Conformational Nonequilibrium Enzyme Kinetics: Generalized Michaelis-Menten Equation
aa r X i v : . [ phy s i c s . b i o - ph ] M a y Conformational Nonequilibrium EnzymeKinetics: Generalized Michaelis–MentenEquation
D. Evan Piephoff, † , ‡ Jianlan Wu, † , ‡ and Jianshu Cao ∗ , † † Department of Chemistry, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, United States ‡ Contributed equally to this work
E-mail: [email protected]
Abstract
In a conformational nonequilibrium steady state (cNESS), enzyme turnover is mod-ulated by the underlying conformational dynamics. Based on a discrete kinetic net-work model, we use the integrated probability flux balance method to derive thecNESS turnover rate for a conformation-modulated enzymatic reaction. The tradi-tional Michaelis–Menten (MM) rate equation is extended to a generalized form, whichincludes non-MM corrections induced by conformational population currents withincombined cyclic kinetic loops. When conformational detailed balance is satisfied, theturnover rate reduces to the MM functional form, explaining its validity for many enzy-matic systems. For the first time, a one-to-one correspondence is established betweennon-MM terms and combined cyclic loops with unbalanced conformational currents.Cooperativity resulting from nonequilibrium conformational dynamics has been ob-served in enzymatic reactions, and we provide a novel, rigorous means of predicting nd characterizing such behavior. Our generalized MM equation affords a systematicapproach for exploring cNESS enzyme kinetics. Conformational dynamics is essential for understanding the biological functions of enzymes.For decades, the framework of enzymatic reactions has been the traditional Michaelis–Menten(MM) mechanism, where enzyme-substrate binding initializes an irreversible catalytic reac-tion to form a product. The average turnover rate v in a steady state (SS) follows a hyperbolicdependence on the substrate concentration [S], v = k [S] / ( K M + [S]) , where the catalytic rate k and the Michaelis constant K M characterize this enzymatic chain reaction. In contrastto the single-conformation assumption for the traditional MM mechanism, recent single-molecule experiments have revealed the existence of multiple enzymatic conformations,spanning a broad range of lifetime scales from milliseconds to hours. Conformational dynam-ics, including hopping between different conformations and thermal fluctuations around asingle-conformation potential well, must be incorporated into enzymatic reaction models fora quantitative study. Slow conformational dynamics modulate the enzymatic reactionand allow the enzyme to exist in a conformational nonequilibrium steady state (cNESS),permitting complex deviations from MM kinetics (the hyperbolic [S] dependence for v ).However, experimental and theoretical studies have shown MM kinetics to be valid in thepresence of slow conformational dynamics under certain conditions, although k and K M be-come averaged over conformations. Is there a unifying theme governing this surprisingbehavior?Non-MM enzyme kinetics have been characterized by cooperativity for many years.
For allosteric enzymes with multiple binding sites, the binding event at one site can alter thereaction activity at another site, accelerating (decelerating) the turnover rate and resultingin positive (negative) cooperativity. Another common deviation from MM kinetics is sub-strate inhibition, where the turnover rate reaches its maximum value at a finite substrateconcentration and then decreases at high substrate concentrations. For a monomeric en-zyme, the above non-MM kinetic behavior, referred to – in this case – as ‘kinetic cooperativ-2ty,’ can be achieved by a completely different mechanism: nonequilibrium conformationaldynamics. Can we characterize and predict this interesting behavior in a cNESS?Recently, theoretical efforts have been applied to study conformation-modulated enzymekinetics by including dynamics along a conformational coordinate. On the basis of the usualrate approach, some previous work has demonstrated certain non-MM kinetics under spe-cific conditions.
