane; ③ their polymerization through end-on annealing; ④ fragmentation of bound septin filaments into shorter ones; ⑤ cooperative binding of bulk septins, and ⑥ unbinding. (F) Ratio of septin adsorption at steady state on 1-μm beads to septin adsorption on 5-μm beads for monodispersed and bidispersed assays.

These results indicate that maximum septin adsorption to different curvatures is not absolute but can vary depending on the presentation of single or multiple curvatures in the same reaction. The time-dependent adsorption of septins is determined by the summation of different processes involved in septin assembly, including membrane binding/unbinding, diffusion, end-on annealing into filaments, fragmentation of filaments, and potentially binding/unbinding of septin–septin interactions that may form layers or lateral assemblies (Fig. 1E). Which of these many steps, if not all, drives the curvature-dependent assembly of septins in these different experimental regimes? We combine experiments with modeling to examine the relationship between membrane curvature and bulk concentration at each of these discrete steps and their potential impact on determining initiation, growth, and saturation periods. Note that the adsorption process typically takes tens of minutes, which is thousands of times longer than the time scale associated with a single oligomer’s dwell time (9). This long time interval makes using time-costly atomistic dynamic simulations and its more coarse-grained variations incompatible with the questions we seek to answer; thus, we use a Smoluchowski reaction kinetic framework (42, 43), informed by the findings of this work and previous ones, to simulate the time-dependent assembly of membrane-bound septins.

Binding/Unbinding Kinetics of Single-Septin Complexes.

Septin assembly begins with single-septin oligomers binding the membrane from the bulk. Our earlier studies show that the binding rate of a single-septin oligomer monotonically increases with curvature (9). Curvature-dependent binding rates have also been reported for the only other characterized micrometer-scale curvature sensor, SpoVM (10). Based on these findings, we hypothesized that curvature-dependent binding of septins might be the main driver for the assembly and curvature sensing of septin assemblies. To test this hypothesis and gain measured parameters for the kinetic model, we used a previously developed assay (9), where solutions of recombinantly purified, nonpolymerizable septin oligomers are mixed with membrane-coated beads of different curvatures (radii). Here, we expand the range of bulk concentrations and bead radii used in ref. 9. The number and rate of binding events as well as the dwell time for the single oligomer were measured using near-TIRF microscopy on beads of different sizes (Fig. 2A). As previously shown for different bead sizes, binding rates change as a function of curvature; however, these lead to a monotonic increase in the binding rate rather than actual curvature sensitivity, suggesting that the binding rate on its own is insufficient to generate a curvature optimum (Fig. 2B).

Fig. 2.

Single-septin complexes can detect only changes in membrane curvature through their binding rate. (A) Representative near-TIRF images showing single oligomers’ binding and unbinding events on various membrane curvatures. (Scale bar, 2 μm.) (B) Measured binding rates (J_on) for a single oligomer binding onto various membrane curvatures, κ = 1/r, at 1 nM bulk concentration. N > 200 events for each curvature. Error bars represent the SE. (C) Measured binding rates of single-septin oligomer onto membrane curvature κ = 2 μm⁻¹ at various septin concentrations. N > 200 events for each concentration. Error bars represent the SE. (D) Violin plots highlighting the distribution of measured dwell times for a single-septin oligomer on different membrane curvatures at 1 nM bulk concentration. N > 200 events per curvature. Filled circles show the experimentally measured mean values.

We incorporated the binding rates into the kinetic model with considerations for what is known about the molecular requirements of curvature sensitivity of septins via conserved amphipathic helices (AH). Other studies of curvature-sensing proteins suggest that membrane binding occurs in part through AH insertion into lipid-packing defects, which are induced through curvature and thermal fluctuations within the membrane (6, 10, 44). Intercalation of AH domains into the defects is driven by attractive electrostatic interactions between the AH domains and the membrane (5, 45).

Following these findings, we model the adsorption/desorption process by dividing it into three subprocesses: i) the formation and healing of lipid-packing defects; ii) the binding of the single septin to a given defect; and iii) the unbinding of the membrane-bound oligomer to the bulk. Following this, the kinetic equations describing septin oligomer’s adsorption with time, n^s, are expressed as

$$\begin{array}{cc}\hfill \frac{d{n}^{\text{d}}}{\mathit{dt}}& =k_{\text{on}}^{\text{d}}-k_{\text{off}}^{\text{d}}{n}^{\text{d}}-k_{\text{on}}^{\text{s}}{n}_{\text{b}}{n}^{\text{d}},\hfill \end{array}$$ [1]

$$\begin{array}{cc}\hfill \frac{d{n}^{\text{s}}}{\mathit{dt}}& =k_{\text{on}}^{\text{s}}{n}_{\text{b}}{n}^{\text{d}}-k_{\text{off}}^{\text{s}}{n}^{\text{s}},\hfill \end{array}$$ [2]

where superscript d and s, respectively, refer to defects and membrane-bound septins; subscript b refers to septins in the bulk; letter n denotes surface density for n^s; and n^d and bulk number density for n_b. The likelihood of having larger and longer-lived defects with higher membrane curvature is captured by an increase in the rate of defect formation, k_on^d, leading to monotonically increasing binding rates with curvature. This is in line with the experimental observations shown in Fig. 2B.

The rate of septin binding per unit area (J_on) is the first term on the right-hand side of Eq. 2: J_on = k_on^sn_bn^d. At steady state (dn^{d, s}/dt = 0),

$$\begin{array}{c}{J}_{\text{on}}=k_{\text{on}}^0\frac{n_{\text{b}}}{1+\beta n_{\text{b}}},\end{array}$$ [3]

where k_on⁰ = k_on^dk_on^s/k_off^d and β = k_on^s/k_off^d. When the binding rate of oligomers to defects is significantly smaller than the healing rate of defects, $k_{\text{on}}^{\text{s}}{n}_{\text{b}}\ll k_{\text{off}}^{\text{d}}\Rightarrow \beta n_{}\text{b}\ll 1$, the model predicts a linear increase of septin binding rate with bulk concentration, which is what we observe experimentally (Fig. 2C) at small bulk concentration n_b ≤ 1 nM. At sufficiently large bulk concentrations ( $k_{\text{off}}^{\text{d}}\ll k_{\text{on}}^{\text{s}}{n}_{\text{b}}\Rightarrow \beta n_b\gg 1$), the net binding rate becomes independent of bulk concentration ($k_{\text{on}}^{\text{eff}}\approx k_{\text{on}}^{\text{d}}$). This is consistent with the observations in Fig. 1 A and B that show that time-dependent adsorption curves at 25 nM and 50 nM are nearly identical for any given bead size.

