Qualitative Description of Model
In this section, we will describe the qualitative features of our model. The quantitative mathematical formulations will be relegated to the Experimental Procedures.
Temporal control and spatial arrangement of proteins and the lipid PI(4,5)P2
at budding yeast endocytic sites are key features in the development of our model (). First, each of the key endocytic proteins appears to localize along the membrane invagination with a distinct spatial profile, predicted by dynamic properties 
and confirmed by EM 
. Second, these proteins can be grouped into four “protein modules” based on their distinct dynamics and functions 
. Lastly, we previously obtained evidence for a PI(4,5)P2
“lipid module” that is dynamically regulated during endocytosis 
Endocytic dynamics in budding yeast.
For this model, we describe clathrin-mediated endocytic dynamics on the level of functional modules, which allows us to look beyond roles of individual molecular players that may vary from one organism to another and to focus upon collective behaviors in membrane shape transformations and local biochemical pathways. Thus, our model can serve as a unified framework for endocytosis across diverse organisms. We propose that the five modules along with their functions are as follows ( provides an overview of the model):
Mechanochemical feedback mechanism for endocytosis in budding yeast.
- Phosphoinositides, e.g., PI(4,5)P2 (PIP2), cover the endocytic membrane and recruit endocytic proteins to the plasma membrane ,,,. PIP2 accumulation driven by lipid kinases, and its hydrolysis by phosphatases, proceeds at the endocytic membrane throughout the course of endocytosis ,,,,,; local PIP2 levels are controlled by the balance between accumulation and hydrolysis.
- Coat proteins (e.g., clathrin and Sla1) accumulate on the vesicle bud via interaction with PIP2 or PIP2-associated adaptor proteins ,,,,,,. The coat proteins anchor and regulate actin filaments while imparting curvature to the bud region .
- Proteins that accumulate in the tubule region, e.g., BDPs ,,, have both membrane-deforming and membrane curvature–sensing power . Taking into account the specific spatial and temporal profile of BDPs during endocytosis  and their binding to PIP2 (Kishimoto and Drubin, unpublished results), we further propose that BDPs generate a lipid phase boundary by protecting the underlying PIP2 from hydrolysis by the phosphatase, as suggested by experiments showing PIP2 clustering by BDPs .
- The actin module proteins are anchored to the bud by binding to coat proteins. Actin and actin-associated proteins (i.e., F-actin and myosin) are responsible for generating the pulling force exerted on the bud ,,,,. The pulling force helps to generate BDP-binding sites, helping to recruit BDPs to the endocytic site .
- Enzymes that hydrolyze PIP2, e.g., synaptojanin or Sjl2p in yeast ,,,,, accumulate late in the vesicle formation process. In vitro experiments show that phosphoinositide hydrolysis rates by phospholipase C critically depend on the local membrane curvature . Here we refer to the mean membrane curvature, which is the average of the curvatures in the tangential and radial directions on the membrane (see ). The mean curvature represents the extent of lipid head group exposure. At higher membrane curvatures, the enzymes have greater access to the lipid head groups, which enhances both the binding to the lipids and the enzyme's hydrolysis activity (see Protocol S1 for details). This curvature sensitivity of enzyme activities may be a general phenomenon, as suggested by the observation that PI3K kinase activity also critically depends on membrane curvature . We propose, therefore, that a similar mechanism applies to PIP2 hydrolysis, which has been corroborated by experiments (Chang-Ileto and Di Paolo, personal communication).
From a mechanical standpoint, the pulling forces generated by the actin/myosin functional module impinge on the bud and invaginate the membrane. The initial pinching force is generated as follows. Because of the protection afforded by BDPs on the tubule, more PIP2 is hydrolyzed at the bud region. This leads to lipid phase segregation—PIP2 levels along the membrane invagination differ, and the resulting interfacial force at the bud-tubule interface squeezes the neck. From a chemical perspective, the local chemical reactions (e.g., actin assembly, PIP2 hydrolysis) control pulling and pinching forces. Equally important, the resulting membrane curvature generated by the mechanical forces also influences the local reaction rates (). In this way, endocytic dynamics are controlled by mechanochemical feedback between endocytic membrane shape changes (membrane curvature) and the local chemical reactions that control the mechanical forces (pulling and pinching forces). This key notion, as we will show below, is essential for the robustness of the sequential endocytic protein recruitment and timely vesicle scission.
