Allostery and Kinetic Proofreading
AAllostery and Kinetic Proofreading
Vahe Galstyan † and Rob Phillips ∗ , ‡ , ¶ , § † Biochemistry and Molecular Biophysics Option, California Institute of Technology,Pasadena, California 91125, United States ‡ Department of Physics, California Institute of Technology, Pasadena, California 91125,United States ¶ Department of Applied Physics, California Institute of Technology, Pasadena, California91125, United States § Division of Biology and Biological Engineering, California Institute of Technology,Pasadena, California 91125, United States
E-mail: [email protected]
Phone: (626) 395-33741 a r X i v : . [ q - b i o . M N ] M a y bstract Kinetic proofreading is an error correction mechanism present in the processes of thecentral dogma and beyond, and typically requires the free energy of nucleotide hydrol-ysis for its operation. Though the molecular players of many biological proofreadingschemes are known, our understanding of how energy consumption is managed to pro-mote fidelity remains incomplete. In our work, we introduce an alternative conceptualscheme called “the piston model of proofreading” where enzyme activation throughhydrolysis is replaced with allosteric activation achieved through mechanical work per-formed by a piston on regulatory ligands. Inspired by Feynman’s ratchet and pawlmechanism, we consider a mechanical engine designed to drive the piston actions pow-ered by a lowering weight, whose function is analogous to that of ATP synthase in cells.Thanks to its mechanical design, the piston model allows us to tune the “knobs” ofthe driving engine and probe the graded changes and trade-offs between speed, fidelityand energy dissipation. It provides an intuitive explanation of the conditions necessaryfor optimal proofreading and reveals the unexpected capability of allosteric moleculesto beat the Hopfield limit of fidelity by leveraging the diversity of states available tothem. The framework that we built for the piston model can also serve as a basis foradditional studies of driven biochemical systems.
Many enzymatic processes in biology need to operate with very high fidelities in order toensure the physiological well-being of the cell. Examples include the synthesis of moleculesmaking up Crick’s so-called “two great polymer languages” (i.e. replication, transcription, and translation ), as well as processes that go beyond those of the central dogma, such asprotein ubiquitylation mediated by the anaphase-promoting complex, signal transductionthrough MAP kinases, pathogen recognition by T-cells, or protein degradation by the26S proteasome. In all of these cases, the designated enzyme needs to accurately selectits correct substrate from a pool of incorrect substrates. Importantly, the fidelity of theseprocesses that one would predict solely based on the free energy difference between correctand incorrect substrate binding is far lower than what is experimentally measured, raising achallenge of explaining the high fidelities that this naive equilibrium thermodynamic thinkingfails to account for.The conceptual answer to this challenge was provided more than 40 years ago in the workof John Hopfield and Jacques Ninio and was coined “kinetic proofreading” in Hopfield’selegant paper entitled “Kinetic Proofreading: A New Mechanism for Reducing Errors inBiosynthetic Processes Requiring High Specificity.” The key idea behind kinetic proofread-ing is to introduce a delay between substrate binding and turnover steps, effectively givingthe enzyme more than one chance to release the incorrect substrate (hence, the term “proof-reading”). The sequential application of substrate filters on the way to product formationgives directionality to the flow of time and is necessarily accompanied by the expenditure offree energy, making kinetic proofreading an intrinsically nonequilibrium phenomenon. In acell, this free energy is typically supplied to proofreading pathways through the hydrolysis ofenergy-rich nucleotides, whose chemical potential is maintained at large out-of-equilibrium2alues through the constant operation of the cell’s metabolic machinery (e.g. the ATP syn-thase).Since its original formulation by Hopfield and Ninio, the concept of kinetic proofreadinghas been generalized and employed in explaining many of the high fidelity processes in thecell.
However, despite the fact that the molecular players and mechanisms of theseprocesses have been largely identified, we find that an intuitive picture of how energy trans-duction promotes biological fidelity is still incomplete. To complement our understandingof how energy is managed to beat the equilibrium limit of fidelity, we propose a concep-tual model called “the piston model of kinetic proofreading” where chemical hydrolysis isreplaced with mechanical work performed by a piston on an allosteric enzyme. Our choice ofallostery is motivated by the fact that in proofreading schemes hydrolysis typically triggersa conformational change in the enzyme-substrate complex and activates it for product for-mation - an effect that our model achieves through the binding of a regulatory ligand tothe enzyme. By temporally controlling the concentration of regulatory ligands which deter-mine the catalytic state of the enzyme, the piston sequentially changes the enzyme’s statefrom inactive to active, creating a delay in product formation necessary for increasing thefidelity of substrate discrimination. The piston actions are, in turn, driven by a Brownianratchet and pawl engine powered by a lowering weight, whose function is akin to that of ATPsynthase. The mechanical design of the piston model allows us to transparently control theenergy input into the system by tuning the “knobs” of the engine and examine the gradedchanges in the model’s performance metrics, intuitively demonstrating the driving conditionsrequired for optimal proofreading.We begin the presentation of our results by first introducing in section 2 the high-levelconcept behind the piston model of proofreading, while at the same time drawing parallelsbetween its features and those of Hopfield’s original scheme. Then, in sections 3 and 3.2 weprovide a comprehensive description of the two key constituents of the piston model, namely,the Brownian ratchet and pawl engine that drives the piston actions, and the allostericenzyme whose catalytic state is regulated by an activator ligand. This is following by buildingthe full thermodynamically consistent framework of the piston model in section 3.3 wherewe couple the external driving mechanism to the enzyme and introduce the expressions forkey performance metrics of the model. In the remaining sections 3.4 and 3.5, we explorehow tuning the “knobs” of the engine leads to graded changes and trade-offs between speed,fidelity and energy dissipation, and probe the performance limits of the piston model as afunction of a select set of key enzyme parameters.3
MODEL
The piston model of kinetic proofreading is designed in analogy with Hopfield’s scheme. Themain idea there was to give the enzyme a second chance to discard the wrong substrateby introducing an additional kinetic intermediate for the enzyme-substrate complex (Figure1A). The difference between substrate binding energies in Hopfield’s original formulation wasbased solely on their unbinding rates (i.e. k Woff > k
Roff and k W on = k R on = k on ) – a convention weadopt throughout our analysis. The first layer of substrate discrimination in Hopfield’sscheme is achieved during the initial binding event where the ratio of right and wrongsubstrate-bound enzymes approaches k Woff /k Roff . The complex then moves into its catalyticallyactive high energy state accompanied by the hydrolysis of an NTP molecule, after which thesecond discrimination layer is realized. Specifically, right and wrong substrates are turnedinto products with an additional bias given by the ratio of their Michaelis constants, namely, ( k Woff + r ) / ( k Roff + r ) . Importantly, for this second layer to be efficiently realized, the rates ofbinding directly to the second kinetic intermediate need to be vanishingly small in order toprevent the incorporation of unfiltered substrates. With this information in mind, consider now the conceptual illustration of the pistonmodel shown in Figure 1B, where we have made several pedagogical simplifications to helpverbally convey the model’s intuition, reserving the full thermodynamically consistent treat-ment to the following sections. The central constituent of the model is an allosteric enzyme,the catalytic activity of which is regulated by activator ligands (the orange circle). Theenzyme is inactive when it is not bound to a ligand, and, conversely, it is active when boundto a ligand. The volume occupied by ligands and hence, their concentration is, in turn,controlled by a piston. The ligand concentration is very low when the piston is expanded,and very high when the piston is compressed in order to guarantee that in those piston statesthe ligand is free and bound to the enzyme, respectively.The active site of the enzyme is exposed to a container filled with right and wrongsubstrates of concentrations [ R ] and [ W ] , respectively, which we take to be equal for the restof our analysis ( [ R ] = [ W ] ). And unlike in Hopfield’s scheme where the substrates exist inenergy-rich and energy-depleted states (e.g. tRNAs first arrive in the EF-Tu · GTP · tRNAternary complex and then release EF-Tu and GDP after hydrolysis), in the piston modelsubstrates exist in a single state and do not carry an energy source. In the expanded pistonstate (Figure 1B, left), substrates can bind and unbind to the inactive enzyme, but do not getturned into products. The highest level of discrimination achievable in this state thereforebecomes η = k Woff k Roff , (1)in analogy to that achieved during the initial binding step of Hopfield’s scheme.After the first layer of substrate discrimination is established in the expanded state ofthe piston, mechanical work is performed to compress it. This increases the ligand con-centration, which, in turn, leads to the activation of the enzyme where catalytic action isnow possible. To prevent the incorporation of unfiltered substrates, we assume that in theactive enzyme state the rate of substrate binding is vanishingly small, similar to Hopfield’s4reatment (Figure 1B, right). If the piston is kept compressed long enough, a filtered sub-strate that got bound earlier when the piston was expanded will experience one of these twooutcomes: it will either turn into a product with a rate r (which is taken to be the samefor the two kinds of substrates) or it will fall off with a rate k off . The product formationreaction will take place with probability r/ ( k off + r ) . Thus, due to the difference in the falloffrate constants between the right and the wrong substrates, the extra fidelity achieved afterpiston compression equals η = k Woff + rk Roff + r . (2)Once this extra fidelity is established, the piston is expanded back, repeating the cycle (thedetailed derivation of the results for η and η is provided in Supporting Information sectionA). Notably, the total fidelity achieved during the piston expansion and compression cycle,namely, η = η η = (cid:18) k Woff k Roff (cid:19) (cid:18) k Woff + rk Roff + r (cid:19) , (3)exceeds the Michaelis-Menten fidelity ( η ) by a factor of η = k Woff /k Roff , demonstrating theattainment of efficient proofreading.The cyclic compressions and expansions of the piston in our model also stand in directanalogy to the hydrolysis-involving transitions between the two enzyme-substrate intermedi-ates in Hopfield’s scheme. In particular, they need to be externally driven for the mechanismto do proofreading. We perform this driving using a mechanical ratchet and pawl enginepowered by a lowering weight. In our pedagogical description of the model’s operation, weimplicitly assumed that this weight was very large in order to enable the mechanism to proof-read, similar to how the hydrolysis energy needs to be large for Hopfield’s scheme to operateeffectively. In the full treatment of the model in section 3, however, we will demonstratehow the tuning of the weight can give us graded levels of fidelity enhancement, and willalso show that in the absence of this weight the equilibrium fluctuations of the piston alonecannot lead to proofreading.In our model introduction we have also made several simplifying assumption for clarityof presentation which do not conform with the principle of microscopic reversibility, andit is important that we relax them in the full treatment of the model to make it thermo-dynamically consistent. In particular, we assumed that the ligand is necessarily unboundand that the enzyme is necessarily inactive when the piston is expanded, with the reverseassumptions made when the piston is compressed. We also assumed that substrate bindingwas prohibited in the active state of the enzyme. These assumptions allowed us to claimthat no premature product formation takes place in the expanded piston state and that nounfiltered substrates bind to the activated enzyme in the compressed piston state, which, inturn, justified the use of long waiting times between the piston actions necessary to establishhigh levels of fidelity in each piston state. In the detailed analysis of our model presentedin section 3, we relax these assumptions and consider the full diversity of enzyme states ateach piston position with reversible transitions between them. This, as we will demonstrate,5ill not only ensure the thermodynamic consistency of our treatment but will also revealthe possibility of doing proofreading more than once by leveraging the presence of multipleinactive intermediates in between enzyme’s substrate-unbound and production states whichwere not accounted for in our conceptual introduction of the model. r k off piston state expandedHopfieldʼsschemepistonmodel compressedunbound boundligand state inactive activeenzyme state allowed not allowedachievedfidelitysubstratebinding k on k on k off k off k off (B)(A) r NTP NDP W k off + r R k off + r W k offR k off c o m p r e s s e x p a n d Figure 1.
