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. 2025 Jul 28;36(8):br20. doi: 10.1091/mbc.E25-03-0147

Force-insensitive myosin-I enhances endocytosis robustness through actin network–scale collective ratcheting

Michael A Ferrin a,†, Ross TA Pedersen a,‡, David G Drubin a,*, Matthew Akamatsu b,*
Editor: Alexander Mogilnerc
PMCID: PMC12367308  PMID: 40434901

Abstract

Force production by type-I myosins influences endocytic progression in many cell types. Because different myosin-I isoforms exhibit distinct force-dependent kinetic properties, it is important to investigate how these properties affect endocytic outcomes, and the mechanisms through which myosin-I contributes to endocytosis. To this end, we adapted our agent-based simulations of endocytic actin networks and incorporated nonprocessive, single-headed myosin motors at the base of the endocytic pit. We varied the unbinding rate and the force dependence of myosin unbinding. Our results revealed that the inclusion of myosin motors facilitated endocytic internalization, but only under kinetic regimes with rapid and less force-sensitive unbinding. Conversely, slow or strongly force-dependent unbinding impeded endocytic progression. As membrane tension increased, the boundary between assistive and inhibitory phases shifted, allowing the myosins to assist over larger regions of the kinetic landscape. Myosin-I's contribution to internalization could not be explained by direct force transduction or increased actin assembly. Instead, the myosins collectively bolstered the robustness of internalization by limiting pit retraction.

  • Type-I myosins with varying force sensitivity levels participate in different membrane deformation pathways, but the mechanistic link between molecular biophysical properties and cellular function remains poorly understood.

  • The authors analyze a computational model of endocytosis with type-I myosins and find that myosins with lower force sensitivity assist endocytosis by reducing backsliding along the internalization trajectory, while myosins with higher force sensitivity stall endocytosis by sequestering actin in nonproductive orientations.

  • These results introduce a new perspective on the function of type-I myosins in membrane reshaping: as a collective emergent property rather than the sum of individual force-generating motors.

INTRODUCTION

Type-I myosins are nonprocessive, membrane-associated molecular motors that generate a sheer force on actin filaments via a mechanochemical cycle that couples ATP hydrolysis to a “power stroke” conformational change (McIntosh and Ostap, 2016). Different myosin-I isoforms function in many distinct membrane trafficking pathways: vertebrate myosin-1b in the generation of post-Golgi carrier vesicles (Almeida et al., 2011), myosin-1c in some examples of compensitory endocytosis (Sokac et al., 2006), and vertebrate myosin-1e and its orthologs (including Myo5 in Saccharomyces cerevisiae) in clathrin-mediated endocytosis (CME) (Geli and Riezman, 1996; Sirotkin et al., 2005; Krendel et al., 2007). Detailed single-molecule studies of several of these isoforms have revealed a rich diversity of mechanochemical properties (Laakso et al., 2008; Laakso et al., 2010; Greenberg et al., 2012; Greenberg and Ostap, 2013; Pedersen et al., 2023). One notably variable property among type-I myosins is the force dependence of their detachment rate from actin filaments (Figure 1, A and B). Vertebrate myosin-1b is so acutely force-sensitive that it forms a “catch bond” that is more suited to function as a tension-sensitive anchor (Laakso et al., 2008), while S. cerevisiae Myo5 has minimal force sensitivity and is better suited to function as a power-generating transporter (Pedersen et al., 2023).

FIGURE 1:

FIGURE 1:

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Weakly force-sensitive myosin-I can assist endocytosis in computational simulations. (A) A diversity of actin-unbinding kinetics exists among type-I myosins involved in different cellular processes, from the force-insensitive motor Myo5 to the force-sensitive anchor Myo1b. (B) Schematic of type-I myosin activity. Membrane-bound type-I myosins can bind to actin filaments, take a single step, and then release the actin at a force-dependent rate. (C) Schematic of components included in agent-based model of CME. (D) Snapshots of simulation endpoints in three different conditions: with weakly force-sensitive myosin (left), without myosin (center), and with strongly force-sensitive myosin (right). (E) Pit internalization over simulated time for the three conditions in C, averaged over 12 simulations each. Error bars are 95% confidence intervals using the t statistic. All simulations were run with a resistance to pit internalization of 0.2 Inline graphic.

