Nonequilibrium chemomechanical transduction of ATP-driven protein unfolding in the 26S proteasome

Abstract

AAA+ ATPases are essential ATP-driven molecular machines with diverse cellular functions, including protein unfolding, active transport, and chromatin remodeling. Despite their broad importance, the precise mechanisms by which energy transduction drives protein unfolding in AAA-ATPase motors remain unclear. Here, we present a probabilistic model that simulates nonequilibrium chemomechanical transduction of ring-like AAA-ATPase motors during substrate unfolding in the 26S proteasome. By capturing the sequential cycling of ATP hydrolysis around the ATPase ring, our model explores a wider range of coordinated conformational transitions than previously observed experimentally. Our simulations reveal multiple high-probability pathways for state transition during hand-over-hand translocation of substrate, elucidating the nonequilibrium dynamics of around-the-ring energy transduction in AAA-ATPase motors. These findings, extensively examined by experiments, provide quantitative insights into chemomechanical coupling that are likely conserved across the AAA+ protease or unfoldase superfamily. This work offers a theoretical framework for understanding ring-like AAA+ translocation motors in general.

Introduction

The AAA+ (ATPases associated with diverse cellular activities) proteins are molecular machines found in all organisms, playing crucial roles in a variety of cellular processes, such as protein degradation, DNA replication and repair, and cell cycle regulation^1,2,3. These proteins contain a conserved AAA+ domain that converts the chemical energy from ATP hydrolysis into mechanical work, which is responsible for a diverse range of cellular activities, including protein quality control, ribosome assembly, genetic regulation, and viral replication^4,5,6,7,8. AAA+ proteases, a subclass of AAA+ superfamily maintaining protein homeostasis in an ATP-dependent manner^9,10,11. They feature a compartmental subcomplex with active proteolytic sites located within its inner chamber and an ATP-hydrolyzing ring-shaped motor subcomplex made of six AAA+ ATPases⁹. AAA+ ATPases are classified into different families, including the proteasomal ATPase, FtsH, Cdc48, ClpA/B/C-Domain 1 and 2, HslU/ClpX, and Lon^9,10.

The 26S proteasome, with a molecular weight of 2.5 million Daltons, is one of the most complex molecular machineries found in all eukaryotes¹¹. It regulates myriad cellular processes, including cell cycle, gene expression, immune responses, tumorigenesis, neurodegeneration, and aging^{12,13,14,15,16}. This multi-subunit complex is composed of a 20S core particle (CP) responsible for proteolysis and a 19S regulatory particle (RP) that regulates substrate processing. The 19S RP contains six distinct ATPase subunits that hydrolyze ATP to unfold and thread substrate proteins into the cylindrical 20S CP chamber. The proteasomal ATPase motor, positioned at the entry of the 20S CP, consists of six RPT subunits (RPT1-RPT6), which form a heterohexameric ring with a central channel for substrate passage. ATP binding and hydrolysis occur at nucleotide-binding pockets at the interfaces between adjacent RPTs, driving the conformational changes of the ATPase motor and allosterically modulating the interaction between pore loops and the substrate.

Recent research on AAA+ proteases and unfoldases has been focused on understanding how ATP hydrolysis in the AAA+ motor drives the conformational changes of ATPases for substrate translocation. High-resolution cryogenic electron microscopy (cryo-EM) structures and biochemical studies propose a qualitative “hand-over-hand” model, where ATP hydrolysis occurs sequentially within the ATPase hexameric ring^17,18,19,20. However, this model remains largely speculative as most high-resolution structures capture only few conformations under specific conditions^21,22,23,24. There is insufficient evidence to determine the temporal order of events along the chemical reaction pathway^18,25. Thus, the exact mechanisms of substrate translocation by AAA+ motors remains unclear¹⁰.

In this work, we conducted extensive computational simulations by developing a nonequilibrium probabilistic model for understanding how the coordinated ATP hydrolysis is coupled with conformational transitions in the proteasomal ATPase motor (Fig. 1). We theoretically model the dynamic process of chemomechanical transduction cycle in the proteasomal ATPase motor during substrate translocation by assuming stochastic transitions among feasible conformations of the ATPase motor. The stochastic model yields predictions that are consistent with a broad spectrum of published experimental observations and are further verified in two accompanying studies^26,27. Our simulations provide a quantitative definition for the hand-over-hand mechanism, weave a self-consistent dynamic relationship among all previously published and newly discovered conformational states of the proteasomal ATPase motor, and offer interpretations for underlying biophysical mechanisms in detail, including substrate translocation rates under varying ATP, ADP, and ATPγS concentrations, as well as the degradation kinetics of substrate proteins with structurally tight domains.