Conceptual introduction to the piston model. (A) Hopfield’s scheme of kineticproofreading where two layers of substrate discrimination take place on the driven pathway – thefirst one during the initial binding of energy-rich substrates ( (cid:10)
NDP accompanying the transition between thetwo intermediates. (B) Pedagogically simplified conceptual scheme of the piston model. Theorange circle represents the activator ligand. Blue and red colors stand for the right and wrongsubstrates, respectively. The closed “entrance door” along with the red cross on the binding arrowin the active state of the enzyme suggests the vanishingly small rate of substrate binding when inthis state. The ratchet with a hanging weight stands for the mechanical engine that drives thepiston actions. Various features of the system in the two piston states, along with the expressionsfor achieved fidelities are listed below the diagram. Transparent arrows between panels A and Bindicate the analogous parts in Hopfield’s scheme and the piston model. RESULTS
To drive the cyclic compressions and expansions of the piston necessary for achieving proof-reading, we use a ratchet and pawl engine whose design is inspired by Feynman’s originalwork. In his celebrated lectures, Feynman presented two implementations of the ratchetand pawl engine – one operating on the temperature difference between two thermal baths,and another driven by a weight that goes down due to gravity. In the piston model weadopt the second scheme as it involves fewer parameters and illustrates the process of energytransduction more transparently.The ratchet and pawl engine coupled to the piston is shown in Figure 2A. The engineis powered by a weight of mass m which is hanging from an axle connected to the ratchet.The free rotational motion of the ratchet is rectified by a pawl; when the pawl sits on aratchet tooth, it prevents the ratchet from rotating in the clockwise (backwards) direction.The mechanical coupling between the engine and the piston is achieved through a crankshaftmechanism which translates each discrete ratchet step into a full compression (up → down)or a full expansion (down → up) of the piston. We assume that the volume regulated by thepiston contains a single ligand – a choice motivated by Szilard’s thermodynamic interpreta-tion of information, where a piston compressing a single gas molecule was considered. The clockwise (backward) and counterclockwise (forward) steps of the microscopic ratchetare enabled through environmental fluctuations. Specifically, a backward step is taken when-ever the pawl acquires sufficient energy from the environment to lift itself over the ratchettooth that it is sitting on, allowing the tooth to slip under it (hence, the name “backward”).Following Feynman’s treatment, we write the rate of such steps as k b = τ − e − βE , (4)where τ − is the attempt frequency, E is the amount of energy needed to lift the pawl overa ratchet tooth, and β = 1 /k B T is the inverse of the thermal energy scale (see SupportingInformation section B.1 for a detailed discussion of the ratchet and pawl mechanism). Everybackward step of the ratchet is accompanied by either a full compression or a full expansionof the piston, as well as the lowering of the weight by an amount of ∆ z , which reduces itspotential energy by ∆ W = mg ∆ z .Unlike in backward stepping, for a forward step to take place, the rotational energyacquired by the ratchet through fluctuations should be sufficient to not only overcome theresistance of the spring pressing the pawl onto the ratchet, but also to lift the weight and toalter the state of the piston. This is a pure consequence of the geometric design of the ratchetand the positioning of the pawl. We assume that piston actions take place isothermally andin a quasistatic way, and therefore, write the changes in ligand free energy upon compression(u → d) and expansion (d → u) as ∆ F u → d = β − ln f, and (5) ∆ F d → u = − β − ln f, (6)7espectively, where f = V u /V d ≥ is the fraction by which the volume occupied by theligand decreases upon compression. The signs of free energy differences suggest that pistoncompressions slow down the forward steps, while expansions speed them up. These featuresare reflected in the two kinds of forward stepping rates that are given by k u → df = τ − e − β ( E +∆ W +∆ F ) = f − k b e − β ∆ W , (7) k d → uf = τ − e − β ( E +∆ W − ∆ F ) = f k b e − β ∆ W , (8)where ∆ F = β − ln f and was used with a “+” and “-” sign in the place of ∆ F u → d and ∆ F d → u ,respectively. The rates of all four kinds of transitions, namely, forward or backward ratchetsteps, accompanied by either a compression or an expansion of the piston are summarizedin Figure 2B.In the presence of a nonzero weight ( ∆ W > ), the ratchet will on average rotate back-wards – a feature reflected in the tilted free energy landscape shown in Figure 2C. As can beseen, the average dissipation per step is ∆ W and it is independent of ∆ F . In addition, thework performed on the ligand upon compression is fully returned upon expansion, which,as we will demonstrate in section 3.4, will generally not be the case when we introduce theenzyme coupling. To further study the nonequilibrium dynamics of the driving mechanism,we map the local minima of the energy landscape corresponding to discrete vertical positionsof the weight (equivalently, discrete ratchet angles) into an infinite chain of transitions shownin Figure 2D. There “d” and “u” stand for the compressed and expanded states of the piston,respectively. The net stepping rate k net at which the weight goes down can be written as k net = (cid:0) k b − k d → uf (cid:1) π d + (cid:0) k b − k u → df (cid:1) π u , (9)where π d and π u are the steady state probabilities of the compressed and expanded pistonstates, respectively. These probabilities can be obtained by considering the equivalent two-state diagram in Figure 2E where the vertical position of the weight has been eliminated,and the nonequilibrium nature of the dynamics is instead captured via the cycle throughtwo alternative pathways connecting the piston states. The driving force ∆ µ in this cycle isgiven by ∆ µ = β − ln (cid:18) k b k d → uf k u → df (cid:19) = 2∆ W, (10)demonstrating the broken detailed balance in the presence of a nonzero weight, and con-firming the dissipation of W per cycle observed in the energy landscape (Figure 2C).We note that this procedure of mapping a linear network onto a cyclic one has also beenused for modeling the processivity of molecular motors, where the linear coordinate corre-sponds to the position of the motor while the alternating states correspond to different motorconformations. At steady state, the net incoming and outgoing fluxes at each piston state in Figure 2Eshould cancel each other (seen more vividly in the collapsed diagram in Figure 2F), namely, (cid:0) k d → uf + k b (cid:1) π d = (cid:0) k u → df + k b (cid:1) π u . (11)8 eightratchet pawl pistonligand (A) (C)(D) duuu ddd d→u k f d→u k fd→u k fd→u k f u→d k fu→d k fu→d k fu→d k f (E)(B) f o r w a r d s t e pb a c k w a r d s t e p compression(u→d) expansion(d→u)ΔF d→u = -β -1 ln(f)ΔF u→d = β -1 ln(f) ΔW f = mgΔzΔW b = -mgΔz k b k b k b d u d→u k fu→d k f k b k b Δμ k b k b k b k net k b k b E +ΔW+ΔF d→u k f u→d k f +ΔW-ΔFE k b k b f r ee e n e r g y z n+4 z n+3 z n+2 z n+1 z n z n-1 z n-2 single cycle e -βE τ1e -βE τ1e -β(E +ΔW+ΔF) τ1 e -β(E +ΔW-ΔF) τ1k b k bu→d k f d→u k f d→u k f +k b u→d k f +k b d u (F) f b Figure 2.
Ratchet and pawl mechanism coupled to the piston. (A) Schematic representation ofthe mechanism. The different radii of the ratchet wheel and the axle of the crankshaft ensure thata single ratchet step translates into a full compression or a full expansion of the piston (i.e. a 180 ◦ rotation of the crankshaft). Arrows with symbols “b” and “f” indicate the directions of backwardand forward ratchet rotation, respectively. (B) Rates of the four kinds of transitions (symbols inshaded boxes with explicit expressions below), along with the accompanying changes in thepotential energy of the weight and the free energy of the ligand. (C) Free energy landscapecorresponding to the non-equilibrium dynamics of the system in the presence of a non-zero weight.Discrete positions of the weight ( z n ) corresponding to energy minima of the landscape are markedon the reaction coordinate. (D) Infinite chain representation of the dynamics of discrete systemstates. k net stands for the net rate at which the weight goes down. (E) Equivalent two-staterepresentation of the engine dynamics where the driving force ∆ µ = 2∆ W breaks the detailedbalance in the diagram. (F) Collapsed representation of the diagram in panel E shown with thenet transition rates from the two pathways. Substituting the expressions for forward stepping rates (eqs 7 and 8) into eq 11 and addi-tionally imposing the probability normalization constraint ( π d + π u = 1 ), we can solve for π d and π u to obtain π d = 1 + e − β (∆ W +∆ F ) β ∆ F ) e − β ∆ W ) , (12)9 u = 1 + e − β (∆ W − ∆ F ) β ∆ F ) e − β ∆ W ) . (13)Notably, in the absence of external drive ( ∆ W = 0 ), piston state occupancies follow theBoltzmann distribution, that is, ( π d /π u ) eq = e − β ∆ F = f − , suggesting that at equilibriumthe piston will predominantly dwell in the expanded state. Conversely, as can be seen inFigure 3A, when the work per step exceeds ∆ F by several k B T , the occupancies of the twopiston states become equal to each other. This happens because at large ∆ W values forwardratchet stepping becomes very unlikely and the dynamics proceeds only through backwardsteps with a rate k b which is identical for both compressive and expansive steps. As will beshown in section 3.4, suppressing this equilibrium bias set by ∆ F is essential for achievingefficient proofreading, analogous to the need for driving the transitions between the twoenzyme-substrate intermediates in Hopfield’s scheme (Figure 1A). ΔW (k B T)0 2 4 6 8 10 12 14 ΔW (k B T)0 2 4 6 8 10 12 14 k n e t / k b π d / π u ΔF = 2 k B TΔF = 4 k B TΔF = 6 k B TΔF = 8 k B T ΔF = 2 k B TΔF = 4 k B TΔF = 6 k B TΔF = 8 k B T (B)(A) ΔW ≈ ΔF ΔW ≈ ΔF-k B Tln2
Figure 3.
Nonequilibrium features of the engine-piston coupling. (A) Steady state probabilityratio of compressed (“d”) and expanded (“u”) piston states and (B) normalized net rate ofbackward stepping ( k net /k b ) as a function of the work per step ( ∆ W ) for different choices of theligand compression energy ( ∆ F ). The ∆ W / expressions stand for the values of ∆ W where thecorresponding value on the y -axis is 0.5 (Supporting Information section B.2). Negative ∆ W values are not considered as they further increase the undesired bias in piston state occupancies. With the steady state probabilities known, we can now substitute them into eq 9 to findthe net rate at which the weight goes down, obtaining k net = (cid:0) − e − β ∆ W (cid:1) k b F ) e − β ∆ W . (14)As expected, k net vanishes at equilibrium ( ∆ W = 0 ), and asymptotes to k b at large ∆ W values, as shown in Figure 3B. The knowledge of k net allows us to calculate the power ( P )dissipated for the maintenance of the nonequilibrium steady state. Specifically, since k net isthe rate at which the weight goes down and ∆ W is the dissipation per step, the power P becomes their product, namely, P = k net ∆ W. (15)The formalism developed in this section for characterizing the steady state behavior ofthe system will be used as a basis for defining the different performance metrics of the modelin section 3.3. 10 .2 Thermodynamic Constraints Make Fidelity Enhancement Unattain-able in the Absence of External Driving In order to implement a thermodynamically consistent coupling between the engine and theallosteric enzyme, we need to consider the full diversity of possible enzyme states, andnot just the dominant ones depicted in Figure 1B. Therefore, in this section we providea comprehensive discussion of the enzyme in an equilibrium setting before introducing itscoupling to the engine.The network diagram of all possible enzyme states is depicted in Figure 4. As can beseen, each of the twelve states are defined by enzyme’s catalytic activity and whether or nota ligand and a right/ wrong substrate are bound to the enzyme. Following the principleof microscopic reversibility, we assign non-zero rate constants to the transitions betweenenzyme states. Only the product formation (with rate r ) is taken to be an irreversiblereaction under the assumption that the system is open where the formed products are takenout and an influx of new substrates is maintained. Since in our model neither the enzymenor the substrates carry an energy source, the choice of the different rate constants cannotbe arbitrary. Specifically, the cycle condition needs to be satisfied for each closed loop ofthe diagram, requiring the product of rate constants in the clockwise direction to equal theproduct in the counterclockwise direction (Supporting Information section C.1). ℓ off A ℓ off A ℓ on [L] I ℓ on [L] I k I L k A L k I R k A R k I R, L k A R, L k I W k A W k I W, L k A W, L k I k A ℓ on [L] A k on [R] A k off R k on [R] A k off R k on [W] A k off W k on [W] A k off W k on [R] I k on [W] I k off R k off W k on [R] I k on [W] I k off R k off W ℓ on [L] A ℓ off I ℓ off I ℓ off A ℓ on [L] I ℓ on [L] A ℓ off I rrrr Figure 4.
Network diagram of enzyme states and transitions between them. Right (“R”) andwrong (“W”) substrates are depicted in blue and red, respectively. The orange circle represents theligand (“L”). Active (“A”) and inactive (“I”) enzymes are shown in green and gray, respectively.