This qualitative picture is captured by Equations 1–6 in the Experimental Procedures. The coupling between the mechanical and chemical processes of endocytosis is specified by the dependence of the reaction rates on membrane curvature and by the dependence of the local membrane curvature and the mechanical force on the local levels and activities of the functional modules. To calculate the dynamics of endocytic events, we numerically integrate Equations 1–6 over time starting from the initial condition: the endocytic membrane is flat and the initial coverage for all of the protein modules is set to zero. The initial PIP2
coverage is set to 2% corresponding to its normal average level 
. At each step, the system is characterized by the instantaneous shape of the endocytic membrane and the local levels of the functional modules as represented in mole fraction. The values of the parameters used in the model are listed in Table 1 with references in Protocol S1
. Below, we first study the endocytic dynamics of budding yeast by choosing the parameter set that quantitatively fits the time-lapse experimental data in . We then vary the parameters to mimic mutant experiments to predict and analyze the associated phenotypes.
As the model dynamics are controlled by many parameters in Equations 1–6, there could in principle be many outcomes depending on parameter choices. To circumvent this problem, 21 of the 25 parameters used in the model were taken from independent experiments (Table 1 in Protocol S1
). The four unmeasured parameters all characterize BDP dynamics; they are the intrinsic BDP recruitment rate, actin-aided recruitment rate, turnover rate, and the relative timescale of BDP dynamics with regard to actin dynamics. With 21 measured parameters being fixed, we only vary the four free value parameters to fit the five time-lapse curves of endocytic dynamics observed in wild-type budding yeast (). The values of these four parameters are constrained because these kinetic rates must be comparable to those experimentally determined for each of the other functional modules. The dynamics of all of the modules are tightly coupled: one sub-process cannot be much faster/slower than the others. In what follows, we use specific proteins or lipids to represent the corresponding functional modules. We stress from outset that the goal of the paper is to illuminate the collective dynamics of endocytosis generated by the interactions among the functional modules, rather than identifying detailed molecular players.
Endocytosis Involves a Precisely Timed and Ordered Sequence of Events
shows that the endocytic dynamics predicted by our model (continuous lines) fit quantitatively with the experimental data (discontinuous lines) 
,[11, and the measurements in this paper]
. shows snap shots of the corresponding computed membrane shape changes (a movie of the process derived from model calculation is provided in Video S1
). Because the fitting parameters are constrained by measurements from independent studies, the agreement between our theoretical results and experimental observations lends validation to our model. An important feature of the process is that each functional module is activated sequentially in step with the membrane shape changes (). We next describe steps in the endocytic process in greater detail based on our model.
Fitting of the results calculated from the model to experimental results.
Early in the process (0–20 s, ) coat proteins begin to accumulate. During this period, the membrane is deformed by the coat proteins, which generate a small dome (less than 50 nm in height and ~50 nm in width, t~20 s in ). However, there is a delay before actin polymerization fully commences, because it takes a while for the nucleation factors to be recruited and activated and because actin assembly is autocatalytic due to Arp2/3 activation by actin filaments. Without the assistance of the actomyosin force, the dome-like membrane deformation would not progress further, which is consistent with observations from recent EM studies 
. Indeed, this dome shape could be the prerequisite for further development of a deep invagination, because the local membrane shape may provide a suitable angle at which the F-actin pulling force can be exerted upon the bud region effectively.
At ~20–25 s (), F-actin polymerization is promoted by nucleation factors recruited by the coat proteins, and the pulling force upon the bud region increases. This drives the endocytic membrane to invaginate further (t~22 s in ). As the membrane invaginates, actin monomers rapidly incorporate into the existing actin filaments with their barbed ends facing the cell cortex 
, while myosin pushes the actin network away from the plasma membrane into the cytoplasm. Meanwhile, the PIP2
phosphatase begins to accumulate all over the endocytic site. Concurrently, BDPs also start to accumulate along the tubule region rapidly, and they increase from 10% to the peak level in only 3 s ().
