Efficient fidelity control by stepwise nucleotide selection in polymerase elongation

Polymerases select nucleotides before incorporating them for chemical synthesis during gene replication or transcription. How the selection proceeds stepwise efficiently to achieve sufficiently high fidelity and speed is essential for polymerase function. We examined step-by-step selections that have conformational transition rates tuned one at time in the polymerase elongation cycle, with a controlled differentiation free energy at each checkpoint. The elongation is sustained at non-equilibrium steady state with constant free energy input and heat dissipation. It is found that error reduction capability does not improve for selection checkpoints down the reaction path. Hence, it is essential to select early to achieve an efficient fidelity control. In particular, for two consecutive selections that reject the wrong substrate back and inhibit it forward from a same kinetic state, the same error rates are obtained at the same free energy differentiation. The initial screening is indispensible for maintaining the elongation speed high, as the wrong nucleotides can be removed quickly and replaced by the right nucleotides at the entry. Overall, the elongation error rate can be repeatedly reduced through multiple selection checkpoints. The study provides a theoretical framework to conduct further detailed researches, and assists engineering and redesign of related enzymes.


I. INTRODUCTION
Polymerases are essential enzymes in charge of gene replication or transcription [1]. A polymerase moves along DNA or RNA while synthesizing a new strand of nucleic acid, largely according to Watson-Crick base pairing with the template strand. Without the polymerase, the templatebased polymerization can also happen, but at an extremely low speed and with low fidelity. The polymerase essentially catalyzes the polymerization and improves the fidelity. Being a nanometer-sized molecular machine, the polymerase works under high viscosity and significant thermal noises. How to achieve sufficiently high fidelity at a sufficiently high elongation speed is thus key to the polymerase function.
From what had been measured experimentally, it was suggested that the polymerase moves as a Brownian ratchet along the DNA/RNA track [2][3][4][5][6][7]. Upon binding and insertion of the incoming nucleotide, backward translocation of the polymerase is inhibited. The nucleotide insertion often accompanies with substantial conformational changes of the polymerase [8,9]. Following the insertion, the nucleotide covalently links to the newly synthesized chain through phosphoryl transfer reaction (see Fig 1a). The catalytic reaction is followed by pyrophosphate ion (PPi) release, which concludes the enzymatic cycle, so that the polymerase can translocate forward and recruit the next nucleotide. From the recruitment to the end of the catalysis, the nucleotide can be selected at multiple kinetic checkpoints.
A wrong nucleotide or an error can be selected against by the polymerase, via destabilizing the intermediate state to increase the backward transition rate (rejection), or via raising the forward activation barrier to decrease the forward transition rate (inhibition) along the reaction path. The nucleotide selection is common to fidelity control of all polymerases. The error rate achieved by the selection can reach as low as one in tens of thousands to one in a million (10 -4~ 10 -6 ) [10]. After the catalysis, or once the nucleotide is covalently added, the error can be further corrected through exo-or endo-nuclease reaction. The enzymatic reaction excises wrong nucleotides, serving for proofreading. In general, the fidelity of the polymerization is controlled through both the nucleotide selection and proofreading [8,9,[11][12][13][14]. The error rate can be lowered by one to three orders of magnitudes further by the proofreading, to as low as one in tens of millions or even lower [10]. The kinetic proofreading had been widely discussed in the context of genetic control [15][16][17][18][19]. In order to achieve high specificity or fidelity, the enzyme and substrate can form multiple intermediate states before generating the final product. The intermediate states are made through driven reactions (breaking the detailed balance) with energy sources. Each kinetic proofreading procedure is implemented through a branching or looping reaction that breaks one of the intermediates back into the apo enzyme and substrate (see Fig 1b). The free energy for differentiation between the right and wrong substrates can thus be repeatedly utilized at those intermediates to achieve high fidelity, through a cascade of the proofreading steps. The proofreading related activities of the polymerases have also been detected at single molecule level in recent years [3,20,21], which inspired further modeling studies [22][23][24][25]. Schematics of polymerase reaction and stepwise nucleotide selection. (a) Polymerase (Pol) enzyme (E) catalyzes a phosphoryl transfer reaction E*NA n + NTP ↔ E*NA n+1 + PPi. The incoming NTP is incorporated according to the template strand. (b) The kinetic scheme for incorporating the nucleotide substrate during polymerase elongation. Nucleotide or substrate selection can happen at any checkpoint prior to chemical catalysis that leads to formation of ES, through backward rejection or forward inhibition (see text). Proofreading happens after formation of ES, while before the final product formation (E+P). Each proofreading step throws errors away via a branched driven reaction, for example, the one destroying ES back into E´+S.