With these equilibrium restrictions imposed on the rate constants, we can show thatwhen the ligand concentration is held fixed ( [ L ]( t ) = const) the fidelity of the enzyme cannotexceed that defined by the ratio of the off-rates, namely, k Woff /k Roff (see Supporting Informationsection C.2). What allows the enzyme to beat this equilibrium limit of fidelity withoutdirect coupling to hydrolysis is the cyclic alteration of the ligand concentration betweenlow and high values (thus, [ L ]( t ) = const). In our model, we achieve this cyclic alterationthrough the ratchet and pawl engine driving the piston actions - a choice motivated byour objective to provide an explicit treatment of energy management. We note, however,that fidelity enhancement can be achieved irrespective of the driving agency as long as thecyclic alteration of ligand concentration is maintained at a certain “resonance” frequency, thepresence of which we demonstrate in section 3.4.11 .3 Coupling the Engine to the Enzyme Gives the Full Descriptionof the Piston Model Having separately introduced the driving mechanism in section 3 and the allosteric enzymewith the full diversity of its states in section 3.2, we now couple the two together to obtainthe full driven version of the piston model, shown in Figure 5A. The coupling is achieved byexposing the ligand binding site of the enzyme to the piston compartment where the activatorligand is present. The enzyme can therefore “sense” the state of the piston (and, thereby,the effects of driving) through the induced periodic changes in the ligand concentration.In the absence of enzyme coupling, the network diagram capturing the nonequilibriumdynamics of the system was an infinite one-dimensional chain (Figure 2D), where each dis-crete state was defined by the vertical position of the weight ( z n ) and the state of the piston(“u” or “d”). In the layout where the engine and the enzyme are coupled, the full specificationof the system state now requires three items: the position of the weight ( z n ), the piston state(“u” or “d”), and the state of the enzyme (one of the 12 possibilities). By converting thethree-dimensional view of the enzyme state network (Figure 4) into its planar equivalent,we represent the nonequilibrium dynamics of this coupled layout again through an infinitechain, but this time each slice at a fixed weight position ( z n ) corresponding to the planarview of the enzyme state network (Figure 5B). The slices alternate between the compressedand expanded piston states (dark and light blue circles, respectively), with high and lowligand concentrations used in the transition network inside each slice.Arrows between the slices (not all of them shown for clarity) represent the forward andbackward steps of the ratchet. Crucially, as a consequence of coupling, the rates of forwardstepping now depend on the state of the enzyme. In particular, when the ligand is boundto the enzyme, it no longer exerts pressure on the piston and therefore, in those cases, theforward stepping rates become simply k u → d , Lf = k d → u , Lf = k b e − β ∆ W , (16)where the superscript “L” indicates that the ligand is bound (orange circles in Figure 5B). Wenote that in the general case with N ligands, the pressure would drop down to that of ( N − ligands upon ligand binding, correspondingly altering the rates of forward stepping (seeSupporting Information section D.1 for details). This adjustment of forward rates is essentialfor the thermodynamic consistency of coupling the engine to the enzyme. Specifically, itensures that any cycle of transitions that brings the enzyme and the weight back into theiroriginal states is not accompanied by dissipation, consistent with the fact that in the pistonmodel energy is spent only when there is a net lowering of the weight. As a demonstrationof this feature, consider the cycle in Figure 5C which is extracted from the larger network.Using the expression of forward stepping rates in eqs 7 and 16, we can write the cyclecondition for this sub-network as k u → d , Lf × ‘ Aoff × k b × ‘ Aon [ L ] u k b × ‘ Aoff × k u → df × ‘ Aon [ L ] d = [ L ] u f [ L ] d = 1 , (17)where the equality f = V u /V d = [ L ] d / [ L ] u was used. The fact that the products of rateconstants in clockwise and counterclockwise directions are identical shows that no dissipation12ccurs when traversing the cycle.Now, to study how driving affects the proofreading performance of the piston model, weneed to obtain the steady state probabilities of the different enzyme states. To that end, weconvert the full network diagram into an equivalent form shown in Figure 5D, where we haveeliminated the position of the weight ( z n ), akin to the earlier treatment of the uncoupledengine in Figure 2E. Note that the transitions between the two slices again represent pistoncompression and expansion events driven by a force ∆ µ = 2∆ W , as in eq 10. The steadystate probabilities π i of the 24 different states in Figure 5D (12 enzyme states × P ), speed of forming right products ( v R ), and fidelity ( η ) as P = X i =1 (cid:16) k b − k ( i )f (cid:17) π i | {z } k net × ∆ W, (18) v R = X i ∈ S AR π i × r, (19) η = v R v W = P i ∈ S AR π i P i ∈ S AW π i , (20)where k ( i )f is the rate constant of making a forward step from the i th state ( ≤ i ≤ ),while S AR and S AW are the sets of catalytically active enzyme states with a right and wrongsubstrate bound, respectively.One significant downside of using these “raw” metrics in the numerical evaluation of themodel performance is their high sensitivity to the particular choices of parameter values. Wetherefore introduce their scaled alternatives which we will use for the numerical studies insections 3.4 and 3.5. Specifically, as a measure of energetic efficiency, we use the dissipationper right product formed, defined as ε = Pv R . (21)This way, the metric of energetics has units of k B T and is independent from the choice ofabsolute timescale. Then, as a dimensionless metric of speed, we introduce the normalizedquantity ν = v R v MMR , (22)which represents the fraction by which the rate of forming right products in the proofreadingsetting ( v R ) is slower than that in the simple Michaelis-Menten scheme ( v MMR ) where theallosteric effects are absent. This normalizing Michaelis-Menten speed is given in terms of13he model parameters via v MMR = k Ion [ R ] k Roff + r k Ion [ R ] k Roff + r + k Ion [ W ] k Woff + r × r. (23)Next, we define the proofreading index α as a fidelity metric which represents the degree towhich the fidelity is amplified in multiples of k Woff /k Roff over its Michaelis-Menten value ( η MM ),that is, η = (cid:18) k Woff + rk Roff + r (cid:19)| {z } η MM (cid:18) k Woff k Roff (cid:19) α , (24) α = ln η − ln η MM ln (cid:16) k Woff k Roff (cid:17) . (25)Note that the proofreading index of Hopfield’s scheme is α Hopfield = 1 , as it involves asingle proofreading realization. Also, since in the absence of external driving the highestfidelity is η maxeq = k Woff /k Roff , the corresponding upper limit in the proofreading index becomes α eq = 1 − ln η MM / ln η maxeq .As a final descriptor of piston model’s nonequilibrium behavior, we introduce the fractionof returned work ( κ ) defined as the ratio of the rate at which the ligand performs work onthe piston upon expansion to the rate at which the piston performs work on the ligand uponcompression. We calculate κ via κ = − P i ∈ S d (cid:16) k b + k ( i )f (cid:17) π i ∆ F ( i ) d → u P i ∈ S u (cid:16) k b + k ( i )f (cid:17) π i ∆ F ( i ) u → d , (26)where S d and S u are the sets of states where the piston is compressed and expanded, respec-tively. The negative sign is introduced to account for the fact that the ligand free energydecreases upon piston expansion (i.e. the system gets the work back). In the absence ofenzyme coupling (section 3), this ratio was 1 because the ligand constantly exerted pressureon the piston. With enzyme coupling, however, the work performed on the ligand uponcompression may not be fully returned since with some probability the ligand will be boundto the enzyme and exert no pressure on the piston during expansion. We therefore expect κ to be generally less than 1, indicating a net rate of performing work on the ligand in thenonequilibrium setting.Having defined analytical expressions for the key model performance metrics, we nowproceed to studying their graded changes and the trade-offs between them numerically.14 eightratchet pawl u→d,L k fd→u,L k f k b k b Δμ (B) (C)(D) k fd→u, L k fd→u, L k fu→d, L k fu→d, L k fd→u, L k b k b k b k b k b k fd→u k fd→u k fu→d k fu→d k fd→u k b k b k b k b k b piston allostericenzymeregulatoryligand substrates ℓ off A ℓ on [L] u A ℓ off A k b k b ℓ on [L] d A k fu→d, L k fu→d zz n+2 z n+1 z n z n-1 k net u→d k fd→u k f k b k b Δμ u→d k fd→u k f k b k b Δμ u→d,L k fd→u,L k f k b k b Δμ (A) ~ [ L ] u ~ [ L ] u ~ [ L ] d ~ [ L ] d ~ [ L ] u ~ [ L ] d Figure 5.
Full and thermodynamically consistent treatment of the piston model of proofreading.(A) Schematic representation of the full model, with the ratchet and pawl engine coupled to theenzyme. (B) Network diagram of the full model. Each slice of the diagram represents the planarview of the enzyme state network, with the alternating colors corresponding to the compressed(dark blue) and expanded (light blue) states of the piston. Ligand-bound enzyme states aremarked with an orange circle. The horizontal arrows connecting the slices stand for forward andbackward ratchet steps. Only those at the outer edges are shown for clarity; however, transitionsare present between all horizontally neighboring enzyme states. Also for clarity, the stepping rateconstants are shown only at two of the outer edges where the ligand is either unbound (bottomedge) or bound to the enzyme (top edge). The hanging weight at different vertical positions isdisplayed below the diagram to symbolize energy expenditure as it gets lowered with a net rate k net . (C) Sub-network of the full diagram in panel B where the state of the system is unchangedafter doing a cyclic traversal. The red arrow with a cross on top indicates that the cycle conditionholds in the sub-network. (D) The finite-state equivalent of the full network in panel B with theweight position ( z n ) eliminated. Red arrows indicate the driving with a force ∆ µ = 2∆ W . .4 Energy-Speed-Fidelity Trade-Off in the Piston Model Because of its mechanical construction, the piston model of proofreading has a distinguishingfeature - in it the external driving mechanism is physically separated from the allostericenzyme. This feature allows us to independently examine how tuning the “knobs” of theengine and varying the kinetic parameters of the enzyme alter the performance of the model. energy per right product, ε (k B T) -2 -2 -3 -4 -5 -4 -6 -2 -4 -6 -2 -4 -6 -8 ΔW (k B T) ΔW (k B T) k b / k o ff k b k b R proofreading index, α ΔW (k B T) Δ W = Δ F Δ W = Δ F p r oo f r e a d i n g i n d e x , α B T) ΔW204681012140.20.4 0.0120.0080.004 (A) k b / k o ff R normalized speed, ν (B)(D) [ L ] d / R d I proofreading index, αf (C) p r oo f r e a d i n g i n d e x , α (E) (F) e q u i l i b r i u m l i m i t H o p f i e l d l i m i t p r oo f r e a d i n g i n d e x , α fraction of work returned, κspeed, ν R k b /k off -6 -4 -2 Figure 6.