Fig. 1: Model for substrate unfolding and translocation in the ATPase motor.

a Cartoon for the structure of the human proteasome and a top view of the ATPase motor composed of six RPT subunits denoted in different colors. H31 is the motor conformation with six substrate-engaged RPTs and closed interfaces. b Schema of state changes of the ATPase motor, including nucleotide changes caused by ATP (also ADP) binding and hydrolysis in the nucleotide-binding pocket, and conformational changes of the RPT complex. c Five types of motor conformations, represented by conformations H1, H2, H3, H4, and H5, considered in our model (left column), and the possible conformations they can transition to in a single next-step change (right column). The distances moved by the substrate as a result of the conformation change are also given, with one step equal to the length of two amino acids. The full list of motor conformations with substrate-disengaged RPT(s), from H1 to H30, is given in Fig. S1. In each conformation, the disengagement of an RPT from the substrate is represented by a displacement of RPT away from the central red dot, and the open interface between adjacent RPTs is represented by a gap.

Results

A nonequilibrium probabilistic model

We expected to develop a minimal stochastic dynamics model that suffices to capture the nonequilibrium physics of chemomechanical transduction in the ATPase motor of the 26S proteasome, based on early experimental high-resolution structural observations^18,25 (Figs. 1 and S1). Details of our modeling are provided in the Methods section. Our computational model was used to simulate the stochastic binding and hydrolysis of ATP in the nucleotide-binding pockets of the ATPase subunits, as well as the conformational transitions of the motor. The pore loops of the ATPase subunits form a central channel in the ring-like ATPase motor, through which they collectively grip and translocate the substrate into the proteasomal core particle for degradation. The substrate moves either forward or backward based on the staircase-like arrangement of the ATPase subunits, which undergo coordinated changes in their nucleotide-binding states. The model incorporates a total of 30 feasible conformational states of the ATPase motor (designated H1-H30), categorized into five main types, each representing different configurations of engagement with the substrate (Fig. 1c). Conformational state transitions follow a hand-over-hand fashion, where only the RPT subunit nearest to the core particle disengages from the substrate and moves to the top of the staircase to re-engage with the substrate during substrate translocation. As ATP hydrolysis irreversibly occurs at closed interfaces between adjacent ATPase subunits, its chemical energy is converted into mechanical energy that drives the conformational changes. This process breaks detailed balance in the microstates of the ATPase motors, outputting mechanical work that propels substrate translocation. Notably, at high ATP concentrations, the ATPase motor can adopt a non-translocating conformation (designated H31), where ATP hydrolysis continues but translocation is suspended, reflecting an inhibitory effect observed in experimental studies of the proteasomal ATPases motor²⁸ (Fig. 1a). We modeled the transitions between different conformational states as Markovian processes, with transition rates based on the binding energy and hydrolytic states of nucleotides in the nucleotide-binding pockets. The stochastic simulation uses the Gillespie algorithm²⁹ to compute over 500,000 random steps for each run, allowing for the calculation of mean translocation rates and standard deviations.

Kinetics of substrate translocation in ATPase motor

We first simulated the stochastic translocation of the substrate polypeptide by the AAA-ATPase motor, by adjusting the concentrations of ATP and ADP. Figure 2a shows the substrate translocation rate as a function of ATP concentration (the line), alongside the experimental observations (symbols with error bar). At ATP concentrations below 1 mM, the translocation rate rises with increasing ATP concentration, indicating that ATP binding is the rate-limiting step. However, this relationship is not monotonic. Notably, once the ATP concentration exceeds 1 mM, the translocation rate peaks and subsequently declines. Under these saturating conditions, the rate-limiting step shifts from ATP binding to downstream events, specifically ATP hydrolysis, ADP release, and the associated conformational changes of ATPases. The inhibitory effect of high ATP concentration on substrate translocation is consistent with the recent experimental findings²⁸, as well as earlier biochemical studies^30,31,32. When ATP concentration is fixed at 0.5 mM and ADP concentration is increased, the simulated translocation rate decreases monotonously, as shown in Fig. 2b. This behavior aligns with the experimental observations reported in ref. ²⁸. Mechanistically, as ADP competes with ATP for binding to the nucleotide pockets of the AAA-ATPase motor, ATP binding and hydrolysis are hindered in the presence of ADP, leading to a continuous decrease in the translocation rate as ADP concentration rises.

Fig. 2: Simulated translocation rates in comparison with experimental observations.

a Substrate translocation rate as a function of ATP concentration. b The depressive effect of ADP on the translocation rate with a fixed ATP concentration of 0.5 mM. The inhibitory effect of ATPγS on substrate translocation, with ATP concentrations of 0.5 mM (c) and 15 mM (d), respectively. The experiment data are from ref. ²⁸.

ATPγS, a common analog of ATP used to investigate the function of motor protein, can also bind to the nucleotide pockets of the proteasomal ATPase motor. Although its structure closely resembles ATP, it is hardly hydrolyzed once bound. The inhibitory effect of ATPγS on substrate translocation is similar to that of ADP, but with a binding affinity more akin to ATP. As a result, ATPγS hinders substrate translocation by locking the ATPase conformations more effectively than ADP. In our simulations, we use the same parameter values for ATPγS as for ATP, with the exception that nucleotide hydrolysis is prohibited. Figure 2c, d depicts the simulated translocation rate as a function of ATPγS concentration with 0.5 mM and 15 mM ATP, respectively. Both sets of results are consistent with experimental observations. From the depressed curves, we estimate the half maximal inhibitory concentration (IC₅₀) of ATPγS to be 0.048 mM for 0.5 mM ATP and 2.1 mM for 15 mM ATP. The IC₅₀ increases at higher ATP concentrations, as more ATPγS molecules are required to compete with ATP for binding to the nucleotide pockets. Additionally, our simulations calculate the translocation rate as a function of the ATP hydrolysis rate, while tuning the ATPγS concentration. These yield linear relationships between translocation rate and hydrolysis rate (Fig. S3), which are also consistent with previous experimental findings³³. Extrapolating the relationship to a zero hydrolysis rate yields a negative value for the translocation rate. This result stems from the application of the external force applied to the AAA-ATPase motor in our simulations. Specifically, when ATP hydrolysis ceases, the motor loses its ability to generate the forward mechanical work necessary to counteract the opposing external force, causing the substrate to be pulled backward.