Next, we measured the dwell times of bound septins on a range of membrane curvatures at low septin concentrations (Fig. 2D). The unbinding rate of a single-septin oligomer, k_off^s, is defined as the inverse of its average dwell time. As shown, the measured dwell times, and thus k_off^s, remain unchanged with curvature variations. This is consistent with the idea that, upon septin insertion, the septin and membrane molecules locally reconfigure to an energy state that is independent of the membrane’s macroscopic energy/curvature; this, then, leads to curvature-independent unbinding rates.

The steady-state adsorption of single oligomers is n^s = J_on/k_off^s. Given that our measurements show a monotonic increase of binding rate with curvature while k_off^s remains roughly constant, the absorption should increase monotonically with curvature. Thus, single-septin binding/unbinding kinetics cannot explain the observed curvature preference of septins. These observations together with the results of time-dependent adsorption on monodispersed and bidispersed assays support the idea that the preferred curvature for septin binding to membranes is determined by the assembly of membrane-bound septins, raising the question: what additional steps in assembly may be controlling the assembly rates? We proceed now with the development of the kinetic model to include multiple steps of assembly along with the experimental measurement of parameters for each step.

Diffusive Motion of Septins on the Membrane.

Once a septin oligomer is bound to the membrane, it undergoes diffusive motion and interacts with nearby oligomers/filaments to form longer filaments in a process referred to as diffusion-driven annealing (Fig. 1E, 2–4). This is a key step in determining the organization of septin filaments on membranes. We hypothesized that differences in diffusion on different curvatures could impact elongation/filament lengths. Using our single-molecule TIRF assay and particle tracking software (46), we tracked the position of bound single nonpolymerizable septin oligomers on curved and flat membranes (Fig. 3A). We were able to obtain the septins’ trajectories only on beads of diameter 3 μm and 5 μm, and not for beads of diameter 1 μm because these beads were too small for sufficient track lengths in the field of view which was limited by the height of the evanescent wave of the TIRF imaging. Fig. 3B–D shows the mean squared displacements (MSD) computed from particles’ trajectories in log-log scale and compares them against the MSD for single septins on planar membranes. We find that the single septins undergo nearly diffusive motion on a planar membrane (MSD ∼ t^α, where α ≈ 1 for purely diffusive motions), consistent with membrane responding as a Newtonian fluid in tangential directions. Interestingly, we also find that single septins exhibit subdiffusive motion (α < 1) on curved membranes, ultimately reaching a nearly constant MSD at long times, which is indicative of an elastic response at long times, as shown in Fig. 3 C and D. Thus, diffusion of single-septin complexes qualitatively changes on flat versus curved membranes, raising the possibility of this being a source of kinetic difference relevant for curvature sensitivity.

Fig. 3.

Single-septin complexes display subdiffusive behavior on curved membranes. (A) Top: Representative near-TIRF images of single-septin complexes on membrane-coated beads of diameter 5 μm and 3 μm, respectively. (Scale bar, 0.25 μm.) Bottom: Individual particle tracks (yellow lines) for beads of diameter 5 μm and 3 μm, respectively. (Scale bar, 0.25 μm.) (B–D) Mean squared displacements (MSDs) of single-septin oligomers vs time on a flat membrane (B) and membrane-coated beads of diameter 3 μm (C) and 5 μm (D). The filled circular symbols denote the mean values computed from analyzing the trajectory of more than 500 particles, and the solid lines represent the fit from Eq. 4 with N = 1. Shaded areas represent spread in experimental data in the interquartile range (darkest, 25% to 75%), interdecile range (next darkest, 10% to 90%), and full range (lightest, 0% to 100%).

We use these measurements to incorporate membrane diffusion into the kinetic model. As discussed previously, AH domain binding to the membrane is driven by electrostatic interactions between the polar residues of the AH and lipid head groups followed by incorporation of nonpolar AH residues into the hydrophobic core of the membrane. These attractive interactions can be modeled, in their simplest form, as spring-like connections between the membrane and a bound septin. These connections lead to an elastic response. We develop a Kelvin–Voigt (a parallel spring and dashpot) phenomenological model to describe MSD of septins in response to thermal forces, where the dashpot element with drag γ models the viscous resistance to a septin’s motion on the fluid membrane, and the spring element with spring coefficient K_sp models the attractive interactions between septin’s AH domain and the membrane. The details of the model development and the discussions related to the variations of the parameters of the model with curvature are provided in SI Appendix, section A. The final expression for MSD of a bound septin polymer made of N oligomers is

$$\begin{array}{c}\hfill \text{MSD}=(4D_0\tau /N)(1-exp(-t/\tau )),\end{array}$$ [4]

where τ = γ/K_sp, and D₀ = k_bT/γ is the diffusion of a single oligomer without AH–membrane interactions. Note that MSD scales inversely with the length of septins (MSD ∼ 1/N). We compute the values of D₀ and τ at different bead curvatures from fitting Eq. 4 to their respective single-septin (N = 1) MSD measurements with very good agreement between the model and experiments (Fig. 3B–D). We then use those computed values in Eq. 4 to compute the MSD of septin filaments made of N oligomers at different curvatures (SI Appendix, Fig. S1).

Unbinding Rates of Septin Filaments.