Parametric studies on the changes in the piston model performance in response totuning the “knobs” of the engine. (A,B) Variations in the proofreading index (A) and speed (B) asthe rate of backward stepping ( k b ) and the work per step ( ∆ W ) are tuned. The dotted linecorresponds to the value of ∆ W equal to the ligand free energy change upon compression ( ∆ F ).(C) Variations in the proofreading index when the high ligand concentration ( [ L ] d ) and thecompression factor ( f ) are tuned. R Id represents the ligand dissociation constant in the inactiveenzyme state. The red dot indicates the pair of [ L ] d and f values used in the studies of the otherpanels. (D) Fidelity-speed trade-off as k b is continuously varied for different choices of ∆ W (gradient arrow shows the direction of increase). The dotted black lines connect the highestfidelity and speed values as ∆ W is tuned. Between these dotted lines fidelity and speed arenegatively correlated. (E) Relation between fidelity and fraction of returned work for discretechoices of ∆ W and continuously tuned k b values (the gradient arrow indicates increasing ∆ W ).(F) Fidelity-dissipation trade-off obtained by continuously tuning ∆ W for discrete choices of thehopping rate ( k b ). The gradient arrow indicates the direction of increasing k b . The red dottedcurve corresponds to the case with resonance k b . We begin our numerical analysis by first exploring the effects of external driving, wherethe tuning “knobs” include the rate of backward stepping ( k b ), the work per step ( ∆ W ),the ligand concentration in the compressed piston state ( [ L ] d ) and the compression factor( f = [ L ] d / [ L ] u ). Choosing a set of enzyme’s kinetic parameters which make proofreading16ossible (see Supporting Information section D.3 for the full list of parameters), we keepthem fixed for the rest of the analysis. We conduct the first parametric study by tuning k b and ∆ W and evaluating the proofreading index (Figures 6A). As anticipated, the proofread-ing index does not exceed its equilibrium limit in the absence of driving ( ∆ W = 0 ). Thisexpected feature can be paralleled by Brownian motors where purely equilibrium fluctuationsof the motor’s energy landscape are unable to generate directed motion. In addition, theproofreading index achieves its highest value if ∆ W is comparable to or larger than the lig-and compression energy ∆ F , and if the backward hopping rate k b is at its “resonance” value.The presence of a “resonance” hopping rate is intuitive since if piston actions take place veryslowly then the fidelity will be reduced due to the small but nonzero rate of forming unfil-tered products (i.e. “leakiness”) in the quasi-equilibrated enzyme states. And, conversely, ifpiston actions take place too rapidly, then the activator ligand will almost always be boundto the enzyme, preventing the realization of multiple substrate discrimination layers throughsequential enzyme activation and inactivation. We note that analogous resonance responseswere also identified for Brownian particles which attain their highest nonequilibrium driftvelocity in a ratchet-like potential landscape when the temperature or the landscape pro-file are temporally modulated at specific resonance frequencies. A similar feature is presentin Hopfield’s model as well; namely, optimal proofreading is attained only when the rate ofhydrolysis is neither too low, nor too high. Interestingly, when the driving is hard enough( ∆ W & ∆ F ) and the backward hopping rate is close to its resonance value, the fidelity ofthe piston model beats the Hopfield limit ( α = 1 ) and raises the question of the largestattainable fidelity, which we discuss in the next section.Trends similar to those for the proofreading index are observed for the speed of formingright products as well (Figure 6B). Specifically, product formation is very slow in the absenceof driving and increases monotonically with ∆ W , until plateauing when ∆ W & ∆ F . Also,the highest speed is achieved at a resonance k b value which is different from that of theproofreading index. The existence of such a resonance frequency is again intuitive, sinceat fast rates of piston actions the enzyme is predominantly active and unable to bind newsubstrates, while at slow rates the enzyme activation for catalysis via piston compressionhappens very rarely. Notably, since the enzyme parameters were chosen in a way so as toyield high fidelities, the largest speed value is substantially lower than the correspondingspeed for a single-step Michaelis-Menten enzyme ( ν max ≈ − ).In the last parametric study, we explore how the choice of the high and low ligand con-centrations affects the performance of the model. To that end, we tune the high ligandconcentration ( [ L ] d ) and the compression factor ( f = [ L ] d / [ L ] u ), and evaluate the highestproofreading index at the resonant k b value with ∆ W > ∆ F . As we can see, large fidelityenhancements are achieved when [ L ] d is comparable to or larger than the ligand dissocia-tion constant in the inactive enzyme state ( R Id ), which is necessary to activate the enzymeupon piston compression. In addition, the compression factor needs to be large enough (or,equivalently, the ligand concentration in the expanded piston state should be low enough)so as to inactivate the enzyme when piston enters its expanded state. This requirement ofhaving a large free energy difference between the compressed and expanded piston states( β ∆ F = ln( f ) (cid:29) ) can be paralleled with a similar condition in Hopfield’s model wherefor optimal proofreading the energy of the activated enzyme-substrate complex needs to bemuch larger than that of the inactive complex.17nowing separately how tuning the engine “knobs” affects the fidelity and speed, wenow explore the trade-offs between the model’s performance metrics as we vary the drivingparameters k b and ∆ W , while holding the high and low ligand concentrations at fixed values(the red dot in Figure 6C). We start with the trade-off between fidelity and speed, depictedin Figure 6D, where we continuously tune the hopping rate k b for different choices of thedriving force ∆ W . As expected from the results of the individual parametric studies inFigure 6A and B, both fidelity and speed increase monotonically with ∆ W . Also, since thevalues of the hopping rate k b that maximize fidelity and speed are not identical, these twoperformance metrics are negatively correlated in the range of k b values defined by the twodifferent resonance rates (region between the dotted lines in Figure 6D), but are positivelycorrelated otherwise. Variations in the metrics in the region of their negative correlation,however, are moderate, suggesting that for an allosteric enzyme that has been optimizedfor doing proofreading, the largest speed and fidelity could be achieved at similar externaldriving conditions.Next, we consider the relation between fidelity and fraction of work returned, shownin Figure 6E. As can be seen, no fidelity enhancement is achieved when κ is close to which happens either in the absence of driving (lighter curves) or in the presence of driving,provided that the hopping rate is very fast. On the other hand, κ is much less than atthe peak fidelity which is achieved when the hopping rate is at its resonance value and whendriving is large ( ∆ W & ∆ F ). Overall, this trade-off study demonstrates that irreversiblework performed on the ligand is a required feature for the attainment of fidelity enhancementin the piston model.Lastly, we look at how fidelity varies with energy dissipation, with the latter characterizedthrough the energy expended per right product ( ε ). The results of the trade-off study areshown in Figure 6F. There, the driving force is continuously tuned for different choices of thehopping rate. As can be seen, there is a minimum dissipation per product required to attainthe given level of performance. This minimum dissipation (the first intercept at a giveny-level) is achieved when the hopping rates are less than the corresponding resonant values(the lighter curves on the left side of the dotted red curve). Additionally, for a given hoppingrate, increasing the driving force ( ∆ W ) could lead to an increased proofreading performanceand a decreased dissipation per product up a critical point where the performance metricreaches its saturating value (horizontal region), demonstrating how increasing the drivingforce could in fact improve the energetic efficiency of proofreading. We note here that theminimum ε values needed for significant proofreading are ∼ − k B T in Figure 6F whichis ∼ orders of magnitude higher than what is calculated for translation by the ribosome. This low energetic efficiency can be a consequence of our particular parameter choice for thestudy as well as the performance limitations of our engine design, the investigation of whichwe leave to future work. 18 .5 Up to Three Proofreading Realizations are Available to the Pis-ton Model
In the previous section, we chose a set of kinetic rate constants for the enzyme and, keepingthem fixed, explored the effects of tuning the external driving conditions on the performanceof the model. In this section, we explore the parameter space from a different angle, namely,we study how tuning the enzyme’s kinetic parameters changes the model performance underoptimal driving conditions. Since there are more than a dozen rates defining the kineticbehavior of the enzyme, it is impractical to probe their individual effects. Instead, wechoose to vary two representative parameters about the effects of which we have a prejudice.These include the rate of substrate binding to the active enzyme ( k Aon ) and the unbindingrate of wrong substrates ( k Woff ). We know already from Hopfield’s analysis that for efficientproofreading the direct binding of substrates to the active enzyme state should be very slow.Therefore, we expect the proofreading performance to improve as k Aon is reduced. We alsoexpect the minimum requirement for k Aon to be lower for larger k Woff values to ensure thatwrong substrates do not enter through the unfiltered pathway. With these expectations in mind, we performed a parametric study to find the highestfidelity, the results of which are summarized in Figure 7A. There we varied k Aon for severalchoices of k Woff , and for each pair numerically optimized over the enzyme’s remaining kineticrates and external driving conditions to get maximum fidelity (see Supporting Informationsection D.4 for implementation details). As expected, the highest attainable fidelity decreasesmonotonically with increasing “leakiness” ( k Aon /k Ion ), and the minimum requirement on k Aon decreases with increasing k Woff .Interestingly, we also see that for small enough leakiness, the piston model manages toperform proofreading (i.e. enhance the fidelity by a factor of k Woff /k Roff ) up to three times, as α max ≈ (Figure 7A). To understand this unexpected feature, we identified the dominanttrajectory that the system would take to form a wrong product for the case where k Aon /k Ion =10 − (Figure 7B, see Supporting Information section D.5 for details). As we can see, afterinitial binding the wrong substrate indeed passes through three different proofreading filters,and these are realized efficiently because the transitions between intermediate states aremuch slower than the rate of substrate unbinding. The first filter occurs right after pistoncompression, while the enzyme is waiting for the activator ligand to bind ( α max ≈ result in Figure 7 represents the theoretical upper limit of the model’s proofreadingindex – a feature that we justify analytically in Supporting Information section D.5.In light of this analysis, we can now explain why the pedagogically simplified versionof the model introduced in section 2 achieved only a single proofreading realization. Therewe made the implicit assumption that ligand binding after piston compression and enzymeactivation after ligand binding took place instantly. Because of this, proofreading filters eakiness, k on /k onA I (A) p r oo f r e a d i n g i n d e x , α off /k off =10 W R k off /k off =100 W R k off /k off =1000 W R -12 -8 -4 (B) k on [W] I k off W substratebinding pistoncompression ligandbinding enzymeactivation k I W, L k A W, L k off W off W off W r b k b ℓ off I ℓ on [L] d I Figure 7.
Proofreading performance of the piston model under optimized enzyme parametersand external driving conditions. (A) The highest proofreading index ( α ) available to the pistonmodel as a function of leakiness ( k Aon /k Ion ) for different choices of k Woff . (B) The dominanttrajectory that the system takes to form a wrong product in the case where k Aon /k Ion = 10 − .Numbers 1, 2, 3 stand for the different proofreading filters along the trajectory. The dotted arrowsindicate that the respective rates are much slower than the substrate unbinding rate k Woff (seeSupporting Information section D.5 for their numerical values for the k Woff /k Roff = 100 case).
A distinctive feature of kinetic proofreading is that it is a nonequilibrium mechanism, byvirtue of which its operation needs to involve energy expenditure.
Mechanical work, be-ing an intuitive representation of energy expenditure, has been used in the past to elucidateimportant physical concepts such as information-to-energy conversion in the thought ex-periment by Szilard, or the mechanical equivalence of heat in Joule’s apparatus. Yet, asimilar demonstration of how mechanical work could be harnessed in a graded fashion tobeat the equilibrium limit in substrate discrimination fidelity has been lacking. Our aim inthis work was to offer such a demonstration through the mechanically designed piston modelof proofreading.We started off by providing the conceptual picture of the piston model, with its con-stituents having direct parallels with Hopfield’s original proofreading scheme (Figure 1).The key idea of the model was to replace the nucleotide hydrolysis step present in Hopfield’sscheme with piston compression which served an identical role of activating the enzyme butin our case achieved through allostery and mechanical work. Just like in the case of bio-logical proofreading, where hydrolysis itself cannot lead to fidelity enhancement unless thenucleotide triphosphates are held at fixed out-of-equilibrium chemical potentials, in the caseof piston model too, the compressive and expansive actions of the piston cannot result inproofreading unless they are driven by an energy-consuming engine. Motivated by Feyn-man’s ratchet and pawl mechanism, we then proposed a dissipative mechanical engine todrive the cyclic piston actions, which maintained the nonequilibrium distribution of enzymestates necessary for achieving proofreading. The function of this engine can be paralleled tothat of the ATP synthase in the cell whose constant operation maintains a finite chemicalpotential of ATP, which different biochemical pathways can then take advantage of.To study how the cyclic variations in ligand concentration generated by the engine alterthe occupancies of enzyme states, we performed a thermodynamically consistent coupling20etween the engine and the enzyme (Figure 5). There we considered the full diversity ofstates that the enzyme could take and, importantly, the feedback mechanism for the engineto “sense” the state of the enzyme. The accounting of this latter feature, which makes thepiston model an example of a bi-partite system, was motivated by our aim of proposinga framework where we could consistently calculate the total dissipation as opposed to onlythe minimum dissipation needed for maintaining the nonequilibrium steady state of theenzyme (without considering the driving engine). Although the dissection of differentcontributions to dissipation and their interconnectedness was not among the objectives ofour work, the framework that we proposed in our model can serve as a basis for additionalstudies of periodically driven molecular systems (e.g. Brownian clocks or artificial molecularmotors) where the driving protocol and thermodynamics are of importance.