Now the question is: what drives the rapid BDP accumulation? We show that curvature-sensing and deforming activities of BDPs form an intrinsic positive feedback loop (see quantitative calculations in Figure S1
). As schematized in , as they bind to the membrane, BDPs deform the adjacent membrane into the preferred curvature for their binding. This leads to a faster recruitment rate, which further promotes BDP recruitment and tubulation of the membrane. This positive feedback also explains and reconciles the two classes of experimental observations, which provided evidence for curvature-sensing and membrane-deforming activities 
. In our scenario, actin assembly and myosin contractile forces invaginate the membrane. The resulting membrane curvature fits relatively favorably to the preferred shape of BDPs and, hence, promotes rapid BDP binding at the right location and at the right time due to the curvature-sensing activity. In turn, BDP binding invaginates the membrane further and generates optimal curvature for BDP binding in the elongating tubule, which self-accelerates BDP accumulation. Thus, the initial membrane invagination generated by the actin/myosin force triggers the positive feedback between BDP binding and membrane tubulation.
Two positive mechanochemical feedback loops between membrane shape changes and local chemical reactions.
During this same period, PIP2
hydrolysis rates are faster on the bud than on the tubule, as the BDPs protect the PIP2
on the underlying tubule from hydrolysis. Lipid-protein interactions involving BDPs could limit PIP2
diffusion in the membrane 
, allowing formation of a lipid-phase boundary. An interfacial force at the bud-tubule boundary thus starts to build up, constricting the neck.
Eventually (t~29 s in ), the interfacial force narrows the neck down to <5 nm, at which distance the opposed bilayers would fuse spontaneously 
, resulting in rapid vesicle scission. Upon vesicle scission, BDPs disassemble from the membrane tubule within 3 s as the tubule retracts due to loss of the actin pulling force. A second crucial effect of the PIP2
phosphatase activity on the vesicle bud is to trigger disassembly of the endocytic coat (t~25–29 s, ). As coat proteins disassemble, the F-actin attachments to the bud weaken, resulting in loss of pulling force on the invagination. We predict that this leads to a small retraction of the endocytic membrane tip concurrent with vesicle scission (see ) and propose that loss of the pulling force on the membrane may be a prerequisite for vesicle scission.
Rapid Vesicle Scission Is Triggered by Lipid Phase Segregation via Curvature-Enhanced PIP2 Hydrolysis
Our description of the endocytic process () raises the following interesting questions: How is the interfacial scission force generated? How does vesicle scission occur so rapidly? And what turns off the positive feedback loop for BDP assembly and drives their extremely fast disassembly? In this section, we propose answers to these questions. Our proposal that an interfacial force can drive vesicle scission is supported by in vitro experiments 
, in which lipid phase segregation is induced by lowering temperature. In vivo, however, cells always maintain constant temperature; instead, lipid-protein interactions could be utilized to yield effective lipid phase segregation. Here, we present two possible scenarios for how the interfacial force is developed in endocytosis (schematized in ): (1) As PIP2
hydrolysis at the bud eliminates hydrogen bonds that had bridged the interfacial boundary, hydrogen bond shielding of the hydrophobic hydrocarbon chains is lost, and at the boundary these aliphatic tails are exposed to water, which is energetically unfavorable. The resulting line tension is proportional to the PIP2
difference across the interface, which contracts to minimize these unfavorable contacts, thus squeezing the neck. (2) The reduced hydrogen bond network at the bud lowers the membrane surface tension of the outer leaflet, which thus tends to expand. Effectively, this is a lateral surface pressure that propagates from the high-lateral pressure region towards the interfacial boundary. Due to the local concavity of the membrane created by the initial interfacial tension, this lateral pressure is directed inwards at the phase boundary and provides an additional pinching force. This additional lateral pressure also increases with the difference in PIP2
levels across the phase boundary (see the detailed derivations in Protocol S1
Development of the interfacial force during endocytosis.
shows the calculated time course for interfacial force development during endocytosis, while shows the calculated profiles for PIP2 levels around the bud-tubule boundary at different time points. show that the interfacial force undergoes rapid changes. During t~0–21 s, PIP2 accumulates uniformly over the entire endocytic site, as promoted by kinase-mediated synthesis. From around t~21 s (), PIP2 levels decline non-uniformly; consequently, the interfacial force starts to build up (). This spatial non-uniformity is because around the same time as the phosphatase is recruited, BDPs start to accumulate at the tubule region of the endocytic membrane (t~21 s in ). As a result of the BDP protection at the tubule, relatively more PIP2 is hydrolyzed on the bud, leading to lipid phase segregation at the BDP–coat protein boundary. This phase separation gives rise to the initial interfacial force at the phase boundary.