Net septin adsorption is dependent not only on binding but also on unbinding rates. Earlier in the paper, we used near-TIRF microscopy (Fig. 2A) to measure the unbinding rate of single septins and found them to be independent of curvature for a single oligomer (Fig. 2D). Next, we ask how does the unbinding rate change with septin’s length? We use a simple kinetic model to answer this question. In our model, a septin composed of N oligomers forms N attachments with the membrane. This septin is released only when all attachments are released. We use the measured single-septin oligomer unbinding rate to describe the unbinding kinetics of each attachment. Futhermore, we assume that once an attachment is released, it can rebind with a rate k_rb. The detachment rate of all N attachments is computed by numerically solving a system of N ordinary differential equations describing the time-dependent probability of observing different attachment states. Through postanalysis of the computed detachment rate for different values of N, k_rb, and k_off^s, we find that the detachment rate exponentially decreases with the number of attachments (length): $k_{\text{off}}=k_{\text{off}}^{s}exp(-\xi (N-1))$, where ξ is an empirical function that monotonically increases with the ratio of k_rb/k_off^s. Thus, our model predicts that as the bound septins merge and form longer filaments, they become exponentially more stable (longer-lived). The details of the model are provided in SI Appendix, section B and Fig. S.2. Values of ξ, and k_rb, are determined through the optimization process.

End-on Annealing and Fragmentation Processes.

Filament formation is required for stable septin–membrane interactions in vitro and essential for septin function in vivo (31). As we discussed in the previous section, end-on annealing of oligomers into filaments stabilizes septins’ interaction with the membrane by dramatically extending their lifetime (reducing their unbinding rates). We define end-on annealing as the process where two bound septins merge and form longer filaments. We assume that end-on annealing can occur only through annealing at either end of each bound septin filament. Fig. 4A shows near-TIRF images of septin polymers prior to and after end-on annealing. Note that bound septins can also increase in length through cooperative binding of bulk septins to either end of the bound septin filament. Conversely, septin filaments can break into smaller filaments or individual oligomers in a process termed fragmentation, thereby reducing the dwell time on the membrane. Fig. 4B shows near-TIRF images of a septin polymer prior to and after fragmentation. For a septin filament composed of N single oligomers, the fragmentation can occur at (N − 1) points along the filaments.

Fig. 4.

The length distribution of septin filaments is curvature dependent. (A and B) Near-TIRF images of septin filament end-on annealing (A) and fragmentation (B) on membrane-coated rods of diameters 46 to 1,508 nm. (Scale bar, 1 μm.) (C) SEM images of septin filaments on rods of different diameters. One of the septins is labeled as a yellow solid line, and the yellow dashed line is the long axis of the rod. The angle between the septin axis and the rod axis is defined as θ. (Scale bars, 0.1 μm.) (D) Probability density of septins’ length (number of oligomers per septin) at different septin curvature intervals. The curvature is defined as ${\kappa }^{\text{s}}=R_{\text{rod}}^{-1}sin\theta$, where R_rod is the radius of the membrane-coated rod and θ is the angle of the septin filament with respect to the long axis of the rod (zero when septin filament lies along the long axis and π/2 when septin filament lies orthogonal to the long axis).

Next, we examined whether membrane curvature could influence the end-on annealing and fragmentation rates, and hence the length distribution of septin filaments. We turned to an assay we previously developed to examine septin filament length and curvature at nanometer resolution. This approach uses scanning electron microscopy (SEM) to image cylindrical rods of radii R_rod = 46 − 1, 508 nm, coated with lipids and incubated with septins (Fig. 4C). Spherical surfaces, such as micrometer beads, are special in that the curvature is constant at any point and along any tangential direction. This is not true for any other surface including the cylindrical geometry of the rod, where the curvature along the filament’s axis is κ^s = 0 when the filament lies along the long axis of the rod, and changes to ${\kappa }^{\text{s}}=R_{\text{rod}}^{-1}$, when it lies perpendicularly across the rod (along the short axis).

In our previous study, we showed that the alignment angle of bound septin filaments with the main axis of the rod (shown as θ in Fig. 4C) is increased from nearly θ = 0 at R_rod = 0.1 μm to θ ≈ π/2 at R_rod > 0.35 μm (9). Given that the curvature of a septin filament with alignment angle θ along its axis is ${\kappa }^{\text{s}}=R_{\text{rod}}^{-1}sin\theta$, we hypothesize that septins sense the membrane curvature in the direction of their main axis and change their alignment angle to achieve optimal curvature along their axis. This hypothesis predicts that the ensemble average curvature of septins on rods is independent of the rod’s radius. We test this prediction by postanalyzing the SEM data, which is outlined in SI Appendix, Fig. S5. We find that the average curvature is indeed independent of the rod’s radius, and, thus, independent of any measure of curvature that is solely determined by the rod’s geometry, such as Gaussian and mean curvatures of the membrane.

We also used the SEM data to determine the probability density distribution of the septins’ length, normalized by the length of a septin oligomer, as a function of the filament’s curvature, κ^s. Fig. 4D shows the exponentially decaying length distribution of septins at different curvature intervals. At the lowest curvature interval (κ < 4 μm⁻¹), i.e., more flat, septins can form filaments that are in average made of up to four oligomers. The average length decreases as the curvature increases, and at the highest interval (κ > 8 μm⁻¹), only oligomers are present. We hypothesize that this could be due to an increased rate of fragmentation or decreased rate of end-on annealing at higher curvatures. Consistent with these observations, we constrain the ratio of fragmentation to end-on annealing rate (k_frag/k_anneal) to be an increasing function of curvature in our modeling of spherical membrane-coated bead assays.

Cooperative Binding.

A major feature of the time-dependent assembly of septin filaments is the superlinear increase of binding with time during the growth period. In writing Eq. 2, we have assumed that the rates of defect formation and septin binding is independent of the density of bound septins. In this case, given that the unbinding rate of septins is a positive constant number, we expect the rate of adsorption (dn^s/dt) to decrease as the adsorption (n^s) is increased over time, which goes against the experimental results (Fig. 1) during the growth period. This prompted us to ask what processes can give rise to the increased slope of adsorption with time? One possible answer is that end-on annealing of bound septins produces longer filaments, thereby decreasing their unbinding rate, k_off(t), resulting in an increase in dn/dt over time. This mechanism is already partially described by the combination of the length-dependent off-rate and fragmentation processes.