The proofreading-free selection, however, has attracted less attention as it appears simple. Early work suggested that the error ratio of the selection cannot be lower than exp(-Ω max /k B T), which is determined by a maximum free energy differentiation Ω max between the right (cognate) and wrong (non-cognate) substrate along the reaction path [17]. Individual steps of the selection had not been further considered. The overall characterization seemed to suffice as if the selection details were not accessible. Nevertheless, stepwise mechanism of transcription fidelity has been revealed recently, for example, from the structure-based mutagenesis and kinetic analysis [26]. Individual steps of the substrate selection have been characterized to contribute to the overall fidelity. On the other hand, modeling and computation technologies allow protein structural dynamics to be captured down to atomistic scale, providing opportunities to characterize detailed selection mechanisms.
Here we present a model framework to study the stepwise substrate selection, in particular, the nucleotide selection during the polymerase elongation. The selection relies on multiple intermediate states prior to the end of the catalysis. Either the backward transition to the previous state is enhanced, when the intermediate structure is bound with the wrong substrate, or the transition toward the next state is inhibited. Each modulated transition constitutes an "elementary selection" addressed below. The elementary selection happens between two consecutive states along the reaction path (see Fig 1b), without branching back to the apo state as that in the proofreading. The selection is sustained at non-equilibrium steady state (NESS), as long as the polymerase elongates at a nonzero speed [27,28].
The key question we want to address in this study is how to conduct stepwise selection efficiently during the polymerase elongation. Being 'efficient' here means to achieve a sufficiently low error rate at a sufficiently high speed, when the free energy differentiation is limited and controlled. With given kinetic parameters for incorporating the right substrates, and Ω max as a control parameter, we wanted to find comparatively 'efficient' selection strategies, or parameter sets for the wrong substrate kinetics that lower the error rate without necessarily lowering much the speed.
One can see in this work how the elongation speed and error rate vary as { Δ i ± } are allocated differently among the checkpoints along the reaction path.
In early studies, an 'efficiency-accuracy' tradeoff was discovered in substrate selection [19,29,30]. The tradeoff means that a selection system operates close to its maximal accuracy when the enzyme efficiency approaches to zero. The maximal accuracy is determined by exp(Ω max /k B T), and both the accuracy and enzyme efficiency vary depending on the kinetic rates of the system. The tradeoff shows a limit of the selection as the system kinetics parameters vary under experimentally designed conditions, and helps to extract kinetic information of the system [19]. In current study, however, we consider how the error rate and speed vary among different selection strategies, without varying kinetics for the right substrate incorporation.
In this work, we adopted polymerase elongation schemes in general, while using data from T7 RNA polymerase (RNAP) [4,7,31] to demonstrate numerical results. T7 RNAP elongates at an error rate ~ 10 -4 without proofreading activities detected [32]. It is an ideal system to study the nucleotide selection. We analyze first a generic three-state kinetic scheme, building connections with previous quantitative studies. The basic findings are reexamined in a more specific elongation scheme with five states. The kinetic schemes apply to most of polymerases, though different rate-limiting steps happen in different cases. Accordingly, how the findings vary as the ratelimiting step varies is also addressed. In addition, implementations of the present framework to exemplary polymerases are introduced as well. The entropy production and heat dissipation during the information acquisition process of elongation are discussed in the end.

II. ELEMENTARY SELECTIONS AND SELECTION STRENGTH
For the template-based nucleotide incorporation, we consider that the polymerases recognize the nucleotides either as right or wrong. Below we use free energy profiles for incorporating both the right and wrong nucleotides to characterize the stepwise selection, starting from the nucleotide binding to the end of the catalytic reaction. Independent of enzymatic activities, the free energy input (>0) upon incorporating a right nucleotide and a wrong one is ΔG c r and ΔG c w , respectively. The overall difference is . When there is no enzyme, the maximum free energy differentiation Ω max = δ G . The enzyme activities modulate intermediate stabilities or barriers along the reaction path to greatly accelerate the right substrate incorporation while deter the wrong substrate incorporation. Consequently, Ω max rises largely above δ G , while ΔG c r , ΔG c w and δ G keep unchanged. First, we consider elementary selections that tune the free energy profile of incorporating the wrong substrates, for only one transition/activation barrier at a time.
comparison with that of the right substrate; the free energy profile thereafter does not distinguish between the right and wrong, except for a difference δ G in the end. Consequently, the backward transition rate from i to i -1 becomes larger for the wrong than that for the right. One can characterize the selection strength as the ratio between the backward rates ( k i− ) of the wrong and right η i indeed measure the accumulative free energy differentiation. For an efficient selection, it is necessary Δ i ± ≥0 so that the stepwise free energy difference Ω i grows along the reaction path. Below, we focus on how the elementary selections impact on the polymerization speed and error rate, since any selection in general applies as the elementary selections combined together.