Multiplicity of chemomechanical coupling

To understand the mechanisms of the AAA-ATPase motor, we analyzed the detailed conformational changes and dynamics of nucleotide-binding pocket during substrate translocation. The state-transition probabilities between the 31 ATPase conformations (H1–H31) were computed from simulation runs at different ATP concentration, as shown in Fig. 3a, b for 0.01 and 1 mM ATP, respectively. Under physiological condition with relatively low ATP concentrations ($\sim$0.01 mM), the AAA-ATPase motor undergoes a variety of conformational changes that align with the hand-over-hand mechanism (Fig. 3a), leading to slower substrate movement. At higher ATP concentrations, with faster translocation rates, the most probable conformational transitions are limited to 24 key changes (Fig. 3b). In these dominant transitions, one or two RPT subunits near the core particle disengage from the substrate, while previously disengaged RPTs re-engage with the substrate, causing the substrate to move forward by one or two steps towards the CP. Figure 3c, d illustrates the nucleotide changes in the six RPT subunits during the 24 key conformational transitions. The heat map displays the frequency of ATP binding (and unbinding), hydrolysis, and ADP release events occurring just before each of these transitions. Consistent with the hand-over-hand mechanism, ATP binding occurs primarily in the RPT subunits near the top of the AAA-ATPase motor, while ATP hydrolysis mainly happens at the bottom subunits, closer to the CP. Additional simulations of conformations transitions and nucleotide exchanges at ATP concentrations of 0.1 and 10 mM further confirm these findings, as shown in Fig. S4.

Fig. 3: Multiplicity of state transitions in the ATPase motor during substrate translocation.

Transition probabilities between 31 conformations H1-H31 obtained from stochastic simulations at ATP concentrations of 0.01 mM (a) and 1 mM (b), respectively. Probabilities of nucleotide state changes obtained by counting the random events of ATP binding, unbinding, and hydrolysis, and ADP unbinding immediately before major conformational transitions occur, at ATP concentrations of 0.01 mM (c) and 1 mM (d), respectively. For each type of nucleotide change listed on the horizontal axis, the six RPTs are arranged according to their distances to the core particle by spreading out the motor subunits anticlockwise, starting with the substrate-engaged RPT at position 1 in the first place. e, f Four predominant pathways of state transitions adopted by the ATPase motor to translocate the substrate peptide, derived from the results in (a–d).

A detailed examination revealed that the 24 predominant transitions can be grouped into four categories, with homologous architectures being considered (Fig. 3e, f). The first two fundamental transitions involve a motor configuration with five substrate-engaged RPTs and one disengaged from the substrate (Fig. 3e). In the first transition, ATP is hydrolyzed by the RPT at position 2, causing the RPT at position 1 to disengage from the substrate. The substrate-disengaged RPT binds ATP, moves to the top of the motor, and re-engaged with the substrate. As the five substrate-engaged RPTs shift counterclockwise by one position, the substrate is advanced by a single step toward the CP. In the second transition, ATP hydrolysis occurs in the RPT at position 3, leading to the simultaneous disengagement of the RPT subunits at positions 1 and 2. The previously disengaged RPT binds ATP, moves to the top of the AAA-ATPase motor top and re-engages with the substrate, moving the substrate forward by two steps. The remaining two predominant transitions (Fig. 3f) occur in structural configurations with two substrate-disengaged and four substrate-engaged RPTs. In a similar manner, ATP hydrolysis at position 2 triggers the movement of the two disengaged RPTs to the top of the motor, where they re-engage with the substrate. Following the disengagement of the RPT at position 1, the substrate advances by one step. In the final case, ATP hydrolysis at position 3, coupled with the disengagement of the RPTs at positions 1 and 2, allowed the disengaged subunits to move to the top and re-engage. The substrate is moved forward by two steps as the RPTs shift counterclockwise, including the repositioning of the RPT at position 3 to position 1. The sequence of ATP binding and hydrolysis events varies stochastically across the four types of transitions. Although both the two-step and one-step transition presumably consume the same amount of chemical energy, the one-step transition performs less work against external forces. Consequently, under conditions of sufficient ATP supply, the one-step transition is more likely to occur, as evidenced by the high probability of the first transition type shown in Fig. 3b. Conversely, ATP depletion disrupts this normal transition mode, potentially leading to the increased frequency of the two-step process observed in Figs. 3a and S4a. These four types of transitions represent the primary modes by which the proteasomal ATPase motor translocates substrate peptides, in line with the hand-over-hand mechanism.

