Microfilament
Microfilaments, also known as actin filaments, are thin, flexible fibers approximately 7 nm in diameter and up to several micrometers in length, formed by the polymerization of globular actin (G-actin) monomers into double-helical filamentous actin (F-actin), serving as a primary component of the eukaryotic cytoskeleton.[1] These structures constitute 5–20% of a cell's total protein content and are highly conserved across eukaryotes, enabling dynamic assembly and disassembly to support cellular architecture and movement.[1][2] The atomic structure of G-actin consists of a 375-amino-acid polypeptide (42–43 kDa) folded into two major α/β-domains divided into four subdomains, forming a compact prism roughly 55 Å × 55 Å × 35 Å in size, with nucleotide-binding (ATP or ADP) at the core.[2] Upon polymerization, G-actin units assemble head-to-tail into polar F-actin filaments, exhibiting a right-handed helical twist with 13 subunits per 36 nm repeat, where the fast-growing barbed (plus) end and slower-growing pointed (minus) end facilitate treadmilling dynamics driven by ATP hydrolysis.[1][2] Actin-binding proteins, such as Arp2/3 complex for branching and cofilin for depolymerization, organize these filaments into bundles, networks, or higher-order arrays like the cortical meshwork beneath the plasma membrane.[1] Microfilaments are essential for maintaining cell shape and mechanical integrity, powering motility through structures like lamellipodia and filopodia, and driving processes such as cytokinesis, phagocytosis, and intracellular transport via interactions with myosin motors.[1] They also contribute to cell adhesion, signaling, and even nuclear functions, including gene regulation, highlighting their versatility beyond structural roles.[2] Dysregulation of microfilament dynamics is implicated in diseases like cancer, where altered polymerization affects invasion and metastasis.[1]Structure and Composition
Actin Monomers
Globular actin, or G-actin, serves as the fundamental monomeric subunit of microfilaments, consisting of a single polypeptide chain with 375 amino acids and a molecular weight of approximately 42 kDa.[1] This globular protein exhibits a high proportion of certain residues, including about 7.5% glycine and 4.9% proline, which contribute to its compact, spherical conformation.[3] Encoded by genes from the actin multigene family, G-actin is highly conserved across eukaryotes, reflecting its essential role in cytoskeletal architecture.[4] The three-dimensional structure of G-actin is organized into two major domains, each subdivided into two subdomains, resulting in four subdomains (I–IV) that surround a central nucleotide-binding cleft. This cleft, located between subdomains II and IV, tightly binds one molecule of ATP or ADP along with a divalent cation, typically Mg²⁺ under physiological conditions, which stabilizes the monomer and coordinates with residues such as Asp¹¹, Asp¹⁵⁴, and Gln¹³⁷.[5] The cation is essential for maintaining the actin's native fold, and its exchange from Ca²⁺ to Mg²⁺ influences polymerization competence.[6] In vertebrates, six principal actin isoforms exist, grouped into alpha, beta, and gamma classes, each encoded by distinct genes and displaying tissue-specific expression patterns.[7] Alpha-actins predominate in muscle tissues, such as alpha-skeletal actin in striated muscle and alpha-cardiac actin in heart tissue, whereas beta- and gamma-actins are primarily cytoplasmic and support non-muscle cell motility and structure.[8] These isoforms share over 90% sequence identity but differ in N-terminal residues, which subtly affect their interactions and functions.[9] Upon initiation of polymerization, G-actin undergoes conformational changes that facilitate filament assembly, including a flattening of the structure and closure of the nucleotide-binding cleft, which activates ATP hydrolysis to ADP and inorganic phosphate (Pi).[10] This hydrolysis, occurring primarily after incorporation into filaments, shifts the monomer to an ADP-bound state, enhancing structural flexibility and enabling dynamic remodeling, though the precise timing and mechanism remain linked to inter-subunit contacts in the polymer.[4] The resulting ADP-actin exhibits increased instability compared to the ATP-bound form, contributing to the overall turnover of actin networks.[5]Filament Organization
Filamentous actin (F-actin) forms a right-handed double-helical polymer from globular actin (G-actin) monomers, featuring 13 subunits per helical repeat of approximately 36 nm.[11][12] This helical architecture arises from the staggered arrangement of two protofilaments, with each monomer rotated by about 166° relative to the previous one.[13] Actin filaments exhibit structural polarity due to the asymmetric addition of G-actin monomers, resulting in a barbed (plus) end that supports rapid polymerization and a pointed (minus) end associated with slower growth and depolymerization rates.[1] The barbed end is defined by the orientation where myosin heads point away during decoration experiments, while the pointed end shows the opposite.[14] F-actin has a diameter of approximately 7-9 nm and achieves variable lengths up to several micrometers within cells, depending on regulatory factors.[1][15] In the absence of stabilizing actin-binding proteins, F-actin displays inherent instability, promoting dynamic turnover through processes like treadmilling.[11]Assembly and Dynamics