III. THREE-STATE ELONGATION SCHEME
We start with a generic three-state kinetic scheme (Fig  3a) to compare two basic nucleotide selection strategies in the elongation cycle, the rejection and the inhibition. The scheme consists of the pre-translocated (I), posttranslocated (II), and substrate state (III). Upon translocation (I→II), an incoming NTP diffuses and binds to the polymerase (II→III), prior to being recognized as right or wrong. A catalytic step then follows (III→I). Correspondingly, recognition and selection of the nucleotide happen either upon substrate binding (III→II) though the rejection, or at the catalytic stage (III→I) through the inhibition.
the free energy difference between the wrong and right species detected at this checkpoint (III→II). As mentioned, one resets δ G at the end of the cycle (see Figure 3b).
Alternatively, the catalytic inhibition S III + raises the activation barrier for the catalysis (III→I) of the wrong nucleotide above that of the right. One can quantify the selection strength by η III one resets in the end. To consider probability fluxes for both the right and wrong species in the three-state scheme, one can define a population vector Π = (P I , P II , P III r , P III w ) T to represent the probability distributions of states I, II and III (for both right and wrong species). The master equation is: (1) where is a 4x4 transition matrix as with k I+ and k II-the forward and backward translocation rate, k II+ and k III-the binding ( ∝ [NTP] ) and unbinding rate of the nucleotide, and k III+ and k I-the catalytic and its reverse rate. i r is the portion of right nucleotides from solution at 'input' (i r =1/4 by default for four equally mixed nucleotides in solution). Err is the 'output' or elongation error rate at the end of the cycle, after nucleotide selection.
η G ≡ e δ G is to keep the overall free energy difference between the right and wrong nucleotide incorporation to δ G .
Using the steady state solution for Eq (1), at k I-→ 0 for simplicity, one obtains the probability flux or the polymerization/elongation rate (or the speed with =1bp), where k max

III.1 Speed modulation by the selection
and Λ = 1 ). Below we show how the selections affect the polymerization rates for three typical cases.
and J 1~J0 , so that the speed keeps high.
ii) When only the catalytic inhibition S III The catalytic rate is effectively reduced , the rate is significantly reduced below J 0 as long as the translocation is not rate limiting. In case that the translocation happens much slower than the catalytic step, Γ →0 holds (see SM Appendix I), then the saturating elongation rate can still keep high under the selection.
In reported or commonly assumed [3,4]. Accordingly, one sees that the strong nucleotide selection through the catalytic inhibition lowers the polymerization rate significantly.
iii) When both selections S III − and S III + work ( η III − > 1 and η III + > 1 ). As both selections get strong, Γ → 0 , the same differentiation free energy Δ / 2 for each. Hence, even a small free energy differentiation (1~2 k B T) at the initial screening can keep the relative speed (J/J 0 ) above 0.5. Indeed, the higher contribution from in the combined strategy, the faster the polymerization rate converges to J 1 .

III.2 Error reduction by the selection
In order to compare the polymerization rates for the right and wrong nucleotides as they compete for binding, one As a result, one obtains the output error rate Err, as the polymerization rate of the wrong nucleotides over that of both the right and wrong: Err is independent of the overall free energy input as the elongation considered is under the strong non-equilibrium limit [27,28].