Among all the intracellular behaviors associated with type-I myosins, their involvement in CME has been examined in the most molecular detail. CME serves as the primary pathway for internalizing extracellular and plasma membrane–associated cargo in eukaryotic cells. As cargo adapters and curvature-generating proteins gather on the intracellular surface of a patch of the plasma membrane, the membrane invaginates into the cytoplasm and eventually pinches off to form a vesicle. Although passive components that induce curvature upon assembly are adequate to transform a relaxed membrane into a vesicle (Dannhauser and Ungewickell, 2012; Saleem et al., 2015), additional active force from actin polymerization is essential to overcome resistance from membrane tension or turgor pressure (Sun et al., 2006; Aghamohammadzadeh and Ayscough, 2009; Boulant et al., 2011; Basu et al., 2014; Hassinger et al., 2017; Kaplan et al., 2022). Type-I myosin additionally aids in actin-based force generation during CME, as its depletion completely halts endocytic internalization in budding yeast (Goodson et al., 1996; Sun et al., 2006; Lewellyn et al., 2015; Pedersen and Drubin, 2019) and partially disrupts internalization in mammalian cell culture (Cheng et al., 2012). Notably, the force-sensitive actin-binding kinetics have been measured for the budding yeast endocytic myosin Myo5 (Pedersen et al., 2023), and homologous myosins in other simple eukaryotes (Ostap and Pollard, 1996; Ostap et al., 2002; Tsiavaliaris et al., 2008; Zhu et al., 2025) and vertebrates (Mezgueldi et al., 2002) have been characterized in detail the absence of resistive force. Given that CME has been studied in rich quantitative detail, we utilized it as a model membrane-budding process to investigate how the various biochemical properties of type-I myosins may influence their mechanistic role in membrane morphogenesis.

A minimal model of actin cytoskeletal assembly in CME has previously demonstrated that the spatial segregation of actin filament linker proteins on the invaginating membrane and branched actin nucleators around the pit's base are sufficient to create a self-organizing actin network that generates an internalizing force (Akamatsu et al., 2020). However, this minimal network was incapable of internalizing the membrane to the degree predicted to be necessary for reaching the scission stage of CME. Given the assumed additional force-generating capacity of type-I myosins, we inquired whether incorporating them could enhance internalization in this model. Additionally, we utilized the power of biophysical modeling to conduct in silico perturbations and take measurements that are not experimentally accessible to predict the design principles and mechanochemical parameters of a type-I myosin optimally tuned for CME.

Simulations of our model across a variety of myosin parameters and mechanical environments revealed that the contribution of type-I myosin to endocytic membrane internalization strongly relies on the force sensitivity of its actin-binding kinetics. Although highly tension-sensitive tether myosins stall endocytosis, myosins that are relatively force-insensitive motors enhance average internalization. We find that tether myosins inhibit internalization by sequestering actin filaments in orientations that are not conducive to transmitting force to the membrane invagination. In contrast, motor myosins subtly influence actin network reorganization and limit pit retraction for more effective internalization.

RESULTS AND DISCUSSION

Type-I myosin can assist internalization in CME simulations

To model the contribution of type-I myosin to CME, we built on our model of branched actin assembly that generates force on an endocytic pit against membrane tension (Akamatsu et al., 2020), utilizing the agent-based cytoskeletal modeling platform Cytosim (Nedelec and Foethke, 2007). Briefly, our model is initialized with the following components and rules. We represent the clathrin-coated pit and membrane neck as a solid sphere and a cylinder with a smaller radius than the sphere, respectively. Because the CME actin module functions following the initiation of membrane curvature development (Kukulski et al., 2012), we initialize our simulations with the pit partially internalized. The actin-binding endocytic coat-linker protein (Hip1R) is randomly distributed over the surface of the pit. The plasma membrane base from which the pit internalizes is a flat surface, where preactivated branched actin nucleators (Arp2/3 complexes) are arranged in a ring around the pit (Almeida-Souza et al., 2018; Mund et al., 2018). A small number of actin mother filaments start with random positions and orientations near the pit, allowing them to diffuse according to three-dimensional (3D) Brownian dynamics. When they come close to an active Arp2/3 complex at the membrane, they can spontaneously bind and nucleate a new daughter filament at a 70° angle (Mullins et al., 1998; Blanchoin et al., 2000). The filaments polymerize and stochastically undergo capping at experimentally estimated rates. Filaments can also stochastically bind to Hip1R when in close proximity. Mechanical force from the spring-like behavior of these transient bonds and collisions with the membrane, along with Brownian fluctuations, can bend an actin filament according to its persistence length (McCullough et al., 2008). Informed by membrane mechanics simulations, we model membrane tension as spring-like resistance of the endocytic pit, with a linear relationship between internalization distance and the force required to maintain it there. As previously shown (Akamatsu et al., 2020), these components alone are sufficient to reproducibly drive internalization up to a stall distance of ∼50 nm (Figure 1, D and E).

In the present work, we added type-I myosins to the base membrane in a ring around the pit (Figure 1C) (Sochacki et al., 2017; Mund et al., 2018). These myosins can stochastically bind to actin filaments, upon which they exert a power stroke to step up to 5 nm toward the barbed end of the filament (Pedersen et al., 2023), and then unbind (Figure 1B). To model a catch bond, the myosin's unbinding rate decreases with increasing applied force of the myosin–actin bond (Greenberg and Ostap, 2013) (Figure 1B). We were able to constrain the majority of the model parameters to available experimental measurements and empirically fit the remainder to recapitulate the behavior of endocytic type-I myosins in reconstitution experiments (Supplemental Table S1; Supplemental Figure S2).