Based on an alternative integrated probability flux balance method,non-MM kinetics were linked to a nonzero conformational population current, i.e., brokenconformational detailed balance, in a two-conformation model, and a general MM expressionwas speculated. However, a generalized theory to systematically analyze cNESS enzymekinetics is still needed. In this Letter, we focus on a monomeric enzyme and apply the inte-grated flux balance method to derive a generalized form for the turnover rate, which includesnon-MM corrections. We show that when conformational detailed balance is satisfied, MMkinetics hold, explaining their general validity. In addition, the deviations from MM kinet-ics are analyzed with reduced parameters from the generalized form of v. For an extendedversion of our derivation, we refer readers to ref 20.To describe the generalized conformation-modulated reaction catalyzed by a monomericenzyme, we introduce a discrete kinetic network model, which is illustrated in Figure 1a.This N × M network consists of a vertical conformation coordinate (1 ≤ i ≤ N ) and ahorizontal reaction coordinate (1 ≤ j ≤ M ) . For the reaction state index, j = 1 denotes theinitial substrate-unbound enzymatic state (E), whereas j ≥ denotes intermediate substrate-bound enzymatic states (ES). Without product states, our network corresponds toa dissipative system. For an arbitrary site R i,j , the reaction rates for the forward ( R i,j → R i,j +1 ) and backward ( R i,j → R i,j − ) directions are given by k i,j and k i, − ( j − , respectively.The rate for enzyme-substrate binding, the only step in our model dependent upon substrateconcentration [S], depends linearly on [S] as k i, = k i, [S] for binding rate constant k i, , with[S] maintained constant in most enzymatic experiments. The conformational dynamics aretreated via a kinetic rate approach, with the interconversion (hopping or diffusion) rates for3 ) flux network R R R R R N,1 R N,2 R R R N,M γ -1, 1 γ γ γ M ,1 γ γ γ γ ,2 γ γ M ,2 γ γ N − γ γ N − γ γ MN ,1 − γ γ a) kinetic network , k , -11 k k , -21 k M- k , -( M- k M ,1 k k , -12 k k , -22 k M-1 ,2 k , -( M- k M ,2 k N , 1 kk N , 2 k N , -2 k N, M- k N, -(M- ) k N, M k N , -1 R R R R R N,1 R N,2 R R R N,M F F F N,M F F F F F F F N,1 F N,2 F N,M-1 F F F N,0 J J J J J J J N-1,1 J N-1,2 J N-1,M -1, 2 -1, M -2, 1 -2, 2 -2, M -( N -1), 1 -( N -1), 2 -( N -1), M E ESE ES
Figure 1: (a) Generalized kinetic network scheme for a conformation-modulated enzymaticreaction. (b) Flux network corresponding to (a) (see text for details).4 i,j → R i +1 ,j and R i,j → R i − ,j given by γ i,j and γ − ( i − ,j , respectively. We note that localdetailed balance results in the constraint k i,j γ − i,j / ( k i, − j γ i,j ) = k i +1 ,j γ − i,j +1 / ( k i +1 , − j γ i,j +1 ) for j ≤ M − . However, for the purposes of our kinetic analysis, it is unnecessary to imposethis relation, as our principal results hold, irrespective of whether it is satisfied. The rateequation for site R i,j is written as ddt P i,j ( t ) = N X i ′ =1 γ i,i ′ ; j P i ′ ,j ( t ) + M X j ′ =1 k j,j ′ ; i P i,j ′ ( t ) (1)where P i,j ( t ) is the probability of an enzyme in site R i,j at time t , i.e., the survival probabilityfor the site. Here, γ i,i ′ ; j = γ i − ,j δ i ′ ,i − + γ − i,j δ i ′ ,i +1 − [ γ i,j + γ − ( i − ,j ] δ i ′ ,i denotes the interconver-sion rates in the j -th reaction state and k j,j ′ ; i = k i,j − δ j ′ ,j − + γ i, − j δ j ′ ,j +1 − [ k i,j + k i, − ( j − ] δ j ′ ,j denotes the reaction rates for the i -th conformation.Within the framework of a dissipative enzymatic network, the average turnover rate v isequivalent to the inverse of the mean first passage time (MFPT) h t i . Using the residencetime τ i,j = ˆ ∞ P i,j ( t ) dt at each site R i,j , we can express the MFPT in the N × M networkas a summation of τ i,j , i.e, h t i = P i,j τ i,j . Instead of inverting the transition matrix, weevaluate τ i,j by inspecting integrated probability fluxes, which