III.3 Connection with previous work
Current formulation can be easily linked to conventions that focus on enzyme efficiency k max / K M . In the absence of the wrong nucleotides or nucleotide selection, the efficiency is written asζ In the presence of both the right and wrong nucleotides, the efficiency is altered to ζ = ζ 0 / Λ ( Λ > 1), according to Eq 2.
From Eq 2a and 2b, one also obtains the polymerase efficiencies for the right and wrong nucleotides as ζ r = ζ 0 , respectively. That says, the efficiency for incorporating the right nucleotide is fixed, while the efficiency for incorporating the wrong approaches zero as the nucleotide selection gets strong ( Λ → 1 / i r ).
To characterize the fidelity level, one could alternatively use the accuracy A, defined as the efficiency of incorporating the right nucleotides over that of the wrong: is to quantify how much the right portion of the nucleotides is at output relative to that at input. In contrast, Err only counts the percentile of wrong nucleotides at output. As where ζ r is the efficiency incorporating the right nucleotide. Since ζ r = ζ 0 (>0) with the given kinetic parameters for the right, the accuracy A only approaches but cannot be equal to the maximum accuracy e Ωmax/k B T when the selection gets strong (as Δ and Ω max increase). In current work, we focus only on how J and Err vary as the selection becomes strong under different selection strategies. Both ζ and A vary with Err but not J.

IV. A FIVE-STATE ELONGATION SCHEME
Next, we use a slightly more specific scheme with five states (Fig 4a) to describe the polymerase elongation cycle. Comparing to the three-state scheme, the essential difference is there are two instead of one kinetic steps (II → III and III → IV) proceeding to the chemical catalysis. In T7 RNAP and some of other polymerases, the two steps are regarded as nucleotide pre-insertion and insertion [33,34], respectively. In particular, the nucleotide insertion likely happens slowly [8,9,31], so we take it a rate-limiting step by default. Variation of the rate-limiting step in the scheme will be addressed later. Upon the nucleotide insertion, Watson-crick base pairing can form between the right nucleotide and the template.
Besides, in this five-state scheme, the catalytic part proceeds in two steps: The covalent linkage of the nucleotide (IV → V) and PPi release (V → I). Since the nucleotide selection happens before the end of the catalysis, splitting the PPi release from the catalysis or not does not matter to the selection. Essentially, one can identify four elementary selections along the reaction path (shown schematically in Fig 4b) prior to the product formation (V).
The first selection strategy, denoted S III − , rejects wrong nucleotides immediately upon binding. The selection strength can be written as η III  4b).

IV.1 Error and speed control by the selection
To compare the speed and error rates under the above selection strategies, each selection is assumed to work at the same strength η , or use the same amount of differential free energy Δ = k B T lnη . The polymerization rate J / J 0 vs. Δ , the error rate Err, and the input-error ratio One can also write down the output error rate, for simplicity, at the irreversible product release condition (k I-→ 0) (4) η G = e δ G is to reset prior to the product formation; η G = 10 is used by default. One Similarly, the latter two selections S IV − and S IV + also perform equally well in the error reduction. However, the performance of the latter two is inferior to the former two. Indeed, when only the first two selections work and are equally strong (η III In contrast, when only the two latter selections work and are equally strong (η III Since κ III < κ IV , it leads to a smaller error rate in the first two selections, starting from state III, than that in the two selections from state IV. The feature is due to the linear reaction topology and is independent of the kinetic parameters (in addition, see SM Appendix III for a sixstate scheme). When the last transition or the full scheme is The catalytic rate is low, and is much lower than the reverse rate of the nucleotide insertion; (h) The translocation is rate limiting.

IV.2 Kinetic impact and variation of the rate-limiting step
From the derivation, one can heuristically write down the elongation error rate (with an irreversible last step in the elongation scheme) in general as: If one lowers the rates of forward transitions involved in the selection (e.g. k III+ , k IV+ or both), or raises the rates of backward transition, one can reduce the value of κ and lower the error rate. Of course, this type of accuracy improvement is at price of lowering the speed, as that indicated in the efficiency-accuracy tradeoff. One can also see how the speed and error rate control varies when the rate-limiting step varies in the elongation cycle. Here the rate-limiting step is determined according to the kinetics of the right substrate. First, if one lowers the NTP concentration, the nucleotide binding can become rate limiting. We see from Fig 4f that all selections significantly lower the speed below that without selection. The first selection S III − also happens after the slow NTP binding, and cannot recover the speed to high. Next, if the catalysis becomes so slow such that the catalytic rate is much smaller than the reversal rate of the nucleotide insertion (k IV+ <<k IV-; see Fig 4g), then the error rate achieved by the former two selections ( S III − and S III + ) become almost identical to that from the latter two ( S IV − and S IV + ). Additionally, if the translocation after the product release happens quite slowly, then the elongation rate is dominated by the translocation rate, and cannot be reduced much by any nucleotide selection (see Fig 4h). Hence, the error and speed control patterns persist but become more or less pronounced at different rate-limiting conditions.

