Subscriber access provided by Caltech Library Article Allostery and Kinetic Proofreading Vahe Galstyan, and Rob Phillips J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.9b08380 • Publication Date (Web): 28 Nov 2019 Downloaded from pubs.acs.org on December 2, 2019

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1 2 3 4 5 6 7 8 Allostery and Kinetic Proofreading 9 10 11 † ∗,‡,¶,§ 12 Vahe Galstyan and Rob Phillips 13 14 15 †Biochemistry and Molecular Biophysics Option, California Institute of Technology, 16 17 Pasadena, California 91125, United States 18 19 ‡Department of Physics, California Institute of Technology, Pasadena, California 91125, 20 21 United States 22 23 ¶Department of Applied Physics, California Institute of Technology, Pasadena, California 24 25 91125, United States 26 27 §Division of Biology and Biological Engineering, California Institute of Technology, 28 29 Pasadena, California 91125, United States 30 31 E-mail: [email protected] 32 33 Phone: (626) 395-3374 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 1 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 2 of 33

1 2 3 Abstract 4 5 Kinetic proofreading is an error correction mechanism present in the processes of the 6 central dogma and beyond, and typically requires the free energy of nucleotide hydrol- 7 ysis for its operation. Though the molecular players of many biological proofreading 8 schemes are known, our understanding of how energy consumption is managed to pro- 9 10 mote fidelity remains incomplete. In our work, we introduce an alternative conceptual 11 scheme called “the piston model of proofreading” where enzyme activation through 12 hydrolysis is replaced with allosteric activation achieved through mechanical work per- 13 formed by a piston on regulatory ligands. Inspired by Feynman’s ratchet and pawl 14 mechanism, we consider a mechanical engine designed to drive the piston actions pow- 15 ered by a lowering weight, whose function is analogous to that of ATP synthase in cells. 16 Thanks to its mechanical design, the piston model allows us to tune the “knobs” of 17 the driving engine and probe the graded changes and trade-offs between speed, fidelity 18 19 and energy dissipation. It provides an intuitive explanation of the conditions necessary 20 for optimal proofreading and reveals the unexpected capability of allosteric molecules 21 to beat the Hopfield limit of fidelity by leveraging the diversity of states available to 22 them. The framework that we built for the piston model can also serve as a basis for 23 additional studies of driven biochemical systems. 24 25 26 27 1 Introduction 28 29 Many enzymatic processes in biology need to operate with very high fidelities in order to 30 ensure the physiological well-being of the cell. Examples include the synthesis of molecules 31 making up Crick’s so-called “two great polymer languages” (i.e. replication,1 transcription,2 32 and translation3), as well as processes that go beyond those of the central dogma, such as 33 protein ubiquitylation mediated by the anaphase-promoting complex,4 signal transduction 34 5 6,7 35 through MAP kinases, pathogen recognition by T-cells, or protein degradation by the 8 36 26S proteasome. In all of these cases, the designated enzyme needs to accurately select 37 its correct substrate from a pool of incorrect substrates. Importantly, the fidelity of these 38 processes that one would predict solely based on the free energy difference between correct 39 and incorrect substrate binding is far lower than what is experimentally measured, raising a 40 41 challenge of explaining the high fidelities that this naive equilibrium thermodynamic thinking 42 fails to account for. 43 The conceptual answer to this challenge was provided more than 40 years ago in the work 44 of John Hopfield9 and Jacques Ninio10 and was coined “kinetic proofreading” in Hopfield’s 45 elegant paper entitled “Kinetic Proofreading: A New Mechanism for Reducing Errors in 46 9 47 Biosynthetic Processes Requiring High Specificity.” The key idea behind kinetic proofread- 48 ing is to introduce a delay between substrate binding and turnover steps, effectively giving 49 the enzyme more than one chance to release the incorrect substrate (hence, the term “proof- 50 reading”). The sequential application of substrate filters on the way to product formation 51 gives directionality to the flow of time and is necessarily accompanied by the expenditure of 52 53 free energy, making kinetic proofreading an intrinsically nonequilibrium phenomenon. In a 54 cell, this free energy is typically supplied to proofreading pathways through the hydrolysis of 55 energy-rich nucleotides, whose chemical potential is maintained at large out-of-equilibrium 56 57 58 59 2 60 ACS Paragon Plus Environment Page 3 of 33 The Journal of Physical Chemistry

1 2 3 values through the constant operation of the cell’s metabolic machinery (e.g. the ATP syn- 4 5 thase). 6 Since its original formulation by Hopfield and Ninio, the concept of kinetic proofreading 7 has been generalized and employed in explaining many of the high fidelity processes in the 8 cell.8,11–17 However, despite the fact that the molecular players and mechanisms of these 9 processes have been largely identified, we find that an intuitive picture of how energy trans- 10 11 duction promotes biological fidelity is still incomplete. To complement our understanding 12 of how energy is managed to beat the equilibrium limit of fidelity, we propose a concep- 13 tual model called “the piston model of kinetic proofreading” where chemical hydrolysis is 14 replaced with mechanical work performed by a piston on an allosteric enzyme. Our choice of 15 allostery is motivated by the fact that in proofreading schemes hydrolysis typically triggers 16 17 a conformational change in the enzyme-substrate complex and activates it for product for- 18–20 18 mation - an effect that our model achieves through the binding of a regulatory ligand to 19 the enzyme. By temporally controlling the concentration of regulatory ligands which deter- 20 mine the catalytic state of the enzyme, the piston sequentially changes the enzyme’s state 21 from inactive to active, creating a delay in product formation necessary for increasing the 22 23 fidelity of substrate discrimination. The piston actions are, in turn, driven by a Brownian 24 ratchet and pawl engine powered by a lowering weight, whose function is akin to that of ATP 25 synthase. The mechanical design of the piston model allows us to transparently control the 26 energy input into the system by tuning the “knobs” of the engine and examine the graded 27 changes in the model’s performance metrics, intuitively demonstrating the driving conditions 28 29 required for optimal proofreading. 30 We begin the presentation of our results by first introducing in section 2 the high-level 31 concept behind the piston model of proofreading, while at the same time drawing parallels 32 between its features and those of Hopfield’s original scheme. Then, in sections 3 and 3.2 we 33 provide a comprehensive description of the two key constituents of the piston model, namely, 34 35 the Brownian ratchet and pawl engine that drives the piston actions, and the allosteric 36 enzyme whose catalytic state is regulated by an activator ligand. This is following by building 37 the full thermodynamically consistent framework of the piston model in section 3.3 where 38 we couple the external driving mechanism to the enzyme and introduce the expressions for 39 key performance metrics of the model. In the remaining sections 3.4 and 3.5, we explore 40 41 how tuning the “knobs” of the engine leads to graded changes and trade-offs between speed, 42 fidelity and energy dissipation, and probe the performance limits of the piston model as a 43 function of a select set of key enzyme parameters. 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 3 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 4 of 33

1 2 3 2 MODEL 4 5 6 The piston model of kinetic proofreading is designed in analogy with Hopfield’s scheme. The 7 main idea there was to give the enzyme a second chance to discard the wrong substrate 8 by introducing an additional kinetic intermediate for the enzyme-substrate complex (Figure 9 1A). The difference between substrate binding energies in Hopfield’s original formulation was 10 based solely on their unbinding rates (i.e. kW > kR and kW = kR = k ) – a convention we 11 off off on on on 12 adopt throughout our analysis. The first layer of substrate discrimination in Hopfield’s 13 scheme is achieved during the initial binding event where the ratio of right and wrong W R 14 substrate-bound enzymes approaches koff /koff . The complex then moves into its catalytically 15 active high energy state accompanied by the hydrolysis of an NTP molecule, after which the 16 second discrimination layer is realized. Specifically, right and wrong substrates are turned 17 18 into products with an additional bias given by the ratio of their Michaelis constants, namely, W R 19 (koff + r)/(koff + r). Importantly, for this second layer to be efficiently realized, the rates of 20 binding directly to the second kinetic intermediate need to be vanishingly small in order to 21 prevent the incorporation of unfiltered substrates.9 22 With this information in mind, consider now the conceptual illustration of the piston 23 24 model shown in Figure 1B, where we have made several pedagogical simplifications to help 25 verbally convey the model’s intuition, reserving the full thermodynamically consistent treat- 26 ment to the following sections. The central constituent of the model is an allosteric enzyme, 27 the catalytic activity of which is regulated by activator ligands (the orange circle). The 28 enzyme is inactive when it is not bound to a ligand, and, conversely, it is active when bound 29 30 to a ligand. The volume occupied by ligands and hence, their concentration is, in turn, 31 controlled by a piston. The ligand concentration is very low when the piston is expanded, 32 and very high when the piston is compressed in order to guarantee that in those piston states 33 the ligand is free and bound to the enzyme, respectively. 34 The active site of the enzyme is exposed to a container filled with right and wrong 35 36 substrates of concentrations [R] and [W], respectively, which we take to be equal for the rest 37 of our analysis ([R] = [W]). And unlike in Hopfield’s scheme where the substrates exist in 38 energy-rich and energy-depleted states (e.g. tRNAs first arrive in the EF-Tu·GTP·tRNA 39 ternary complex and then release EF-Tu and GDP after hydrolysis), in the piston model 40 substrates exist in a single state and do not carry an energy source. In the expanded piston 41 42 state (Figure 1B, left), substrates can bind and unbind to the inactive enzyme, but do not get 43 turned into products. The highest level of discrimination achievable in this state therefore 44 becomes 45 46 kW η = off , (1) 47 1 kR 48 off 49 in analogy to that achieved during the initial binding step of Hopfield’s scheme. 50 51 After the first layer of substrate discrimination is established in the expanded state of 52 the piston, mechanical work is performed to compress it. This increases the ligand con- 53 centration, which, in turn, leads to the activation of the enzyme where catalytic action is 54 now possible. To prevent the incorporation of unfiltered substrates, we assume that in the 55 active enzyme state the rate of substrate binding is vanishingly small, similar to Hopfield’s 56 57 58 59 4 60 ACS Paragon Plus Environment Page 5 of 33 The Journal of Physical Chemistry