As notedearlier, however, in the presence of a periodically changing ligand concentration, the allostericenzyme could perform proofreading irrespective of the driving agency, which suggests apossible biochemical mechanism of fidelity enhancement without the direct coupling of theenzyme state transitions to hydrolysis.Having explicit control over the “knobs” of the mechanical engine, we then probed theperformance of the model under different driving conditions. We found that both speed andfidelity increased as we tuned up the mass of the hanging weight, until plateauing at a pointwhere the free energy bias of the expanded piston state was fully overcome ( ∆ W & ∆ F ),beyond which increasing the weight only increased the dissipation without improving themodel performance (Figure 6A,B). This result can be paralleled with the presence of aminimum threshold for the strength of driving in Hopfield’s model, past which the highestfidelity becomes attainable. In addition, we found that in the piston model there is a“resonance” rate of piston actions which maximizes fidelity, analogous to the similar featureof Hopfield’s scheme where both very fast and very slow rates of hydrolysis reduce the qualityof proofreading. The tunable control over the driving parameters also allowed us to study the trade-offbetween fidelity, speed and energy spent per right product. These studies revealed that thecorrelation between speed and fidelity could be both positive and negative when varyingthe rate of driving. Notably, theoretical investigations of translation by the
Escherichia coli ribosome under Hopfield’s scheme identified a similar behavior for the fidelity-speed correla-tion in response to tuning the GTP hydrolysis rate, with the experimentally measured valuesbeing in the negative correlation (i.e. trade-off) region. In contrast to the ribosome study,however, where the two metrics vary by several orders of magnitude in the trade-off region,in the piston model the variations in fidelity and speed in the negative correlation region aremoderate (Figure 6D), calling for additional investigations of the underlying reasons behindthis difference and search for the realization of the latter advantageous behavior in biolog-ical proofreading systems. Furthermore, our studies showed that the minimum dissipationrequired to reach the given level of fidelity was achieved for hopping rates necessarily lowerthan their resonance values, and that increasing the work performed per step (analogously,the chemical potential of ATP) could actually improve the energetic efficiency of the model– features that again motivate the identification of their realization in biochemical systems.In the end, we explored the limits in the proofreading performance of the piston modelfor various choices of the allosteric enzyme’s “leakiness” ( k Aon /k Ion ) and the ratio of the wrongand right substrate off-rates ( k Woff /k Roff ). We found that the trends for the highest available21delity matched analogously with the features of Hopfield’s original scheme, suggesting theirpossible ubiquity for general proofreading networks. More importantly, our analysis revealedthat the piston model could do proofreading not just once but up to three times in thelimit of very low leakiness, despite the fact that energy consumption takes place during asingle piston compression. This is in contrast to the typical involvement of several energyconsumption instances in multistep proofreading schemes which manage to beat the Hopfieldlimit of fidelity, as, for example, in the cases of the T-cell or MAPK activation pathwayswhich require multiple phosphorylation reactions.
Our finding therefore suggests thepossibility of achieving several proofreading realizations with a single energy consumingstep by leveraging the presence of multiple inactive intermediates intrinsically available toallosteric molecules. We would like to mention here that the presence of a similar featurewas also experimentally demonstrated recently for the ribosome which was shown to use thefree energy of a single GTP hydrolysis to perform proofreading twice after the initial tRNAselection – first, at the EF-Tu · GDP-bound inactive state and second, at the EF-Tu-freeactive state. In the presentation of the piston model we focused on the thermodynamic consistencyof the framework for managing the energy dissipation and did not consider strategies forimproving the performance of the mechanism. One such possibility that can be consideredin future work is to use a more elaborate design for the ratchet and pawl engine withalternating activation barriers for pawl hopping which would allow to have different rates ofpiston compression and expansion, analogous to how hydrolysis and condensation reactionsgenerally occur with different rates in biological proofreading.
Another avenue is toconsider alternative ways of allocating the mechanical energy dissipation across the differentratchet transition steps, similar to how optimization schemes of allocating the free energy ofATP hydrolysis were studied for molecular machine cycles. Incorporating these additionalfeatures would allow us to probe the performance limits of the piston model and comparethem with the fundamental limits set by thermodynamics. Acknowledgements
We thank Tal Einav, Erwin Frey, Christina Hueschen, Sarah Marzen, Arvind Murugan,Manuel Razo-Mejia, Matt Thomson, Yuhai Tu, Jin Wang, Jerry Wang, Ned Wingreen andFangzhou Xiao for fruitful discussions. We also thank Haojie Li and Dennis Yatunin fortheir input on this work, Alexander Grosberg, David Sivak, and Pablo Sartori for providingvaluable feedback on the manuscript, and Nigel Orme for his assistance in making the illus-trations. This work was supported by the National Institutes of Health through the grant1R35 GM118043-01 (MIRA), and the John Templeton Foundation as part of the Boundariesof Life Initiative grants 51250 and 60973. 22 eferences (1) Kunkel, T. A. DNA Replication Fidelity.
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Proc. Natl. Acad. Sci. U.S.A. , , 11057–11062.(42) Qian, H. Reducing Intrinsic Biochemical Noise in Cells and Its Thermodynamic Limit. J. Mol. Biol. , , 387–392. 25 llostery and Kinetic ProofreadingSupporting Information Vahe Galstyan and Rob Phillips ∗ E-mail: [email protected]
Contents
A Discrimination Fidelity in the Conceptual Scheme of the Piston Model S2B Ratchet and Pawl Engine S4
B.1 Details of the Ratchet and Pawl Mechanism in the Absence of Piston Coupling S4B.2 Derivation of ∆ W / Expressions . . . . . . . . . . . . . . . . . . . . . . . . S6
C Equilibrium Properties of the Allosteric Enzyme S8
C.1 Constraints on the Choice of Enzyme’s Rate Constants . . . . . . . . . . . . S8C.2 Enzyme Fidelity at a Fixed Ligand Concentration . . . . . . . . . . . . . . . S10
D Full Description of the Piston Model with Engine-Enzyme Coupling S13
D.1 Equilibrium Fidelity of the Piston Model in the Absence of External Driving S13D.2 Obtaining the Steady-State Occupancy Probabilities . . . . . . . . . . . . . S16D.3 Enzyme’s Kinetic Parameters Used for the Numerical Study in Section 3.4 . S18D.4 Details of the Numerical Optimization Procedure for Finding the HighestFidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S19D.5 Investigation of the α max ≈ Result for the Highest Available ProofreadingIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S20
References S24 S1 a r X i v : . [ q - b i o . M N ] M a y Discrimination Fidelity in the Conceptual Scheme ofthe Piston Model
In this section, we derive the expressions for the fidelities achieved at the two piston stepsintroduced in section 2 of the main text, namely, η = k Woff /k Roff when the piston is expanded,and η = ( k Woff + r ) / ( k Roff + r ) when the piston is compressed. In our discussion, we retain thesimplifying assumptions made during the presentation of the model concept.As discussed in section 2, the first level of substrate discrimination occurs in the ex-panded piston state. If the waiting time for compression is long enough for the substrates toequilibrate with the inactive enzyme, we can impose the detailed balance condition at thetwo pairs of edges in Figure S1A to obtain p RI k Roff = p I k on [ R ] , (S1) p WI k Woff = p I k on [ W ] . (S2)Here p I , p RI and p WI stand for the probabilities of the empty, right substrate-bound andwrong substrate-bound inactive states of the enzyme, respectively. Taking the substrateconcentrations to be identical ( [ R ] = [ W ] ), we can equate the left sides of eqs S1 and S2 tofind p RI p WI = k Woff k Roff . (S3)The above ratio of probabilities represents the proportion in which right and wrong substrate-bound inactive enzymes enter the active state, and therefore, becomes equivalent to thefidelity η achieved in the first discrimination step. k on [W]k on [R] k off rfirst level of discrimination second level of discrimination R k off W k off (A) (B) Figure S1.
The two substrate discrimination levels in the conceptual scheme of the piston model.(A) The first level is achieved when the piston is expanded and a roughly equilibrium distributionof substrate-bound and free enzyme states is established. (B) The second level is achieved in thecompressed state of the piston where the enzyme is active and can either release the boundsubstrate or turn it into a product.
The second level of substrate differentiation takes place when the piston gets compressed,leading to the activation of the enzyme. We assume that in its active state the enzyme canno longer bind new substrates. If we wait long enough, a substrate that was already boundbefore piston compression will either unbind with a rate k off or get turned into a productS2ith a rate r . The probability that a product is formed can be written as p prod = Z ∞ d t p bound ( t ) × r, (S4)where p bound ( t ) is the probability that the substrate is still bound by time t . Using the factthat the waiting time distribution of substrate release (either through unbinding or productformation) is P release ( t ) = ( k off + r ) e − ( k off + r ) t , the probability p bound ( t ) can be found as p bound ( t ) = Z ∞ t d t P release ( t ) = e − ( k off + r ) t . (S5)Substituting this result into eq S4 and performing the integration, we obtain p prod = rk off + r . (S6)Due to the difference in the off-rates of the right and wrong substrates, their respectiveprobabilities of production will also be different, resulting in the second level of fidelity givenby the ratio of these probabilities, namely, η = p Rprod p Wprod = k Woff + rk Roff + r . (S7)S3 Ratchet and Pawl Engine
In this section, we first provide a detailed discussion of the ratchet and pawl mechanismin the absence of piston coupling. Then, for the case of piston coupling, we derive of theexpressions for the work per step ( ∆ W / ) shown in Figure 3 at which the ratio of pistonstate probabilities ( π d /π u ) and the net rate of backward stepping ( k net ) reach 50% of theirrespective saturation values. B.1 Details of the Ratchet and Pawl Mechanism in the Absence ofPiston Coupling
The ratchet and pawl mechanism was originally proposed by Richard Feynman with anaim to demonstrate the validity of the second law of thermodynamics. In his description,the mechanism had an additional element, namely, vanes that were connected to the ratchetthrough a massless axle (Figure S2A). The purpose of the vanes was to induce forward ratchetsteps through thermal fluctuations. When the temperature in the vane compartment wasmaintained at a higher value than that in the ratchet compartment ( T > T ), the mechanismcould utilize this difference to operate as a heat engine and lift a weight hanging from theaxle.In the piston model, instead of running the ratchet and pawl mechanism as a heat engine,we drive it at a constant temperature through the expenditure of the gravitational potentialenergy of the hanging weight. We have therefore removed the vane compartment from ourdescription of the engine and ascribed forward stepping to random rotational fluctuations ofthe ratchet instead (Figure S2B).As mentioned in section 3.1, backward stepping takes place whenever the pawl borrowssufficient energy from the environment to overcome the potential energy barrier E of thespring and lift itself over the ratchet tooth that it is sitting on, allowing the tooth to slipunder it (Figure S2C). Once the pawl gets over the ratchet tooth (step 2 in Figure S2C), thehanging weight and the recovering pawl start applying torque on the ratchet, causing it torotate in the clockwise direction (step 3 in Figure S2C). Following Feynman’s treatment, weassume that when the pawl hits the bottom of the next tooth (step 4 in Figure S2C), thetotal kinetic energy of the system, which is the sum of the energy borrowed by the pawl andthe change in the potential energy of the weight per step ( ∆ W = mg ∆ z ), gets dissipateddue to the perfectly inelastic collision of the pawl with the ratchet. Therefore, as a resultof a single backward step, the net heat dissipated into the environment becomes ∆ W , asreflected in the free energy landscape in Figure S2E.A similar set of arguments for forward stepping would imply that initially the mechanismneeds to borrow enough energy from the environment to overcome the spring barrier andto lift the weight by