V. DISCUSSION
In current work, we show how stepwise nucleotide selection could proceed efficiently for fidelity control in polymerase elongation. Basically, we want to identify selection strategies that achieve comparatively low error rates for certain free energy differentiation, without significantly lowering the polymerization speed. From previous studies, various ways of nucleotide selection had been reported for different polymerases [12]. As ratelimiting steps also vary among different systems, it is hard to identify common selection mechanisms. In this study, we demonstrate that for efficient selections, there exist some general features in the selection systems, as summarized below. However, we want to make it clear that polymerases are not necessarily evolved to be highly efficient in the selectivity. Their functional development has to meet various internal and external requirements.
To characterize stepwise nucleotide selection, one needs to consider the free energy differentiation between the right and wrong substrates at every checkpoint along the reaction path. The free energy differentiation Δ (>0) at any particular checkpoint relies on physical properties of the enzyme and ambient conditions. For example, to differentiate the substrate species, some structural or electrostatic characters of the protein have to be developed, while water molecules need to be more or less excluded, and certain ions may also be required for coordination [11,35,36]. Accordingly, the selectivity demands very specific and fine-tuning of molecular interactions, and the differentiation capacity is restricted at any one checkpoint ( Δ cannot be very large).
Indeed, an elementary selection is either to inhibit the forward transition or to enhance the backward transition in the wrong substrate incorporation, by modulating the transition activation barrier by Δ in comparison to that of the right species. When Δ is fixed at every selection point, while the system kinetics varies in controlled conditions, the selection accuracy can be improved at compensation of the reaction efficiency.
On the other hand, if the reaction kinetics of the right substrates is given, while Δ is allowed to increase at one checkpoint, the error rate can be continuously lowered while the speed converges to a constant value. Depending on which selection checkpoint is exploited on the reaction path, the error rate and speed vary for a same value of Δ .
Our study shows that early selections on the reaction path outperform the late ones on the error reduction, and the initial selection is indispensible for maintaining the speed high. Here, the early selection starts upon the substrate binding, and the late one ends once the catalysis finishes. We essentially show that the error rate can be repeatedly lowered through the stepwise selection. That is to say, multiple kinetic checkpoints along the reaction path do improve the fidelity level as the free energy for differentiation accumulates. Mathematically, this property is similar to the amplifying effects in kinetic proofreading, as the elongation cycle is essentially maintained at the NESS with the detailed balance broken.
In previous sections, we compared error reduction and speed modulation of the elementary nucleotide selections in the three-and five-step elongation scheme. The three-state scheme is characterized by one-step NTP binding and two potential kinetic checkpoints, while two-step NTP insertion and four potential checkpoints apply in the five-state scheme. For mathematical simplicity, we assumed in both schemes that the PPi dissociation step is irreversible, as if PPi concentration is quite low around the active site of the polymerase. Similar results show as well in the fully reversible scheme on the speed and error control (see SM Appendix II). We also checked four-state and six-state kinetic schemes (see SM Appendix III). The four-state scheme is similar to the three-state scheme if there is only one-step NTP binding prior to the chemical catalysis; the scheme becomes similar to the five-state case if the NTP insertion happens in two steps. In the six-state scheme, we made a three-step pre-chemical NTP insertion process, with six potential checkpoints in total. The variations of the kinetic scheme lead to no essential changes in the speed and error control in the corresponding elementary selections.
Below, we summarize crucial aspects of the efficient fidelity control, which is to achieve low error rate without lowering much the speed. In the end, we apply current framework to describe some particular polymerases, and use this framework as well to analyze the information acquisition features of the selection system.