1 2 3 treatment (Figure 1B, right). If the piston is kept compressed long enough, a filtered sub- 4 5 strate that got bound earlier when the piston was expanded will experience one of these two 6 outcomes: it will either turn into a product with a rate r (which is taken to be the same 7 for the two kinds of substrates) or it will fall off with a rate koff. The product formation 8 reaction will take place with probability r/(koff +r). Thus, due to the difference in the falloff 9 rate constants between the right and the wrong substrates, the extra fidelity achieved after 10 11 piston compression equals 12 W 13 koff + r η2 = R . (2) 14 koff + r 15 16 Once this extra fidelity is established, the piston is expanded back, repeating the cycle (the 17 detailed derivation of the results for η1 and η2 is provided in Supporting Information section 18 19 A). Notably, the total fidelity achieved during the piston expansion and compression cycle, 20 namely, 21 kW  kW + r 22 η = η η = off off , (3) 23 1 2 R R koff koff + r 24 25 exceeds the Michaelis-Menten fidelity (η ) by a factor of η = kW /kR , demonstrating the 26 2 1 off off 27 attainment of efficient proofreading. 28 The cyclic compressions and expansions of the piston in our model also stand in direct 29 analogy to the hydrolysis-involving transitions between the two enzyme-substrate intermedi- 30 ates in Hopfield’s scheme. In particular, they need to be externally driven for the mechanism 31 to do proofreading. We perform this driving using a mechanical ratchet and pawl engine 32 33 powered by a lowering weight. In our pedagogical description of the model’s operation, we 34 implicitly assumed that this weight was very large in order to enable the mechanism to proof- 35 read, similar to how the hydrolysis energy needs to be large for Hopfield’s scheme to operate 36 effectively.9 In the full treatment of the model in section 3, however, we will demonstrate 37 how the tuning of the weight can give us graded levels of fidelity enhancement, and will 38 39 also show that in the absence of this weight the equilibrium fluctuations of the piston alone 40 cannot lead to proofreading. 41 In our model introduction we have also made several simplifying assumption for clarity 42 of presentation which do not conform with the principle of microscopic reversibility, and 43 it is important that we relax them in the full treatment of the model to make it thermo- 44 45 dynamically consistent. In particular, we assumed that the ligand is necessarily unbound 46 and that the enzyme is necessarily inactive when the piston is expanded, with the reverse 47 assumptions made when the piston is compressed. We also assumed that substrate binding 48 was prohibited in the active state of the enzyme. These assumptions allowed us to claim 49 that no premature product formation takes place in the expanded piston state and that no 50 51 unfiltered substrates bind to the activated enzyme in the compressed piston state, which, in 52 turn, justified the use of long waiting times between the piston actions necessary to establish 53 high levels of fidelity in each piston state. In the detailed analysis of our model presented 54 in section 3, we relax these assumptions and consider the full diversity of enzyme states at 55 each piston position with reversible transitions between them. This, as we will demonstrate, 56 57 58 59 5 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 6 of 33

1 2 3 will not only ensure the thermodynamic consistency of our treatment but will also reveal 4 5 the possibility of doing proofreading more than once by leveraging the presence of multiple 6 inactive intermediates in between enzyme’s substrate-unbound and production states which 7 were not accounted for in our conceptual introduction of the model. 8 9 (A) #1 10 Hopfieldʼs scheme 11 #2 12 koff kon koff r 13 NTP NDP 14 15 16 17 (B) pr 18 piston com ess 19 model 20 e 21 xpand 22 23 24 r 25 kon koff koff 26 27 28 29 piston state expanded compressed 30 ligand state unbound bound 31 enzyme state inactive active 32 substrate allowed not allowed 33 binding W W 34 achieved koff koff + r 35 fidelity R R koff koff + r 36 37 Figure 1. Conceptual introduction to the piston model. (A) Hopfield’s scheme of kinetic 38 proofreading where two layers of substrate discrimination take place on the driven pathway – the 39 first one during the initial binding of energy-rich substrates (#1 in the diagram) and the second 40 one upon the release of the energy-depleted substrates (#2 in the diagram). Energy consumption 41 42 takes place during the hydrolysis reaction NTP NDP accompanying the transition between the 43 two intermediates. (B) Pedagogically simplified conceptual scheme of the piston model. The 44 orange circle represents the activator ligand. Blue and red colors stand for the right and wrong 45 substrates, respectively. The closed “entrance door” along with the red cross on the binding arrow 46 in the active state of the enzyme suggests the vanishingly small rate of substrate binding when in 47 this state. The ratchet with a hanging weight stands for the mechanical engine that drives the 48 piston actions. Various features of the system in the two piston states, along with the expressions 49 for achieved fidelities are listed below the diagram. Transparent arrows between panels A and B 50 51 indicate the analogous parts in Hopfield’s scheme and the piston model. 52 53 54 55 56 57 58 59 6 60 ACS Paragon Plus Environment Page 7 of 33 The Journal of Physical Chemistry

1 2 3 3 RESULTS 4 5 6 3.1 Ratchet and Pawl Engine Enables a Tunable Control of Piston 7 Actions 8 9 To drive the cyclic compressions and expansions of the piston necessary for achieving proof- 10 reading, we use a ratchet and pawl engine whose design is inspired by Feynman’s original 11 21 12 work. In his celebrated lectures, Feynman presented two implementations of the ratchet 13 and pawl engine – one operating on the temperature difference between two thermal baths, 14 and another driven by a weight that goes down due to gravity. In the piston model we 15 adopt the second scheme as it involves fewer parameters and illustrates the process of energy 16 transduction more transparently. 17 18 The ratchet and pawl engine coupled to the piston is shown in Figure 2A. The engine 19 is powered by a weight of mass m which is hanging from an axle connected to the ratchet. 20 The free rotational motion of the ratchet is rectified by a pawl; when the pawl sits on a 21 ratchet tooth, it prevents the ratchet from rotating in the clockwise (backwards) direction. 22 The mechanical coupling between the engine and the piston is achieved through a crankshaft 23 24 mechanism which translates each discrete ratchet step into a full compression (up → down) 25 or a full expansion (down → up) of the piston. We assume that the volume regulated by the 26 piston contains a single ligand – a choice motivated by Szilard’s thermodynamic interpreta- 27 tion of information, where a piston compressing a single gas molecule was considered.22 28 The clockwise (backward) and counterclockwise (forward) steps of the microscopic ratchet 29 30 are enabled through environmental fluctuations. Specifically, a backward step is taken when- 31 ever the pawl acquires sufficient energy from the environment to lift itself over the ratchet 32 tooth that it is sitting on, allowing the tooth to slip under it (hence, the name “backward”). 33 Following Feynman’s treatment,21 we write the rate of such steps as 34 35 k = τ −1e−βE0 , (4) 36 b 37 −1 38 where τ is the attempt frequency, E0 is the amount of energy needed to lift the pawl over 39 a ratchet tooth, and β = 1/kBT is the inverse of the thermal energy scale (see Supporting 40 Information section B.1 for a detailed discussion of the ratchet and pawl mechanism). Every 41 backward step of the ratchet is accompanied by either a full compression or a full expansion 42 of the piston, as well as the lowering of the weight by an amount of ∆z, which reduces its 43 44 potential energy by ∆W = mg∆z. 45 Unlike in backward stepping, for a forward step to take place, the rotational energy 46 acquired by the ratchet through fluctuations should be sufficient to not only overcome the 47 resistance of the spring pressing the pawl onto the ratchet, but also to lift the weight and to 48 alter the state of the piston. This is a pure consequence of the geometric design of the ratchet 49 50 and the positioning of the pawl. We assume that piston actions take place isothermally and 51 in a quasistatic way, and therefore, write the changes in ligand free energy upon compression 52 (u → d) and expansion (d → u) as 53 54 −1 ∆Fu→d = β ln f, and (5) 55 −1 56 ∆Fd→u = −β ln f, (6) 57 58 59 7 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 8 of 33