Unfolding of substrates with tight folds

When the proteasomal ATPase motor encounters a substrate protein with tightly folded domains, it must exert increased mechanical work to overcome the energetic barrier of substrate unfolding during substrate translocation into the CP. In our model, this resistance is represented by the parameter F_sub in Eq. 6 (see “Methods”). The time required to unfold such a protein domain depends on the stability of the substrate³⁴. Experimental studies have shown that the proteasomal ATPase motor does not stall or lose functionality when encountering substrates with domains that resist unfolding. Instead, the motor actively releases the substrate to maintain the integrity of proteasome function³⁵. Similar behaviors have been observed in other AAA+ motors, such as E. coli ClpX³⁶.

Figure 4 illustrates the degradation rate of a substrate protein with and without obstacles, based on our Gillespie simulations. The degradation rate was calculated by dividing the peptide length (with 445 amino acids, as used in experiments²⁸) by the simulated translocation rate. Our model predictions are consistent with recent experimental observations²⁸. Fang et al. measured the degradation rate of substrates containing the dihydrofolate reductase (DHFR) domain, which can be stabilized by the addition of folic acid (FA)²⁸. In the presence of FA, the AAA-ATPase motor required more time to unfold the DHFR domain. As shown in Fig. 4a, the addition of FA significantly reduces the degradation rate. In our simulations, the AAA-ATPase motor had to exert greater force to overcome the resistance caused by FA. After normalizing the degradation rates by their respective maximum rates (with and without the obstacle), the plots of degradation rates overlap with each other (Fig. 4b), indicating that our model effectively reproduces the experimental observations.

Fig. 4: Degradation of substrate proteins with tight domains.

a Degradation rates obtained from simulations (lines) and from experiments (dot symbols). b Normalization of the rates in (a) showing an overlap. The simulated degradation rate was calculated by dividing the peptide length (445 amino acids, as used in experiments²⁸) by the simulated translocation rate. The parameters in our simulation are as listed in Table S2, except F_sub = 1.984 pN (or 0.2 kcal/(mol∙step)) for the green line without folic acid (FA), and F_sub = 6.44 pN (or 0.65 kcal/(mol∙step)) for the red line with FA. The experiment data are from ref. ²⁸.

Energy transduction efficiency

The energy for the entire AAA-ATPase complex during conformational changes can be calculated using the parameter defined in our model (see Eq. 2 in “Methods” and Table S2). Figure 5a presents the energy landscape of the AAA-ATPase motor throughout a cycle of work output as the motor undergoes conformational changes. ATP hydrolysis raises the total energy of the motor, which is then reduced as mechanical work is performed during conformational changes that move the substrate forward. Our analysis confirms that this energy landscape remains consistent across the four primary operating modes of the ATPase motor as depicted in Fig. 3e, f.

Fig. 5: Energy transduction in the ATPase motor during substrate unfolding and translocation.

a The free energy changes during the cycle of ATP hydrolysis and nucleotide exchange. The substrate translocation (b), the ATP hydrolysis rate (c), and the normalized efficiency of the ATPase motor in converting the chemical energy of ATP hydrolysis into mechanical work (d), respectively, as function of the resistance (F_sub) the motor experiences during substrate unfolding and translocation. The experiment data in (b) is from ref. ³⁶.

The functionality of the AAA+ motor is further checked in our model with varied resistance the motor experiences during translocation. Figure 5b–d shows the substrate translocation rate, ATP hydrolysis rate, and the efficiency of the AAA-ATPase motor in converting the chemical energy of ATP hydrolysis into mechanical work, respectively, as the resistance F_sub is adjusted. As expected, the translocation rate decreases with increasing resistance, indicating that higher resistance reduces the “power stroke” of the motor during ATP hydrolyses. Although experimental measurements of substrate translocation rates in the 26S under varying driving forces are not yet available, our simulation results reveal a trend similar to that experimentally observed in the translocation rate changes of ClpX, another important member of the AAA+ protease family, under external forces. Similarly, as shown in Fig. 5c, the ATP hydrolysis rate decreases as resistance increases. The efficiency of the AAA-ATPase motor in converting chemical energy into mechanical work can be estimated as following

$$Efficiency=\frac{Resistance\,force\,\times \,Translocation\,Rate}{{Rate}\,{of}\,{ATP}\,hydrolysis\,\times \,Chemical\,energy\,of\,single\,ATP\,hydrolysis}$$

(1)

Figure 5d depicts the normalized efficiency of energy conversion as a function of the resistance F_sub. At a low resistance, efficiency is low due to high rate of ATP hydrolysis and minimal work output. The efficiency increases as F_sub grows, reaching a peak at an intermediate level of resistance. However, as the resistance F_sub continues to rise, the efficiency drops because of the rapid decline in translocation rates. Eventually, the motor stalls under a sufficiently high resistance.