In Vitro Self-Assembly
The in vitro self-assembly of microfilaments from globular actin (G-actin) monomers follows a classical nucleation-elongation mechanism. The nucleation phase is rate-limiting and involves the thermodynamically unfavorable formation of small oligomers, primarily dimers and trimers, which serve as nuclei for filament growth.[16] Once a stable trimer nucleus forms, elongation occurs rapidly through the addition of ATP-bound G-actin monomers to the filament ends.[17] Due to the inherent polarity of actin filaments—with a fast-growing barbed end and a slow-growing pointed end—monomer addition predominantly favors the barbed end during this phase.[18] A key parameter governing net polymerization is the critical concentration (C_c), defined as the equilibrium G-actin concentration above which filament assembly exceeds disassembly. This value differs between filament ends owing to asymmetric on- and off-rates: approximately 0.1 μM at the barbed end and 0.6 μM at the pointed end, yielding an overall steady-state C_c of about 0.2–0.5 μM under physiological salt conditions.[18] Below C_c, filaments depolymerize, while above it, the system shifts toward net elongation until equilibrium is reached. Actin assembly in vitro is tightly coupled to ATP hydrolysis, which introduces vectorial dynamics. ATP-G-actin monomers incorporate into the filament preferentially at the barbed end, followed by rapid hydrolysis to ADP-P_i within the filament lattice; subsequent release of inorganic phosphate (P_i) destabilizes subunits at the pointed end, enhancing dissociation rates there.[19] This nucleotide-dependent process underlies the treadmilling model, where, at steady state above the overall C_c but between the end-specific values, subunits flux unidirectionally from the barbed end (net assembly) to the pointed end (net disassembly), resulting in filament treadmilling without net length change.[20] Kinetics of in vitro assembly are commonly monitored using pyrene-labeled actin fluorescence assays, in which the pyrenyl moiety exhibits a 20-fold increase in fluorescence upon incorporation into filaments, enabling real-time tracking of polymerization rates and half-times with high sensitivity. These assays have confirmed the sigmoidal time course of assembly, reflecting the lag in nucleation followed by exponential elongation.[21]In Vivo Dynamics and Regulation
In living cells, microfilament dynamics are tightly regulated by a suite of actin-binding proteins that control nucleation, polymerization, depolymerization, and filament stability, enabling rapid remodeling in response to cellular needs. Unlike in vitro conditions where actin filaments exhibit simple treadmilling, in vivo assembly involves coordinated protein interactions that dictate filament architecture and turnover rates. This regulation maintains a large pool of monomeric G-actin while allowing localized bursts of F-actin formation essential for processes like cell migration and cytokinesis. Recent cryo-EM studies (as of 2024) have provided atomic-level insights into Arp2/3 branch formation and pointed-end regulation, enhancing models of filament turnover.[22] Actin nucleation in vivo is primarily mediated by two major families of nucleators: the Arp2/3 complex, which generates branched actin networks, and formins, which promote linear filament elongation. The Arp2/3 complex binds to the sides of existing filaments and nucleates new branches at a 70-degree angle, facilitating the formation of dense, dendritic arrays that drive lamellipodia protrusion.[23] Activated by nucleation-promoting factors such as WASP and WAVE proteins, Arp2/3 lowers the energy barrier for actin dimer formation, accelerating nucleation by over 100-fold compared to spontaneous assembly.[24] In contrast, formins, particularly diaphanous-related formins (DRFs), associate processively with the barbed end of growing filaments via their FH2 domain, enabling unbranched elongation while protecting against capping.[25] This linear polymerization supports structures like stress fibers and filopodia, with formins increasing the rate of barbed-end addition by recruiting profilin-bound G-actin through their FH1 domain.[26] Capping proteins further refine filament dynamics by terminating growth at filament ends, promoting stability or disassembly. At the barbed end, CapZ (also known as capping protein) binds with high affinity (Kd ~0.1 nM) to block further monomer addition, thereby limiting filament length in branched networks and redirecting monomers for new nucleation events.[27] This activity is crucial for maintaining network density, as uncapped barbed ends would otherwise lead to uncontrolled elongation. At the pointed end, tropomodulin (Tmod) caps filaments in cooperation with tropomyosin, preventing depolymerization and stabilizing structures like muscle sarcomeres and erythrocyte membranes.