1 2 3 respectively, where f = V /V ≥ 1 is the fraction by which the volume occupied by the 4 u d 5 ligand decreases upon compression. The signs of free energy differences suggest that piston 6 compressions slow down the forward steps, while expansions speed them up. These features 7 are reflected in the two kinds of forward stepping rates that are given by 8 9 u→d −1 −β(E0+∆W +∆F ) −1 −β∆W kf = τ e = f kbe , (7) 10 d→u −1 −β(E0+∆W −∆F ) −β∆W 11 kf = τ e = fkbe , (8) 12 −1 13 where ∆F = β ln f and was used with a “+” and “-” sign in the place of ∆Fu→d and ∆Fd→u, 14 respectively. The rates of all four kinds of transitions, namely, forward or backward ratchet 15 steps, accompanied by either a compression or an expansion of the piston are summarized 16 17 in Figure 2B. 18 In the presence of a nonzero weight (∆W > 0), the ratchet will on average rotate back- 19 wards – a feature reflected in the tilted free energy landscape shown in Figure 2C. As can be 20 seen, the average dissipation per step is ∆W and it is independent of ∆F . In addition, the 21 work performed on the ligand upon compression is fully returned upon expansion, which, 22 23 as we will demonstrate in section 3.4, will generally not be the case when we introduce the 24 enzyme coupling. To further study the nonequilibrium dynamics of the driving mechanism, 25 we map the local minima of the energy landscape corresponding to discrete vertical positions 26 of the weight (equivalently, discrete ratchet angles) into an infinite chain of transitions shown 27 in Figure 2D. There “d” and “u” stand for the compressed and expanded states of the piston, 28 29 respectively. The net stepping rate knet at which the weight goes down can be written as 30 d→u
u→d
31 knet = kb − kf πd + kb − kf πu, (9) 32 33 where πd and πu are the steady state probabilities of the compressed and expanded piston 34 states, respectively. These probabilities can be obtained by considering the equivalent two- 35 state diagram in Figure 2E where the vertical position of the weight has been eliminated, 36 37 and the nonequilibrium nature of the dynamics is instead captured via the cycle through 38 two alternative pathways connecting the piston states. The driving force ∆µ in this cycle is 39 given by23 40 41  k2  ∆µ = β−1 ln b = 2∆W, (10) 42 kd→uku→d 43 f f 44 demonstrating the broken detailed balance in the presence of a nonzero weight, and con- 45 46 firming the dissipation of 2∆W per cycle observed in the energy landscape (Figure 2C). 47 We note that this procedure of mapping a linear network onto a cyclic one has also been 48 used for modeling the processivity of molecular motors, where the linear coordinate corre- 49 sponds to the position of the motor while the alternating states correspond to different motor 50 24,25 51 conformations. 52 At steady state, the net incoming and outgoing fluxes at each piston state in Figure 2E 53 should cancel each other (seen more vividly in the collapsed diagram in Figure 2F), namely, 54 55 d→u
u→d
kf + kb πd = kf + kb πu. (11) 56 57 58 59 8 60 ACS Paragon Plus Environment Page 9 of 33 The Journal of Physical Chemistry

1 2 3 (A) pawl 4 5 6 b 7

8 f 9 piston 10 11 12 ratchet weight ligand 13 14 (B) (C) u→d E0+ΔW+ΔF compression expansion kf 15 kb (u→d) (d→u) E0+ΔW-ΔF 16 d→u 17 kf kb E0 kb kb 2ΔW

18 free energy

step 1 -βE 1 -βE single cycle e 0 e 0 Δz 19 τ τ backward z 20 ΔWb = -mgΔz zn-2 zn-1 zn zn+1 zn+2 zn+3 zn+4 21 (D) u→d d→u u→d d→u u→d d→u u→d d→u u→d d→u kf kf kf kf kf kf kf kf 22 kf kf knet

23 step 1 -β(E0+ΔW+ΔF) 1 -β(E0+ΔW-ΔF) d u d u d du

forward e e 24 τ τ ΔWf = mgΔz 25 kb kb kb kb kb kb kb kb 26 d→u (E) kf (F) 27 d→u kf +kb 28 kb -1 -1 d Δμ u d u 29 ΔFu→d = β ln(f) ΔFd→u = -β ln(f) kb u→d 30 kf +kb u→d 31 kf 32 33 Figure 2. Ratchet and pawl mechanism coupled to the piston. (A) Schematic representation of 34 the mechanism. The different radii of the ratchet wheel and the axle of the crankshaft ensure that 35 a single ratchet step translates into a full compression or a full expansion of the piston (i.e. a 180◦ 36 rotation of the crankshaft). Arrows with symbols “b” and “f” indicate the directions of backward 37 and forward ratchet rotation, respectively. (B) Rates of the four kinds of transitions (symbols in 38 shaded boxes with explicit expressions below), along with the accompanying changes in the 39 potential energy of the weight and the free energy of the ligand. (C) Free energy landscape 40 41 corresponding to the non-equilibrium dynamics of the system in the presence of a non-zero weight. 42 Discrete positions of the weight (zn) corresponding to energy minima of the landscape are marked 43 on the reaction coordinate. (D) Infinite chain representation of the dynamics of discrete system 44 states. knet stands for the net rate at which the weight goes down. (E) Equivalent two-state 45 representation of the engine dynamics where the driving force ∆µ = 2∆W breaks the detailed 46 balance in the diagram. (F) Collapsed representation of the diagram in panel E shown with the 47 net transition rates from the two pathways. 48 49 50 Substituting the expressions for forward stepping rates (eqs 7 and 8) into eq 11 and addi- 51 52 tionally imposing the probability normalization constraint (πd + πu = 1), we can solve for 53 πd and πu to obtain 54 55 1 + e−β(∆W +∆F ) π = , (12) 56 d 2(1 + cosh(β∆F ) e−β∆W ) 57 58 59 9 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 10 of 33

1 2 3 1 + e−β(∆W −∆F ) 4 πu = −β∆W . (13) 5 2(1 + cosh(β∆F ) e ) 6 7 Notably, in the absence of external drive (∆W = 0), piston state occupancies follow the −β∆F −1 8 Boltzmann distribution, that is, (πd/πu)eq = e = f , suggesting that at equilibrium 9 the piston will predominantly dwell in the expanded state. Conversely, as can be seen in 10 Figure 3A, when the work per step exceeds ∆F by several kBT , the occupancies of the two 11 piston states become equal to each other. This happens because at large ∆W values forward 12 13 ratchet stepping becomes very unlikely and the dynamics proceeds only through backward 14 steps with a rate kb which is identical for both compressive and expansive steps. As will be 15 shown in section 3.4, suppressing this equilibrium bias set by ∆F is essential for achieving 16 efficient proofreading, analogous to the need for driving the transitions between the two 17 enzyme-substrate intermediates in Hopfield’s scheme (Figure 1A).9 18 19 20 (A) (B) 1.0 1.0 21 22 0.8 0.8 ΔF = 2 kBT ΔF = 2 kBT u b

23 π ΔF = 4 kBT ΔF = 4 kBT 0.6 /k 0.6

/ ΔF = 6 k T ΔF = 6 k T d B B 24 net k π 0.4 0.4 25 ΔF = 8 kBT ΔF = 8 kBT 0.2 0.2 26 ΔW1/2 ≈ ΔF ΔW1/2 ≈ ΔF-kBTln2 27 0.0 0.0 28 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 ΔW (k T) ΔW (k T) 29 B B 30 Figure 3. Nonequilibrium features of the engine-piston coupling. (A) Steady state probability 31 ratio of compressed (“d”) and expanded (“u”) piston states and (B) normalized net rate of 32 backward stepping (knet/kb) as a function of the work per step (∆W ) for different choices of the 33 ligand compression energy (∆F ). The ∆W expressions stand for the values of ∆W where the 34 1/2 corresponding value on the y-axis is 0.5 (Supporting Information section B.2). Negative ∆W 35 36 values are not considered as they further increase the undesired bias in piston state occupancies. 37 38 With the steady state probabilities known, we can now substitute them into eq 9 to find 39 the net rate at which the weight goes down, obtaining 40 41 −2β∆W
1 − e kb 42 knet = −β∆W . (14) 43 1 + cosh(∆F )e 44 45 As expected, knet vanishes at equilibrium (∆W = 0), and asymptotes to kb at large ∆W 46 values, as shown in Figure 3B. The knowledge of knet allows us to calculate the power (P ) 47 dissipated for the maintenance of the nonequilibrium steady state. Specifically, since knet is 48 the rate at which the weight goes down and ∆W is the dissipation per step, the power P 49 50 becomes their product, namely, 51 52 P = knet∆W. (15) 53 54 The formalism developed in this section for characterizing the steady state behavior of 55 the system will be used as a basis for defining the different performance metrics of the model 56 in section 3.3. 57 58 59 10 60 ACS Paragon Plus Environment Page 11 of 33 The Journal of Physical Chemistry