correspond to stationarypopulation fluxes normalized by v , and these will be shown to directly reflect conformationalnonequilibrium. Along the horizontal reaction coordinate, the integrated flux for R i,j → R i,j +1 is given by F i,j = k i,j τ i,j − k i, − j τ i,j +1 . Along the vertical conformation coordinate, theintegrated flux for R i,j → R i +1 ,j is given by J i,j = γ i,j τ i,j − γ − i,j τ i +1 ,j . In addition, we needto specify the initial condition P i,j ( t = 0) for calculating h t i . For a monomeric enzyme, eachturnover event begins with the substrate-unbound state, and P i, ( t = 0) defines the initialflux F i, . With the definition of { F i,j , J i,j } , we map the original kinetic network to a fluxnetwork as shown in Figure 1b. For each site R i,j , the rate equation in eq 1 is replaced by aflux balance relation, F i,j − + J i − ,j = F i,j + J i,j (2)5hich is generalized to the probability conservation law: the total input integrated probabilityflux must equal the total output integrated probability flux . This conservation law can beextended to complex first-order kinetic structures including the N × M network. The fluxbalance method thus provides a simple means of calculating the MFPT.To evaluate the MFPT, we begin with the final reaction state ( j = M ) and propagateall the fluxes back to the initial reaction state ( j = 1) based on eq 2. For each site R i,j ,the physical nature of the first-order kinetics determines that all three variables, τ i,j , J i,j and F i,j , are linear combinations of terminal fluxes F i,j = M . The first two variables areformally written as τ i,j = P Ni ′ =1 a i,j,i ′ F i ′ ,M and J i,j = P i ′ c i,j,i ′ F i ′ ,M , where a i,j,i ′ and c i,j,i ′ are coefficients depending on rate constants { k, γ } . For example, the coefficients for thefinal reaction state are a i,M,i ′ = 1 /k i,M δ i ′ ,i and c i,M,i ′ = γ i,M /k i,M δ i ′ ,i − γ − i,M /k i +1 ,M δ i ′ ,i +1 .Because of the direction of our reversed flux propagation, only the coefficients for the initialreaction state are [S] dependent, and they can be explicitly written as a i, ,i ′ = b i,i ′ / [S] and c i, ,i ′ = d i,i ′ / [S] . The substrate-unbound (E i = R i, ) and substrate-bound ( ES i = P Mj =2 R i,j )states are distinguished by the different [S] dependence of the coefficients. The MFPT isthus given by h t i = N X i ′ =1 " P Ni =1 b i,i ′ [S] + N X i =1 M X j =2 a i,j,i ′ F i ′ ,M (3)The essential part of our derivation is then to solve for the terminal fluxes F i,M . TheSS condition can be interpreted as follows: after each product release, the enzyme returnsto the same conformation for the next turnover reaction, i.e., F i,M = F i, . Applying theprobability conservation law to each horizontal chain reaction with a single conformationand considering the boundary condition at conformations i = 1 and N , we express the SScondition as a flux constraint, J i, E + J i, ES = 0 for i = 1 , , · · · , N − , where J i, E = J i, and J i, ES = P Mj =2 J i,j . For each combined cyclic loop E i → E i +1 → ES i +1 → ES i → E i ,there may exist a stabilized nonequilibrium conformational population current (see Figure1b), with J i, E representing this stationary current normalized by v . However, under certaincircumstances, J i, E can vanish, and the SS condition is further simplified to J i, ES = 0 . We6ote that satisfaction of the aforementioned constraint resulting from local detailed balancestill permits nonzero J i, E . In general, we assume that there exist N c ( ≤ N − nonzeroconformational currents and ( N − − N c ) zero ones. In addition to these ( N − currentconditions, the normalization condition P Ni =1 F i, = 1 is needed for fully determining theinitial fluxes (due to F i, = F i,M ). As a result, we derive an N -equation array for F i, , U · F = · · · C , + (cid:16) d , [S] (cid:17) C , + (cid:16) d , [S] (cid:17) · · · C , + (cid:16) d , [S] (cid:17) C , + (cid:16) d , [S] (cid:17) · · · ... ... ... · F , F , ... = ... (4)with C i,i ′ = P Mj =2 c i,j,i ′ . Notice that for the ( i + 1) -th row of matrix U in eq 4, d i,i ′ / [S] onlyexists when J i, E = 0 , and N c rows are [S] dependent for this matrix. We solve for the initialfluxes by the matrix inversion F i, = [ U − ] i, . After a tedious but straightforward derivation, F i, is written as F i, ([S]) = f i, + N c X n =1 f i,n / ([S] + s n ) (5)where each s n is assumed to be distinct, and constraints hold for P i f i, = 1 and P i f i,n = 0 for n ≥ .Substituting eq 5 into eq 3, we obtain the key result of this Letter: the cNESS turnoverrate for the N × M network with N c unbalanced conformational currents is given by ageneralized Michaelis–Menten equation, v = " A + B [S] + N c X n =1 B n [S] + s n − (6)where the reduced parameters are A = h /k eff2 i [S] →∞ , B = h K effM /k eff2 i [S]=0 , and B n = P i [1 /k eff i, − K eff i, M / ( k eff i, s n )] f i,n . For each conformational channel, we introduce an effective cat-7lytic rate k eff i, = ( P Ni ′ =1 P Mj =2 a i ′ ,j,i ) − and an effective Michaelis constant K eff i, M = k eff i, P i ′ b i ′ ,i ,which describe the kinetics within that channel in the decomposed representation of thescheme, wherein the N two-state chain reactions are effectively independent, each withprobability F i, . The conformational average is defined as h x i [S] = P i x i F i, ([S]) for aconformation-dependent variable x i . In the right hand side of eq 6, the first two termsretain the traditional MM form, whereas the remaining N c terms introduce non-MM ratebehavior, with a 1:1 correspondence between non-MM terms and combined cyclic loops withnonzero conformational currents. Our derivations clearly show that these non-MM terms areinduced by the [S]-dependent conformational distribution F resulting from nonequilibriumconformational currents. Therefore, MM kinetics are valid when conformational detailedbalance is satisfied, where all B n vanish due to F i, = f i, . positive cooperativitynegative cooperativitysubstrate inhibition Figure 2: (a) Three non-MM turnover rates for the single-loop model with A = B = s = 1 . The circles ( B = − and the up-triangles ( B = 2) exhibit positive and negativecooperativity, respectively. The two solid lines are the fit using the Hill equation. Thedashed line ( B = − shows substrate inhibition behavior. (b) Phase diagram of enzymekinetics for the single-loop model. Two lines, B = 0 and B = − B , separate three regimesof kinetics.With nonzero conformational currents, the enzyme kinetics are expected to exhibit coop-erative non-MM behavior. As a demonstration, the single-loop model with only one current J , E and one non-MM term B / ([S] + s ) is first considered. With other parameters fixed,we calculate turnover rates v for the three values of B in Figure 2a. For the two turnoverrates monotonically increasing with [S] ( B = − and ), we fit them with the Hill equation, v/v max = [S] n H / ( κ + [S] n H ) , where the Hill constant n H > n H < indicates positive8negative) cooperativity. The fitting results show that cooperativity is completely deter-mined by the sign of B : positive for B < and negative for B > . This result is alsoreflected in eq 6, where negative (positive) B increases (decreases) the MM turnover rate ( A + B / [S]) − . The dashed line in Figure 2a shows that a largely negative B + B leads tosubstrate inhibition behavior. The cNESS substrate inhibition shows positive cooperativityat low substrate concentrations, and then the turnover rate decreases to a nonzero value A − in the substrate-saturation limit. Next, we plot the phase diagram of enzyme kinetics for thesingle-loop model in Figure 2b, which only depends on B and B . From this phase diagram, α = B /B is defined as a unique non-MM indicator for single-loop systems, with negativecooperativity for α > , positive cooperativity for − ≤ α < , and substrate inhibition for α < − . Figure 3: (a)–(c) Three cases in which a current J , E circulating counterclockwise