an amount of ∆ z (step 3 in Figure S2D). We again assume, that oncethe pawl passes over the next tooth and inelastically hits the ratchet, it dissipates all itsaccumulated potential energy. Therefore, in the end of a single forward step, the totalenergy extracted from the environment is equal to the increase in the potential energy of theweight per forward step (Figure S2E).Implicit in our treatment of ratchet stepping has been the assumption that we coulddiscretize the possible configurations of the mechanism into states where the pawl fully restsS4 C)(D) E s = 0E s = E s Working details of the ratchet and pawl mechanism. (A) Feynman’s original ratchetand pawl mechanism operating as a heat engine. (B) Ratchet and pawl engine driven by ahanging weight that is used in the piston model. Arrows with symbols “b” and “f” indicate thedirections of backward and forward ratchet rotation, respectively. (C)-(D) Breakdown of backward(C) and forward (D) steps of the ratchet, accompanied by the lowering or the lifting of the weight,respectively. E s stands for the potential energy of the spring. (E) Free energy landscapecorresponding to the directionally biased rotations of the ratchet due to a net lowering of theweight. Discrete positions of the weight ( z n ) corresponding to the energy minima of the landscapeare marked on the reaction coordinate. (F) Infinite chain representation of the discrete statedynamics. When a non-zero weight is hung from the axle, the ratchet makes backward steps witha net rate k net . on ratchet teeth. Within this formalism, we took E to be the activation energy of backwardstepping and ( E + ∆ W ) to be the activation energy of forward stepping, resulting in rateconstants given by k b = τ − e − βE , (S8) k f = τ − e − β ( E +∆ W ) , (S9)where τ − is the attempt frequency. The choice of identical attempt frequencies for forwardand backward steps is, in a way, a requirement in the discretization formalism to ensureS5hat in the absence of driving ( ∆ W = 0 ) no net rotation of the ratchet is generated, since k net = k b − k f (Figure S2F). We note that a more rigorous treatment of ratchet steppingkinetics would need to account for the precise shape of the energy landscape, defined bothby the position of the weight (equivalently, the ratchet angle) and the angular position ofthe fluctuating pawl, similar to the analysis done by Magnasco and Stolovitzky. B.2 Derivation of ∆ W / Expressions We begin by deriving the ∆ W / expression at which the ratio π d /π u is / . From eqs 12and 13, this ratio, evaluated at ∆ W / , can be written as π d π u (cid:12)(cid:12)(cid:12)(cid:12) ∆ W / = 1 + e − β (∆ W / +∆ F ) e − β (∆ W / − ∆ F ) = 12 . (S10)Solving for ∆ W / , we obtain ∆ W / = ∆ F + ln (cid:0) e − β ∆ F (cid:1) = ∆ F + ln (cid:0) f − (cid:1) , (S11)where in the last step we used the expression for the ligand free energy change written interms of the compression factor, that is, ∆ F = β − ln f . Since for efficient proofreading thecompression factor needs to be large ( f (cid:29) ), the ∆ W / expression reduces into ∆ W / ≈ ∆ F. (S12)To estimate how much the work per step needs to exceed ∆ F in order for the ratio π d /π u to reach its saturation value of , we calculate the derivative of the ratio at ∆ W / ≈ ∆ F ,namely, ∂∂ ∆ W (cid:18) π d π u (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∆ W / = − β e − β (∆ W / +∆ F ) (cid:0) e − β (∆ W / − ∆ F ) (cid:1) + (cid:0) e − β (∆ W / +∆ F ) (cid:1) β e − β (∆ W / − ∆ F ) (1 + e − β (∆ W / − ∆ F ) ) = β (1 − e − β ∆ F )4 ≈ k B T , (S13)where we again employed the e − β ∆ F (cid:28) approximation. These results indicate that inorder to overcome the equilibrium bias in piston state probabilities caused by the higherligand entropy in the expanded state, the work per step needs to exceed the ligand freeenergy change upon compression ( ∆ F ) by several k B T values.Now, we perform a similar set of calculations for the net rate of backward stepping ( k net ).Using its expression in eq 14, we obtain k net (cid:12)(cid:12) ∆ W / = (cid:0) − e − β ∆ W / (cid:1) k b β ∆ F ) e − β ∆ W / = k b (S14)S6earranging the terms, we obtain a quadratic equation for e β ∆ W / , namely,e β ∆ W / − cosh( β ∆ F ) e β ∆ W / − . (S15)Since e β ∆ W / > , we take the positive solution and obtaine β ∆ W / = cosh( β ∆ F ) + q cosh ( β ∆ F ) + 82 . (S16)For large degrees of compression (e β ∆ F (cid:29) ), we can make the approximation cosh( β ∆ F ) ≈ e β ∆ F / and ignore the constant term in the square root, which yieldse β ∆ W / ≈ e β ∆ F , (S17) ∆ W / ≈ ∆ F − β − ln 2 . (S18)Like in the treatment of the ratio π d /π u , we now estimate how much the work per step needsto exceed ∆ W / in order for the backward stepping rate ( k net ) to reach its saturating value k b . To that end, we calculate the derivative of k net /k b at ∆ W / , namely, ∂∂ ∆ W (cid:18) k net k b (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∆ W / = 2 β e − β ∆ W / (1 + cosh( β ∆ F ) e − β ∆ W / ) + (1 − e − β ∆ W / ) cosh(∆ F ) β e − β ∆ W / (cid:0) β ∆ F ) e − β ∆ W / (cid:1) ≈ β e − β ∆ W / (1 + e β ∆ F e − β ∆ W / ) + (1 − e − β ∆ W / ) e β ∆ F β e − β ∆ W / (1 + e β ∆ F e − β ∆ W / ) = 4 β e − β ∆ W / + β (1 − e − β ∆ W / )4 ≈ k B T . (S19)Here we made the approximation cosh(∆ F ) ≈ e β ∆ F / in the first step, used the result fromeq S17 to write e β ∆ F e − β ∆ W / = 1 in the second step, and in the last step ignored thee − β ∆ W / terms since from eq S17 we have e − β ∆ W / = 4 e − β ∆ F (cid:28) for large degrees ofcompression.As we can see, when the work per step exceeds ∆ F by several k B T values, the chancesof forward stepping become vanishingly small compared with backward stepping, resultingin a net backward stepping rate k net ≈ k b . S7 Equilibrium Properties of the Allosteric Enzyme In this section, we introduce the constraints on the choices of rate constants for the enzymestemming from the cycle condition (based on the fact that is does not consume energy), andalso, discuss the fidelity available to it when the ligand concentration is held at a fixed value. C.1 Constraints on the Choice of Enzyme’s Rate Constants Consider the network of enzyme states and transitions in the absence of engine couplingredrawn in Figure S3A for convenience. Because the transitions between the states of theenzyme are not coupled to external energy consuming processes, the choice of the rate con-stants is constrained by the cycle condition which states that the products of rate constantsin the clockwise and counterclockwise directions should be equal to each other in all the loopsof the network diagram. Imposing the cycle condition results in the constraint equations forthe different loops shown in Figure S3B. In our analysis, we choose substrate unbinding tobe the only process whose rate is different between right and wrong substrates ( k Woff > k Roff ).Therefore, the rate constants of all other identical processes are chosen to be the samebetween the substrates, i.e. k RA = k WA ≡ k SA , (S20) k RI = k WI ≡ k SI , (S21) k R , LA = k W , LA ≡ k S , LA , (S22) k R , LI = k W , LI ≡ k S , LI , (S23)where the superscript “S” stands for both right (“R”) and wrong (“W”) substrates. Using thisgeneral notation, we write the full set of constraint conditions on the rate constants obtainedfrom the different loops (mid, front, and back) asmid: k LI k LA = ‘ Aoff /‘ Aon ‘ Ion /‘ Ioff k I k A , (S24)front: k SI k SA = k Ion k Aon k I k A , (S25)back: k S , LI k S , LA = k Ion k Aon k LI k LA . (S26)Note that the conditions imposed on the side loops follow directly from those of the otherthree loops via side = mid × backfront , (S27)which is why the sides loop do not contribute a unique condition.When writing the cycle conditions we did not include the product formation rate constant r , despite the fact that production takes the enzyme into its substrate-free state, just likewhat unbinding through k off reactions does. The reason for this is that r is an effective rateS8 offA ℓ offA ℓ on [L] I ℓ on [L] I k IL k AL k IR k AR k IR,L k AR,L k IW k AW k IW,L k AW,L k I k A ℓ on [L] A k on [R] A k offR k on [R] A k offR k on [W] A k offW k on [W] A k offW k on [R] I k on [W] I k offR k offW k on [R] I k on [W] I k offR k offW ℓ on [L] A ℓ offI ℓ offI ℓ offA ℓ on [L] I ℓ on [L] A ℓ offI rrfrontfront backback m i d s i d e s i d e k A = R k IR,L ℓ onA ℓ offI k IR k AR,L ℓ onI ℓ offA k A = k IL ℓ onA ℓ offI k I k AL ℓ onI ℓ offA k A = W k IW,L ℓ onA ℓ offI k IW k AW,L ℓ onI ℓ offA = k IR k A k onA k AR k I k onI = k IW k A k onA k AW k I k onI = k IR,L k onA k AL k AR,L k onI k IL = k IW,L k onA k AL k AW,L k onI k IL rr (A)(B) Figure S3. The allosteric enzyme in the absence of engine coupling. (A) Network diagram ofenzyme states and transitions between them at a fixed ligand concentration. The diagram isredrawn identically from Figure 4 for convenience. (B) Cycle condition on rate constants appliedfor the different loops of the diagram. The lighter color of the side loop conditions indicates thatthey are redundant and follow from the conditions on the other three loops. constant for the process E : S r −→ E + P representing the coarse-grained version of the fullbiochemical pathway of enzymatic production, namely, E : S (cid:29) E : P → E + P, which isdistinct from the k off pathway of emptying the enzyme. In our treatment we assume thatproduct formation is practically irreversible which will be true if the product concentrationis kept low and, optionally, if the reverse reaction P → S is energetically highly unfavorable(e.g. requires a spontaneous formation of a covalent bond).If the product formation rate is nonzero ( r > ), the enzyme will be out of equilibriumdespite the fact that its individual transitions are not coupled to an energy source. This isdue to the implicit assumption of having the right and wrong substrate concentrations fixed,which makes the system open (i.e. new substrates enter and products exit the system). Wediscuss the implications of this open system feature on the fidelity of the enzyme in theabsence of driving in the next section. S9 .2 Enzyme Fidelity at a Fixed Ligand Concentration As mentioned in the previous section, the presence of a nonzero production rate ( r > )makes the system open and thereby takes the enzyme out of equilibrium even at a fixedligand concentration where the engine-enzyme coupling is absent. For Hopfield’s scheme,it can be shown that in an analogous situation where driving is absent but the system isopen, the “equilibrium” (un-driven) fidelity is confined in a range defined by the ratio of theMichaelis and dissociation constants, which, for equal on-rates ( k R on = k W on ), becomes η eq ∈ (cid:20) k Woff + rk Roff + r , k Woff k Roff (cid:21) . (S28)We hypothesize that the same holds true for the allosteric enzyme as well despite themuch wider diversity of states available to it. To demonstrate that, we first consider thelimiting r → case where the product formation is so slow that the system effectively existsin a thermodynamic equilibrium. All possible enzyme states along with their statisticalweights in this equilibrium setting are shown in Figure S4. Fidelity can be found by addingthe statistical weights of the right and wrong substrate-bound active states and dividingthem, yielding η eq ( r → 0) = [ R ][ W ] K W,AD K R,AD = k Woff k Roff , (S29)where we used [ R ] = [ W ] and the equal on-rate assumption to go from dissociation constantsto unbinding rates. This corresponds to the upper limit in eq S28. state weight W,A K D [W] e -ϵA W,A K D [W] A R D [L] e -ϵA W,I K D [W] I R D [L] e -ϵI W,I K D [W] e -ϵI state weight e -ϵA A R D [L] e -ϵA I R D [L] e -ϵI e -ϵI R,A K D [R] e -ϵA R,A K D [R] A R D [L] e -ϵA R,I K D [R] I R D [L] e -ϵI R,I K D [R] e -ϵI state weight W,A K D [W] e -ϵA A R D [L]1 + ( ) R,A K D [R] e -ϵA A R D [L]1 + ( ) Figure S4. Table of possible enzyme states and their statistical weights in the r → limit wherethe system is effectively at equilibrium. Here (cid:15) A and (cid:15) I stand for the energies of the enzyme in itsactive and inactive states, respectively. The dissociation constants of the ligand and the substratesare denoted by R D and K D , respectively. Intuitively, the presence of a nonzero production rate ( r > ) should reduce the fidelityS10ince the enzyme would have less time to perform substrate filtering in its active state beforeproduct formation takes place. To study how large this reduction can be, let us first considera limiting case where the enzyme is exclusively in its active state – a setting where we expectthe reduction effect to be manifested the most. The active “slice” of the full network diagramcorresponding to this limiting case is depicted in Figure S5A. Since product formation is justanother path to substrate unbinding, we can derive a corresponding reduced network diagramby adding the production rate to the off-rates, as shown in Figure S5B.A peculiar feature of this network is that the cycle condition holds in its two loops, despitethe fact that the system is open ( r > ). This means that at steady state the detailed balancecondition will hold on all edges of the network (cf. Schnakenberg, section X), allowing usto assign effective statistical mechanical weights to the different states (Figure S5C). ℓ offA ℓ offA ℓ on [L] A k on [R] A k offR k on [R] A k offR k on [W] A k offW k on [W] A k offW ℓ on [L] A ℓ offA ℓ on [L] A rrrr (A) ℓ offA ℓ offA ℓ on [L] A k on [R] A k on [R] A k off + r R k off + r W k off + r R k off + r W k on [W] A k on [W] A ℓ on [L] A ℓ offA ℓ on [L] A (B)(C) state weight W,A K M [W] W,A K M [W] A R D [L] state weight A R D [L] R,A K M [R] R,A K M [R] A R D [L] state weight W,A K M [W] A R D [L]1 + ( ) R,A K M [R] A R D [L]1 + ( ) Figure S5. The enzyme in the limit of constant activity and in the absence of engine coupling.