From t~21–27 s (), the interfacial force grows sharply. Such rapid growth of the interfacial force is the result of another positive feedback loop involving curvature-enhanced PIP2 hydrolysis. We schematize the qualitative mechanism in . As the initial interfacial force squeezes the neck, it creates a higher mean curvature at the interface. The higher the mean curvature of the membrane, the more PIP2 is exposed and susceptible to phosphatase activity. Consequently, more PIP2 is depleted at the interface region along the membrane invagination. Thus, a larger difference in local PIP2 levels bounding this location is induced (~21–27 s, ), which in turn speeds up the growth of the interfacial force and, hence, further squeezes the interface. This is a self-accelerating process.
The sharp dip of the PIP2
levels around the bud-tubule interface compared to the smaller difference between those of tubule and bud (t~23 s and 27 s in ) suggests that curvature-dependent PIP2
hydrolysis is the predominant driving force for generating the interfacial force. Our model thus predicts that the pinching force arises as a result of differential phosphatase activity along the membrane invagination. This prediction is consistent with the observations that phosphatase activity is essential for endocytic vesicle scission in yeast, that the phosphatase concentrates at the endocytic site during the late stages of endocytic vesicle internalization, and that it moves into the cell with the forming vesicle, possibly suggesting enrichment at the vesicle tip 
During t~27–29 s (), as the pinching force squeezes the neck, the membrane curvature in the radial direction of the tubule deviates from the optimal shape for BDP binding (t~28 s and 29 s in ). This deviation acts as a “disassembly signal” and invokes the intrinsic positive feedback loop between curvature sensing and curvature deforming of BDPs (), triggering the rapid BDPs turnover (~27–29 s in ). Meanwhile, PIP2 gets hydrolyzed not only at the bud but also on the tubule due to the lack of BDP protection (t~29 s, ). Although this leads to a fast decrease in the interfacial force (~27–29 s, ), the pinching force is still sufficient to drive rapid vesicle scission according to our calculations.
We need to point out that, while the in vitro systems on lipid phase segregation are crucial for identifying mechanical forces that might be involved in vesicle scission, the experimental conditions used are quite different from the in vivo conditions during endocytosis. Once the lipid phase segregation takes place in the in vitro systems, the resulting interfacial force persists and there is no time limit for the vesicle scission process. All that matters is that the interfacial force needs to be sufficiently large to overcome the membrane bending resistance 
. In cells, the timing of the lipid phase segregation is predicted to be critical for successful endocytosis. The threshold interfacial force value required for scission can be determined by force-balance calculations 
. A rapid nonlinear time course for interfacial force development in endocytosis means that successful scission in vivo can only occur within a short time window (the shaded region in ).
Successful Endocytosis Depends on the Feedback between Local Chemical Reactions and Membrane Shape Changes
In this section, we will explore in detail how mechanochemical feedback ensures the precise timing and sequence of endocytic events and guarantees rapid endocytic vesicle scission. In phase diagrams for endocytosis are computed for different pairs of model parameters. These diagrams serve several purposes. First, they show that the model is robust: it can generate successful endocytosis over a large range of the parameters. Second, equipped with these phase diagrams, we can vary the parameters to mimic the conditions of mutant experiments. Third, they constitute an independent experimental test of the model. This is because the identities of the functional modules were in part derived from mutant experiments, but we did not explicitly take into account the mutant phenotypes in the model. That is, we used the five time-lapse curves and membrane shape changes to determine the four free parameters in the model, and then used these parameter values to predict mutant phenotypes. Thus, these predictions are independent of the parameter set, and consequently the agreement between predicted and observed phenotypes constitutes cross-validation of the model. Finally, based on the calculated phase diagrams, we can predict endocytic phenotypes for mutants that have not yet been made, thus guiding further experiments.
Phase diagrams for endocytic dynamics.