within atwo-conformation loop can be modulated by ∆∆ τ eff (see text for details); such modulationunderlies the emergence of kinetic cooperativity. In each conformational channel, a horizontalarrow proceeds from the state with the faster effective characteristic residence time (see textfor details) to the state with the slower one, with J , E superimposed onto this view. Notethat there are also analogous cases for J , E proceeding in the clockwise direction.The direction of a conformational current alone does not predict its influence on thecooperativity, which raises the question of how currents are modulated to govern cooperativebehavior. For the two-conformation network, the simplest single-loop model, we can rewritethe non-MM term as B [S] + s ∝ ∆∆ τ eff × J , E ([S]) (7)where ∆∆ τ eff = ∆ τ eff 1 − ∆ τ eff 2 , with ∆ τ eff i = τ effE i − τ effES i . Here, the E i and ES i residencetimes in the decomposed representation, each independent of the non-MM term [and thus9f J , E ([S]) ], are given by τ effE i ([S]) = K eff i, M / ( k eff i, [S]) and τ effES i = 1 /k eff i, , respectively. Also, τ effE i = τ effE i ([S] = s ) , where s is the value of [S] at which | J , E ([S]) | is at half its maxi-mum and thus represents a characteristic non-MM substrate concentration. Therefore, τ effE i represents a characteristic value of τ effE i ([S]) , with corresponding characteristic residence timegradient ∆ τ eff i . Thus, ∆∆ τ eff represents the difference in characteristic residence time gradi-ent between the two decomposed conformational channels. Cooperativity depends upon J , E modulated by ∆∆ τ eff , i.e., it is governed by the relative modulation of the current betweenthe two decomposed chain reactions. In the two-conformation model, J , E proceeds from E i to ES i in one conformational channel and from ES i to E i in the other, as illustrated inFigure 3a–c for a counterclockwise current, which corresponds to J , E > based upon ouroriginal definition of J i,j . In each two-state chain reaction, enzyme turnover is accelerated(decelerated) when J , E proceeds from the state with the slower (faster) effective character-istic residence time to the state with the faster (slower) one. In Figure 3a (b), turnover isaccelerated (decelerated) in both chain reactions, resulting in overall turnover acceleration(deceleration), i.e., positive cooperativity or substrate inhibition (negative cooperativity). InFigure 3c, turnover is accelerated in conformation 1 and decelerated in conformation 2 (theopposite [not shown] is possible as well), with the cooperativity depending upon the relativemodulation of the current between the two decomposed chains. Kinetic cooperativity isthus explained as follows: when J , E proceeds in the direction that, on average, correspondsto decreasing (increasing) effective characteristic residence time, positive cooperativity orsubstrate inhibition (negative cooperativity) occurs.Interestingly, when the effective characteristic residence time gradient is conformationinvariant, modulation of the conformational current is balanced, resulting in MM kinetics,even in the presence of circulating current (i.e., when ∆∆ τ eff = 0 and J , E = 0 for thetwo-conformation network). This scenario represents a unique type of nonequilibrium sym-metry in multidimensional kinetic networks and is not precluded by the satisfaction of theaforementioned constraint resulting from local detailed balance. Additionally, we note that10or the × model, J , E vanishes under a simple conformational detailed balance condition, γ , γ , K , M = γ − , γ − , K , M (8)where K i, M = ( k i, − + k i, ) /k i, . Explicit calculations for this model are provided in theSupporting Information. a) b)