1 2 3 3.2 Thermodynamic Constraints Make Fidelity Enhancement Unattain- 4 5 able in the Absence of External Driving 6 7 In order to implement a thermodynamically consistent coupling between the engine and the 8 allosteric enzyme, we need to consider the full diversity of possible enzyme states,26 and 9 not just the dominant ones depicted in Figure 1B. Therefore, in this section we provide 10 a comprehensive discussion of the enzyme in an equilibrium setting before introducing its 11 coupling to the engine. 12 13 The network diagram of all possible enzyme states is depicted in Figure 4. As can be 14 seen, each of the twelve states are defined by enzyme’s catalytic activity and whether or not 15 a ligand and a right/ wrong substrate are bound to the enzyme. Following the principle 16 of microscopic reversibility,27 we assign non-zero rate constants to the transitions between 17 18 enzyme states. Only the product formation (with rate r) is taken to be an irreversible 19 reaction under the assumption that the system is open where the formed products are taken 20 out and an influx of new substrates is maintained. Since in our model neither the enzyme 21 nor the substrates carry an energy source, the choice of the different rate constants cannot 22 be arbitrary. Specifically, the cycle condition needs to be satisfied for each closed loop of 23 24 the diagram, requiring the product of rate constants in the clockwise direction to equal the 23 25 product in the counterclockwise direction (Supporting Information section C.1). 26 A A k [W] k [R] 27 r on on r kW kR 28 A off A off A ℓ A ℓ A ℓ A 29 on[L] ℓ on[L] ℓ on[L] ℓ off kA off kA off on[W] on[R] 30 r r kW, L W L L R kR, L 31 W, L k k k k R, L k I off A I off k I 32 A A

33 W W I I R R k k k [W] k k k [R] k k 34 A I on A I on A I 35 kW kR ℓI off ℓI off ℓI 36 on[L] ℓI on[L] ℓI on[L] ℓI off k I off kI off 37 on[W] on[R] kW kR 38 off off 39 40 Figure 4. Network diagram of enzyme states and transitions between them. Right (“R”) and 41 wrong (“W”) substrates are depicted in blue and red, respectively. The orange circle represents the 42 ligand (“L”). Active (“A”) and inactive (“I”) enzymes are shown in green and gray, respectively. 43 44 With these equilibrium restrictions imposed on the rate constants, we can show that 45 46 when the ligand concentration is held fixed ([L](t) = const) the fidelity of the enzyme cannot W R 47 exceed that defined by the ratio of the off-rates, namely, koff /koff (see Supporting Information 48 section C.2). What allows the enzyme to beat this equilibrium limit of fidelity without 49 direct coupling to hydrolysis is the cyclic alteration of the ligand concentration between 50 low and high values (thus, [L](t) 6= const). In our model, we achieve this cyclic alteration 51 52 through the ratchet and pawl engine driving the piston actions - a choice motivated by 53 our objective to provide an explicit treatment of energy management. We note, however, 54 that fidelity enhancement can be achieved irrespective of the driving agency as long as the 55 cyclic alteration of ligand concentration is maintained at a certain “resonance” frequency, the 56 presence of which we demonstrate in section 3.4. 57 58 59 11 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 12 of 33

1 2 3 3.3 Coupling the Engine to the Enzyme Gives the Full Description 4 5 of the Piston Model 6 7 Having separately introduced the driving mechanism in section 3 and the allosteric enzyme 8 with the full diversity of its states in section 3.2, we now couple the two together to obtain 9 the full driven version of the piston model, shown in Figure 5A. The coupling is achieved by 10 exposing the ligand binding site of the enzyme to the piston compartment where the activator 11 ligand is present. The enzyme can therefore “sense” the state of the piston (and, thereby, 12 13 the effects of driving) through the induced periodic changes in the ligand concentration. 14 In the absence of enzyme coupling, the network diagram capturing the nonequilibrium 15 dynamics of the system was an infinite one-dimensional chain (Figure 2D), where each dis- 16 crete state was defined by the vertical position of the weight (zn) and the state of the piston 17 18 (“u” or “d”). In the layout where the engine and the enzyme are coupled, the full specification 19 of the system state now requires three items: the position of the weight (zn), the piston state 20 (“u” or “d”), and the state of the enzyme (one of the 12 possibilities). By converting the 21 three-dimensional view of the enzyme state network (Figure 4) into its planar equivalent, 22 we represent the nonequilibrium dynamics of this coupled layout again through an infinite 23 24 chain, but this time each slice at a fixed weight position (zn) corresponding to the planar 25 view of the enzyme state network (Figure 5B). The slices alternate between the compressed 26 and expanded piston states (dark and light blue circles, respectively), with high and low 27 ligand concentrations used in the transition network inside each slice. 28 Arrows between the slices (not all of them shown for clarity) represent the forward and 29 30 backward steps of the ratchet. Crucially, as a consequence of coupling, the rates of forward 31 stepping now depend on the state of the enzyme. In particular, when the ligand is bound 32 to the enzyme, it no longer exerts pressure on the piston and therefore, in those cases, the 33 forward stepping rates become simply 34 35 ku→d,L = kd→u,L = k e−β∆W , (16) 36 f f b 37 38 where the superscript “L” indicates that the ligand is bound (orange circles in Figure 5B). We 39 note that in the general case with N ligands, the pressure would drop down to that of (N −1) 40 ligands upon ligand binding, correspondingly altering the rates of forward stepping (see 41 Supporting Information section D.1 for details). This adjustment of forward rates is essential 42 for the thermodynamic consistency of coupling the engine to the enzyme. Specifically, it 43 44 ensures that any cycle of transitions that brings the enzyme and the weight back into their 45 original states is not accompanied by dissipation, consistent with the fact that in the piston 46 model energy is spent only when there is a net lowering of the weight. As a demonstration 47 of this feature, consider the cycle in Figure 5C which is extracted from the larger network. 48 Using the expression of forward stepping rates in eqs 7 and 16, we can write the cycle 49 50 condition for this sub-network as 51 ku→d,L × `A × k × `A [L] [L] f 52 f off b on u = u = 1, (17) A u→d A 53 kb × `off × kf × `on[L]d [L]d 54 55 where the equality f = Vu/Vd = [L]d/[L]u was used. The fact that the products of rate 56 constants in clockwise and counterclockwise directions are identical shows that no dissipation 57 58 59 12 60 ACS Paragon Plus Environment Page 13 of 33 The Journal of Physical Chemistry

1 2 3 occurs when traversing the cycle. 4 5 Now, to study how driving affects the proofreading performance of the piston model, we 6 need to obtain the steady state probabilities of the different enzyme states. To that end, we 7 convert the full network diagram into an equivalent form shown in Figure 5D, where we have 8 eliminated the position of the weight (zn), akin to the earlier treatment of the uncoupled 9 engine in Figure 2E. Note that the transitions between the two slices again represent piston 10 11 compression and expansion events driven by a force ∆µ = 2∆W , as in eq 10. The steady 12 state probabilities πi of the 24 different states in Figure 5D (12 enzyme states × 2 piston 13 states) can be obtained from the set of all rate constants, the details of which we discuss in 14 Supporting Information section D.2. With these probabilities known, we calculate the rate 15 of energy dissipation (P ), speed of forming right products (v ), and fidelity (η) as 16 R 17 24 18 X  (i) P = k − k π ×∆W, (18) 19 b f i 20 i=1 | {z } 21 knet 22 X 23 vR = πi × r, (19) A 24 i∈SR 25 P A πi vR i∈SR 26 η = = P , (20) 27 vW A πi i∈SW 28 29 (i) th where kf is the rate constant of making a forward step from the i state (1 ≤ i ≤ 24), 30 A A 31 while SR and SW are the sets of catalytically active enzyme states with a right and wrong 32 substrate bound, respectively. 33 One significant downside of using these “raw” metrics in the numerical evaluation of the 34 model performance is their high sensitivity to the particular choices of parameter values. We 35 therefore introduce their scaled alternatives which we will use for the numerical studies in 36 37 sections 3.4 and 3.5. Specifically, as a measure of energetic efficiency, we use the dissipation 38 per right product formed, defined as 39 40 P 41 ε = . (21) vR 42 43 This way, the metric of energetics has units of kBT and is independent from the choice of 44 absolute timescale. Then, as a dimensionless metric of speed, we introduce the normalized 45 46 quantity 47 vR 48 ν = MM , (22) 49 vR 50 51 which represents the fraction by which the rate of forming right products in the proofreading MM 52 setting (vR) is slower than that in the simple Michaelis-Menten scheme (vR ) where the 53 allosteric effects are absent. This normalizing Michaelis-Menten speed is given in terms of 54 55 56 57 58 59 13 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 14 of 33

1 2 3 the model parameters via 4 5 I kon[R] 6 R MM koff +r vR = I I × r. (23) 7 kon[R] kon[W] 1 + R + W 8 koff +r koff +r 9 10 Next, we define the proofreading index α as a fidelity metric which represents the degree to 11 W R which the fidelity is amplified in multiples of koff /koff over its Michaelis-Menten value (ηMM), 12 that is, 13 14 kW + r kW α 15 off off η = R R , (24) 16 koff + r koff 17 | {z } η 18 MM ln η − ln η 19 α = MM . (25)  W  20 koff ln kR 21 off 22 23 Note that the proofreading index of Hopfield’s scheme is αHopfield = 1, as it involves a 24 single proofreading realization. Also, since in the absence of external driving the highest 25 max W R fidelity is ηeq = koff /koff , the corresponding upper limit in the proofreading index becomes 26 max αeq = 1 − ln ηMM/ ln η . 27 eq 28 As a final descriptor of piston model’s nonequilibrium behavior, we introduce the fraction 29 of returned work (κ) defined as the ratio of the rate at which the ligand performs work on 30 the piston upon expansion to the rate at which the piston performs work on the ligand upon 31 compression. We calculate κ via 32 33 P  (i) (i) kb + k πi∆F 34 i∈Sd f d→u 35 κ = −   , (26) P k + k(i) π ∆F (i) 36 i∈Su b f i u→d 37 38 where Sd and Su are the sets of states where the piston is compressed and expanded, respec- 39 tively. The negative sign is introduced to account for the fact that the ligand free energy 40 decreases upon piston expansion (i.e. the system gets the work back). In the absence of 41 42 enzyme coupling (section 3), this ratio was 1 because the ligand constantly exerted pressure 43 on the piston. With enzyme coupling, however, the work performed on the ligand upon 44 compression may not be fully returned since with some probability the ligand will be bound 45 to the enzyme and exert no pressure on the piston during expansion. We therefore expect 46 κ to be generally less than 1, indicating a net rate of performing work on the ligand in the 47 48 nonequilibrium setting. 49 Having defined analytical expressions for the key model performance metrics, we now 50 proceed to studying their graded changes and the trade-offs between them numerically. 51 52 53 54 55 56 57 58 59 14 60 ACS Paragon Plus Environment Page 15 of 33 The Journal of Physical Chemistry