[28] Tmod binding reduces the off-rate of ADP-actin subunits by over 100-fold, ensuring length constancy in stable cytoskeletal arrays.[29] Severing and depolymerization are orchestrated by the ADF/cofilin family, which preferentially binds ADP-bound actin in older filament regions, inducing twists that weaken inter-subunit contacts and promote filament fragmentation.[30] This severing generates new barbed ends for polymerization while accelerating pointed-end disassembly, recycling actin during motility.[31] Cofilin activity is regulated by phosphorylation, which inhibits binding, allowing spatial control of disassembly. A significant portion of G-actin (~50% in non-muscle cells) is sequestered by thymosin β4 (Tβ4), which binds monomers in a 1:1 complex with low micromolar affinity (Kd ≈ 1 μM), preventing spontaneous nucleation and maintaining a readily available pool for rapid assembly.[32] Tβ4 competes with profilin for G-actin binding but facilitates exchange, enabling quick transfer to polymerizing factors like formins.[32] Spatial regulation of these dynamics is achieved through signaling pathways, notably Rho GTPases, which locally activate nucleators via guanine nucleotide exchange factors (GEFs) and GTPase-activating proteins (GAPs). RhoA GTPase stimulates formins like mDia1 to drive linear actin assembly in stress fibers, while Rac1 and Cdc42 activate the Arp2/3 complex through WAVE and N-WASP, respectively, promoting branched networks in lamellipodia and filopodia.[24] This compartmentalized signaling ensures that actin remodeling occurs precisely where needed, integrating extracellular cues with cytoskeletal responses.[26]Cellular Functions
Force Generation
Microfilaments generate protrusive forces primarily through the polymerization of actin monomers at the barbed ends of filaments, a process described by the elastic Brownian ratchet model. In this model, thermal fluctuations create transient gaps between the growing filament tip and an obstructing surface, such as a cell membrane, allowing ATP-bound G-actin monomers to add to the barbed end and rectify Brownian motion into directed force. The elastic bending of the filament tip stores and releases energy to push against the obstacle, enabling individual filaments to produce forces up to approximately 10 pN.[33] This mechanism exploits the inherent polarity of actin filaments, with assembly favoring the barbed end to drive protrusion.[34] Depolymerization and severing of microfilaments contribute to force generation via elastic recoil, releasing stored mechanical energy that promotes contractility within the cytoskeleton. When filaments are severed, the sudden release of tension causes rapid recoil of the segments, reflecting underlying contractile stresses and facilitating network reorganization. This disassembly-driven recoil can dominate over polymerization forces in certain contexts, pulling structures inward and aiding in processes like cytokinesis by harnessing the elastic energy accumulated during prior assembly.[35][36] ATP hydrolysis plays a crucial role in powering these dynamics by coupling chemical energy to conformational changes in actin subunits, which drive filament instability and force production. Upon incorporation into the filament, ATP-bound actin hydrolyzes to ADP-Pi, and subsequent phosphate release induces structural changes that weaken inter-subunit bonds, promoting depolymerization at the pointed end while maintaining barbed-end growth. This vectorial hydrolysis provides the thermodynamic driving force for treadmilling and enhances the efficiency of force generation by modulating critical concentrations at filament ends.[19][37] The force-velocity relationships during actin assembly have been directly measured using optical tweezers, revealing how external loads stall polymerization at specific thresholds. In these experiments, actin filaments polymerize against a trapped bead, producing a linear force-velocity curve where growth rates decrease with increasing opposing force, stalling at loads of several hundred piconewtons for small filament ensembles. Such measurements confirm the Brownian ratchet predictions, showing that polymerization velocity drops to zero under forces equivalent to 1-10 pN per filament, depending on conditions.[38][39] In cellular applications, these polymerization-driven forces enable lamellipodia protrusion during cell migration, where branched actin networks at the leading edge push the plasma membrane forward. The probabilistic nature of filament-membrane contacts in the Brownian ratchet leads to intermittent bursts of force, collectively generating sufficient protrusive power—on the order of hundreds of piconewtons—to overcome membrane tension and drive invasive motility.[40][41]Motility and Transport