To understand how the resistance F_sub impacts the nucleotide states, we computed and compared the proportion of bound ADPs in the AAA-ATPase motor under varying resistance levels. As shown in Fig. 6a, at a low resistance (0.1 pN), the motor predominantly binds zero to two ADPs. However, as the resistance increases (6.5 pN) and approaches substrate translocation stalling (13 pN), the proportion of AAA-ATPase motors binding zero or one ADP significantly decreases, while the probability of motors binding two or more ADPs increases markedly, especially for those binding three or four ADPs. These computational predictions are strongly supported by our experimental observations²⁷. In cryo-EM studies of the human 26S proteasome engaged with a largely less ordered substrate Sic1^PY that presents low resistance, only one or two ADP molecules were typically observed within the AAA-ATPase motor^18,25,37. In contrast, in the latest structures of the substrate-engaged proteasome in complex with mutant USP14 and inhibited RPN11 described in an accompanying paper²⁷, there is a notable increase in the number of bound ADPs, distributed from three to five ADPs. As RPN11 inhibition stalls the substrate translocation, it presents virtually an infinite high resistance, which interferes with ADP release and leads to the accumulation of ADP within the AAA-ATPase motor. The quantitative agreement between our computational modeling and high-resolution cryo-EM experiments demonstrates that our model effectively predicts the coupling between the load force and ATP hydrolysis of AAA-ATPase motor.

Fig. 6: Effects of nucleotide variation on substrate translocation.

a The coupling between resistance of substrate and the number of bound ADPs within the AAA-ATPase motor. The simulations were conducted at an ATP concentration of 1 mM. b Normalized translocation rates plotted as a function of the [ATP$\gamma$S]/[ATP] ratio in human and yeast proteasome and E. coli ClpX, obtained from experiments and our simulations. The rates are normalized by the translocation rate without the inhibition of ATPγS. The experiment data are from refs. ^28,33,38.

Discussion

In this study, we develop a stochastic model that simulates the temporal nonequilibrium progression of chemomechanical transduction within the AAA-ATPase motor of the 26S proteasome, which are largely supported and verified by the latest experimental findings on the structures and dynamics of the 26S proteasome^26,27. Our stochastic model elaborates the hand-over-hand mechanism in a quantitative manner, which assumes a sequential, unidirectional cycling of ATP hydrolysis around the ATPase ring. We simulate how translocation rates are influenced by various factors such as the concentrations of ATP, ADP, and ATPγS, as well as the addition of folic acid. The model’s predictions are closely reproduced by the experimental observations in two accompanying studies^26,27. Through these simulations, we identified four primary modes of ATP-driven chemomechanical transductions. Typically, as ATP is hydrolyzed, ATPase subunits disengage from the substrate at the bottom of the motor and regrip it at the top as new ATP binds.

Interactions between the ATPase and other RP subunits, including RPN1, RPN2, RPN5-7, as well as certain deubiquitylating enzymes (DUBs), introduce asymmetric regulation on the dynamic behavior of the proteasomal ATPase motor²⁵. In the absence of the DUB ubiquitin-specific protease 14 (USP14), 20 conformational states were identified in the substrate-engaged human 26S proteasome^18,26. Recent cryo-EM studies^25,27 revealed 29 distinct substrate-engaged conformational states regulated by USP14 or its mutant, providing finer observations of intermediate conformations of the substrate-engaged ATPase motors. While some major intermediates were not observed under specific condition, the combined experimental observations captured all six major conformational states, completing a full hand-over-hand cycle around the ATPase ring and numerous intermediate substates illustrating most conformational transitions between these major states. These intermediate conformers are in line with and validate our predictions regarding primary modes of chemomechanical transduction, as shown in Fig. 3. Particularly, our simulations recapitulate and explain the experimental observations of intermediate conformers of the ATPase motor during the process of a single nucleotide exchange that drives the initiation of substrate translocation in the 26S proteasome, where two consecutive ATPases are simultaneously disengaged from the substrate (Fig. 3f). Although certain predictions from our computational modeling require further experimental studies, these findings are largely corroborated by our accompanying experimental studies^26,27, providing the final pieces in the “jigsaw puzzle” to fully characterize the nonequilibrium dynamics of substrate translocation in the human 26S proteasome.

Given that proteolytic AAA+ ATPases typically exhibit a conserved hexameric ring structure, it may suggest that the mechanochemical coupling mechanisms for force production and substrate translocation simulated in our model can be applicable across the AAA+ protease family. As shown in Fig. 6b, the normalized curves of substrate translocation rate as a function of ATPγS concentration—observed in yeast proteasome³³, human proteasome²⁸, E. coli ClpX³⁸ and in our simulations—align closely with one another. This consistent behavior of motor inhibition by ATPγS suggests that the hexameric motors in the AAA+ ATPase family might well share a common hand-over-hand mechanism for substrate translocation. Recent studies on the Clp family have employed molecular dynamics simulations to elucidate the translocation mechanisms of ATPase motors^39,40,41. Some of these works have challenged the hand-over-hand mechanism⁴⁰. However, these studies have not accounted for the coupling effects between ATP hydrolysis/exchange processes and conformational changes on ATPase translocation. Given the structural variations among AAA+ proteases, subtle differences in their dynamics cannot be ruled out. For example, ClpX exhibits larger step sizes and higher translocation rates compared to the proteasomes³⁶. These variations might explain the discrepancies between the simulated translocation rates of proteasomal ATPases under variable load and experimental observations for ClpX in Fig. 5b. Whether other AAA+ ATPases share identical or partially conserved translocation mechanisms with proteasomal ATPases remains to be further investigated.