V.1 Achieve low error rates -select early, properly, and repeatedly
We have examined elementary selections that tune only one transition barrier forward or backward at a time when incorporating the wrong nucleotides. The elementary selections are arranged sequentially along the reaction path of the elongation cycle. Our results highlight two interesting findings (i) The error rate achieved by the selection that rejects the wrong nucleotides from state i to i-

S i+1
± becomes insignificant. That is, when k IV+ << k IV-in the five-state scheme, the error rates becomes almost the same for all four elementary selections. In brief, the error reduction performance of the selection does not improve following down the reaction path. The property would persist in general to any elongation scheme. As any nucleotide selection strategy can be regarded as a combination of the elementary selections, the above results give some rules of thumb on identifying a proper selection strategy. First, as a direct consequence of (i) above, one to achieve an error reduction. Here the anti-selection indicates an operation with a 'selection' strength less than one, which favors the wrong substrate rather than the right one. The futile strategy is illustrated in Fig 5a left, . It shows that an 'improperly' combined selection strategy cannot improve the fidelity over that of an elementary selection. It also indicates that to make a proper selection strategy to take advantage of every kinetic checkpoint, the separation between the free energy files of the wrong and right has to gradually expand along the reaction path. Indeed, we obtained a general expression of the elongation error rate as a function of individual selection strength in Eq 4c. The properties (i) and (ii) summarized above are the natural consequences of this expression (due to the term ± ∑ /k B T for any combined selection at the strong selection limit ( e Δ i ± /k B T >> 1 ). Importantly, it indicates that the error reduction can be amplified through multiple steps along the reaction path. Since κ >0 , it is easy to see that the error rate cannot be lower than becomes the maximum accuracy as suggested previously [17]. Note that the maximum free energy differentiation Δ i ± ∑ = Ω max over the reaction path is well defined as long as  right, but is indeed irrelevant to the selectivity). Hence, an overall description of the stepwise selection using only the maximum free energy differentiation can be insufficient or even misleading.

V.2 Maintain high polymerization speedinitial screening is indispensible
Our results indicate that the overall elongation rate is more or less reduced upon any nucleotide selection. It is consistent with the understanding that raising the accuracy is achieved at the price of lowering the speed. Indeed, the sort of tradeoff idea applies for a certain differentiation capacity. When the energy differentiation increases for any selection checkpoint, the error rate can be continuously lowered while the speed converges to a certain value. Nevertheless, for a constant energy differentiation capacity, varying the selection checkpoint can possibly improve both the fidelity and speed. For the very first selection, it outperforms the later selections not only in achieving the low error rate, but also in maintaining high speed. When Δ increases, the wrong fluxes uniformly decrease to zero. On the other hand, the right fluxes diminish to small values for all but the initial screening ( S III − ). The more stringent the initial screening, the higher the polymerization flux of the right nucleotides, as most of the wrong species are expelled away soon at entry and replaced by the right species. However, the initial free energy differentiation can be quite limited: When the nucleotide is just recruited, the site is relatively open and water solvent is not well excluded yet. Hence, it is unlikely to achieve nucleotide selection largely at the beginning. Nevertheless, when initial screening is combined with selections performed later in the cycle, the polymerization speed would still be maintained high, approaching to the speed under the initial screening alone. This is because the presence of the initial screening makes it efficient to throw away the wrong substrates in the proofreading-free system. Hence, even a small portion of initial screening in the combined selection can lead to fairly robust polymerization rates insensitive to input error rates from the solution. The properties highlight the importance of including the substrate screening at the beginning for an efficient selection, even there is only a limited amount of free energy differentiation.
When the nucleotide concentration is very low, however, even the initial selection can lower the elongation rate significantly. This is because the selection happens after the rate-limiting nucleotide binding, and the replacement of the right nucleotides slows down. On the other hand, if the ratelimiting step happens far behind all selection checkpoints, such as at the translocation, then the selection hardly impacts on the speed. Anyhow, it has not been reported the translocation being the single rate-limiting event, though it can be similarly slow as some catalytic or pre-catalytic event [37]. In brief, for polymerase elongation at high nucleotide concentration, the initial selection always helps to maintain the speed high, while later selections lower the speed. This gives some clues to determine essential amino acids for the initial screening through mutagenesis: For mutations that adversely affect the accuracy and speed, the original amino acids at the mutation sites likely contribute to the initial screening; for those that lower the accuracy but not the speed, either the original amino acids do not contribute to the initial selections, or the mutations still keep the initial screening on.