Microfilaments, composed primarily of actin, play a pivotal role in cellular motility through the sliding of actin filaments driven by non-muscle myosin II (NMII), which forms bipolar filaments to generate contractile forces essential for processes such as cytokinesis and cell shape maintenance. In cytokinesis, NMII assembles into minifilaments that cross-link and slide actin filaments within the contractile ring, constricting the cell equator to separate daughter cells after mitosis.[42] This mechanism relies on the ATPase activity of NMII heads to translocate actin toward the filament minus ends, producing tension that coordinates with membrane ingression for efficient division.[43] Similarly, in non-muscle cells, NMII-mediated actin sliding facilitates dynamic shape changes, such as bleb retraction or stress fiber contraction, enabling adaptation to mechanical cues in tissues.[42] Beyond contraction, microfilaments serve as tracks for myosin-based transport of vesicles and organelles, with myosin V and VI enabling processive movement toward specific cellular destinations. Myosin V, a plus-end-directed motor, walks along actin filaments to transport cargos like melanosomes in melanocytes, dispersing them peripherally for pigmentation; this involves dimerization of myosin V tails binding to adaptors such as melanophilin and Rab27a, achieving velocities up to 0.3 μm/s under physiological loads.[44] In contrast, myosin VI moves toward minus ends, supporting endocytosis and Golgi trafficking by pulling vesicles inward, often in coordination with clathrin adaptors.[45] For melanosome biogenesis, myosin VI interacts with branched actin networks to constrict tubular carriers, facilitating their fission and release from organelles, which underscores its role in short-range, force-dependent transport.[46] In cell migration, actin polymerization at the leading edge drives lamellipodial protrusion, where Arp2/3-mediated branching creates a dendritic network that pushes the plasma membrane forward, coupled with integrin-based adhesions that anchor the structure to the extracellular matrix. This polymerization, nucleated by WAVE complex activation downstream of Rac GTPases, generates protrusive forces of approximately 1-10 pN per filament, enabling directional sensing via chemotactic gradients.[47] Adhesion dynamics, involving focal adhesion kinase (FAK) signaling, regulate turnover: nascent adhesions form at the lamellipodium-lamella junction, maturing into stress fibers that transmit traction forces while disassembling at the rear to allow retraction.[48] This integrated cycle, balancing polymerization push with adhesion pull, sustains persistent migration speeds of 0.1-1 μm/min in fibroblasts and leukocytes. Pathogenic bacteria like Listeria monocytogenes exploit microfilament dynamics for intracellular motility, using the surface protein ActA to mimic host nucleation-promoting factors and induce actin polymerization at one pole, forming a comet tail that propels the bacterium at speeds of approximately 0.1 μm/s through the cytosol.[49][50] ActA recruits the Arp2/3 complex and Ena/VASP proteins to assemble branched F-actin networks, with tail architecture featuring Y-junctions oriented toward the bacterium for efficient force transmission.[49] This actin-based propulsion allows Listeria to evade immune detection by spreading cell-to-cell via membrane protrusions, highlighting microfilaments' vulnerability to microbial hijacking.[51] Actin-based motility mechanisms exhibit remarkable evolutionary conservation, from yeast endocytic actin patches to mammalian lamellipodia, where core components like Arp2/3 and formins drive pseudopod extension across eukaryotes. In budding yeast, type I myosins and profilin facilitate actin comet tails for vesicle trafficking, analogous to myosin V transport in mammals, reflecting ancient adaptations for force generation and cargo delivery.[52] This conservation extends to crawling migration, with WASP-family proteins regulating branched networks in amoebae, dictyostelium, and human cells, underscoring actin microfilaments' fundamental role in eukaryotic locomotion since the last eukaryotic common ancestor.[53]Associated Proteins
Actin-Binding Proteins
Actin-binding proteins (ABPs) are a diverse group of proteins that interact directly with actin monomers or filaments to regulate microfilament structure, stability, and dynamics in eukaryotic cells.[54] Humans express over 100 such ABPs, which are classified by their primary functions, including monomer sequestration, filament nucleation, cross-linking, bundling, capping, and severing.[55] Many ABPs contain conserved structural domains, such as the WH2 domain, which binds G-actin monomers with high affinity to control availability for polymerization.[54] Cross-linking proteins connect actin filaments into networks or bundles, thereby influencing cytoskeletal architecture and mechanical properties. Filamin, a dimeric ABP, cross-links actin filaments at flexible angles to form orthogonal, gel-like networks that provide structural support in dynamic cellular regions like the cell cortex.[56] In contrast, α-actinin cross-links filaments in a more rigid, parallel manner to assemble contractile bundles, particularly in stress fibers where it stabilizes actin-myosin interactions essential for cell adhesion and force transmission.