Alternatively, this superlinear growth may be driven by cooperative binding of bulk septin to the membrane, which we define as any mechanism where the bound septins facilitate the binding of additional bulk septins to the membrane. Regardless of the particular physical underpinning of the cooperative binding process, from a mathematical standpoint, all scenarios produce a binding rate that scales positively with the density of bound septins: dn/dt ∼ n_bn^s. We model the cooperative binding kinetics of defect formation and healing in the same manner as the direct binding mechanism discussed earlier. We account for the defect formation enhanced by septin binding by changing $k_{\text{on}}^0$ in Eq. 3 for direct binding to $k_{\text{coop}}^{\text{d}}{n}^{\text{s}}$ for cooperative binding.

Furthermore, Fig. 1 shows that steady-state adsorptions at the two largest bulk concentrations (n_b = 25 nM and 50 nM) remain nearly unchanged for a given bead curvature. This maximum saturation is larger in 5 μm in monodispersed assays. This may be due to its relatively slower organization kinetics, compared to 1-μm beads, allowing sufficient time for the filaments to align thereby increasing its packing efficiency. We phenomenologically model the effect of having maximum surface density by multiplying the binding rate by (1 − n/n^sat), where n^sat is the maximum surface density on a particular radius. After these modifications, we have:

$$\begin{array}{c}{J}_{}{}_{\text{on}}^{\text{coop}}=k_{\text{on}}^{\text{coop}}{n}^{\text{s}}\frac{n_{\text{b}}}{1+\beta n_{\text{b}}}\left(1-\frac{n^{\text{s}}}{n^{\text{sat}}}\right),\end{array}$$ [5]

where we have assumed the same value of β = k_on^s/k_off^d.

The cooperative flux shown in Eq. 5 can either increase the length of bound septins by adding the bulk septins to either of the two free ends of bound septins (modeled as J_on^{coop, i} in Table 1) or it can bind single oligomers from the bulk to the membrane near preexisting filaments (modeled as J^{coop, 1} in Table 1). The second scenario leads to polymerization of new septins in the vicinity of preexisting filaments.

Table 1.

Different processes involved in septin assembly on the membrane and their associated mathematical models

Kinetic process	Mathematical model
Effective bulk density	${\displaystyle n_{\text{b}}^{\text{eff}}=\frac{n_{\text{b}}}{1+\beta n_{\text{b}}}(1-\frac{n^{\text{s}}}{n^{\text{sat}}})}$
Direct binding	${\displaystyle J_{}{\text{on}}^{\text{direct}}=k_{\text{on}}^0{n}_{\text{b}}^{\text{eff}}}$
Cooperative binding	${\displaystyle J_{}{\text{on}}^{\text{coop},1}=k_{\text{on}}^{\text{coop}}{n}_{\text{b}}^{\text{eff}}(\sum _{j=3}(j-2)n_j-2n_1)}$
	${\displaystyle J_{}{\text{on}}^{\text{coop},i}=2k_{\text{on}}^{\text{coop}}{n}_{\text{b}}^{\text{eff}}\left(n_{i-1},-,n_i\right)}$
End-on annealing	${\displaystyle J_{\text{anneal}}^1=-n_1\sum _j{k}_{\text{anneal}}{n}_j}$ ${\displaystyle J_{\text{anneal}}^i=\frac{1}{2}\sum _{p+q=i}{k}_{\text{anneal}}{n}_p{n}_q}$      − ni∑ p kannealnp
Fragmentation	${\displaystyle J_{\text{frag}}^1=\sum _{p=2}{k}_{\text{frag}}{n}_p}$
	${\displaystyle J_{\text{frag}}^i=-\frac{1}{2}{n}_i\sum _{p=1}^{i-1}{k}_{\text{frag}}+\sum _{p=i+1}{k}_{\text{frag}}{n}_p}$
Unbinding	${\displaystyle J_{\text{off}}^i=k_{\text{off}}^i{n}_i,k_{\text{off}}^i=k_{\text{off}}^{\text{s}}\exp (-\xi (i-1))}$
Bulk depletion	nb = nb0 − h−1ns
Number of bound oligomers	$n^{\text{s}}={\displaystyle \sum }_{j}j{n}_j$

Note that all experiments are carried out in a closed system with a fixed total mass of septins, which all appear initially as oligomers in the bulk. Thus, the septins in the bulk are depleted as the adsorption is increased with time: $n_{\text{b}}=n_{\text{b}}^0-h^{-1}{n}^{\text{s}}$. Here, $n_{\text{b}}^0$ is the bulk concentration of septins at the start of the experiments, n^s is the time-dependent surface density on each bead assay, and h⁻¹ = A/V is the experimentally known ratio of the total membrane surface area of the beads, A, to the volume of the chamber containing septin oligomers in the bulk, V. As we shall see in the modeling section, this depletion effect must be included to predict the observed variations of adsorption with curvature on monodispersed vs. bidispersed assays. Note also that in experiments involving depolymerizable single-septin complexes shown in Fig. 2, the net adsorption and depletion effects are negligible.

Finally, we note that while the measured fluorescent intensity from confocal imaging, referred to here as adsorption, is proportional to the total mass of bound septins, it is not equal to the septin surface density: n^s = Ω × Adsorption, where Ω is the factor that relates adsorption with arbitrary units to surface density with the unit of number/area. In the absence of bulk depletion, Ω cannot be determined independently, i.e., any choice of Ω would simply rescale the adsorption curves. Including depletion effects and the observed reversal in the steady-state adsorption ratio of 1- and 5-μm beads in monodispersed and bidispersed assays resolves this issue and allows us to uniquely determine parameter Ω by matching the predictions against experiments.