Recently, there have been a few computational modeling approaches that have been applied to investigate the functionality of the 26S proteasome. Saha et al.⁴² conducted coarse-grained molecular dynamics (MD) simulations to calculate the energy landscape of the human 26S proteasome based on seven different conformations¹⁸. A discrete kinetic model for peptide translocations coordinated with ATP hydrolysis was proposed²⁸. This model, based on an empirical nucleotide-dependent free-energy landscape of the proteasome’s heterohexameric ATPase complex, simulated the dynamics of the proteasome and evaluated its structural and kinetic properties. Unlike the broadly proposed hand-over-hand model of sequential ATP hydrolysis, Fang et al. assumed all possible transitions between conformations²⁸. Despite advancements, these computational models did not address the energy transduction of AAA-ATPase motor in this work and fall short of making testable predictions that can be verified in experiments such as those newly reported in our accompanying studies^26,27.

Computational and theoretical modeling plays a crucial role in complementing experimental approaches, offering deeper insights into complex, nonequilibrium, functional dynamics of molecular machinery like the 26S proteasome. This work establishes a quantitative framework to further understanding the complexity of dynamic regulation of the proteasome and how such a single entity can be harnessed to control the turnover of the entire proteome in cells in a precisely regulated, spatiotemporally coordinated fashion. Advancing our understanding of the detailed molecular mechanisms of AAA+ proteolytic systems is crucial, as they are found across all kingdoms of life. Given that highly conserved hexameric ring structure of proteolytic AAA+ ATPases, it is anticipated that the chemomechanical coupling patterns simulated here for force generation and substrate translocation are potentially conserved and widely utilized throughout the AAA+ protease family.

Methods

Stochastic model for substrate unfolding by proteasome ATPase motor

The 26S proteasome is a large multi-subunit complex comprising a 20S core particle (CP) responsible for proteolysis and a 19S regulatory particle (RP) that recognizes, unfolds, and translocates substrates into the CP. The 19S RP is composed of six RPT subunits with ATPase activity that use energy from ATP hydrolysis to unfold and thread substrate proteins into the 20S CP. The 20S CP is a cylinder-shaped chamber in which substrates are cleaved. As schematically depicted in Fig. 1a, the ATPase motor, anchored on the 20S CP, consists of six RPT subunits forming a hexameric ring in a right-handed, staircase-like structure. In the top view of the ATPase motor in Fig. 1a, the RPT subunits are depicted in different colors. The central part of the hexameric ring forms a channel that allows the substrate polypeptide chain (denoted by the red dot in the top view of motor) to pass through. Each RPT subunit has a pore loop located on the surface of the channel, which can grip the substrate peptide. The nucleotide-binding pocket of each RPT, referred to as the cis-RPT of this pocket, is located at the contact interface with the adjacent RPT, referred to as the trans-RPT of the pocket. ATP binding and hydrolysis facilitated by trans-RPTs affect the interaction between adjacent RPTs, thereby altering the conformation of the hexameric ATPase complex and the interaction between the pore loop and the substrate.

In our model, ATP binding and hydrolysis in the nucleotide-binding pockets, as well as transitions between different ATPase conformations, occur stochastically (Fig. 1b). The RPT subunits that grip the substrate form a staircase-like structure. The ordered numbers 1, 2, 3, etc. in Fig. 1c denote the RPTs ascending along the staircase. Depending on whether the pore loop is in contact with the substrate, each RPT subunit can be in a substrate-engaged state (i.e., RPTs marked with numbers in Fig. 1c) or a substrate-disengaged state (i.e., RPTs without labeling numbers in Fig. 1c). Previous experiments revealed that two adjacent substrate-engaged RPTs always have a relatively small gap and large solvent-inaccessible buried interfacial area, called a closed interface. The substrate-disengaged RPTs, on the other hand, have a large void gap with least or no interfacial contact, called an open interface, with the adjacent substrate-engaged RPTs (see H1 in Fig. 1). We assume that for two adjacent RPTs simultaneously in the substrate-disengaged state, the interface between them can be either opened or closed, and that neighboring substrate-disengaged RPTs can form a united segment through closed interfaces (e.g., H2). As shown in Fig. 1c, the ATPase conformations we consider are limited to those that have one (such as H1), two (such as H2, H3), and three substrate-disengaged RPTs (such as H4, H5). The ATPase conformations with substrate-disengaged RPTs have been categorized into five classes (see the left column in Fig. 1c), with each class encompassing six different but homologous ATPase conformations which are simply rotations of the RPT complex. Thus, a total of 30 conformations with substrate-engaged RPTs were considered, labeled from H1 to H30 in Fig. S1. Among the ATPase conformations considered in our model, eleven conformations exhibit correspondence with the cryo-EM structures observed in the substrate-engaged human 26S proteasome experiments^18,25,27. The full correspondence is provided in Table S1. Although the remaining conformations have not been experimentally observed, their possible existence can be reasonably inferred based on the structural symmetry of the ATPase motor.