V.3 The nucleotide selection in some exemplary polymerase systems
In this work, we have used elongation kinetic data of single-subunit T7 RNAP [4,7,31] for numerical demonstration (see kinetic parameters in Table S1 from SM Appendix IV). Though the three-state scheme had been employed in early work for experimental data fitting [4], later on studies supported a five-state elongation scheme in this system [31]. In the five-state scheme, the pre-chemical nucleotide insertion following the initial nucleotide binding/pre-insertion is regarded rate limiting [31]. Our recent molecular dynamics simulations show that substantial nucleotide selection happens prior to the full insertion of the nucleotide into the active site in T7 RNAP [38]. That is, both the initial screening S III − upon NTP perinsertion and the second selection S III + during the NTP insertion play essentials roles in the nucleotide selection. Hence, T7 RNAP seems to be a quite efficient selection system that can fully employ the early selections on the reaction path. Since the error rate achieved by T7 RNAP is ~ 10 -4 [32], one can estimate the maximum or accumulate free energy differentiation at ~ 10 k B T. Actually, T7 RNAP achieves the error rate without proofreading detected. This likely explains why the nucleotide selection has to be efficient in this RNAP.
Next, we examined selection kinetics in T7 DNAP as the kinetic rates for incorporating both the right and wrong nucleotides had been reported in this system [14] (see SM Appendix IV). The chemical catalysis proceeds more slowly than the nucleotide insertion in T7 DNAP, while the reversal of the nucleotide insertion happens extremely However, the initial screening does not seem to be strong enough to support very high speed (see SM Appendix IV); the first two selections also do not appear strong enough to make the overall selection highly efficient. The full selection gives an error rate ~ 10 -3 in T7 DNAP. The performances seem to leave room for further improvements by proofreading, which is indeed required and substantial in this DNAP.
In multi-subunit RNAPs, recent mutagenesis studies nicely show that discrimination against wrong nucleotides proceeds via a stepwise mechanism, and each step contributes differently to the overall fidelity [26]. In these systems, at least two steps happen prior to the chemical catalysis [39], starting from the NTP entry. In particular, the non-complementary NTPs are discriminated efficiently through both the first and second checkpoints, at the open active center and through a trigger loop folding process, respectively. The discrimination of the deoxy-NTPs does not happen until after the first checkpoint, hence, appearing less efficient. Indeed, the regulation of the enzyme activities is largely controlled through the trigger loop folding, providing possibilities that several selection checkpoints are coordinated in the fidelity control. It is not clear which step is rate limiting in the multi-subunit RNAPs. Likely one slow event takes place before or during catalysis, and another slow event is around translocation stage [37]. In that case, the speed modulation of the nucleotide selection may not be significant.

V.4 Entropy production and heat dissipation under nucleotide selection
Last, we use current framework to quantify entropy production and energy dissipation under the nucleotide selection during the elongation NESS [40,41]. These quantities are physically important to the selection system at nano-scale, but are not well defined in conventional simplified kinetic studies. Consider that the selection can never be perfect to prevent all errors, one expects at least two species (right and wrong) identified during the elongation. In solution of the mixed nucleotides, however, there is no way to identify the nucleotide species, so all input substrates are regarded as one species. Hence, during the template-based polymerization process, one always expects entropy variations upon the nucleotide species recognition.
The entropy variations can be decomposed into two components, the entropy production and heat dissipation [40,42], with their respective rates denoted  Ξ p and  H d .
We derived both quantities in Appendix V in SM as: is the overall free energy input. Hence, the net entropy change rate can be written Here the protein-solution contribution to the entropy production was not counted in. We focus only on the information content of the polymer chain. The chain disorder can indeed drive the polymer growth [28]. The entropy production and heat dissipation rates vs. the differentiation free energy (as in the five-state scheme) are provided in Fig S5 in Appendix V. In current example, the highest overall entropy change rate is close to ~80 k B T/s at Δ ~ 2 k B T under the initial screening S III − . The corresponding entropy production rate is close to 1500 k B T/s, with most part being dissipated as heat.
The entropy production and heat dissipation per nucleotide cycle ( /J or /J) are shown in Fig 5c left. The entropy production increases with the selection strength and then decreases. The heat dissipation always increases with the selection strength, until it converges to the entropy production. As Err ->0, the net entropy change approaches zero (one unidentified species in, and one right species out).
In Fig 5c right,