Modeling and Simulation Methods

We have now developed the required pieces for modeling binding, unbinding, surface diffusion, end-on annealing, and fragmentation processes and their curvature dependence that ultimately determine the time-dependent adsorption and assembly of septin filaments on membranes. The equations describing these processes are summarized in Table 1. The kinetic equations describing the general assembly process are

$$\begin{array}{cc}\hfill \frac{d{n}_1}{dt}& =J_{\text{on}}^{\text{direct}}+J_{\text{on}}^{\text{coop,1}}+J_{\text{anneal}}^1+J_{\text{frag}}^1+J_{\text{off}}^1,\end{array}$$ [6a]

$$\begin{array}{cc}\hfill \frac{d{n}_i}{dt}& =J_{\text{on}}^{\text{coop,}i}+J_{\text{anneal}}^i+J_{\text{frag}}^i+J_{\text{off}}^i,i>1,\end{array}$$ [6b]

where i denotes the number of oligomer units in the filament.

For filaments to polymerize, their ends must be within a critical distance of each other, which is accomplished through their subdiffusive motion. The parameters that appear in modeling these processes are listed in Table 2. The parameters labeled as “S” under the source column (except for n_i and l_ave, which are determined by specifying the other parameters) are determined through a least-square optimization that minimizes the difference between the predicted and measured time-dependent adsorption curves at different bulk concentrations and curvatures. In order to solve this inverse problem, we first need a method to solve the direct problem: given all the parameters in Table 2, predict the time-dependent adsorption.

Table 2.

Parameters that appear in the simulations

Symbol	Description	Value (1 μm)	Value (5 μm)	Dimension	Source
n s	Number density (ND) of bound septins			μm−2	E (Fig. 1)
$n_{\text{b}}^0$	ND of bulk oligomers	6.25,  12.5,  25,  50	6.25,  12.5,  25,  50	nM	E (Fig. 1)
$k_{\text{on}}^0$	Oligomer’s binding rate	4.41 ± 0.30	0.86 ± 0.06	μm−2s−1nM−1	E (Fig. 2)
$k_{\text{off}}^{\text{s}}$	Oligomers’ unbinding rate	9.72 ± 0.92	11.54 ± 1.39	s−1	E (Fig. 2)
D 0	Single oligomer diffusivity	7.86 × 10−1 (3 μm)	1.07	μm2s−1	E (Fig. 3)
τ	Relaxation time of a bound oligomer	1.0 × 10−1 (3 μm)	5.4 × 10−2	s	E (Fig. 3)
h	${\displaystyle \frac{\text{Chamber’s volume}}{\text{Total area of the beads}}}$	2 × 104	2 × 104	μm	E (Mat & Method)
n i	ND of filaments made of i oligomers			μm−2	S
l ave	Septins’ average length at steady state	3.28 × 102	4.67 × 102	nm	S
β	$k_{\text{on}}^s$/$k_{\text{off}}^{\text{d}}$ in Eqs. 1 and 2	4.4 × 10−2	1.0 × 10−1	nM−1	S
ξ	Effective rebinding exponent	3.0	2.0	1	S
$k_{\text{on}}^{\text{coop}}$	Septins’ cooperativity rate	3.1 × 10−3	3.1 × 10−3	s−1nM−1	S
n sat	Maximum ND of bound septins	1.2 × 105	2.9 × 105	μm−2	S
k anneal	Septins’ end-on annealing rate	2.0 × 10−7	2.0 × 10−7	μm2s−1	S
k frag	Septins’ fragmentation rate	5.1 × 10−4	2.5 × 10−4	s−1	S
Ω	Surface density to adsorption ratio	6.1 × 105	6.1 × 105	μm−2a.u.−1	S

Parameters labeled as “E” under the source column denote variables that are known experimentally and those labeled as “S” (simulation) are determined through least-square optimization of the predictions against experiments.

The typical choice for solving the direct problem is to integrate the overdamped Langevin equation to compute the displacements of septin on the membrane using Eq. 4 in combination with Eq. 6 to describe the kinetic processes. The main challenge in using this method is that the timescale for reaching steady-state adsorption (minutes) is significantly larger than a single oligomer dwell time (≈0.1 s), during which the oligomers are undergoing subdiffusive displacements. As a result, an exceedingly large number of time steps are needed to reach steady state and resolve the subdiffusive dynamics of the septins, making this method computationally prohibitive.

As an alternative, we consider the two limits for the annealing process: 1) reaction-limited, where the annealing kinetics are controlled by the slower reaction rate when the septins meet and 2) diffusion-limited reaction, where the annealing kinetics are solely determined by the time it takes the septins to meet. We separately consider these two limits as approximations to the kinetics of annealing. Our results strongly suggest that septin assembly is reaction limited. The detailed analysis is provided in SI Appendix, section C. In the reaction-limited case, the annealing reaction of bound septin filaments of length i and j oligomers is simply given by dn_{i + j}/dt = k_anneal n_i n_j, where n_i denotes surface density of a filament of length i oligomers, and k_anneal is only a function of bead’s curvature and independent of bulk concentration, filaments’ lengths, and surface densities.

Table 1 shows the set of equations that describe the assembly process. These equations can now be numerically integrated to compute the density of septins of different lengths, n_i, as a function of time. As a result, we can compute the probability density distribution of septins of different lengths and, thus, the total surface density of bound septins (marked as adsorption here) vs. time for a given set of parameters. We can then search for a combination of parameters that gives the least error between the experiments and simulation results given a fitness function. We use the nonlinlsq function in Matlab for our optimization process.

Modeling and Simulation Results

Our kinetic model involves several processes and parameters to be determined through comparison with experiments. The principle of parsimony requires the models to contain only parameters/processes that are necessary for predicting the observations and nothing more. Overfitting occurs when the model does not meet this standard. In SI Appendix, section H, we provide an argument as to answer why our model is not overfitting the data.

Fig. 5 A and B, Top row compares the predictions of the model against the measured time-dependent adsorption on monodispersed beads of diameters 1 μm and 5 μm, each at different bulk concentrations, where we have used a reaction-limited model for the annealing process. Fig. 5C, Top row shows the same comparison for bidispersed assays at n_b = 25 nM bulk concentration. The predictions are displayed with thick solid lines of the same color as their associated experimental data. The computed values of the model are presented in Table 2. Our model gives a reasonable prediction of all three aspects of the assembly process: initiation, growth, and saturation in both assays.

Fig. 5.