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Figure 4: Enzyme kinetics for the two-loop model with A = B = s = 1 and s = 4 . (a)Three turnover rates v that are non-monotonic functions of [S]. Each line shows a typicaltype of non-MM kinetic behavior from the regime labeled by the same number in (b). (b)Phase diagram determined by two non-MM parameters B and B . There are five regimesof non-MM behavior (see text for details).For the two-loop models with two non-MM terms, the cNESS enzyme kinetics becomemore complicated, as illustrated in a typical phase diagram in Figure 4b. Except for anunphysical regime where v shows divergence and negativity, five regimes of enzyme kineticscan be characterized in the phase space composed of B and B . Similar to the single-loop model, when monotonically increasing to the maximum value v max in the substrate-saturation limit ([S] → ∞ ) , v can exhibit negative (Regime 1) and positive (Regime 2)cooperativity. The separation line between these two kinetic regimes, however, is hard torigorously define. The dashed separation line in Figure 4b corresponds to n H = 1 , where theHill constant is empirically calculated using n H = log 81 / log([S] . v max / [S] . v max ) , and [S] v is the substrate concentration for v . In Regimes 3-5, the turnover rate v is a non-monotonicfunction of [S] (examples shown in Figure 4a). In Regime 3, with v max occurring at a finite [S] v max , the turnover rate exhibits the same substrate inhibition behavior as the single-loop11odel. Alternatively, an additional local minimum of v can appear at [S] v min ( > [S] v max ) ,and v increases at high substrate concentrations instead. Two examples are shown by thedashed and dotted lines in Figure 4a. Based on a criterion whether the global v max appearsas [S] → ∞ or at the finite [S] v max , this non-MM kinetic behavior is further divided intoRegimes 4 and 5, respectively.For the generalized N c -loop model, cNESS enzyme kinetics can be similarly analyzedusing reduced parameters from eq 6. In the case that all the non-MM parameters B n arepositive (negative), the turnover rate exhibits negative cooperativity (positive cooperativityor substrate inhibition). With the coexistence of positive and negative non-MM parameters,cooperativity can be qualitatively determined by the small- [S] expansion of the turnoverrate in eq 6, v ∼ B − [S] − ( A B − + P N c n =1 B n s − n B − )[S] + O ([S] ) . For a largely negative P n B n /s n , the positive quadratic [S] term dominates in v , resulting in positive cooperativity.When this summation becomes largely positive, the cancellation between linear and nonlinearterms can slow down the increase of v with [S], inducing negative cooperativity. The signof P n B n /s n is thus a qualitative indicator of cooperativity. To investigate the substrateinhibition behavior, we expand v in the substrate-saturation limit as v ∼ A − − ( B + P n B n ) A − [S] − + O ([S] − ) . For P n B n < − B , v is a decreasing function of [S], and themaximum turnover rate v max must appear at a finite [S]. The investigation of other types ofnon-monotonic behavior for v needs the explicit rate form in eq 6.In summary, we study cNESS enzyme kinetics induced by population currents fromconformational dynamics. Applying the flux balance method to a discrete N × M kineticmodel, we derive a generalized Michaelis–Menten equation to predict the [S] dependenceof the turnover rate. Using reduced non-MM parameters, B n in eq 6, our generalized MMequation provides a systematic approach to explore cNESS enzyme kinetics. Compared tothe typical rate matrix approach, our flux method characterizes non-MM enzyme kineticsin a much simpler way. For example, a unique kinetic indicator α = B /B is defined forthe single-loop model, and phase diagrams are plotted for the single- and two-loop models.12ur study can be extended to other important biophysical processes following the MMmechanism, e.g., the movement of molecular motors induced by ATP binding. Acknowledgments
This work was supported by the NSF (Grant No. CHE-1112825) and the Singapore-MITAlliance for Research and Technology (SMART). D. E. P. acknowledges support from theNSF Graduate Research Fellowship Program.