Simulating this minimal CME model with type-I myosins showed that the modulation of pit internalization depended on myosin parameters (Figure 1, D and E). With highly force-sensitive myosins present, internalization was significantly hindered compared with simulations that omitted myosins under the same conditions (Figure 1E). Conversely, including mildly force-sensitive myosins led to greater internalization than in simulations without myosin (Figure 1E). After observing this relationship between myosin's kinetic parameters and endocytic efficiency, we aimed to systematically investigate how the myosin's force-sensitivity parameter impacts endocytic outcomes.

The kinetics of myosin's force-sensitive unbinding determine its assistive capacity during endocytic pit internalization

Force-sensitive detachment kinetics in myosin are commonly modeled with the Bell equation (Bell, 1978) (schematized in Figure 2A and plotted in B):

FIGURE 2:

FIGURE 2:

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Kinetics of myosin's force-sensitive unbinding determine its assistive versus resistive capacity in endocytosis. (A) Cartoon schematic highlighting the relevant parameters for force-sensitive unbinding. The unloaded detachment rate k0 is defined as myosin's detachment rate from actin filaments in the absence of applied force F. With applied force, the predicted detachment rate becomes a function of k0, F, and the distance parameter d. (B) Plots of the Bell equation provide a theoretical prediction of how the actin–myosin detachment rate k(F) responds to applied force F at different values of the distance parameter d. (C) Demonstration of the 95th percentile of internalization relative to the plot of internalization over time for a single simulation. This serves as a summary statistic for comparing many simulations. (D) Contour plot of the percentage of change in mean 95th percentile of internalization relative to simulations with 0 myosins, over a range of parameter values for d and k0. At low d and high k0, myosins cause an increase in internalization, while they cause a decrease in internalization at high d and low k0. Type-I myosins with experimentally measured values (Laakso et al. 2008; Greenberg et al. 2012; Pedersen et al. 2023) are mapped in the plot. (E) Plots with 95% confidence interval of the percentage of change in mean 95th percentile of internalization over d for selected values of k0 (different lines). All simulations were run with a resistance to pit internalization of 0.2 Inline graphic.

graphic file with name mbc-36-br20-e002.jpg

where k(F) is the effective detachment rate at applied force F, k0 is the unloaded detachment rate, d is the distance parameter characterizing force sensitivity, kB is Boltzmann's constant, and T is temperature. The outcome of varying distance parameter d can be visualized in Figure 2B: the detachment rate is completely force-insensitive at d = 0, and the detachment rate becomes more dependent on applied force F as d increases (see Materials and Methods for implementation details).

We use the 95th percentile statistic of internalization as a single value summarizing endocytic success for comparison across various conditions (see Figure 2C). Visualizing the percentage change in the 95th percentile of internalization relative to simulations with no myosins revealed clear boundaries between myosin kinetic parameters that facilitate endocytosis and those that inhibit it (Figure 2D). We observed that internalization, on average, increased with myosins that have fast unloaded detachment (high k0) and weak force sensitivity (low d) (Figure 2D). Notably, overlaying the single-molecule measurements of these parameters for known type-I myosins (Laakso et al., 2008; Greenberg et al., 2012; Pedersen et al., 2023) demonstrated that the kinetic parameters for Myo5, a myosin involved in CME, lie in the facilitative regime, while the two type-I myosins unrelated to CME, Myo1c and Myo1b, fall within the nonfacilitative regime (Figure 2D). Accounting for variability between simulations, the distinction between assistive and stall-inducing regimes was sharpest between d of 5 and 50 nm, for most values of k0 (Figure 2E).

The mechanical environment influences the optimal myosin parameters for endocytosis

The necessity for actin-based force generation in CME strongly depends on the magnitude of forces opposing internalization: although impairing actin assembly has a minimal effect on endocytic success in mammalian cells with low membrane tension, the same perturbation almost completely stalls internalization when membrane tension is high (Boulant et al., 2011). Similarly, reducing osmotic pressure can even partially relieve the requirement for actin assembly in budding yeast CME (Soheil Aghamohammadzadeh and Ayscough, 2009). Given that the depletion of Myo1e from mammalian cells results in similar endocytic defects as actin drug treatment (Cheng et al., 2012), we hypothesized that the assistive role of myosin-I in CME should depend on the magnitude of mechanical resistance from membrane tension or pressure in a manner qualitatively mirroring that of actin (Figure 3A).