1 2 3 (A) pawl 4 5 6 7 8 9 piston 10 11 12 ratchet weight 13 allosteric regulatory enzyme 14 ligand 15 substrates 16 (B) (C) 17 u→d, L 18 kf k 19 b 20 A A A A ℓ [L] ℓ ℓ [L] ℓ d→u, L u→d, L d→u, L u→d, L d→u, L on u off on d off 21 kf kf kf kf kf 22 u→d kb kb kb kb kb kf 23 k u d u d b 24 ~[L] ~[L] ~[L] ~[L] u→d,L 25 (D) kf 26 Δμ kb 27 kb d→u,L 28 kf knet u d u→d 29 ~[L] ~[L] kf 30 Δμ kb 31 kb 32 d→u kf u d,L 33 d→u u→d d→u u→d d→u → kf 34 kf kf kf kf kf kb kb kb kb kb kb Δμ 35 kb 36 d→u,L z kf 37 zn-1 zn zn+1 zn+2 u→d kf 38 Δμ kb 39 kb 40 d→u kf 41 42 Figure 5. Full and thermodynamically consistent treatment of the piston model of proofreading. 43 (A) Schematic representation of the full model, with the ratchet and pawl engine coupled to the 44 enzyme. (B) Network diagram of the full model. Each slice of the diagram represents the planar 45 view of the enzyme state network, with the alternating colors corresponding to the compressed 46 (dark blue) and expanded (light blue) states of the piston. Ligand-bound enzyme states are 47 marked with an orange circle. The horizontal arrows connecting the slices stand for forward and 48 49 backward ratchet steps. Only those at the outer edges are shown for clarity; however, transitions 50 are present between all horizontally neighboring enzyme states. Also for clarity, the stepping rate 51 constants are shown only at two of the outer edges where the ligand is either unbound (bottom 52 edge) or bound to the enzyme (top edge). The hanging weight at different vertical positions is 53 displayed below the diagram to symbolize energy expenditure as it gets lowered with a net rate 54 knet. (C) Sub-network of the full diagram in panel B where the state of the system is unchanged 55 after doing a cyclic traversal. The red arrow with a cross on top indicates that the cycle condition 56 57 holds in the sub-network. (D) The finite-state equivalent of the full network in panel B with the 58 weight position (zn) eliminated. Red arrows indicate the driving with a force ∆µ = 2∆W . 59 15 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 16 of 33

1 2 3 3.4 Energy-Speed-Fidelity Trade-Off in the Piston Model 4 5 Because of its mechanical construction, the piston model of proofreading has a distinguishing 6 feature - in it the external driving mechanism is physically separated from the allosteric 7 8 enzyme. This feature allows us to independently examine how tuning the “knobs” of the 9 engine and varying the kinetic parameters of the enzyme alter the performance of the model. 10 11 (A) (B) (C) 6 proofreading index, α 6 normalized speed, ν 8 proofreading index, α 12 10 10 -2 10 10 1.35 1.23 13 imit 6 4 m l 4 10 1.20 10 riu 1.08 10 ib 14 il 4 1.05 qu -3 10 2 e 2 10 10 0.93 10

I 2 15 d 0.90 ield limit 10 R pf R off Ho 0.78 off 0 0 /R 0.75 16 d 0 /k 10 /k 10 10 b 0.63 b -4

> Hopfield limit [L]

k k 0.60 17 10 -2 -2 -2 10 10 0.48 10 0.45 18 -4 10 -4 0.33 -4 0.30 10 10 -5 19 ΔW=ΔF ΔW=ΔF 10 -6 0.18 10 0.15 20 -6 -6 10 10 -8 0.03 10 0 2 4 6 8 0.00 21 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 10 10 10 10 10 22 ΔW (kBT) ΔW (kBT) f k /kR ΔW (kBT) ΔW (kBT) b off 23 (D) 14 (E) 14 (F) 1.4 1.4 kb ΔW 4 24 1.4 10 12 1.2 12 1.2 1.2 25 2 10 10 10 1.0 1.0 1.0 26 kb 0.8 0.8 0 27 8 8 0.8 10 0.6 0.6 6 0.6 28 6 -2 10 0.4 0.4 29 4 4 0.4 -4 0.2 0.2 0.2 10 30 index, α proofreading 2 index, α proofreading 2 index, α proofreading 0.0 0.0 0.0 -6 31 0 0 10 0 2 4 6 8 10 12 32 0.000 0.004 0.008 0.012 0.0 0.2 0.4 0.6 0.8 1.0 10 10 10 10 10 10 10 speed, ν 33 fraction of work returned, κ energy per right product, ε (kBT) 34 Figure 6. Parametric studies on the changes in the piston model performance in response to 35 tuning the “knobs” of the engine. (A,B) Variations in the proofreading index (A) and speed (B) as 36 37 the rate of backward stepping (kb) and the work per step (∆W ) are tuned. The dotted line 38 corresponds to the value of ∆W equal to the ligand free energy change upon compression (∆F ). 39 (C) Variations in the proofreading index when the high ligand concentration ([L]d) and the I 40 compression factor (f) are tuned. Rd represents the ligand dissociation constant in the inactive 41 enzyme state. The red dot indicates the pair of [L]d and f values used in the studies of the other 42 panels. (D) Fidelity-speed trade-off as kb is continuously varied for different choices of ∆W 43 (gradient arrow shows the direction of increase). The dotted black lines connect the highest 44 fidelity and speed values as ∆W is tuned. Between these dotted lines fidelity and speed are 45 46 negatively correlated. (E) Relation between fidelity and fraction of returned work for discrete 47 choices of ∆W and continuously tuned kb values (the gradient arrow indicates increasing ∆W ). 48 (F) Fidelity-dissipation trade-off obtained by continuously tuning ∆W for discrete choices of the 49 hopping rate (kb). The gradient arrow indicates the direction of increasing kb. The red dotted 50 curve corresponds to the case with resonance kb. 51 52 53 We begin our numerical analysis by first exploring the effects of external driving, where 54 the tuning “knobs” include the rate of backward stepping (kb), the work per step (∆W ), 55 the ligand concentration in the compressed piston state ([L]d) and the compression factor 56 (f = [L]d/[L]u). Choosing a set of enzyme’s kinetic parameters which make proofreading 57 58 59 16 60 ACS Paragon Plus Environment Page 17 of 33 The Journal of Physical Chemistry

1 2 3 possible (see Supporting Information section D.3 for the full list of parameters), we keep 4 5 them fixed for the rest of the analysis. We conduct the first parametric study by tuning kb 6 and ∆W and evaluating the proofreading index (Figures 6A). As anticipated, the proofread- 7 ing index does not exceed its equilibrium limit in the absence of driving (∆W = 0). This 8 expected feature can be paralleled by Brownian motors where purely equilibrium fluctuations 9 of the motor’s energy landscape are unable to generate directed motion.28 In addition, the 10 11 proofreading index achieves its highest value if ∆W is comparable to or larger than the lig- 12 and compression energy ∆F , and if the backward hopping rate kb is at its “resonance” value. 13 The presence of a “resonance” hopping rate is intuitive since if piston actions take place very 14 slowly then the fidelity will be reduced due to the small but nonzero rate of forming unfil- 15 tered products (i.e. “leakiness”) in the quasi-equilibrated enzyme states. And, conversely, if 16 17 piston actions take place too rapidly, then the activator ligand will almost always be bound 18 to the enzyme, preventing the realization of multiple substrate discrimination layers through 19 sequential enzyme activation and inactivation. We note that analogous resonance responses 20 were also identified for Brownian particles which attain their highest nonequilibrium drift 21 velocity in a ratchet-like potential landscape when the temperature29 or the landscape pro- 22 30 23 file are temporally modulated at specific resonance frequencies. A similar feature is present 24 in Hopfield’s model as well; namely, optimal proofreading is attained only when the rate of 25 hydrolysis is neither too low, nor too high.12 Interestingly, when the driving is hard enough 26 (∆W & ∆F ) and the backward hopping rate is close to its resonance value, the fidelity of 27 the piston model beats the Hopfield limit (α = 1) and raises the question of the largest 28 29 attainable fidelity, which we discuss in the next section. 30 Trends similar to those for the proofreading index are observed for the speed of forming 31 right products as well (Figure 6B). Specifically, product formation is very slow in the absence 32 of driving and increases monotonically with ∆W , until plateauing when ∆W & ∆F . Also, 33 the highest speed is achieved at a resonance k value which is different from that of the 34 b 35 proofreading index. The existence of such a resonance frequency is again intuitive, since 36 at fast rates of piston actions the enzyme is predominantly active and unable to bind new 37 substrates, while at slow rates the enzyme activation for catalysis via piston compression 38 happens very rarely. Notably, since the enzyme parameters were chosen in a way so as to 39 yield high fidelities, the largest speed value is substantially lower than the corresponding 40 −2 41 speed for a single-step Michaelis-Menten enzyme (νmax ≈ 10 ). 42 In the last parametric study, we explore how the choice of the high and low ligand con- 43 centrations affects the performance of the model. To that end, we tune the high ligand 44 concentration ([L]d) and the compression factor (f = [L]d/[L]u), and evaluate the highest 45 proofreading index at the resonant k value with ∆W > ∆F . As we can see, large fidelity 46 b 47 enhancements are achieved when [L]d is comparable to or larger than the ligand dissocia- I 48 tion constant in the inactive enzyme state (Rd), which is necessary to activate the enzyme 49 upon piston compression. In addition, the compression factor needs to be large enough (or, 50 equivalently, the ligand concentration in the expanded piston state should be low enough) 51 so as to inactivate the enzyme when piston enters its expanded state. This requirement of 52 53 having a large free energy difference between the compressed and expanded piston states 54 (β∆F = ln(f)  1) can be paralleled with a similar condition in Hopfield’s model where 55 for optimal proofreading the energy of the activated enzyme-substrate complex needs to be 56 much larger than that of the inactive complex. 57 58 59 17 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 18 of 33