(A) The active “slice” of the enzyme’s full network diagram depicted in Figure S3A. (B) Thereduced diagram corresponding to the network in panel A with the production and off-ratescombined under the same reaction arrow. (C) Table of the different enzyme states and theireffective statistical weights. K M stands for the Michaelis constant. Total weights of the wrong andright substrate-bound states are shown below the left and right columns, respectively. Dividing the total weights of the right and wrong substrate-bound states, we obtain theS11delity in this special limit where the enzyme is exclusively in its active state, namely, η activeeq ( r > 0) = [ R ][ W ] K W,AM K R,AM = k Woff + rk Roff + r . (S30)Here we again used [ R ] = [ W ] and the equal on-rate assumption. Note that this correspondsto the lower fidelity limit in the un-driven Hopfield model (eq. S28).We now hypothesize that the enzyme’s fidelity falls between these two limits in thegeneral case where the system is open ( r > ) and when the states are not constrained tobe in the active “slice” of the full network diagram. Since obtaining the exact expressionof fidelity in the general case is highly complicated due to the presence of a large numberof states and loops in the network diagram, and since a paper-and-pencil approach wherethe symmetries existing between the left and right “wings” of the network could potentiallybe taken advantage of to provide an analytical proof is also not straightforward, we use anumerical method instead to justify our hypothesis.To that end, we fixed the ratio of the wrong and right substrate off-rates to be k Woff /k Roff = 100 ,sampled values for enzyme’s remaining transition rate constants from the [10 − k Roff , k Roff ] range (generating 20,971,520 independent sets in total), and evaluated the fidelity for eachparameter set. The results of the numerical study are summarized in Figure S6. As can beseen, despite the wider diversity of allosteric enzyme’s states, its fidelity in the absence ofengine coupling still falls between the “equilibrium” limits of Hopfield’s model (eq S28). -4 -2 -0 f i d e li t y , η production rate, r/k offR Figure S6. Allosteric enzyme’s fidelity in the absence of engine coupling (fixed [ L ] ) for differentchoices of transition rates. The upper and lower red curves correspond to the ratios of thedissociation (eq S29) and Michaelis constants (eq S30), respectively. Only the data points withfidelity values different up to the third significant digit were used in the plot. S12 Full Description of the Piston Model with Engine-Enzyme Coupling In this section, we provide details on the analytical and numerical explorations of the fullmodel. In D.1 we discuss the thermodynamics of coupling the engine to the allosteric enzyme.Then, in D.2 we present the methodology for obtaining the steady state probabilities ofsystem states under external drive. In D.3 and D.4 we provide the parameters used in thenumerical study of section 3.3 and describe the fidelity optimization strategy used in studyof the section 3.4, respectively. Lastly, in D.5 we investigate in detail the α max ≈ result forthe highest fidelity of the model. D.1 Equilibrium Fidelity of the Piston Model in the Absence ofExternal Driving In Supporting Information section C.2 we showed that at a fixed ligand concentration thefidelity of the allosteric enzyme was constrainted within the range given in eq S28. Here wedemonstrate that the same result holds also for the full model in the absence of externaldriving when a thermodynamically consistent coupling is made between the engine and theenzyme.In the absence of driving, the finite-state equivalent of the full network (Figure 5D) canbe reduced into the one shown in Figure S7 where we have combined the ratchet transitionsthrough forward and backward pathways under a single arrow – a procedure allowed when thetransitions are not driven. Because of the equilibrium constraints imposed on the enzyme’stransition rates discussed in Supporting Information section C.1, the cycle condition willhold for the loops in the left and right “layers” of the diagram in Figure S7. The loops wherethe cycle condition could possibly be violated are the ones that involve transitions betweenthe two layers, i.e. piston compressions and expansions.The first class of such loops does not involve ligand binding and unbinding events (forexample, the shaded vertical rectangle in Figure S7) and therefore, the cycle condition is au-tomatically satisfied in such loops since for each clockwise transition there is a correspondingcounterclockwise transition with an identical rate. The second class of loops that connect thetwo layers involves ligand binding and unbinding events which affect the rate of switchingbetween the layers (e.g. the shaded horizontal rectangle in Figure S7). The driving force inthese loops is given by ∆ µ = β − ln ( k u → df + k b )( k d → u , Lf + k b )[ L ] d ( k d → uf + k b )( k u → d , Lf + k b )[ L ] u , (S31)where we used the fact that the ligand binding rates are proportional to ligand concentrations.Now, in the general case where there are N ligands under the piston (one of which can bebound to the enzyme), the different forward stepping rates become k u → df = k b e − β (∆ W eq + β − N ln f ) = k b f − N , (S32) k u → d , Lf = k b e − β (∆ W eq + β − ( N − 1) ln f ) = k b f − ( N − , (S33)S13 f + k b u→d, L k f + k b d→u, L k f + k b ~[L] u ~[L] d u→d k f + k b d→u k f + k b u→d k f + k b d→u k f + k b u→d, L k f + k b d→u, L Figure S7. The effective network diagram of the piston model in the absence of driving. Theforward and backward pathways connecting the two layers of the diagram are combined to yieldeffective rates. The two kinds of cycles where ligand binding events are present or absent areshown as horizontal and vertical shaded rectangles, respectively. The crossed cycling arrowsindicate the absence of driving forces in the shaded loops. k d → uf = k b e − β (∆ W eq − β − N ln f ) = k b f N , (S34) k d → u , Lf = k b e − β (∆ W eq − β − ( N − 1) ln f ) = k b f N − . (S35)Here we set ∆ W eq = 0 to account for the absence of driving and used the fact that the freeenergy change of N ligands upon isothermal compression is β − N ln f (with a negative signupon expansion) and that N should be replaces with N − when one of the ligands is boundto the enzyme.Substituting these expressions into eq S31 and using the identity [ L ] d = f [ L ] u , we find ∆ µ = β − ln ( f − N + 1)( f N − + 1) f ( f N + 1)( f − N + 1)= β − ln (cid:18) f − N (1 + f N ) f N + 1 × f N − (1 + f − N ) f − N + 1 × f (cid:19) = β − ln (cid:0) f − N × f N − × f (cid:1) = β − ln 1 = 0 . (S36)This shows that in the absence of external driving ( ∆ W = 0 ) the cycle condition holds for allloops of the network, demonstrating the thermodynamic consistency of the coupling betweenthe engine and the enzyme.As in our separate treatment of the allosteric enzyme in Supporting Information sec-tion C.2, here too in the r → limit the system will approach a thermodynamic equilibrium.Since we already know that in the equilibrium limit the fidelity of the enzyme at a fixedligand concentration is given by the ratio of the wrong and right off-rates, we can apply thisresult to the left and right layers of the diagram in Figure S7 and obtain a relation betweenS14he net statistical weights of the right and wrong substrate-bound active states, namely, w Ru w Wu = w Rd w Wd = k Woff k Roff . (S37)Here “u” and “d” stand for the expanded (left layer) and compressed (right layer) states ofthe piston. We can then write the fidelity of the full network in terms of these weights as η eq ( r → 0) = w Ru + w Rd w Wu + w Wd = w Wu (cid:16) k Woff k Roff (cid:17) + w Wd (cid:16) k Woff k Roff (cid:17) w Wu + w Wd = k Woff k Roff , (S38)which corresponds to the upper limit of the equilibrium fidelity range in eq S28.To demonstrate that in the absence of driving the coupled system meets also the lowerfidelity limit given by ( k Woff + r ) / ( k Roff + r ) , we again use a numerical approach and sample theparameter space, evaluating the fidelity at each of the 10,628,820 sampled sets of parameters.As in the study of Figure S7, here too we set the ratio of off-rates to be k Woff /k Roff = 100 . Theresults are summarized in Figure S8, where it can be seen that all points lie between thelimits of eq S28. Overall, this study shows that in the absence of driving, the coupling ofthe engine to the enzyme alone cannot lead to fidelity enhancement. -4 -2 -0 f i d e li t y , η production rate, r/k offR Figure S8. Fidelity of the full piston model in the absence of driving ( ∆ W = 0 ) for differentchoices of model parameters. The upper and lower red curves correspond to the ratios of thedissociation (eq S29) and Michaelis constants (eq S30), respectively. Only the data points withfidelity values different up to the third significant digit were used in the plot. S15 .2 Obtaining the Steady-State Occupancy Probabilities The kinetics of the full piston model is characterized by a × transition rate matrix Q ,which has the block form Q = Q enzymeu Q d → u Q u → d Q enzymed ! . (S39)Here the non-diagonal elements of the × matrices Q enzymeu and Q enzymed represent thetransition rates between the different enzyme states when the ligand concentration is [ L ] =[ L ] u and [ L ] = [ L ] d , respectively. These non-diagonal terms at a given ligand concentration [ L ] are depicted in Figure S9. -Σ -Σ -Σ -Σ -Σ -Σ -Σ -Σ -Σ -Σ -Σ -Σ off A ℓ off A ℓ off A k I L k A L k I R k A R k I R, L k A R, L k I W k A W k I W, L k A W, L k I A k on [R] A k on [R] A k off R k off R k off + r R k off + r R k off + r W k on [W] A k on [W] A k off W k off + r W k off W k on [R] I k on [R] I k on [W] I k on [W] I ℓ on [L] A ℓ on [L] A ℓ on [L] A ℓ off I ℓ off I ℓ off I ℓ on [L] I ℓ on [L] I ℓ on [L] I Figure S9. Transition rates between the enzyme states. The element Q enzyme i,j stands for the rateof transitioning from the j th into the i th state of the enzyme ( i = j ) at a fixed ligandconcentration [ L ] . Red- and blue-colored cells show transitions involving the binding or release ofincorrect and correct substrates, respectively. Green- and gray-colored cells show inactivating andactivating enzyme transitions, respectively. The diagonals are shaded to indicate that they are notused when constructing Q . S16he other two matrices, namely, Q d → u and Q u → d , are diagonal whose elements stand forthe net piston compression (d → u) and expansion (u → d) rates that alter the state of thepiston but leave the state of the enzyme unchanged. They are given by Q d → u = diag (cid:18) (cid:0) k b + k d → uf (cid:1) , ..., (cid:0) k b + k d → uf (cid:1)| {z } , (cid:0) k b + k d → u , Lf (cid:1) , ..., (cid:0) k b + k d → u , Lf (cid:1)| {z } (cid:19) , (S40) Q u → d = diag (cid:18) (cid:0) k b + k u → df (cid:1) , ..., (cid:0) k b + k u → df (cid:1)| {z } , (cid:0) k b + k u → d , Lf (cid:1) , ..., (cid:0) k b + k u → d , Lf (cid:1)| {z } (cid:19) . (S41)Note that since the forward stepping rates depend on whether the ligand is bound or not,they appear without a superscript “L” in the first 6 terms (where the ligand unbound), andwith a superscript “L” in the last 6 terms (where the ligand is bound). Lastly, the diagonalelements of Q are assigned such that Q jj = − P i = j Q ij , ensuring that the columns sum tozero.The dynamics of the coupled engine-enzyme system is described via d ~p d t = Q ~p, (S42)where ~p is a column vector whose 24 elements stand for the probabilities of the differentsystem states (12 enzyme states × d ~p/ d t = ~ . Since the exact analytical expressions forthe steady state probabilities ~p ss ≡ ~π are highly convoluted, in our parametric studies weuse numerical methods to find ~π from Q ~π = ~ and P i π i = 1 , where the latter equationguarantees that the probabilities sum to 1. S17 .3 Enzyme’s Kinetic Parameters Used for the Numerical Studyin Section 3.4 Here we provide the list of enzyme’s transition rates used for numerically studying the effectsof tuning the engine “knobs” in section 3.4. Since none of the performance metrics used inthe study depend on the absolute timescale of the model’s dynamics, we set the unbindingrate of the right substrate to be unity ( k Roff = 1 ), and defined all other rates relative to it.Specifically, we chose k Woff = 100 so that the fidelity after a single proofreading realizationroughly matched that of the ribosome ( η translation ∼ ). Also, we chose the catalysis rate tobe much slower compared with the off-rates ( r = 0 . ) - a condition for high fidelity suggestedin Hopfield’s original paper.The remaining rate constants were assigned values that meet the intuitive expectationsfrom the conceptual introduction