FIGURE 3:

FIGURE 3:

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The mechanical environment influences the optimal myosin parameters for endocytosis. (A) These simulations explore how myosin's influence on CME depends on the magnitude of force resisting internalization, which could arise from membrane tension or pressure. (B) We varied myosin's distance parameter d and unloaded detachment rate k0 for simulations with varying resistance to internalization. The myosins were universally nonassistive to endocytosis at very low resistance, while they exhibited a boundary at intermediate resistance that levels out at high resistance.

To investigate for this yet experimentally unexplored impact of myosin under varying opposing force, we ran simulations with the same combinations of force-dependent kinetic parameters as described above, across a wide range of internalization resistances (Figure 3B). We found that, under increasing values of membrane tension, the boundary between assistive and inhibitory contributions from myosin lessened and shifted, such that myosin became assistive over larger ranges of kinetic parameters. Although the size of the myosin-assistive regime increased with increasing membrane tension, the magnitude of assistance generally decreased.

This increase in the myosin-assistance regime with internalization resistance can be mapped to experimental measurements of membrane tension and myosin-I endocytic phenotypes. Previous theoretical analysis demonstrated linear scaling between internalization resistance of 0.6 to 1.6 Inline graphic to plasma membrane tension of 0.15 to 0.45 Inline graphic (Akamatsu et al., 2020). AFM measurements of SK-MEL-2 cells in isotonic media estimated a plasma membrane tension of 0.14 Inline graphic (corresponding to a resistance of ∼0.6 Inline graphic in our model), while hypotonic shock could elevate the tension to 0.29 Inline graphic (Kaplan et al., 2022) (corresponding to a resistance of ∼1.0 Inline graphic). As our simulations span this range, these results are useful for predicting the yet unmeasured kinetic parameters of the mammalian endocytic Myo1e for maximizing robustness of CME. Interestingly, a myosin-I with force sensitivity and unloaded detachment rates corresponding to those of the budding yeast endocytic Myo5 is assistive to simulated internalization throughout three orders of magnitude around the resistance closest to a “relaxed” mammalian plasma membrane tension (Figure 3B, 0.2 Inline graphic). Yet other experimentally characterized non-CME type-I myosins inhibit internalization at all resistance values, demonstrating that CME requires a fundamentally different myosin activity than the myosins that function in tension-sensitive anchoring can provide. Myosins with any parameters universally inhibit internalization at the lowest resistance, which is lower than the estimated tension for any cells measured to date (Sitarska and Diz-Muñoz, 2020).

Myosins provide resilience against pit retraction

In seeking a mechanistic explanation for myosin's binding kinetics-dependent function in endocytic outcomes, we observed that assistive myosins reduced pit retraction events at the nanometer scale. An inspection of individual internalization trajectories revealed that the pit did not steadily internalize throughout the simulation; rather, it transitioned between phases of internalization and retraction (Figure 4A). Unlike the >100-nm scale retraction events observed experimentally for endocytic sites lacking scission machinery (Kishimoto et al., 2011), these sub-10-nm scale retractions would fall below the resolution limit of time-lapse fluorescence microscopy. We investigated whether myosins modulate these retractions by quantitatively classifying retraction events and measuring their durations and distances. By comparing these retraction metrics across various myosin-I kinetic parameters, we identified a boundary of myosin parameters that closely mirrored our previous internalization measurements (Figure 4, B and C, compared with Figure 2D). In the low d (weak force sensitivity) and high k0 (fast unloaded detachment) regime, where the 95th percentile of internalization was higher than in no-myosin controls (Figure 2D), endocytic events spent less time in the retraction phase per simulation (Figure 4B) and retracted shorter distances overall per simulation (Figure 4C). These results suggested a mechanistic link between myosin's control over pit retractions and endocytic outcomes.

FIGURE 4:

FIGURE 4:

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Assistive capacity of myosins for endocytic internalization can be explained by robustness to retraction. (A) Plot of raw and filtered internalization over time for a single simulation with retraction events highlighted in red (see for details). We analyzed retraction events in the first 10 simulated seconds of simulations. (B and C) Change in mean total retraction time (B) and total retraction distance (C) relative to no-myosin simulations. Both values were higher with high d (strongly force-sensitive) and low k0 (low unbinding rate) myosins, while the retraction measurements were lower with low d and high k0. (D and E) Change in mean 95th percentile of internalization relative to no-myosins (y-axis) as a function of the change in mean total retraction time (D) and total retraction distance (E) (x-axes). Both metrics inversely correlated with change in internalization. We used a 2D lookup table of myosin-kinetic parameters to color code the data points. Each point with error bars represents the mean and 95% confidence interval of all replicate simulations of the same myosin parameter combinations indicated by color from the lookup table. (F) Cartoon conceptual model of type-I myosin function in CME. Endocytic internalization is like a ratchet with force from actin assembly pulling the ratchet toward the right (pit internalization). The kinetic parameters of myosin are analogous to the behavior of the pawl (brown). (Top) Myosins in the assistive parameter regime collectively behave as a balanced pawl spring, flexible enough to allow forward movement but strong enough to limit backsliding. (Bottom left) In the absence of myosin, the pawl spring is removed, allowing back-sliding of the ratchet. (Bottom right) If d is too high and k0 too low, the myosins behave like an immobilized pawl, stalling movement in any direction. All simulations were run with a resistance to pit internalization of 0.2 Inline graphic.