Dependence on Fast PIP2 Hydrolysis Rate on Membrane Curvature
shows that endocytosis can only be successful when the curvature-dependent PIP2
hydrolysis rate is sufficiently fast. Otherwise, the PIP2
level difference across the interfacial boundary will not have had sufficient time to grow before the membrane bending energy resists squeezing and quickly balances the interfacial force without triggering the positive feedback loop (). Accordingly, the absence of positive feedback between the interfacial force and the local membrane curvature leads to a distinct phenotype (phenotype 1
, wherein the PIP2
hydrolysis rate k2
is reduced from 20 (nm) per second to zero): F-actin associated forces could still drive membrane invagination; the interfacial force, however, would not squeeze the neck effectively, because the force cannot grow large enough. Thus, the whole system would eventually reach a mechanochemical equilibrium wherein a slightly curved membrane invagination could persist for a time without vesicle scission. This phenotype is consistent with the budding yeast mutant sjl1Δ sjl2Δ 
, wherein the PIP2
hydrolysis is dramatically reduced.
If the PIP2
hydrolysis rate is very fast but independent of the local membrane curvature, then the positive feedback between the interfacial force and the local membrane curvature is ablated (see and 45). Without this positive feedback, the interfacial force would always remain at its initial basal level, which is insufficient to pinch off the vesicle (see details in Figure S2
). Successful endocytosis, therefore, requires the positive feedback between interfacial force and curvature-dependent PIP2
hydrolysis activity. This is further dictated by two conditions: first, the PIP2
hydrolysis rate must be faster than the typical response time scale of the membrane, and second, PIP2
hydrolysis must be curvature-dependent. The former can be tuned by the local concentration of phosphatases, and the latter is intrinsic to the mechanism of enzyme activity.
Proper Protection of Tubule PIP2 by BDPs Is Essential for Endocytosis
shows that, even with a sufficiently high curvature-dependent PIP2
hydrolysis rate, endocytosis may not be successful unless the protection of PIP2
at the tubule by BDPs is sufficiently effective (large K2
). Otherwise (small K2
), the resulting interfacial force would be too small to drive vesicle scission. On the other hand, if the protection is too effective, then PIP2
levels at the tubule would be maintained at a high level, which in turn would lead to persistent BDP accumulation. As BDPs tend to deform the membrane to a specific, preferred shape (diameter ~30 nm), persistence of the BDPs would effectively hold the neck and prevent any further narrowing of the membrane tubule, hindering vesicle scission. This leads to prediction of a unique phenotype (phenotype 2
, wherein the protection strength of PIP2
hydrolysis at the tubule region
increases from 0.5 μM−1
to 2.5 μM−1
), in which the absolute levels and the lifetimes of the BDPs would increase significantly as compared with the wild-type situation. Furthermore, a long and narrow membrane invagination could persist without vesicle scission. This is because BDPs have their own preferred shape (a tubule of ~30 nm in diameter), and their persistence would tend to preserve the shape of membrane tubule, preventing any further squeezing in response to the interfacial force.
Dependence on the Timing of Phosphatase Recruitment
Our model predicts that within the successful endocytosis region in , increasing the curvature-dependent PIP2
hydrolysis rate k2
will speed up endocytosis and that this effect will saturate at large k2
. This is because in this case endocytic dynamics are limited by the phosphatase recruitment rate, instead of by its activity. As shown in , positive feedback between interfacial force development and local membrane curvature will not develop if the phosphatase activity is not sufficient. Insufficient phosphatase results in a phenotype similar to those observed when PIP2
hydrolysis curvature dependence is insufficient, as shown in , and/or when PIP2
hydrolysis is independent of curvature, as shown in Figure S2
On the other hand, endocytosis will also be impeded if the phosphatase is overexpressed or overactive at the endocytic site, which leads to phenotype 3 (where the curvature-dependent factor of phosphatase recruitment rate α increases from 100 nm to 500 nm). Here scission fails because the excessive phosphatase diminishes the initial PIP2 level difference across the bud-tubule boundary, thus preventing the development of the initial squeezing force. As a result, the membrane at the interface cannot be deformed sufficiently to invoke positive feedback between interfacial force development and the curvature-dependent PIP2 hydrolysis activity.