Modeling predictions are in good agreement with time-dependent adsorption experimental data. Top row: Experimental data repeated from Fig. 1 A and B) Septin adsorption over time in monodispersed assays on 1-μm (A) and 5-μm (B) beads. The filled symbols denote experimental data, and solid lines represent the predictions of our kinetic model. Error bars correspond to SE of the experimental data. (C) Septin adsorption on 1-μm (circle symbols) and 5-μm (square symbols) beads over time in bidispersed assays at 25 nM bulk septin concentration. Bottom row: Variations of the average length of septins vs. time, corresponding to the same conditions as the top row.

In SI Appendix, Fig. S8, we compare the predicted values for lag time, inflection slope, and steady-state adsorption for each curve against their experimental values. We see a general agreement between the predicted and measured values. The predicted values for the mentioned parameters all behave monotonically with bulk concentration. The experimentally measured lag times for 5-μm assays show abrupt fluctuations with concentration, which may be due to insufficient sample size (For the same total area of beads, the number of beads, and thus the sample size for 5 μm, is about 25 times smaller than that of 1-μm assays) and experimental noise.

Another output of our simulations is the time-dependent length distribution of septins. In SI Appendix, Fig. S6, we show the predicted time-dependent length distribution functions in monodispersed and bidispersed assays at different points in time for a selected number of bulk concentrations and curvatures. We observe that after initial times, for all concentrations and for both monodispersed and bidispersed assays, the length distributions resemble a gamma distribution. This is also consistent with the length distribution of bound septins on membrane-coated rod assays shown in SI Appendix, Fig. S3. Fig. 5, Bottom row presents the variations of septins’ average length with time (computed from the length distribution function) corresponding to the same conditions at the Top row. In 1-μm monodispersed assays, the average length shows a local maximum very near the inflection point (maximum slope) in its corresponding time-dependent adsorptions. This can be understood by noting that beyond the inflection point, the net binding rates start to decrease due to a combination of approaching the saturation concentration and further depletion of bulk septins. Meanwhile, the fragmentation and unbinding rates remain unchanged, leading to a net decrease in septins’ length with time.

Focusing now on the steady-state values of the average length, we see that in both monodispersed assays, the average length increases weakly with septin bulk concentration, with these variations being more pronounced in 5-μm beads. Furthermore, the average steady-state length of septins on 5-μm beads is larger than that of 1-μm beads at the same bulk concentration for both monodispersed and bidispersed assays. This is in line with the trend observed for length distribution of septins on rod assays, measured from SEM images (Fig. 4). Recall that the underlying reason for this reduction in length is larger fragmentation rates at higher curvatures (1 μm). Indeed, we observe that the predicted rates increase from 2.5 × 10⁻⁴s⁻¹ at 5 μm to double the value 5.1 × 10⁻⁴s⁻¹ at 1 μm. Interestingly, the average steady state values for filament length on 1-μm beads at all septin concentrations are 300 to 400 nm, consistent with previous measurements in budding yeast (39).

Competition Between Beads of Different Curvatures.

We now return to our surprising observation in Fig. 1 that motivated the current study: The density of bound septins on 1-μm beads is larger than 5-μm beads in bidispersed assays, and septins localize to near 1-μm diameter curvatures in cellular contexts (31), while the opposite is true in monodispersed assays when only one curvature is present. As it can be seen in Fig. 5, our model can correctly predict this change in behavior between monodispersed and bidispersed assays. The observed trend can be explained as follows. We know that the adsorption rate on 1-μm beads at early and intermediate times is larger than that of 5-μm beads, leading to shorter initiation and saturation times. Given that 1-μm beads are competing with the slower kinetics on 5-μm beads in bidispersed assays, the majority of bulk septins will bind to 1-μm beads, leading to steady-state adsorption levels that are slightly higher than those in monodispersed assays in comparison. We predict that the relatively fast binding events of 1-μm beads in bidispersed assays result in considerable depletion of septin oligomers in the bulk. This, in turn, reduces the effective binding flux of bulk septins to 5-μm beads, ultimately leading to lower steady-state adsorption compared to 5 μm monodispersed beads and lower than the adsorption of 1-μm beads in the bidispersed experiments.

Since this reversal in adsorption in monodispersed and bidispersed assays in our model is driven by the depletion of bulk septins, the model predicts that reducing the extent of depletion levels in bidispersed assays would favor maximum adsorption on 5-μm beads, leading to a decrease in the ratio of steady-state adsorptions of 1-μm to 5-μm beads. To test this prediction, we manipulated the degree of bulk depletion by changing the total surface area (total number of beads in the assay) in the reactions. Specifically, we performed the bidispersed assay experiment, with 25%, 50%, and 75% of the total beads’ surface area of the first experiment. The ratio of steady-state adsorption values of 1-μm to 5-μm beads is shown in Fig. 6. Consistent with our model predictions, the adsorption ratio approaches that of monodispersed assays as the depletion extent is reduced at lower surface areas.

Fig. 6.

Reducing the degree of septin depletion from the bulk changes septins’ apparent curvature sensitivity. The ratio of steady-state septin adsorption on 1-μm to 5-μm beads in bidispersed assays, using different total membrane surface areas. The surface area is indicated as a percentage of the standard surface area we used in these assays (5 mm²). The bulk septin concentration is 6.25 nM. The blue violin plot shows the distribution of the ratio, and the black diamond indicates the average value. The orange line represents the ratio of adsorption between 1-μm and 5-μm beads at steady state in monodispersed assays; the orange belts represent spread in experimental data in the interquartile range (darkest, 25% to 75%), interdecile range (next darkest, 10% to 90%), and full range (lightest, 0% to 100%).

As we mentioned, the measured fluorescent intensity, labeled as adsorption here, is proportional to the surface density of bound septins: n^s = Ω × Adsorption. Including depletion effects in our model and the change of behavior we observe between monodispersed and bidispersed assays allow us to compute the proportionality constant Ω and, thus, n^s. If we approximate the area occupied by each septin oligomer as a rectangle that is 32 nm in length and 4 nm in width (35), we can also calculate the total area covered by septins. Dividing these numbers by the known total beads’ area gives the area fractions of septins in the experiments. Interestingly, all predicted numbers of layers are larger than one and in the range 3 to 21, suggesting a multilayered assembly of septin filaments. In fact, if we assume a maximum packing fraction of 1, these computed surface fractions serve as the lower-bound estimates of the number of septin layers on a membrane surface.