To relate the effects of myosin-I on pit retractions to net internalization more quantitatively, we plotted the percentage change in the 95th percentile of internalization as a function of retraction time (Figure 4D) and distance (Figure 4E) for each myosin parameter combination. We identified a clear trend in which myosins that reduced retractions increased net internalization (Figure 4, D and E). In contrast, there was no apparent correlation between 95th percentile of internalization and retraction time (Supplemental Figure S3A) and distance (Supplemental Figure S3B) in simulations with no myosins included. Therefore, myosin's influence on overall internalization may be explained by its ability to limit pit retraction.

These results lead us to a mesoscale conceptual model in which myosins modulate the ratchet-like behavior of the overall endocytic pit-actin system (Figure 4F). CME can be envisioned as a ratchet driven by actin forces. Movement in the direction of the actin-based force is countered by membrane tension. In the presence of weakly force-sensitive myosins (Figure 4F, top), the actomyosin network collectively acts like a spring-loaded pawl that enables the ratchet to be pulled by actin assembly in the membrane internalization direction while blocking retraction. In the absence of these myosins (Figure 4F, bottom left), no equivalent spring exists to keep the pawl engaged with the ratchet, allowing the actin force to pull it toward internalization but making it susceptible to retractions due to fluctuations in the system. With strongly force-sensitive myosins (Figure 4G, bottom right), the actomyosin network effectively pins the pawl, resisting any displacement of the ratchet. This “collective ratchet,” at the scale of the cytoskeletal network, differs from the Brownian ratchet mechanisms as described for single molecules (Peskin et al., 1993; Mogilner and Oster, 1996; Ait-Haddou and Herzog, 2003).

On a detailed molecular level, this ratcheting action may originate from myosin's effect on actin rearrangement. As the system's geometry continually evolves over time with pit internalization, the actin network must continually reorganize to sustain a pulling force. Because of the stochastic nature of filament movements, this self-organizing network inevitably transitions through intermediate structures that do not maintain the same internalization force, leading to pit retractions. The weak force-sensitivity of myosin bonds might bias the actin network against rearranging in ways that hinder force production, thus reducing retractions. Indeed, we noticed subtle variations in individual actin filament organization due to the supportive myosins, such as an increase in radial distance from the pit and a flatter angle of actin-barbed ends relative to the base membrane (Supplemental Figure S4). This organization may indicate a mechanism that allows shallower filament angles to enhance forces from actin polymerization (Dmitrieff and Nédélec, 2016). Future modeling and experimental efforts are needed to explore myosin's impact on actin reorganization more fully.

Importantly, many outcomes of these simulations have also been observed experimentally. In vitro bead motility assays with branched actin assembly showed that Drosophila Myo1d inhibited bead motility in some conditions, including high-filament capping, high concentrations of myosin, or “rigor” myosins that cannot unbind actin (Xu et al., 2024). Our simulations recapitulated this inhibition in conditions of little or no-myosin unbinding (Figure 2D). In experiments, the presence of myosin led to a decrease in actin assembly by 20 to 30%, which was recapitulated quantitatively by in silico bead motility simulations as well as our endocytic actin simulations (Xu et al., 2024; Supplemental Figure S5, A–C). Each of these similarities with experimental observations help to validate the model.

Several other mechanisms for the function of myosin-Is in force-generating branched actin networks have been proposed, but these were not supported by our observations. Correlations between the presence of myosin motors, the size of endocytic actin networks, and pit internalization rates in live budding yeast suggest a model in which myosins assist CME by pulling actin filament barbed ends away from the plasma membrane, thereby reducing the “stall” force on barbed ends that would otherwise limit filament elongation (Manenschijn et al., 2019). Single-molecule characterization of Myo5's binding kinetics indicates that myosin assists CME by directly applying force to actin filaments (Pedersen et al., 2023). The aforementioned joint biochemical reconstitution and computational modeling study proposes a conceptually similar model in which myosins enhance branched actin force production by pushing actin filaments away from the membrane surface, strengthening the network by modulating the branching frequency along filament length (Xu et al., 2024). However, we observed minimal changes to actin filament elongation rates due to the presence of myosin in the model (Supplemental Figure S5D). Furthermore, we found that myosin's assistance to internalization persisted even without motor activity (Supplemental Figure S6, A and B).