A surprising conclusion from our model is that coat proteins will still assemble at the endocytic site in the presence of excessive phosphatase and will disassemble slowly. This conclusion is based on the linear dependence of the PIP2 hydrolysis rate on the local membrane curvature, which is in accordance to experimental observations. PIP2 hydrolysis is relatively slow despite high phosphatase levels because the membrane is not highly curved (e.g., phenotype 3). Thus, even though the phosphatase recruitment is very fast in phenotype 3, its action is limited by the lack of membrane curvature, which is low because a pronounced phase boundary does not develop.
Endocytosis Critically Depends on Coordination between BDP Recruitment and F-Actin Polymerization
shows that successful endocytosis also critically depends on the coordinated dynamics of BDP recruitment and F-actin polymerization. Without actin polymerization, the endocytic membrane cannot become deeply invaginated. Failure to invaginate the membrane prevents BDP accumulation and the ensuing development of the interfacial force. Consequently, the membrane cannot deform into a deep invagination, nor proceed to vesicle scission. This situation is similar to having excessive phosphatase at the endocytic site, leading to phenotype 3 in , consistent with actin-assembly inhibition phenotype in budding yeast 
When actin polymerizes normally, efficient endocytosis requires sufficiently fast BDP accumulation. Insufficient BDP recruitment would lead to phenotype 4
(wherein the BDP recruitment rate drops to zero): the endocytic membrane will be pulled out and will then retract without vesicle scission (a movie of the process is given in Video S2
). This is because although the peak interfacial force is large enough to squeeze the neck in phenotype 4, the force declines so rapidly that the membrane does not have time to undergo deformation and, hence, the vesicle cannot be successfully pinched off. A large interfacial force can develop in the absence of the BDPs in phenotype 4 because the actin filaments contact actin-binding proteins associated with the coat so that the actin pulling force impinges on the entire bud region of the endocytic membrane, including the bud-tubule boundary. Although very small, the force from the actin module can still deform the membrane at the neck slightly, which activates the curvature-dependent PIP2
phosphatase activity. Hence, the positive feedback loop is triggered, leading to generation of a large interfacial force. However, without BDP protection, this large interfacial force is too short-lived and vesicle scission does not occur.
On the other hand, in the absence of sufficient numbers of BDPs, the high curvature of the membrane invagination generated by F-actin polymerization would still induce phosphatase recruitment, which would result in disassembly of the entire endocytic apparatus and retraction of the membrane invagination. This predicted phenotype is consistent with the phenotype of a budding yeast rvs167
(a BDP) knockout mutant 
and a lipid-binding defective rvs167
point mutant (Kishimoto and Drubin, unpublished).
Interplay between the Interfacial Force and BDP Turnover
The lifetime of BDPs at endocytic sites is extremely short (~10 s) in wild-type budding yeast 
. We have shown for phenotype 2 of that prolonged accumulation of BDPs could prevent endocytosis. A key message emerging from these two observations is that the interplay between the interfacial force and BDP turnover is critical for successful endocytosis. As the interfacial force squeezes the interface, it tends to narrow the adjacent membrane tubule, which deviates from the shape preferred by BDPs. This deviation leads to a curvature mismatch and acts as a “disassembly” signal for the BDPs as dictated by the BDP sensitivity factor (the exponential term χ in Equation 5). Accordingly, upon narrowing of the tubule, the higher the sensitivity factor χ, the faster the turnover of the BDPs, and hence the more that vesicle scission is facilitated. As shows, when the interfacial force is very large (>60 pN), it is capable of squeezing the interfacial boundary even if the BDPs are not disassembled; endocytosis would proceed normally even with prolonged BDP accumulation at the tubule (χ
0). On the other hand, when the interfacial force is in an intermediate range (e.g., 30–60 pN), its action could be insufficient to overcome the bending resistance of the preferred membrane shape set by the BDPs. Given that the interfacial force will also dissipate in a short period of time (~5 s, ), a minimal level of curvature-dependent sensitivity in BDP accumulation is required to induce fast BDP turnover upon squeezing of the membrane tubule, relieving the bending resistance, and hence facilitating vesicle scission. This sets the lower threshold of the curvature-dependent sensitivity of BDP dynamics for successful vesicle scission. Note that the curvature sensitivity, χ, is central to the positive feedback between BDP recruitment and the local membrane deformation (). The above results imply that successful endocytosis requires that BDP binding feeds back positively with the underlying membrane shape.