Recent analysis of transmission electron microscopy images of fly and mammalian septins on planar membranes (47) and the results of high-speed atomic microscopy of yeast septins (48) reveal that septins can indeed form multilayered assemblies. However, having 21 layers seems unlikely. Another possibility is that, while the current model assumes that all bulk septins have identical binding kinetics, they may consist of subpopulations with different binding kinetics. It is straightforward to modify the model so that, for example, only a fraction ϵ of bulk septins can bind and polymerize on the membrane. The adsorption predictions remain unchanged under this modification, i.e., we can still predict the depletion effects and the reverse trends observed for absorption in monodispersed and bidispersed assays by simply rescaling the kinetic parameters. The most important difference in predictions is $n_{\text{new}}^{\text{s}}=ϵn_{\text{old}}^{\text{s}}$. In such a condition, we may observe a single layer of septins, while still observing the novel behaviors that arise from a combination of the depletion of reactive bulk septins and the competition between different curvatures in recruiting these septins from the bulk.

Septin Polymerization Is Driven by Cooperative Binding.

Next, we asked whether including all the elements incorporated in the model, such as cooperativity, annealing, and fragmentation, is essential for predicting the correct time-dependent adsorption curves. To test this, we inactivated each of these individual processes in our model and solved for the best agreements between the predictions and experiments. SI Appendix, Figs. S9–S14 show the results of time-dependent adsorption and steady-state length distribution under these numerical experiments for both monodispersed and bidispersed assays. Below we summarize the main observations.

In all instances, aside from end-on annealing, eliminating the processes leads to poor predictions of the time-dependent adsorption curves in the monodispersed system and even poorer predictions of bidispersed adsorptions. Furthermore, excluding fragmentation leads to all bound septins polymerizing to their maximum allowed length in the simulations. (The maximum length in our simulations was set to 60 oligomers for 5-μm beads and 40 oligomers for 1-μm beads; changing this maximum length produced almost exactly the same results.) The same trend is observed when we exclude the cooperative binding. As expected, simulations without depletion completely fail to predict the results of bidispersed systems. Setting β = 0, which models the competition between septin oligomer binding and defect healing, results in poor predictions of the adsorption curve, particularly, the steady-state adsorptions.

Interestingly, we find that our model gives fairly good predictions of time-dependent adsorption after excluding end-on annealing process and including only direct and cooperative binding, fragmentation, and unbinding processes. Recall that in the absence of end-on annealing, bound septins can still polymerize through cooperative binding of bulk septin oligomers. Our modeling results suggest that septin polymerization on curved membranes may be predominantly driven by cooperative binding of subunits from the bulk.

Note that in the end-on annealing process, filament pairs merge and form longer filaments, without changing the total density of bound septins. Thus, we expect that for a given surface density of septins, the simulations with end-on annealing should produce longer filaments than those without end-on annealing. SI Appendix, Fig. S15 compare the variations of the average length of septins at steady-state against the steady-state adsorption at different bulk concentrations for 1-μm and 5-μm assays, in the presence and absence of end-on annealing. We see that when end-on annealing is included, in both assays, the average length increases roughly linearly with the steady-state adsorption. In contrast, when end-on annealing is excluded, the average length decreases with adsorption in both curvatures. These observations show that the length distribution of bound septins is more sensitive to the details of the assemblies than their net surface density. Experimental measurements of the length distribution at different bulk concentrations would allow us to inspect the relative importance of these assembly processes in more depth and to further constrain the range of modeling parameters.

Finally, we perform sensitivity analysis on the parameters computed through optimization. Specifically, we compute the change in the relative errors in the computed values of time lag, inflection slope, and steady-state adsorption as the parameters of the model are varied around their optimal values. These results are presented in SI Appendix, Figs. S16–S21 for both monodispersed and bidispersed assays. We observe that the relative errors are significantly increased for varying the parameters around their optimized value, for at least one of the three variables (lag time, maximum slope, and steady-state adsorption). This analysis shows that all parameters except the annealing rate have a significant quantitative effect on the time-dependent adsorption, and this combination of computed parameters minimizes the error between the experiments and predictions in the wide range of parameter variations explored here.

Discussion

Micrometer-curvature sensing by nanometer-scale proteins presents a challenge due to the mismatch in length scales. Our previous work (9, 31) led to a working model where differences in the rates of a single oligomer binding to beads of different curvatures were the basis for septins distinguishing among membrane-coated beads of different sizes. This mechanism is also consistent with studies of another micrometer-scale membrane curvature sensor, SpoVM, that binds the micrometer-scale forespore membrane in bacteria (10). However, the additional experiments presented here along with the modeling reveal that while the binding rate is likely a key factor in determining initial kinetic differences among different curvatures, it is insufficient to support a preference of septins for an optimal curvature. This realization emerged from a simple change in an experimental assay, where we compared curvature-dependent adsorption in regimes where septins must choose between two curvatures or where there is a single curvature present.

Remarkably, depending on available curvature options, different curvature preferences manifest, suggesting that the curvature sensitivity of septins is not a fixed property of the polymer but instead is sensitive to the process and context of assembly. A series of results presented here reveal that it is the hierarchical septin assembly and the physicochemical parameters that tune this assembly and determine specific curvature preferences of septins: 1) We showed that single oligomer kinetics alone cannot describe the observed curvature preference of septins, and the assembly of bound septin oligomers must be considered. Though single oligomer kinetics are insufficient, the curvature-dependent binding/unbinding still impacts the curvature preference. 2) Our confocal microscopy data show that septin binding vs. time is composed of initiation, growth, and saturation periods (Fig. 1D). 3) Using our current and previous experimental results, we developed a kinetic model that describes the multistep time-dependent septin assembly in terms of direct and cooperative binding of septin oligomers and diffusion, end-on annealing, fragmentation, and unbinding kinetics of bound septins. 4) We found very good agreements between the kinetic model and experiments, and surprisingly, we found that we can do so without the need to include diffusion and end-on annealing kinetics. 5) We observed that the steady-state adsorption on 5-μm beads at bulk concentrations larger than that n_b = 12.5 nM is larger than 1-μm beads in monodispersed assays, and this is reversed in bidispersed assays of the same bulk concentration. Our model can reproduce this behavior by accounting for competition between fast kinetics on 1-μm beads and slow kinetics on 5-μm beads for a finite pool of septins in the bulk that becomes depleted over time. 6) The predicted values of depletion in both assays suggest that septins may be organized into a multilayered structure on the membrane.