It is important to note that our model omits several microscopic properties of myosin that may account for some of the discrepancies between our simulation results and previously proposed mechanistic models. These omissions include endocytic myosins' nucleation-promoting factor (NPF) activity (Geli et al., 2000; Sun et al., 2006), recruitment of downstream NPF proteins (Cheng et al., 2012), the torque on actin resulting from myosin's power stroke (Chavali et al., 2025), a threshold force where detachment kinetics transition from a catch bond regime (decreasing unbinding rate with increasing force) to a slip bond regime (increasing unbinding rate with increasing force), and the capacity of myosins to reversibly attach to and diffuse along the membrane. The model also does not incorporate the directional dependence of force on detachment rate that has been reported for some myosins (Veigel et al., 2003; 2005; Sung et al., 2015), which may be a property of myosin-I though not yet experimentally observed to our knowledge. We anticipate that each of these features will be systematically incorporated into future computational models and assessed for their emergent influence on the entire system, much like this work expands on an earlier CME model that lacked any myosins (Akamatsu et al., 2020).

MATERIALS AND METHODS

Cytosim model

We adapted the agent-based model developed in Akamatsu et al., (Akamatsu et al., 2020) to run 3D stochastic simulations of endocytic pit internalization via a minimal branched actin network. In short, we model the endocytic pit as a solid sphere connected to a flat membrane base by a solid cylindrical neck. The pit experiences a spring-like resistance to internalization set by the parameter “bud confine.” Actin filament–binding linker proteins are randomly distributed over the surface of the pit. Preactivated branched actin nucleators are randomly distributed in a ring around the pit, where they can freely diffuse along the flat membrane surface until they bind actin filaments. Myosins, when included, are also randomly distributed in a ring around the pit, though they cannot diffuse along the membrane. To initialize the event with a small number of actin filaments, linear actin filament nucleators are randomly distributed in a ring around the base of the pit, and subsequently are free to diffuse throughout the cytoplasm. Filaments can elongate, become stochastically capped, bind linker proteins emanating from the surface of the membrane at the nascent pit, bind branch nucleators to nucleate daughter filaments, and bind to myosins. Filaments also have a bending rigidity, so they store and release mechanical energy upon bending and straightening, respectively. The forces from actin polymerization against the membrane while bound to membrane-associated linker proteins are sufficient to internalize the pit 50 to 60 nm against the resisting force (Akamatsu et al., 2020).

For main results, CME simulations were run for 15 simulated seconds, with simulation steps every 0.00005 s, and outputs reported every 0.1 s. Most model configurations analyzed in this work were run in replicates of 12, though simulations that terminated prematurely due to numerical instabilities were excluded from all analyses. See Supplemental Figure S1 for a comprehensive replicate count for each configuration. Simulations for validating myosin nonprocessivity (Supplemental Figure S2B) and force-dependent attachment lifetime (Supplemental Figure S2C) were run for 1 simulated second with outputs reported at every 0.00005 s simulation step. Except where explicitly noted, we ran CME simulations with the pit resistance to internalization set at 0.2 Inline graphic.

For simulated gliding motility assays analyzed in Supplemental Figure S2D, we initialized several linear capped actin filaments (no polymerization, depolymerization, or nucleation activity) on a surface of randomly distributed type-I myosins in a two-dimensional (2D) system. Specifically, we included 10 filaments 1 µm in length, 10 filaments 5 µm in length, and 5 filaments 10 µm in length. These were run for 60 simulated seconds with simulation steps every 0.001 s, and outputs reported every 0.1 s.

The modifications made from (Akamatsu et al., 2020) were changing the Arp2/3 branching angle from 77° to 70° to better fit recent experimental evidence (Serwas et al., 2022) and adding myosin-I motors, as detailed below.

For a full list of parameters used in the model, see Supplemental Table S1.

Myosin-I model and parameter choices

Because the default model motor proteins defined in Cytosim are processive, that is, they continuously displace along filaments while attached, we developed a new nonprocessive motor class for our model. To create the new motor, we modified the “mighty” class of Cytosim's source code, which is a copy of the “motor” class allocated for such customized functionalities. This motor stops moving along the filament and remains attached after its initial displacement along the filament, thereby modeling a single power stroke for a single-headed myosin (Supplemental Figure S2, A and B). We modeled myosins with abolished motor activity (Supplemental Figure S6A) by setting the displacement speed to 0.