[57] Bundling proteins organize actin filaments into tightly packed, parallel arrays that contribute to specialized protrusions. Fascin, a compact ABP, bundles actin filaments into straight, rigid bundles that form the core of microspikes and filopodia, facilitating cell motility and sensory functions.[58] Similarly, espin bundles actin in stereocilia of inner ear hair cells, promoting elongation and parallel organization critical for mechanotransduction in hearing and balance.[59] Stabilizing proteins enhance filament persistence by modulating polymerization and depolymerization rates. Tropomyosin wraps around actin filaments in a cooperative manner, inhibiting depolymerization from the pointed end and blocking access of severing proteins like cofilin, thereby promoting filament stability in structures such as stress fibers and lamellipodia.[60][61] Nucleation-promoting factors initiate new filament formation by activating the Arp2/3 complex. The WASP (Wiskott-Aldrich syndrome protein) and Scar (also known as WAVE) families recruit and activate Arp2/3 to nucleate branched actin filaments, generating dendritic arrays that drive processes like lamellipodia extension.[62]Myosin Interactions
Myosins are actin-based motor proteins characterized by a modular structure consisting of a globular head domain, a neck region, and a tail domain. The head, also known as the motor domain, encompasses the actin- and ATP-binding sites, enabling ATP hydrolysis to drive conformational changes that generate force along actin filaments.[63] The neck serves as a lever arm, binding essential and regulatory light chains that amplify small structural changes in the head into larger displacements of the tail.[64] The tail domain facilitates dimerization via coiled-coil formation in certain classes or binds cargo for transport, linking the motor to cellular structures. Several myosin classes interact with microfilaments to perform specialized functions in eukaryotic cells. Myosin II, the conventional class, forms bipolar filaments that drive contractile events, such as muscle contraction and cytokinesis, by sliding antiparallel actin filaments relative to one another.[65] Myosin V operates as a processive dimer, taking multiple steps along actin to transport vesicles and organelles over long distances within the cell.[66] In contrast, myosin I functions primarily as a monomer or dimer that links actin filaments to plasma membranes, facilitating tension sensing and membrane remodeling.[67] The interaction between myosin and microfilaments occurs through the cross-bridge cycle, an ATP-dependent process that powers filament sliding. In this cycle, ATP binding to the myosin head induces detachment from actin, followed by hydrolysis to ADP and inorganic phosphate (Pi), which cocks the lever arm into a high-energy state.[68] Rebinding to actin triggers Pi release, initiating the power stroke—a conformational change that propels the lever arm forward, displacing actin by a step size of approximately 5 nm for myosin II and up to 36 nm for myosin V, matching the helical repeat of actin filaments.[69] ADP release completes the cycle, resetting the head for another attachment, with the entire process repeating rapidly to achieve continuous motion.[70] Myosin motility along microfilaments exhibits load-dependent velocity, where increasing opposing force reduces stepping speed until reaching a stall force beyond which net backward motion occurs. For individual myosin heads, stall forces typically range from 1 to 5 pN, with myosin V achieving around 2-3 pN per dimer under physiological loads, enabling robust intracellular transport.[71] These forces scale with the number of engaged motors in ensembles, such as in myosin II filaments, to generate contractile tensions in cellular structures.[72] Regulation of myosin II activity primarily involves phosphorylation of its regulatory light chains (RLCs) at serine-19 by kinases like myosin light-chain kinase, which relieves an autoinhibited state by disrupting head-tail interactions and promoting filament assembly and ATPase activity.[73] This post-translational modification is essential for activating contraction in both muscle and non-muscle cells, with dephosphorylation by phosphatases reversing the process to inhibit motility.[74]Mechanistic Models
Actoclampin Model