Supporting Information Available
Four-site, single-loop model calculations
References (1) Michaelis, L.; Menten, M. L. The kinetics of the inversion effect.
Biochem. Z. , ,333–369.(2) Noji, H.; Yasuda, R.; Yoshida, M.; Kinosita, K. Direct observation of the rotation ofF1-ATPase. Nature , , 299–302.(3) English, B. P.; Min, W.; van Oijen, A. M.; Lee, K. T.; Luo, G.; Sun, H.; Cherayil, B. J.;Kou, S. C.; Xie, X. S. Ever-fluctuating single enzyme molecules: Michaelis-Mentenequation revisited. Nat. Chem. Biol. , , 87–94.(4) Lu, H. P. Sizing up single-molecule enzymatic conformational dynamics. Chem. Soc.Rev. , , 1118–1143.(5) Frieden, C. Slow Transitions and Hysteretic Behavior in Enzymes. Annu. Rev. Biochem. , , 471–489. 136) Cornish-Bowden, A.; Cárdenas, M. L. Co-operativity in monomeric enzymes. J. Theor.Biol. , , 1–23.(7) Cao, J. Event-averaged measurements of single-molecule kinetics. Chem. Phys. Lett. , , 38–44.(8) Gopich, I. V.; Szabo, A. Statistics of transitions in single molecule kinetics. J. Chem.Phys. , , 454.(9) Xue, X.; Liu, F.; Ou-Yang, Z. Single molecule Michaelis-Menten equation beyond qua-sistatic disorder. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. , .(10) Min, W.; Xie, X. S.; Bagchi, B. Two-Dimensional Reaction Free Energy Surfaces of Cat-alytic Reaction: Effects of Protein Conformational Dynamics on Enzyme Catalysis † . J. Phys. Chem. B , , 454–466.(11) Lomholt, M. A.; Urbakh, M.; Metzler, R.; Klafter, J. Manipulating Single Enzymes byan External Harmonic Force. Phys. Rev. Lett. , .(12) Xing, J. Nonequilibrium Dynamic Mechanism for Allosteric Effect. Phys. Rev. Lett. , .(13) Qian, H. Cooperativity and Specificity in Enzyme Kinetics: A Single-Molecule Time-Based Perspective. Biophys. J. , , 10–17.(14) Chaudhury, S.; Igoshin, O. A. Dynamic Disorder-Driven Substrate Inhibition and Bista-bility in a Simple Enzymatic Reaction. J. Phys. Chem. B , , 13421–13428.(15) Cao, J. Michaelis-Menten Equation and Detailed Balance in Enzymatic Networks. J.Phys. Chem. B , , 5493–5498.(16) Kolomeisky, A. B. Michaelis–Menten relations for complex enzymatic networks. J.Chem. Phys. , , 155101. 1417) Ochoa, M. A.; Zhou, X.; Chen, P.; Loring, R. F. Interpreting single turnover catalysismeasurements with constrained mean dwell times. J. Chem. Phys. , , 174509.(18) Barato, A. C.; Seifert, U. Universal Bound on the Fano Factor in Enzyme Kinetics. J.Phys. Chem. B , , 6555–6561.(19) Fersht, A. Enzyme Structure and Mechanism ; W. H. Freeman, New York, 1985.(20) Wu, J.; Cao, J. Generalized Michaelis-Menten Equation for Conformation-ModulatedMonomeric Enzymes.
Adv. Chem. Phys. , , 329.(21) Cao, J.; Silbey, R. J. Generic Schemes for Single-Molecule Kinetics. 1: Self-ConsistentPathway Solutions for Renewal Processes. J. Phys. Chem. B , , 12867–12880.(22) Moffitt, J. R.; Bustamante, C. Extracting signal from noise: kinetic mechanisms froma Michaelis-Menten-like expression for enzymatic fluctuations. FEBS J. ,281