The force-sensitive ADP release and ATP-binding transitions of type-I myosin's ATPase cycle result in its force-sensitive detachment property (Greenberg and Ostap, 2013) (Figure 1B). We implemented this force-sensitive detachment by interconverting between the Bell equation (Eq. 1) and Kramers' theory used in Cytosim (Eq. 2):

graphic file with name mbc-36-br20-e013.jpg

where k(F), F, and k0 are defined the same as in Eq. 1, and F0 is the “unbinding force” parameter in Cytosim. By setting the remaining quantity from Eq. 1 T = 303.15K, we fully defined the mathematical relationship to convert between myosin distance parameter (d) and unbinding force (F0). The unloaded detachment rate (k0) and distance parameter (d) of the Bell equation (Eq. 1) can be fit from single-molecule optical trapping experiments, thus providing a direct map between simulations and experiments. We modeled this force-sensitive detachment rate as a single exponential expression rather than two exponentials that would allow for catch bond activity at low force and slip bond activity at high force (Belyaev and Fedotova, 2023) because this transition to slip bond–unbinding kinetics has not been experimentally characterized for myosin-I, to our knowledge. To validate this implementation, we simulated 2D filament gliding motility powered by myosin motors with parameters corresponding to measurements from the budding yeast endocytic Myo5. The resultant simulations qualitatively recapitulated its experimentally observed force-dependent attachment lifetime (Supplemental Figure S2C) and gliding actin motility rates (Supplemental Figure S2D) (Pedersen et al., 2023).

We set the parameters of myosin-I that were not the primary subject of investigation in this work based on experimental measurements of the budding yeast endocytic Myo5 where possible (Supplemental Table S1). The actin-binding rate of 3 s-1 comes from the steady-state ATPase rate of phosphorylated Myo5 in saturating ATP, which measures the rate-limiting step of the Myo5 ATPase cycle for actin binding (Pedersen et al., 2023). Myosin's “max speed” sets the step size of a single power stroke, which we fit to 5 nm for Myo5's total step size per ATPase cycle (Pedersen et al., 2023) (Supplemental Figure S2B). Except where explicitly noted, we set the baseline number of myosin-I molecules at 100, based on estimations of the total number of budding yeast type-I myosins recruited per CME site from live-cell fluorescence microscopy (Manenschijn et al., 2019; Sun et al., 2019). We set the radial distance of myosins around the pit based on super-resolution microscopy from budding yeast (Mund et al., 2018) and correlative super-resolution light and transmission electron microscopy in HeLa cells (Sochacki et al., 2017).

For other myosin-I simulation parameters that could not be directly constrained by experimental measurements, we empirically determined appropriate values by performing parameter sensitivity analyses. Although the myosin-I motor domain is connected to the membrane through a lever arm and tail domain, the Cytosim model myosin is only composed of the motor domain that cannot move in space. To model this spatial separation between the motor domain and the membrane, we placed the myosins at a fixed offset distance from the membrane (Supplemental Figure S2A). Inspection of myosin-I structural models informed our hypothesized 8-nm typical offset distance (Chavali et al., 2025), and parameter scans revealed no significant impact on internalization around this range (Supplemental Figure S2E). The model myosin has a binding range, which is the maximum distance between the myosin and actin in which a bond can be established (Supplemental Figure S2A). Though this quantity may be related to myosin's radius of gyration around its membrane-immobilized tail domain, there is no obvious direct physical complement to this parameter. We hypothesized a binding range of 4 nm, and found no significant impact on internalization around this range (Supplemental Figure S2F). Molecular bonds in Cytosim are modeled as springs with a stiffness parameter (Supplemental Figure S2A). As there is no direct physical correlate to this parameter of myosin, we scanned several orders of magnitude and chose a value of 100 Inline graphic (Supplemental Figure S2G). Below this value, bonds stretched physically unrealistic distances of 10 nm in CME simulations. Above this value, internalization was severely hindered for myosins with parameters tuned to Myo5 (Supplemental Figure S2G). Furthermore, this approximate value resulted in the simulated gliding motility assay velocity closest to experimental measurements (Supplemental Figure S2D). Myosins also have a stall force, which is the force required to decrease their step size (Supplemental Figure S2A). This has not been measured for Myo5, so we picked an arbitrarily high value in order to guarantee stepping and subsequent unbinding in our simulations. The greatest pit internalization occurred with a high stall force for force-insensitive myosins, but we observed no significant trend for Myo5-tuned myosins (Supplemental Figure S2H).

Simulation analysis

We wrote and modified custom Python scripts to parse simulation outputs, analyze results, and generate plots. Major concepts used in analysis are explained below.

95th percentile internalization

We used the 95th percentile internalization as a metric summarizing the extent of pit internalization for a single simulation. This value, less sensitive to noise than the final timepoint or maximum internalization, is defined as the 95th percentile of internalization values at all timepoints throughout the 15-s simulation.

Statistics summarizing differences relative to no-myosin

To quantify the effect of a given myosin-I configuration on a simulation summary measurement (such as 95th percentile internalization), we calculate the percentage of change in the mean measurement relative to no-myosins with the function f (Eq. 3):

graphic file with name mbc-36-br20-e015.jpg

where A is the mean of the measurement in replicate simulations with myosin and B is the mean of the measurement in replicate simulations without myosin.

We propagate the uncertainty of the two means in the function by calculating the SD σf Eq. 4:

graphic file with name mbc-36-br20-e016.jpg

in which σA is the SD of the measurement in replicate simulations with myosin and σB is the SD of the measurement in replicate simulations without myosin.