VI. CONCLUSIONS
In this work we have studied how stepwise nucleotide selection could proceed efficiently during template-based polymerase elongation. Basically, the selection can happen at multiple checkpoints prior to the end of chemical catalysis or the product formation. At each state, conformational transition of the enzyme backward or forward is enhanced or inhibited when the enzyme is bound with a wrong nucleotide. The selection through a single backward or forward transition is regarded as an elementary selection, and any selection in general can be regarded as a combination of the elementary selections. An efficient selection strategy takes advantage of multiple selection checkpoints to reduce the error rate repeatedly, and selects as early along the reaction path. At the same time, any selection checkpoint is subject to structural and energetic constraints to differentiate the nucleotide species. The efficient selection strategy achieves a low error rate with a limited amount of differentiation free energy accumulated along the reaction path, while minimally perturbs the overall elongation rate or speed.
We found that at the sufficiently high nucleotide concentration, the initial screening selecting against wrong nucleotides immediately upon their arrival perturbs the elongation rate slightly, while selections thereafter on the reaction path can significantly diminish the elongation rate.
Importantly, combing the initial screening with selections afterwards keeps the speed similarly high as that under the initial screening alone. Hence, for polymerases that need high cycling rates, the initial screening seems indispensible, and even a small differentiation there can help. Interestingly, we found that the early selections along the reaction path outperform the late ones in the error reduction, as lower error rates are achieved under the early rather than the late selections at the same free energy differentiation. In particular, for a pair of neighboring elementary selections, the one rejects the wrong substrate state back to the previous state and the one inhibits it toward the next state give a same error rate at the same free energy differentiation. These properties persist but become more or less pronounced at different rate-limiting conditions. In counting the entropy production rate of the selection system, we notice that a large portion of the entropy production is dissipated as heat in maintaining the elongation far from equilibrium. Comparing to the late selections, the early selections promote the information entropy production when the selection is weak, while quench the entropy production when the selection gets strong. Based on this framework, we compared a proofreadingfree T7 RNAP with a proofreading T7 DNAP. We found T7 RNAP to be an efficient selection system while T7 DNAP does not seem so. The current work of the stepwise nucleotide selection supports further quantitative researches to reveal underlying mechanisms of the selection. It may further help molecular engineering and redesign of efficient selection systems.

Supplementary Material for "Efficient fidelity control by stepwise nucleotide selection in polymerase elongation" Appendix I: The generic three--state scheme and the efficiency--accuracy tradeoff
In Eq 2 in main text, the polymerization flux or rate J is obtained in the Michaelis--Menten form. The maximum rate constant k max 0 and the Michaelis constant K M 0 are written as: ). Correspondingly, the exact forms of Γ and Λ are: The approximation in Eq S3 is taken for kIII+ << kI+, that is, the translocation rate kI+ is much larger than the catalytic rate kIII+. It had been measured that the translocation is much faster than other kinetic steps in the elongation cycle 1, 2 . If the translocation slows down, the value of Γ will decrease, and the impact from Γ weakens. As kIII+ >> kI+, Γ →0, the selection strength η can only modulate K M rather than k max in the Michaelis--Menten form of the elongation rate. In that case, the nucleotide insertion happens very fast and the selection cannot affect much the elongation rate at fairly high nucleotide concentration. To make it clear how the individual selection strength η = e Δ/k B T or the differentiation free energy Δ affects Γ and Λ to modulate the rate and efficiency of the polymerization, Fig S1 below   As in the efficiency--accuracy tradeoff as discussed early 3--5 , we can calculate straightforward the efficiency ζ r for incorporating the right nucleotide substrates: where κ a ≡ k T 0 k I + k I + + k II − = k T 0 1+ k II − k I + is the effective binding constant in the three--state cycle, κ a~kT 0 / 2 as polymerases translocate in a Brownian ratchet fashion, with equal forward and backward rates (kI+ ~ kII--) 6 . From Eq 3a in main, we write

Appendix III: A six--state scheme with three pre--chemistry transitions
One can also build a six--state kinetic scheme by putting another 'insertion' step (see Fig S3a  below) prior to the chemical catalysis, in addition to the pre--insertion and insertion step in the five--state scheme. Solving an equation similar to Eq 1 but in a ten--state vector space: =(PI, PII, PIII r , PIII w , PIV r , PIV w , PV r , PV w , PVI r , PVI w ) T ('r' and 'w' labeling for probabilities of the wrong and right nucleotide bound states, respectively), one obtains the error rate: (S7) The diagrams on polymerization rates and error rates under the selections are:  Fig 4 (c--e). From the above results we see that the conclusions listed in the previous section still hold. In particular, one sees that two neighboring selections against wrong nucleotides, S i − and S i + from the same initial state i, give the same error rates at the same selection strength (asη i − = η i + ).