1 2 3 Knowing separately how tuning the engine “knobs” affects the fidelity and speed, we 4 5 now explore the trade-offs between the model’s performance metrics as we vary the driving 6 parameters kb and ∆W , while holding the high and low ligand concentrations at fixed values 7 (the red dot in Figure 6C). We start with the trade-off between fidelity and speed, depicted 8 in Figure 6D, where we continuously tune the hopping rate kb for different choices of the 9 driving force ∆W . As expected from the results of the individual parametric studies in 10 11 Figure 6A and B, both fidelity and speed increase monotonically with ∆W . Also, since the 12 values of the hopping rate kb that maximize fidelity and speed are not identical, these two 13 performance metrics are negatively correlated in the range of kb values defined by the two 14 different resonance rates (region between the dotted lines in Figure 6D), but are positively 15 correlated otherwise. Variations in the metrics in the region of their negative correlation, 16 17 however, are moderate, suggesting that for an allosteric enzyme that has been optimized 18 for doing proofreading, the largest speed and fidelity could be achieved at similar external 19 driving conditions. 20 Next, we consider the relation between fidelity and fraction of work returned, shown 21 in Figure 6E. As can be seen, no fidelity enhancement is achieved when κ is close to 1 22 23 which happens either in the absence of driving (lighter curves) or in the presence of driving, 24 provided that the hopping rate is very fast. On the other hand, κ is much less than 1 at 25 the peak fidelity which is achieved when the hopping rate is at its resonance value and when 26 driving is large (∆W & ∆F ). Overall, this trade-off study demonstrates that irreversible 27 work performed on the ligand is a required feature for the attainment of fidelity enhancement 28 29 in the piston model. 30 Lastly, we look at how fidelity varies with energy dissipation, with the latter characterized 31 through the energy expended per right product (ε). The results of the trade-off study are 32 shown in Figure 6F. There, the driving force is continuously tuned for different choices of the 33 hopping rate. As can be seen, there is a minimum dissipation per product required to attain 34 35 the given level of performance. This minimum dissipation (the first intercept at a given 36 y-level) is achieved when the hopping rates are less than the corresponding resonant values 37 (the lighter curves on the left side of the dotted red curve). Additionally, for a given hopping 38 rate, increasing the driving force (∆W ) could lead to an increased proofreading performance 39 and a decreased dissipation per product up a critical point where the performance metric 40 41 reaches its saturating value (horizontal region), demonstrating how increasing the driving 42 force could in fact improve the energetic efficiency of proofreading. We note here that the 3 4 43 minimum ε values needed for significant proofreading are ∼ 10 −10 kBT in Figure 6F which 44 is ∼ 2 orders of magnitude higher than what is calculated for translation by the ribosome.12 45 This low energetic efficiency can be a consequence of our particular parameter choice for the 46 47 study as well as the performance limitations of our engine design, the investigation of which 48 we leave to future work. 49 50 51 52 53 54 55 56 57 58 59 18 60 ACS Paragon Plus Environment Page 19 of 33 The Journal of Physical Chemistry

1 2 3 3.5 Up to Three Proofreading Realizations are Available to the Pis- 4 5 ton Model 6 7 In the previous section, we chose a set of kinetic rate constants for the enzyme and, keeping 8 them fixed, explored the effects of tuning the external driving conditions on the performance 9 of the model. In this section, we explore the parameter space from a different angle, namely, 10 we study how tuning the enzyme’s kinetic parameters changes the model performance under 11 optimal driving conditions. Since there are more than a dozen rates defining the kinetic 12 13 behavior of the enzyme, it is impractical to probe their individual effects. Instead, we 14 choose to vary two representative parameters about the effects of which we have a prejudice. 15 A These include the rate of substrate binding to the active enzyme (kon) and the unbinding 16 rate of wrong substrates (kW ). We know already from Hopfield’s analysis that for efficient 17 off 18 proofreading the direct binding of substrates to the active enzyme state should be very slow. A 19 Therefore, we expect the proofreading performance to improve as kon is reduced. We also A W 20 expect the minimum requirement for kon to be lower for larger koff values to ensure that 21 wrong substrates do not enter through the unfiltered pathway.9 22 With these expectations in mind, we performed a parametric study to find the highest 23 A 24 fidelity, the results of which are summarized in Figure 7A. There we varied kon for several W 25 choices of koff , and for each pair numerically optimized over the enzyme’s remaining kinetic 26 rates and external driving conditions to get maximum fidelity (see Supporting Information 27 section D.4 for implementation details). As expected, the highest attainable fidelity decreases 28 A I A monotonically with increasing “leakiness” (kon/kon), and the minimum requirement on kon 29 W 30 decreases with increasing koff . 31 Interestingly, we also see that for small enough leakiness, the piston model manages to W R 32 perform proofreading (i.e. enhance the fidelity by a factor of koff /koff ) up to three times, as 33 αmax ≈ 3 (Figure 7A). To understand this unexpected feature, we identified the dominant 34 A I trajectory that the system would take to form a wrong product for the case where kon/kon = 35 −12 36 10 (Figure 7B, see Supporting Information section D.5 for details). As we can see, after 37 initial binding the wrong substrate indeed passes through three different proofreading filters, 38 and these are realized efficiently because the transitions between intermediate states are 39 much slower than the rate of substrate unbinding. The first filter occurs right after piston 40 compression, while the enzyme is waiting for the activator ligand to bind (#1). We note 41 42 that this particular filter is made possible due to the presence of alternative piston states 43 (equivalently, alternative environments that the enzyme could “sense”). The remaining two 44 filters (#2 and #3) take place while the ligand-bound enzyme is waiting to get activated and 45 while the active enzyme is waiting to turn the wrong substrate into a product, respectively. 46 The presence of these two filters is purely a consequence of allostery. Importantly, the 47 48 αmax ≈ 3 result in Figure 7 represents the theoretical upper limit of the model’s proofreading 49 index – a feature that we justify analytically in Supporting Information section D.5. 50 In light of this analysis, we can now explain why the pedagogically simplified version 51 of the model introduced in section 2 achieved only a single proofreading realization. There 52 we made the implicit assumption that ligand binding after piston compression and enzyme 53 54 activation after ligand binding took place instantly. Because of this, proofreading filters #1 55 and #2 were not realized, leaving filter #3 as the only available one which we showed in 56 Figure 1B. 57 58 59 19 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 20 of 33