We can then calculate the 95% confidence interval CI with Eq. 5:

graphic file with name mbc-36-br20-e017.jpg

where t is the t statistic for a 95% confidence interval (two-tailed) and n is the minimum number of replicates between the with- or without-myosin measurement.

Internalization retraction analysis

We carried out the following steps to measure internalization retraction time and distance per simulation:

  1. We applied a Savitzky–Golay convolution filter to smooth each internalization trajectory. We used a window size of 10 timepoints (1 s simulated time) and a 4th-order polynomial for fitting.

  2. We used a peak-finding algorithm to identify all local minima and maxima in the smoothed trajectory. The duration between a local maximum and the next local minimum is defined as a retraction event. The length of this duration is the retraction time, and the difference in smoothed internalization between the start and end of the retraction event is the retraction distance.

  3. We summed the retraction times and distances for all retraction events in the first 10 s of the simulation to arrive at the final measurements per simulation.

2D-Lookup table

To create a quantitative colormap that represents both myosin parameters d and k0, we created a 2D lookup table that varies two color axes corresponding to different combinations of myosin parameters.

2D-Gliding assay analysis

To compare simulated gliding velocity with experimental measurements, we tracked the 2D displacement of each filament barbed end from its starting position over time. The slope fit from a linear regression of this relationship provided the velocity of each barbed end. We reported the mean and SEM of all 25 filaments in each simulation.

Myosin-binding event analyses

The binding state of each individual myosin was tracked at each simulation timepoint to identify all binding events. The information associated with the actin filament participating in each binding event was used for subsequent analyses.

We subtracted the distance along the actin contour from the pointed end (defined as the abscissa in Cytosim) at the end of each binding event from the distance from the pointed end at the beginning to analyze the displacement per binding event in Supplemental Figure S2B. We calculated the mean force on the myosin–actin bond among all timepoints in the lifetime of a binding event for the mean force measurement in Supplemental Figure S2C.

For individual binding event analyses in Supplemental Figure S4, C to E, we calculated the difference in the given actin measurement (barbed end localization or angle) from the simulation timepoint immediately before each binding event to the first timepoint of the binding event. We performed the same calculation immediately before and after the end of the bond lifetime for unbinding events as well.

Actin density and localization analysis

Density of assembled actin subunits at an endocytic site was calculated as the total count of actin model points within a given volume. The density for Supplemental Figure S5A was calculated for the volume of the whole simulation system. The density calculated for Supplemental Figure S4, A and B was filtered just for actin barbed ends, within the volume of each radial distance and z-distance bin. The count of daughter filaments for Supplemental Figure S5B is analogous to the density of bound Arp2/3 complexes in the branched actin network. Filament length for Supplemental Figure S5C was calculated as the contour length for actin filaments in the simulation. Elongation attenuation for Supplemental Figure S5D was defined as the difference between measured per-filament elongation rate and the theoretical maximum elongation rate. All measurements for Supplemental Figure S5 were filtered for filaments either directly bound to Hip1R linkers or indirectly bound through mother or daughter filaments.

Bending energy analysis

Per-filament actin bending energy for Supplemental Figure S6C was calculated from the difference in tangential direction between the barbed end and the closest Hip1R attachment by contour length. We use the theory of deforming elastic beams (Boal, 2001) in which bending energy is defined by Inline graphic. Here, kB is the Boltzmann constant, T is temperature, kB is the filament persistence length (10 mm for actin), θ is the bending angle, and l is the free filament length (contour length of the filament between the attachment site and barbed end of the filament). Only filaments that were directly or indirectly bound to Hip1R were included in this analysis.

Supplementary Material

mbc-36-br20-s001.pdf (8.1MB, pdf)

ACKNOWLEDGMENTS

We thank Sam Smith, Leanna Owen, Jennifer Hill, Tom Pollard, and members of the Drubin/Barnes laboratory for helpful discussions. We thank the Berkeley Research Computing team for maintaining the computing resources essential to carrying out this work and for technical support. This work was supported by the NSF Graduate Research Fellowship Program (to M.A.F.), as well as NIH Grants F32GM142145 (to R.T.A.P.), R35GM118149 (to D.G.D.), and R00GM132551 (to M.A). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Abbreviations used:

AFM

atomic force microscopy

Arp2/3 complex

actin related protein 2/3 complex

ATP

adenosine triphosphate

CME

clathrin-mediated endocytosis

HeLa

human cervical cancer cell line

Hip1R

huntingtin-interacting protein 1-related

nm

nanometer

pN

piconewton

SEM

standard error of the mean

SD

standard deviation

SK-MEL-2

human melanoma cell line.

Footnotes

This article was published online ahead of print in MBoC in Press (http://www.molbiolcell.org/cgi/doi/10.1091/mbc.E25-03-0147) on May 28, 2025.

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