There are contexts where septins associate with negative membrane curvature such as in rings early and late in the cell cycle of yeast and in vitro on wavy substrates (49), where septin appears to also bind to negative curvature. In cells, this is coincident with the activities of regulators and also posttranslational modifications added to septins that could tune any one of the parameters relevant for assembly. This negative curvature preference is described in ref. 49 by considering an attractive septin–membrane interaction. The total length of septin on the side that interacts with the membrane (contact length) is increased when the septin is curved and bound to a negative curvature, compared to the straight septin binding to a planar membrane. The combination of this attractive force that increases with negative curvature and the resisting elastic forces that also increase with the absolute curvature ultimately determines the preferred curvature in their model. However, this model assumes that the ratio of the rate of binding to unbinding is independent of curvature, which is inconsistent with our measurements and modeling results. The methodology developed here can be applied to study septin assembly on negative curvatures as well. Just as septins can distinguish among different positive curvatures depending on features of the whole system, we predict that there are contexts where parameters such as membrane properties, additional proteins, or the septins themselves can be changed to lead to association with negative curvature.

One interesting prediction of our model is that the average length of septins remains within the narrow range of 300 to 500 nm over the large range of septin bulk concentrations and curvatures in both assays. This average length is very similar in magnitude to the observed length of septin hourglass structures (39). This raises the possibility that septins control their curvature preference by modulating their length. One issue to note is that in the model, we can still see good fits to the data for binding adsorptions even when highly variable lengths arise as a function of changing other parameters. This suggests that length control may be more complex than simply the considerations of the septin proteins themselves.

Based on all of these findings and observations, we propose the following model for binding and organization of septins on membranes (Fig. 7). The initiation period begins with septin oligomers directly binding to the membrane, and through reaction-limited end-on annealing, they form longer and more stable filaments. When the densities of bulk and membrane-bound septins (or cooperative binding rates) are sufficiently large and the unbinding rates are sufficiently small, the bulk septins will bind to the membrane through interactions with the bound filaments giving rise to a superlinear increase in adsorption in the growth state. Our predictions in monodispersed and bidispersed assays suggest that this cooperative binding controls the polymerization process.

Fig. 7.

A schematic model of septin curvature sensitivity in competition case. Septin adsorption on 1-μm (pink) and 5-μm (blue) beads is plotted over time. Insets illustrate how septins are depleted from the bulk solution primarily by 1-μm beads, leading to faster initiation and growth phases on 1 μm and ultimately larger adsorptions. Because 5-μm beads recruit from a more depleted bulk, compared to monodispersed assays, their growth rate and final adsorption are reduced. Shades of green represent septin bulk concentration.

In sum, these data point to a key polymer–polymer interaction that builds hierarchical structures of septins as an essential step of curvature-dependent assembly on the micrometer scale. Importantly, it is possible that control of different binding equilibria in cells is regulated by other proteins or modifications to the septins that could further tune the multistep and multiscale septin assembly process and ultimately the emergent curvature preference of septins. Although in many contexts, septins associate with micrometer-scale membrane curvature, there are wide ranges of curvatures where they bind within this scale that could be dependent on both the binding affinities set by local membrane composition and/or the presence of multiple different curvatures competing for a common pool of limiting septins. This raises the possibility that control of curvature sensing could also emerge through very general mechanisms that change bulk protein concentrations through regulation of transcription/translation/degradation.

Materials and Methods

Monitoring Adsorption of Septins Onto Supported Lipid Bilayers.

Measuring septin adsorption onto lipid-coated silica microspheres was performed as previously described (9, 31). In short, SLBs were mixed with septins in reaction buffer. Images were acquired using a custom-built spinning disc (Yokogowa) confocal microscope equipped with a Ti-82 Nikon stage, 100 x Plan Apo 1.49 NA oil objective, and a Zyla sCMOS (Andor) camera. Analysis of septin adsorption over time was carried out in Imaris 8.1.2 (Bitplane, AG). Adsorption is defined as the ratio of the intensity sum of the septin surface divided by the intensity sum of the lipid surface.

Kinetics and Diffusion of Single-Septin Complexes on SLBS.

Kinetic measurements were measured as previously described in ref. 9. Briefly, two polyethylene glycol-coated coverslips were sandwiched together using double-sided adhesive tape to make flow chambers. Nonpolymerizable septins and SLBs were added to the chamber and imaged using a custom-built total internal reflection fluorescence microscope (Nikon) equipped with a Ti-82 Nikon stage, 100 x Plan Apo 1.49 NA oil objective, and a Prime 95B CMOS (Photometrics) camera. The number and duration of binding events were preformed manually, whereas particle position over time was determined using Trackmate (46).

Generation and Imaging of Septin-Rod Supported Lipid Bilayer Mixture.

Preparation of septin-bound lipid-coated rods is described in refs. 9 and 40. Septin-rod mixtures were placed onto a 12 mm coverslip and mixed with fixative buffer followed by two rinses in washing buffer. The mixture was postfixed in 0.5% osmium tetroxide and washed with postfixed washing buffer. This was followed by a second 0.5% osmium tetroxide fix. Samples were dehydrated in ethanol. Transition fluid (hexamethyldisilazane) was added to the sample, followed by drying and desiccation until sputter coating. Samples were imaged on a Zeiss Supra 25 Field Emission Scanning Electron Microscope.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

The main study data are included in the article and/or SI Appendix. Additional data and materials are available from the corresponding authors upon request (50).