Figure 1c lists all possible one-step transitions from the current five basic types of conformations. The transitions follow the hand-over-hand model, which is generally hypothesized to qualitatively interpret previous experiments on hexameric ATPase motor of the AAA+ family^19,21,25,43. In this model, only the substrate-engaged RPT that is nearest to the core particle and is adjacent to a substrate-disengaged RPT can be released from the substrate peptide. The substrate-disengaged RPT can then move to the top of the staircase and re-engage with the substrate. RPT subunits disengage from the substrate sequentially, with the disengaged state transmitted across all six RPTs in turn, without skipping any engaged RPTs. The pore loops of the RPTs that remain engaged before and after the conformational change do not undergo relative slippage on the substrate during translocation. Therefore, the movement distance of the substrate can be determined by the changes in positions of the pore loops that remain in contact with it. The pore loops of adjacent substrate-engaged RPTs are offset by two amino-acid residues along the substrate, corresponding to the 0.7 nm step size of a single forward or backward translocation event in our model. In Fig. 1c, the movement distance is noted at the bottom of each conformation that H1, H2, H3, H4 and H5 can transition to. The substrate can move forward or backward by one or two steps, or it may not move at all. We list all possible transitions between the 30 conformations, along with the corresponding movement distances in line with the hand-over-hand mechanism, in Fig. S2. In addition to these 30 conformations, we also consider the case where all six RPTs are engaged, denoted as H31 and shown in Fig. 1a.

The conformational changes are coordinated with ATP hydrolysis. In our model, ATP in the nucleotide-binding pocket is irreversibly hydrolyzed to ADP when the interface between the two involved adjacent RPTs is closed. Additionally, we assume that after ATP hydrolysis, the trans-RPT is temporarily elevated to a higher-energy unstable state and then relaxes back as it disengages from the substrate. We assume that only when all six nucleotide-binding pockets are occupied by ATP can the ATPase motor enter the non-translocating conformation H31, in which ATP can be hydrolyzed but does not translocate the substrate peptide. Due to a lack of direct evidence, the assumption is based on previous experimental observations of prokaryotic translocases that belong to the same AAA+ ATPase family as the human 26S proteasome. Atomic force microscopy experiments observed that the ring-like structure in ClpB motor was closed at high ATP concentrations⁴⁴. Similarly, the ClpX motor has been shown to have a more compact structure that lead to an inhibitory effect on translocation at high ATP concentrations⁴⁵. In yeast, the proteasome motor was found to undergo additional changes at high ATPγS concentrations⁴⁶. In view of these observations, we assume that conformation H31 can transition to and back from each of the conformations H1 to H30, and that the substrate peptide is not moved during these idle transitions.

Transition rates in the stochastic model for proteasome ATPase motor

All the changes in the motor conformations and nucleotide-binding pockets are assumed to be Markovian processes. The transition rates for random changes in nucleotide states and ATPase conformations are as follows.

In the nucleotide-binding pockets of RPTs, the changes include binding and dissociation of ATP, ATPγS, and ADP, as well as ATP hydrolysis within the pockets. The energy of an RPT subunit in the motor consists of three parts: the binding energies of the nucleotide to the trans-RPT (${{\rm{e}}}_{{\rm{t}}}^{({\rm{i}})}$) and to the cis-RPT (${{\rm{e}}}_{{\rm{c}}}^{({\rm{i}})}$), and the energy change (${{\rm{e}}}_{{\rm{act}}}^{({\rm{i}})}$) due to ATP hydrolysis,

$${E}_{{\rm{RPT}}}^{\left({\rm{i}}\right)}={{\rm{e}}}_{{\rm{t}}}^{\left({\rm{i}}\right)}+{{\rm{e}}}_{{\rm{c}}}^{\left({\rm{i}}\right)}+{{\rm{e}}}_{{\rm{act}}}^{\left({\rm{i}}\right)},{i}=\mathrm{1,2},\ldots 6.$$

(2)

Specifically, the energy ${{\rm{e}}}_{{\rm{t}}}^{({\rm{i}})}$ can take the value of ${{\rm{e}}}_{{\rm{t}},{\rm{ATP}}}$, ${{\rm{e}}}_{{\rm{t}},{\rm{ADP}}}$ or ${{\rm{e}}}_{{\rm{t}},{\rm{ATP}}{\rm{\gamma }}{\rm{S}}}$, depending on whether the bound nucleotide is ATP, ADP or ${\rm{ATP}}{\rm{\gamma }}{\rm{S}}$ with a closed interface. In cases of empty pocket or when the interface is open, ${{\rm{e}}}_{{\rm{t}}}^{({\rm{i}})}$ is zero. Similarly, ${{\rm{e}}}_{{\rm{c}}}^{({\rm{i}})}$ takes the value ${{\rm{e}}}_{{\rm{c}},{\rm{tight}}}$ or ${{\rm{e}}}_{{\rm{c}},{\rm{loose}}}$ depending on whether the cis-RPT is engaged or disengaged with the substrate. It is zero when the pocket is empty without a bound nucleotide. The term ${{\rm{e}}}_{{\rm{act}}}^{({\rm{i}})}$ takes the value ${{\rm{e}}}_{{\rm{act}}}$ when the RPT energy is increased due to ATP hydrolysis and is zero otherwise. The total energy ${E}_{{\rm{tot}}}$ for the whole ATPase complex is obtained by summing up the energies for the six RPTs,

$${E}_{\mathrm{tot}}=\mathop{\sum }\limits_{{\rm{i}}=1}^{6}{E}_{\mathrm{RPT}}^{\left({\rm{i}}\right)}$$