1 2 3 (A) (B) 4 3.0 substrate piston ligand enzyme 5 2.5 binding compression binding activation 6 2.0 7 1.5 8 WR koff/koff =10 1.0 I 9 WR I W, L k /k =100 k k ℓ k off off on[W] b on[L]d A r 10 0.5 WR proofreading index, α proofreading koff/koff =1000 kW k ℓI kW, L 11 0.0 off b off I -12 -8 -4 0 kW #1 kW #2 kW #3 12 10 10 10 10 off off off AI 13 leakiness, kon/kon 14 15 Figure 7. Proofreading performance of the piston model under optimized enzyme parameters 16 and external driving conditions. (A) The highest proofreading index (α) available to the piston A I W 17 model as a function of leakiness (kon/kon) for different choices of koff . (B) The dominant 18 A I −12 trajectory that the system takes to form a wrong product in the case where kon/kon = 10 . 19 Numbers 1, 2, 3 stand for the different proofreading filters along the trajectory. The dotted arrows 20 indicate that the respective rates are much slower than the substrate unbinding rate kW (see 21 off Supporting Information section D.5 for their numerical values for the kW /kR = 100 case). 22 off off 23 24 25 4 Discussion and Conclusions 26 27 A distinctive feature of kinetic proofreading is that it is a nonequilibrium mechanism, by 28 virtue of which its operation needs to involve energy expenditure.9,10 Mechanical work, be- 29 ing an intuitive representation of energy expenditure, has been used in the past to elucidate 30 important physical concepts such as information-to-energy conversion in the thought ex- 31 22 31 32 periment by Szilard, or the mechanical equivalence of heat in Joule’s apparatus. Yet, a 33 similar demonstration of how mechanical work could be harnessed in a graded fashion to 34 beat the equilibrium limit in substrate discrimination fidelity has been lacking. Our aim in 35 this work was to offer such a demonstration through the mechanically designed piston model 36 of proofreading. 37 38 We started off by providing the conceptual picture of the piston model, with its con- 9 39 stituents having direct parallels with Hopfield’s original proofreading scheme (Figure 1). 40 The key idea of the model was to replace the nucleotide hydrolysis step present in Hopfield’s 41 scheme with piston compression which served an identical role of activating the enzyme but 42 in our case achieved through allostery and mechanical work. Just like in the case of bio- 43 44 logical proofreading, where hydrolysis itself cannot lead to fidelity enhancement unless the 45 nucleotide triphosphates are held at fixed out-of-equilibrium chemical potentials, in the case 46 of piston model too, the compressive and expansive actions of the piston cannot result in 47 proofreading unless they are driven by an energy-consuming engine. Motivated by Feyn- 48 man’s ratchet and pawl mechanism,21 we then proposed a dissipative mechanical engine to 49 50 drive the cyclic piston actions, which maintained the nonequilibrium distribution of enzyme 51 states necessary for achieving proofreading. The function of this engine can be paralleled to 52 that of the ATP synthase in the cell whose constant operation maintains a finite chemical 53 potential of ATP, which different biochemical pathways can then take advantage of. 54 To study how the cyclic variations in ligand concentration generated by the engine alter 55 56 the occupancies of enzyme states, we performed a thermodynamically consistent coupling 57 58 59 20 60 ACS Paragon Plus Environment Page 21 of 33 The Journal of Physical Chemistry

1 2 3 between the engine and the enzyme (Figure 5). There we considered the full diversity of 4 5 states that the enzyme could take and, importantly, the feedback mechanism for the engine 6 to “sense” the state of the enzyme. The accounting of this latter feature, which makes the 7 piston model an example of a bi-partite system,32,33 was motivated by our aim of proposing 8 a framework where we could consistently calculate the total dissipation as opposed to only 9 the minimum dissipation needed for maintaining the nonequilibrium steady state of the 10 12,34,35 11 enzyme (without considering the driving engine). Although the dissection of different 12 contributions to dissipation and their interconnectedness was not among the objectives of 13 our work, the framework that we proposed in our model can serve as a basis for additional 14 studies of periodically driven molecular systems (e.g. Brownian clocks or artificial molecular 15 motors) where the driving protocol and thermodynamics are of importance.36–38 As noted 16 17 earlier, however, in the presence of a periodically changing ligand concentration, the allosteric 18 enzyme could perform proofreading irrespective of the driving agency, which suggests a 19 possible biochemical mechanism of fidelity enhancement without the direct coupling of the 20 enzyme state transitions to hydrolysis. 21 Having explicit control over the “knobs” of the mechanical engine, we then probed the 22 23 performance of the model under different driving conditions. We found that both speed and 24 fidelity increased as we tuned up the mass of the hanging weight, until plateauing at a point 25 where the free energy bias of the expanded piston state was fully overcome (∆W & ∆F ), 26 beyond which increasing the weight only increased the dissipation without improving the 27 model performance (Figure 6A,B). This result can be paralleled with the presence of a 28 29 minimum threshold for the strength of driving in Hopfield’s model, past which the highest 9 30 fidelity becomes attainable. In addition, we found that in the piston model there is a 31 “resonance” rate of piston actions which maximizes fidelity, analogous to the similar feature 32 of Hopfield’s scheme where both very fast and very slow rates of hydrolysis reduce the quality 33 of proofreading.12 34 35 The tunable control over the driving parameters also allowed us to study the trade-off 36 between fidelity, speed and energy spent per right product. These studies revealed that the 37 correlation between speed and fidelity could be both positive and negative when varying 38 the rate of driving. Notably, theoretical investigations of translation by the Escherichia coli 39 ribosome under Hopfield’s scheme identified a similar behavior for the fidelity-speed correla- 40 41 tion in response to tuning the GTP hydrolysis rate, with the experimentally measured values 42 being in the negative correlation (i.e. trade-off) region.17 In contrast to the ribosome study, 43 however, where the two metrics vary by several orders of magnitude in the trade-off region, 44 in the piston model the variations in fidelity and speed in the negative correlation region are 45 moderate (Figure 6D), calling for additional investigations of the underlying reasons behind 46 47 this difference and search for the realization of the latter advantageous behavior in biolog- 48 ical proofreading systems. Furthermore, our studies showed that the minimum dissipation 49 required to reach the given level of fidelity was achieved for hopping rates necessarily lower 50 than their resonance values, and that increasing the work performed per step (analogously, 51 the chemical potential of ATP) could actually improve the energetic efficiency of the model 52 53 – features that again motivate the identification of their realization in biochemical systems. 54 In the end, we explored the limits in the proofreading performance of the piston model 55 A I for various choices of the allosteric enzyme’s “leakiness” (kon/kon) and the ratio of the wrong 56 and right substrate off-rates (kW /kR ). We found that the trends for the highest available 57 off off 58 59 21 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 22 of 33

1 2 3 fidelity matched analogously with the features of Hopfield’s original scheme, suggesting their 4 5 possible ubiquity for general proofreading networks. More importantly, our analysis revealed 6 that the piston model could do proofreading not just once but up to three times in the 7 limit of very low leakiness, despite the fact that energy consumption takes place during a 8 single piston compression. This is in contrast to the typical involvement of several energy 9 consumption instances in multistep proofreading schemes which manage to beat the Hopfield 10 11 limit of fidelity, as, for example, in the cases of the T-cell or MAPK activation pathways 5,6,34 12 which require multiple phosphorylation reactions. Our finding therefore suggests the 13 possibility of achieving several proofreading realizations with a single energy consuming 14 step by leveraging the presence of multiple inactive intermediates intrinsically available to 15 allosteric molecules. We would like to mention here that the presence of a similar feature 16 17 was also experimentally demonstrated recently for the ribosome which was shown to use the 18 free energy of a single GTP hydrolysis to perform proofreading twice after the initial tRNA 19 selection – first, at the EF-Tu · GDP-bound inactive state and second, at the EF-Tu-free 20 active state.39 21 In the presentation of the piston model we focused on the thermodynamic consistency 22 23 of the framework for managing the energy dissipation and did not consider strategies for 24 improving the performance of the mechanism. One such possibility that can be considered 25 in future work is to use a more elaborate design for the ratchet and pawl engine with 26 alternating activation barriers for pawl hopping which would allow to have different rates of 27 piston compression and expansion, analogous to how hydrolysis and condensation reactions 28 17,40 29 generally occur with different rates in biological proofreading. Another avenue is to 30 consider alternative ways of allocating the mechanical energy dissipation across the different 31 ratchet transition steps, similar to how optimization schemes of allocating the free energy of 32 ATP hydrolysis were studied for molecular machine cycles.41 Incorporating these additional 33 features would allow us to probe the performance limits of the piston model and compare 34 42 35 them with the fundamental limits set by thermodynamics. 36 37 38 Supporting Information 39 40 Details on the discrimination steps in the conceptual scheme of the piston model; in-depth 41 discussion of the operation of the ratchet and pawl engine; equilibrium constraints on the 42 43 enzyme’s transition rates; details on the engine-enzyme coupling; procedure for implementing 44 the numerical studies; and investigation of the highest discriminatory capacity of the model 45 (PDF); supplementary Python scripts and Jupyter notebooks which can be used to reproduce 46 the results of the different numerical studies (ZIP). 47 48 49 50 Acknowledgements 51 52 We thank Tal Einav, Erwin Frey, Christina Hueschen, Sarah Marzen, Arvind Murugan, 53 Manuel Razo-Mejia, Matt Thomson, Yuhai Tu, Jin Wang, Jerry Wang, Ned Wingreen and 54 Fangzhou Xiao for fruitful discussions. We also thank Haojie Li and Dennis Yatunin for 55 their input on this work, Alexander Grosberg, David Sivak, and Pablo Sartori for providing 56 57 58 59 22 60 ACS Paragon Plus Environment Page 23 of 33 The Journal of Physical Chemistry