The actoclampin model proposes a mechanism for actin-based motility in which hypothetical proteins, termed actoclampins, bind tightly to the barbed ends of growing microfilaments, enabling processive tracking and force generation without detachment during polymerization. This model, introduced by Dickinson and Purich in 2002, exploits the intrinsic ATPase activity of actin to drive a cyclic reaction that couples filament elongation to directed movement.[75] Actoclampins are envisioned as multi-subunit complexes that maintain a high-affinity clamp on the terminal ATP-bound actin subunit at the filament end, allowing sustained motility observed in systems like Listeria propulsion.[76] Key features of the model include affinity modulation triggered by ATP hydrolysis, which ensures the motor remains attached to the advancing barbed end while facilitating subunit addition. The process begins with the binding of an ATP-actin terminal subunit to the actoclampin clamp on a surface or cargo.[75] Subsequent steps involve the addition of new actin.ATP monomers, inducing a conformational change and ATP hydrolysis in the rear subunit, which lowers its affinity and promotes forward stepping of the entire complex by approximately 5.4 nm per cycle.[77] This is followed by the release of the ADP-bound actin from the rear, resetting the cycle for continued elongation and translocation.[76] Stochastic simulations of this "lock, load, and fire" mechanism demonstrate how it can generate flexural or tensile forces on tethered filaments, explaining pause fluctuations and stepwise motion in experimental motility assays.[75] The model draws from biophysical observations of end-tracking behaviors in actin comet tails, such as those powering bacterial pathogens, though the specific actoclampin proteins remain hypothetical.[78] It references foundational 1980s studies by Pollard on actin polymerization kinetics, which highlighted the role of nucleotide states in filament dynamics.[79] While the actoclampin concept has been influential in conceptualizing barbed-end tracking, it has largely been superseded by the identification of real proteins like formins, which perform similar processive capping and elongation functions without requiring the hypothetical clamping machinery.[80] Nonetheless, the model illustrates early theoretical insights into how microfilament ends could drive comet tail formation and cellular propulsion.[78]Polymerization-Driven Models
Polymerization-driven models explain how the addition of actin monomers to growing microfilament ends generates mechanical force for cellular protrusion, primarily through rectification of thermal fluctuations during assembly.[34] The Brownian ratchet model posits that under load, such as from a cell membrane, thermal fluctuations create transient gaps between the filament tip and the barrier, allowing insertion of new globular actin (G-actin) subunits; polymerization then rectifies these fluctuations by preventing backward diffusion, thereby producing directed force.[34] This mechanism, first proposed for actin-based motility, predicts that force generation depends on monomer concentration and load, with higher loads reducing polymerization rates by limiting gap formation.[34] Building on the Brownian ratchet, the elastic tether model incorporates flexible linkers, such as formins or Arp2/3 complex attachments, that connect growing filaments to the load (e.g., plasma membrane).[81] These tethers store elastic energy as the filament tip bends under load, facilitating subunit addition even when fluctuations alone are insufficient; release of this energy contributes to pushing forces, particularly in structures where direct tip-barrier contact is intermittent.[81] Formins, for instance, processively add monomers while remaining tethered, enhancing force output compared to untethered growth.[81] In cellular contexts like lamellipodia, multi-filament cooperativity arises from ensembles of parallel or branched microfilaments acting collectively to generate sustained force against the membrane.[81] Individual filaments produce modest forces (∼1-5 pN), but cooperativity amplifies total force through load sharing and synchronized growth across multiple filaments, enabling protrusion rates of 0.1-1 μm/min and network stall forces of ∼100-500 pN depending on conditions; in dendritic networks, Arp2/3-mediated branching further distributes force across hundreds of filaments.[81][82] This ensemble behavior stabilizes protrusions by compensating for filament variability in length and orientation.[81] A key mathematical framework for these models derives the force-velocity relationship from polymerization kinetics, given byv = \delta (k_{\text{on}} [G\text{-actin}] - k_{\text{off}})
where v is protrusion velocity, \delta is the size of the added subunit (∼2.7 nm), k_{\text{on}} is the on-rate constant, [G\text{-actin}] is free monomer concentration, and k_{\text{off}} is the off-rate constant modulated by load.[34] Under zero load, v maximizes at high monomer levels; increasing force exponentially decreases k_{\text{on}} via reduced gap probability, yielding a nonlinear stall force of ∼1-5 pN per filament depending on conditions.[34][82] Computational simulations of these models, using Brownian dynamics to track filament growth against barriers, reproduce in vitro pushing assays where actin networks propel beads or liposomes at forces up to 50 pN and velocities matching experimental data (e.g., 0.05-0.2 μm/s at 1-10 μM G-actin).[38] These validate the ratchet mechanisms by showing load-dependent velocity curves and enhanced force from tethering or multi-filament arrays, consistent with optical trap measurements of individual and ensemble polymerization.[38]