of the model in section 2. Specifically, the rate of substratebinding to the active enzyme was chosen to be much less than the rate of binding to theinactive enzyme in order to yield low leakiness ( k Aon /k Ion = 10 − (cid:28) ). Next, the enzyme waschosen to be predominantly inactive in its native state to allow for new substrate bindingevents ( k I /k A = 50 (cid:29) ). Lastly, the rates of ligand binding and unbinding were assignedvalues that ensure that the ligand acts as a strong activator ( ‘ Ioff /‘ Ion ‘ Aoff /‘ Aon = 10 (cid:29) ).The values of the independent parameters k LA and k SA were assigned after manually in-specting the effect of different numerical choices on the model performance. Finally, thevalues of the remaining four parameters (lower section in Table S1) were calculated from thecycle conditions in eqs S24-S26 under the assumption that ligand binding does not alter theratio of inactivation or activation rates in the substrate-bound and substrate-unbound states(i.e. k S , LI /k LI = k SI /k I and k S , LA /k LA = k SA /k A ). Table S1. Values of enzyme’s different transition rates used in the studies of Figure 6. Transition rate Value k Roff , k Woff , r . k Aon [ S ] 10 − k Ion [ S ] 1 k A k I ‘ Aon . ‘ Aoff ‘ Ion . ‘ Ioff k LA k SA . k LI . k SI k S , LA k S , LI S18 .4 Details of the Numerical Optimization Procedure for Findingthe Highest Fidelity In our optimization scheme, we first chose the values of rates which were kept fixed forthe rest of the study. These include the unbinding rate of right substrates ( k Roff = 1 ), thecatalysis rate ( r = 0 . ), and the effective first-order rate of substrate binding to the inactiveenzyme state ( k Ion [ S ] = 1 ). Also, since no limits were imposed on the amount of energyexpenditure, we chose large values for the compression factor ( f = 10 ) and the work perstep ( ∆ W = 1000 k B T ) to maximize the quality of proofreading.Then, we considered a set of 144 different initialization options for the remaining param-eters to be used in our numerical optimization procedure. To avoid the completely indepen-dent tuning of related enzyme activation and inactivation rates, we considered three possibleoptions that met the cycle condition. Namely, 1) k SA = k A and k SI = k I /γ , 2) k SA = √ γk A and k SI = k I / √ γ , 3) k SA = γk A and k SI = k I , where γ = k Aon /k Ion . All of these three optionssatisfy the cycle constraint k Aon k SI k A = k Ion k SA k I (Figure S3B). Options for the transition ratesbetween ligand-bound enzyme states (i.e. k S , LA and k S , LI ) were chosen analogously.In our custom-written maximization algorithm we iteratively perturbed all the parame-ters for multiple rounds with decreasing amplitudes until the convergence criterion was metor until the number of iterations exceeded the specified threshold (at most 20 iterations foreach of the 6 decreasing amplitudes). The results from each of these local maximizationprocedures are summarized in Figure S10. We chose the largest among the different localmaxima to represent the highest fidelity available for the given (cid:0) k Aon , k Woff (cid:1) pair. (A) p r oo f r e a d i n g i n d e x , α leakiness, k on /k onA I -12 -8 -4 k off /k off =10 W R (C) p r oo f r e a d i n g i n d e x , α leakiness, k on /k onA I k off /k off =1000 W R -12 -8 -4 (B) p r oo f r e a d i n g i n d e x , α leakiness, k on /k onA I k off /k off =100 W R -12 -8 -4 Figure S10. Fidelity optimization results for each of the 144 parameter initialization options.(A) k Woff /k Roff = 10 . (B) k Woff /k Roff = 100 . (C) k Woff /k Roff = 1000 . The dotted lines in each panelrepresent the trends for the globally optimal fidelities. S19 .5 Investigation of the α max ≈ Result for the Highest AvailableProofreading Index Our numerical scheme for optimizing the fidelity (Supporting Information section D.4) re-vealed that the piston model could perform proofreading up to three times ( α max ≈ ). Togain intuition on how this is possible, let us consider the wrong “wing” of the full reactionnetwork (Figure S11A). Each system state is characterized by the piston position (up ordown) and the state of the enzyme (one of 8 possibilities). To turn a wrong substrate into aproduct, the system needs to traverse a trajectory that starts at a substrate-unbound stateon the right side of the diagram and reaches one of the substrate-bound active states onthe left side, at which point catalysis can take place. Using the terminology introduced inMurugan, et al. , we can say that a proofreading filter can be realized every time the systemmakes a transition parallel to the “discriminatory fence” of the network (Figure S11A). Rateswhich are on either side of the fence do not discriminate between the two kinds of substrates;only those that cross the fence do, which in our case are the off-rates ( k Woff > k Roff ). Thus,the number of such parallel transitions that the system makes before reaching the catalyti-cally active state represents the largest number of proofreading filters available to the giventrajectory.Figure S11B shows the full list of unique trajectories that start on the right side of thenetwork, cross the discriminatory fence and eventually reach an active enzyme state aftertraversing through a series of inactive states. The trajectories are grouped by the number ofthese inactive states visited on the left side of the wing prior to reaching the active state. Forexample, entries of the first group represent trajectories where the substrate binds directlyto the active enzyme and hence, undergoes zero proofreading filtrations. The discriminatorycapacity of the piston model will therefore depend on which of the trajectories dominates inproduct formation.To compare the contributions from different trajectories, we assign each of them a proba-bility flux, which approximates the average rate of product formation through the trajectory.We define this flux via J ~s = π s k s → s N − Y i =1 p s i → s i +1 ! p cat s N − , (S43)where ~s is the set of N states in the trajectory, π s k s → s is the substrate binding flux thatcrosses the fence at the start of the trajectory, p s i ,s i +1 are the probabilities of staying onthe trajectory during traversal, and, lastly, p cat s N − is the probability of catalysis once thesystem has reached the active enzyme state s N − . Note that the flux expression does notaccount for backtracking events whose contribution we expect to be insignificant for efficientproofreading trajectories, since for them p s i → s i +1 (cid:28) .Having defined a flux metric for each trajectory, we then calculated its value for alltrajectories listed in Figure S11B in the case where k Woff /k Roff = 100 and k Aon /k Ion = 10 − (lowleakiness). Figure S11C shows the fluxes normalized by the highest one and grouped by thenumber of proofreading filters. As we can see, the dominant trajectory indeed contains threefilters. This dominant trajectory is highlighted in red in Figure S11B and also correspondsto the one shown in Figure 7B of the main text.S20 A)(C) (D)(B) k off W k off W k off W k off W k off W k off W k off W k off W discriminatory fencenumber of proofreading filters r e l a t i v e p r o d u c t i o n f l u x ~ [ L ] u ~ [ L ] u ~ [ L ] u ~ [ L ] u ~ [ L ] d ~ [ L ] d ~ [ L ] d ~ [ L ] d substratebinding pistoncompression ligandbinding enzymeactivation Figure S11. (A) The wrong “wing” of the full reaction network along with the discriminatoryfence. Ligand concentrations that enter the ligand binding rates are shown to indicate thedifference between the upper and lower halves of the diagram. (B) The full set of uniquetrajectories that start on the right side of the network and end up at an active enzyme state onthe left side. Numbers of proofreading filters available to trajectories are shown on the side.Piston state transitions are marked with dotted lines for clarity. The dominant trajectory in panel(C) is highlighted in red. (C) The relative product formation fluxes of all possible trajectoriescalculated for the case where k Woff /k Roff = 100 and k Aon /k Ion = 10 − , and grouped by the number offilters. The red dot indicates the dominant one. (D) Schematic illustration of the dominanttrajectory in panel (C) along with the numerical values of the rates. The dotted arrows suggestthat the intermediate transitions are much slower than the off-rate. S21e would like to note here that the model parameters inferred from the unconstrainedfidelity optimization were degenerate, and there was an alternative set with α ≈ proofread-ing index whose corresponding dominant trajectory was different from the one highlightedin Figure S11B. Some of the parameters of this set, however, contradicted our model cri-teria (e.g. the binding rate in the expanded piston state was very high), which is why wedid not use this alternative set for our main discussion. Parameters that did satisfy ourmodel criteria are shown for the k Woff /k Roff = 100 case in Figure S11D. The transition rates be-tween intermediates are much slower compared with the off-rate, as expected for an efficientproofreading performance.Lastly, as can be seen in Figure S11B, the highest number of filters that a unique trajec-tory could, in principle, realize is 4 and not 3. This raises the question of why a trajectorywith 4 potential filters cannot be a dominant one, as our numerical results in Figure S11Chave suggested. We answer this question for three representative cases and invite the readerto work through the remaining examples. Our approach will be to show that the flux througha given 4-filter trajectory is necessarily less than that of some other trajectory with fewerfilters, which would suggest that it cannot be a dominant one.Figure S12 shows three different 4-filter trajectories next to corresponding trajectorieswith fewer filters, flux through which, as we will show, will necessarily be greater. Through-out our analysis we will be making use of the fact that the rates of piston expansion andcompression are identical (and equal to k b ) in the large driving limit considered here. Letus start from the first example. Using Eq S43, we can write the fluxes through 4-filter ( J (4) )and 3-filter ( J (3) ) trajectories respectively as J (4) = π k → × p → p → p → p → × p cat , (S44) J (3) = π k → × p → p → p → × p cat . (S45)Since the states (3) and (4) correspond to the same enzyme state and have identical outgoingrates, we have p → = p → . From the same argument for states (5) and (6) we find p cat = p cat .With these identities at hand, we can write the ratio of the two fluxes as J (4) J (3) = p → < . (S46)Therefore, the 4-filter trajectory is necessarily slower than the 3-filter one and cannot dom-inate the dynamics of wrong product formation.Now let us look at the slightly more complicated second example. There the fluxes ofthe 4-filter and 1-filter trajectories are J (4) = π k → × p → p → p → p → × p cat , (S47) J (1) = π k → × p → × p cat , (S48)respectively. The full expression of the transition probability p → is p → = k → k → + k → + k → + k → . (S49)S22 012 36012 345 01 234 7 45 0 61 2345 4501 86 Figure S12. Three representative 4-filter trajectories paired with ones which have a lower filternumber and, necessarily, a higher product formation flux. The state indices are added to facilitatethe comparison between the corresponding trajectories. Similarly, the expression for p → is p → = k → k → + k → + k → + k → . (S50)All corresponding rates in the above probability expressions are equal to each other (i.e. k → = k → = k SA , k → = k → = k b , k → = k → = k Woff ), with the exception of k → = ‘ Ion [ L ] d and k → = ‘ Ion [ L ] u . Now, since [ L ] d > [ L ] u , we obtain p → < p → . With an identicalreasoning we can also find that p cat < p cat . Therefore, the ratio of the 4-filter and 1-filtertrajectory fluxes becomes J (4) J (1) = p → p → p → (cid:18) p → p → (cid:19)| {z } < (cid:18) p cat p cat (cid:19)| {z } < < , (S51)proving our claim.Lastly, we consider the third example in Figure S12. We again start off by writing thetrajectory fluxes, namely, J (4) = π k → × p → p → p → p → × p cat , (S52) J (1) = π k → × p → × p cat . (S53)The two rates appearing in the flux expression represent the substrate binding rate andare equal to each other, that is, k → = k → = k Ion [ S ] . Now, note that π is the steadystate probability of the inactive ligand-unbound enzyme state at a low ligand concentration,whereas π is the probability of the same enzyme state at a high ligand concentration. Sincethese are ligand-unbound states, the one in the presence of a lower ligand concentration willhave a higher probability, i.e. π > π . Thus, taking the ratio of the two fluxes, we obtain J (4) J (1) = (cid:18) π π (cid:19)| {z } < p → p → p → < , (S54)suggesting that the 4-filter trajectory in the third example too cannot be the dominant one.S23 eferences (1) Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman Lectures on Physics ; Addison-Wesley: Reading, MA, 1963; Vol. 1.(2) Magnasco, M. O.; Stolovitzky, G. Feynman’s Ratchet and Pawl. J. Stat. Phys. , , 615–632.(3) Hill, T. L. Free Energy Transduction in Biology: The Steady State Kinetic and Thermo-dynamic Formalism ; Academic Press, 1977.(4) Schnakenberg, J. 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