1 2 3 valuable feedback on the manuscript, and Nigel Orme for his assistance in making the illus- 4 5 trations. This work was supported by the National Institutes of Health through the grant 6 1R35 GM118043-01 (MIRA), and the John Templeton Foundation as part of the Boundaries 7 of Life Initiative grants 51250 and 60973. 8 9 10 References 11 12 (1) Kunkel, T. A. DNA Replication Fidelity. J. Biol. Chem. 2004, 279, 16895–16898. 13 14 (2) Sydow, J. F.; Cramer, P. RNA Polymerase Fidelity and Transcriptional Proofreading. 15 16 Curr. Opin. Struc. Biol. 2009, 19, 732–739. 17 18 (3) Rodnina, M. V.; Wintermeyer, W. Fidelity of Aminoacyl-tRNA Selection on the Ribo- 19 some: Kinetic and Structural Mechanisms. Annu. Rev. Biochem. 2001, 70, 415–435. 20 21 (4) Lu, Y.; Wang, W.; Kirschner, M. W. Specificity of the Anaphase-Promoting Complex: 22 a Single-Molecule Study. Science 2015, 348, 1248737. 23 24 (5) Swain, P.; Siggia, E. The Role of Proofreading in Signal Transduction Specificity. Bio- 25 phys. J. 2002, 82, 2928–2933. 26 27 (6) Mckeithan, T. W. Kinetic Proofreading in T-cell Receptor Signal Transduction. Proc. 28 29 Natl. Acad. Sci. U.S.A. 1995, 92, 5042–5046. 30 31 (7) Goldstein, B.; Faeder, J. R.; Hlavacek, W. S. Mathematical and Computational Models 32 of Immune-Receptor Signalling. Nat. Rev. Immunol. 2004, 4, 445. 33 34 (8) Bard, J. A.; Bashore, C.; Dong, K. C.; Martin, A. The 26S Proteasome Utilizes a 35 Kinetic Gateway to Prioritize Substrate Degradation. Cell 2019, 92, 5042–5046. 36 37 (9) Hopfield, J. J. Kinetic Proofreading: A New Mechanism for Reducing Errors in Biosyn- 38 thetic Processes Requiring High Specificity. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 39 40 4135–4139. 41 42 (10) Ninio, J. Kinetic Amplification of Enzyme Discrimination. Biochimie 1975, 57, 587– 43 595. 44 45 (11) Murugan, A.; Huse, D. A.; Leibler, S. Speed, Dissipation, and Error in Kinetic Proof- 46 reading. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 12034–12039. 47 48 (12) Wong, F.; Amir, A.; Gunawardena, J. Energy-Speed-Accuracy Relation in Complex 49 Networks for Biological Discrimination. Phys. Rev. E. 2018, 98, 012420. 50 51 (13) Sartori, P.; Pigolotti, S. Kinetic versus Energetic Discrimination in Biological Copying. 52 53 Phys. Rev. Let.. 2013, 110, 188101. 54 55 (14) Depken, M.; Parrondo, J. M.; Grill, S. W. Intermittent Transcription Dynamics for the 56 Rapid Production of Long Transcripts of High Fidelity. Cell Rep. 2013, 5, 521–530. 57 58 59 23 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 24 of 33

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1 2 3 TOC Graphic 4 5 binding work proofreading 6 7 piston ompres 8 c s engine 9 activator ligand e 10 xpand 11 12 allosteric r 13 kon koff koff enzyme 14 15 substrates 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 26 60 ACS Paragon Plus Environment (A) #1 PageHopfieldʼs 27The of Journal 33 of Physical Chemistry scheme #2

koff kon koff r 1 NTP NDP 2 3 4 5

(B) pr 6 piston com ess 7 model 8 e 9 xpand 10 11 12 r 13 kon koff koff 14 15 16 17 piston state expanded compressed 18 ligand state unbound bound 19enzyme state inactive active 20 substrate allowed not allowed 21 bindingACS Paragon Plus Environment 22 W W achieved koff koff + r 23 fidelity R R koff koff + r 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 (A) pawl The Journal of Physical Chemistry Page 28 of 33

b 1

2 f 3 piston 4 5 6 ratchet weight ligand 7 8 (B) (C) u→d E0+ΔW+ΔF compression expansion kf 9 kb (u→d) (d→u) E0+ΔW-ΔF 10 d→u 11 kf kb E0 kb kb 2ΔW

12 free energy

step 1 -βE 1 -βE single cycle e 0 e 0 Δz 13 τ τ backward z 14 ΔWb = -mgΔz zn-2 zn-1 zn zn+1 zn+2 zn+3 zn+4 15 (D) u→d d→u u→d d→u u→d d→u u→d d→u u→d d→u kf kf kf kf kf kf kf kf 16 kf kf knet

17 step 1 -β(E0+ΔW+ΔF) 1 -β(E0+ΔW-ΔF) d u d u d du

forward e e 18 τ τ ΔWf = mgΔz 19 kb kb kb kb kb kb kb kb 20 d→u (E) kf (F) 21 d→u kf +kb 22 kb ACS Paragon Plus Environment -1 -1 d Δμ u d u 23 ΔFu→d = β ln(f) ΔFd→u = -β ln(f) kb u→d 24 kf +kb u→d 25 kf 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 (A) (B) Page1.0 29 of 33 The Journal of Physical Chemistry1.0

0.8 0.8 ΔF = 2 kBT ΔF = 2 kBT u b

π ΔF = 4 kBT ΔF = 4 kBT 0.6 1 /k 0.6

/ ΔF = 6 k T ΔF = 6 k T d B B 2 net k π 0.4 0.4 3 ΔF = 8 kBT ΔF = 8 kBT 40.2 ACS Paragon Plus Environment0.2 ΔW1/2 ≈ ΔF ΔW1/2 ≈ ΔF-kBTln2 50.0 0.0 6 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 7 ΔW (kBT) ΔW (kBT) 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A A k [W] k [R] r Theon Journal of Physical Chemistry on Page 30 of 33r kW kR A off A off A ℓ A ℓ A ℓ A on[L] ℓ on[L] ℓ on[L] ℓ off kA off kA off on[W] on[R] r r 1 W, L W L L R R, L W, L k k k k k R, L k k I off A I off k I 2 A A 3 I I 4 kW kW k k k k kR kR A I on[W] A I on[R] A I 5 kW kR 6 ℓI off ℓI off ℓI on[L] ℓI ACS Paragonon[L] PlusℓI Environment on[L] ℓI 7 off k I off kI off on[W] on[R] 8 kW kR 9 off off 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 (A) pawl Page 31 of 33 The Journal of Physical Chemistry

1 2 3 piston 4 5 6 ratchet weight 7 allosteric regulatory enzyme 8 ligand 9 substrates (B)10 (C) 11 u→d, L 12 kf k 13 b 14 A A A A ℓ [L] ℓ ℓ [L] ℓ d→u, L u→d, L d→u, L u→d, L d→u, L on u off on d off 15 kf kf kf kf kf 16 u→d kb kb kb kb kb kf 17 k u d u d b 18 ~[L] ~[L] ~[L] ~[L] u→d,L 19 (D) kf 20 Δμ kb 21 kb d→u,L 22 kf knet u d u→d 23 ~[L] ~[L] kf 24 Δμ kb 25 kb 26 d→u kf u d,L 27d→u u→d d→u u→d d→u → kf 28kf kf kf kf kf kb kb kb kb kb kb Δμ 29 kb 30 d→u,L z kf 31 zn-1 zn zn+1 zn+2 u→d kf 32 ACS Paragon Plus Environment Δμ kb 33 kb 34 d→u kf 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 (A) (B) (C) 6 proofreading index, α 6 normalized speed, ν 8 proofreading index, α 10 The10 Journal of Physical Chemistry -2 10 Page 32 of 33 10 1.35 1.23 imit 6 4 m l 4 10 1.20 10 riu 1.08 10 ib il 4 1.05 qu -3 10 2 e 2 10 1 10 0.93 10 I 2 d 0.90 ield limit 10 R pf R off Ho 0.78 off 2 0 0 /R 0.75 d 0 /k 10 /k 10 10 b 0.63 b -4

> Hopfield limit [L] 3k k 10 -2 0.60 -2 -2 10 10 0.48 10 0.45 4 -4 10 -4 0.33 -4 0.30 510 10 -5 ΔW=ΔF ΔW=ΔF 10 -6 0.18 10 0.15 6 -6 -6 10 10 -8 0.03 10 0 2 4 6 8 0.00 7 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 10 10 10 10 10 8 ΔW (kBT) ΔW (kBT) f k /kR 9 ΔW (kBT) ΔW (kBT) b off (D) 14 (E) 14 (F) 1.4 1.4 kb ΔW 4 10 1.4 10 12 1.2 12 111.2 1.2 2 10 10 10 121.0 1.0 1.0 kb 0.8 0.8 0 13 8 8 0.8 10 0.6 0.6 6 0.6 14 6 -2 10 0.4 0.4 15 4 4 0.4 -4 160.2 0.2 0.2 10 proofreading index, α proofreading 2 index, α proofreading ACS Paragon Plus Environment2 index, α proofreading 170.0 0.0 0.0 -6 0 0 10 0 2 4 6 8 10 12 18 0.000 0.004 0.008 0.012 0.0 0.2 0.4 0.6 0.8 1.0 10 10 10 10 10 10 10 speed, ν 19 fraction of work returned, κ energy per right product, ε (kBT) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 (A) (B) Page3.0 33 of 33 The Journal of Physical Chemistry substrate piston ligand enzyme 2.5 binding compression binding activation

2.0 1 1.5 2 WR koff/koff =10 1.0 I 3 WR I W, L k /k =100 k k ℓ k off off on[W] b on[L]d A r 4 0.5 WR proofreading index, α proofreading koff/koff =1000 ACS ParagonkW Plus Environmentk ℓI kW, L 5 0.0 off b off I -12 -8 -4 0 kW #1 kW #2 kW #3 6 10 10 10 10 off off off AI 7 leakiness, kon/kon 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60