(3)

The dissociation rate ${r}_{{\rm{off}}}$ of ATP, ATPγS or ADP from RPTs is given by

$${r}_{{\rm{off}}}={k}_{{\rm{off}}}\exp \left(-\frac{{E}_{{\rm{tot}},{\rm{n}}}-{E}_{{\rm{tot}},{\rm{c}}}}{{k}_{{\rm{B}}}T}\right),$$

(4)

where ${E}_{{\rm{tot}},{\rm{c}}}$ and ${E}_{{\rm{tot}},{\rm{n}}}$ are the energies of the current motor state and the next state the current state transitions to, respectively. Here, ${k}_{{\rm{off}}}$ is the coefficient of the dissociation rate, ${k}_{{\rm{B}}}$ is the Boltzmann constant, and $T$ is the temperature. The binding rate of nucleotide to RPTs takes the form

$${r}_{{\rm{on}},{\rm{X}}}={k}_{{\rm{on}}}\left[{\rm{X}}\right],$$

(5)

where ${k}_{{\rm{on}}}$ is a constant and [X] represents the molar concentration of ATP, ATPγS, or ADP. The model does not account for the influence of ATPase conformations on nucleotide diffusion, which may affect the values of ${k}_{{\rm{on}}}$ and ${k}_{{\rm{off}}}$ under different conformations.

According to experimental cryo-EM observations⁴⁷, the three RPT subunits at the top of the motor spiral are usually ATP-bound, while the nucleotide state of the subunits close to the core particle varies between ATP-bound, ADP-bound, or apo states. In view of this, we assume that ATP hydrolysis occurs only in the two ATP-bound RPTs that are close to the core particle. The hydrolysis rate ${r}_{{\rm{h}},{\rm{tr}}}$ in the nucleotide-binding pocket is set as a constant. In the non-translocating conformation H31, only the total ATP hydrolysis rate of the ATPase motor is considered, which is given by

$${r}_{{\rm{h}},{\rm{ntr}}}={k}_{{\rm{h}},{\rm{ntr}}}{N}_{{\rm{ATP}}},$$

(6)

where ${N}_{{\rm{ATP}}}$ is the number of ATP molecules bound to the ATPase motor.

For state changes involving in conformational reconfigurations of the ATPase complex, the transition rate ${r}_{{\rm{c}}}$ is similarly given as

$${r}_{{\rm{c}}}={k}_{{\rm{c}}}\exp \left(-\frac{{E}_{{\rm{tot}},{\rm{n}}}-{E}_{{\rm{tot}},{\rm{c}}}+{F}_{{\rm{sub}}}\cdot d}{2{k}_{{\rm{B}}}T}\right),$$

(7)

where ${E}_{{\rm{tot}},{\rm{n}}}-{E}_{{\rm{tot}},{\rm{c}}}$ is the energy difference between the current RPT conformation and the next conformation it changes to. ${F}_{{\rm{sub}}}$ is the force required to overcome during translocating the substrate, $d$ is the moving distance of the substrate when the conformational change occurs, as shown in Fig. S2. The transition from conformations H1-H30 to the non-translocating conformation H31 and the reverse process occur with constant rates ${r}_{{\rm{tr}}\to {\rm{ntr}}}$ and ${r}_{{\rm{ntr}}\to {\rm{tr}}}$, respectively.

For the model parameters, the values for ${{\rm{e}}}_{{\rm{c}},{\rm{loose}}}$, ${{\rm{e}}}_{{\rm{c}},{\rm{tight}}}$, ${{\rm{e}}}_{{\rm{t}},{\rm{ATP}}}$ (${{\rm{e}}}_{{\rm{t}},{\rm{ATP}}{\rm{\gamma }}{\rm{S}}}$), ${{\rm{e}}}_{{\rm{t}},{\rm{ADP}}}$, ${k}_{{\rm{on}}}$(${k}_{{\rm{off}}}$) and temperature $T$ were adopted from the measurements reported in ref. ²⁸. The remaining parameters were determined by fitting the experimental data shown in Fig. 2. The complete set of parameters used in our model is provided in Table S2. To verify the sensitivity of the computational results to the fitted parameter values, we varied each parameter obtained from the fitting by ±10% and performed corresponding calculations. The results demonstrate that moderate adjustments to the fitted parameters do not significantly affect the model’s ability to explain the experimental observations.

The Markovian jump processes are simulated numerically using the Gillespie algorithm²⁹. A total of 500,000 random steps is computed for each simulation run with fixed ambient nucleotide concentrations. The mean translocation rate and standard deviation are calculated from ten simulation runs.