T he severe acute respiratory syndrome coronavirus (SARS- CoV) belongs to the Nidovirales order of viruses, which includes the family of coronaviruses (CoVs). 1 SARS-CoV has an unusually long RNA genome with ∼29.7 kb. [1][2][3][4] The SARS-CoV genome encodes 16 nonstructural proteins (nsps1-16) that all assemble into one huge replication-transcription complex. 1,5 This large multiprotein complex is comprised of an RNA-dependent RNA polymerase (nsp12 RdRp) and a collection of other cofactor enzymes that all cooperate with the RdRp during replication and transcription of the SARS-CoV genome. 1 The SARS-CoV RdRp complex adds an incorrect nucleotide into the nascent RNA chain once in every one million nucleotides, and thus, has a very low error rate on the order of 10 -6 . 1,4 In contrast, other RNA viruses such as the measles virus are more error-prone with an error rate on the order of 10 -4 . 6 The strikingly low error rate of the SARS-CoV RdRp complex suggests that the virus most likely removes the wrongly incorporated nucleotides via a proofreading process. Another major clue that the SARS-CoV RdRp complex had proofreading ability was revealed when patients infected with SARS-CoV did not respond to treatments with ribavirin (Rbv), a drug that mimics naturally occurring nucleotides. 7 This lead to the hypothesis that the SARS-CoV RdRp complex could incorporate nucleotide analogues such as Rbv into the nascent RNA chain and then remove them with proofreading. [7][8][9] Kinetic and crystallographic experiments have established that the SARS-CoV RdRp complex does indeed possess proofreading activity. [10][11][12] It has been shown specifically through in vitro experiments that the nsp12 RdRp (in complex with cofactors nsp7 and nsp8) directly binds to nsp14. 11,[13][14][15][16] The bifunctional nsp14 enzyme (in complex with its cofactor nsp10) consists of an exonuclease (ExoN) domain that confers 3′-5′ proofreading ability to the SARS-CoV RdRp complex, and an N7-methyltransferase (N7-MTase) domain that adds a 5′ methylguanosine cap to the viral mRNA transcripts produced by the nsp12 RdRp. 7,[11][12][13]17,18 Despite the extensive body of experimental evidence demonstrating that the SARS-CoV RdRp complex does have the proofreading ability due to the presence of the ExoN domain, 7,[10][11][12][13]17,18 it is still not clear why the RdRp complex needs to detect and remove wrongly inserted nucleotides. This is especially surprising since the majority of other RNA viruses do not utilize proofreading. In addition, for biological systems with the proofreading ability, it has been shown that all of the so-called characteristic properties cannot be optimized at the same time because there are trade-offs that exist between them. [19][20][21][22][23][24][25] For example, it is possible that the RdRp complex operates at a slower speed during genome replication in order to achieve its low error rate on the order of 10 -6 . Moreover, trade-offs may also arise between the speed or error rate and the energetic cost of proofreading (fraction of extra ATP molecules consumed during futile cycles per step). [23][24][25] In this work, we address three important questions regarding the molecular mechanisms and the interplay between the characteristic properties of the SARS-CoV RdRp complex: (i) Why does the virus need proofreading to make copies of its unusually large genome? (ii) What would be the functionally tolerable and optimal rate of proofreading? (iii) How does the RdRp complex prioritize among the error rate, speed, and proofreading cost? To answer these questions, we applied our recently developed methodology 21,23,24 to analytically compute the speed, accuracy, and cost and see how variations in the rate constants of underlying chemical transitions affect their tradeoffs. To gain additional insights into the interplay between the error rate and accumulation of mutations, the fraction of nonfunctional virus copies from the SARS-CoV genome analysis was determined. 26,27 To model the kinetic proofreading (KPR) mechanism 28,29 for the SARS-CoV RdRp complex, we used the scheme shown Figure 1A. The KPR mechanism consists of quasi-first-order transitions for the reversible elementary reactions with the rate constants (kinetic parameters) k ±i,R/W as indicated. The RdRp complex replicates the SARS-CoV genome by adding RNA nucleotides to a nascent strand in the 5′-3′ direction using the complementary strand as a template. 1,5 The KPR mechanism gives the RdRp complex the ability to preferentially discriminate between cognate (right R) and noncognate (wrong W) nucleotides. 28,29 Following the insertion of one nucleotide, the RdRp complex can either incorporate another nucleotide into the growing chain or remove the newly incorporated nucleotide during proofreading in the 3′-5′ direction via the nsp14 ExoN domain. 7,10,[10][11][12]18 The removal of a nucleotide from the nascent strand via the proofreading pathway consists of two steps, namely, polymerase-exonuclease (Pol-Exo) sliding and hydrolysis. 30 In the first step, the RNA nucleotide slides out of the nsp12 RdRp into the nsp14 ExoN domain. After the RNA nucleotide has entered the nsp14 ExoN domain, it is removed via hydrolysis during the second step. This KPR mechanism enables the RdRp complex to reset back to its original state prior to the addition of the new nucleotide. Thus, the proofreading activity of the RdRp complex leads to the futile hydrolysis of one nucleotide triphosphate (NTP) molecule that is not added to the growing RNA chain. 31 The kinetic parameters k ±i,R and the corresponding selectivities (discrimination factors) f ±i = k ±i,W /k ±i,R for each elementary reaction of the KPR mechanism were determined from different kinetic experiments and theoretical studies on the SARS-CoV RdRp complex and on other viral enzymatic systems (see the Supporting Information Table S1). To this end, the polymerization rate constant k 1,R = k p,R comes from a recent kinetic experiment on the SARS-CoV RdRp complex. 32 However, due to the lack of kinetic experiments on the SARS-CoV RdRp complex and the rapidly evolving nature of this field, we had to extrapolate the other kinetic parameters from kinetic experiments and/or theoretical studies on the Poliovirus RdRp (3Dpol) 33 and the T7 DNA polymerase. 21,23,34,35 We used the kinetic parameters from these enzymes due to the proposed kinetic and structural similarities between these viral polymerases and the SARS-CoV RdRp complex. 5,10,33 Moreover, we constrained the KPR model by setting the error rate of the SARS-CoV RdRp complex to the experimentally measured one: η nat = 3.4 × 10 -6 . 3,4,27 Then, this experimental error rate was used to determine the catalytic rate k p,W for incorporation of a mismatched nucleotide (i.e., for the EW → EWR step). This was done because the error rate is primarily controlled by the discrimination factor f p = k p,W /k p,R of the product formation (see the Supporting Information for more details).

The characteristic properties of the SARS-CoV RdRp complex, namely, error rate η, kinetic speed V, and proofreading cost C, are all defined in the so-called linear framework 36 (see the Supporting Information for the mathematical derivations from the forward master equations formalism). The forward master equations for probability distributions of the 5 states in the KPR scheme (Figure 1A) are solved exactly at steady state, and the solution can be used to obtain the stationary fluxes for each transition. Thus, it is straightforward to examine how perturbations of the kinetic parameters affect the interplay among the three characteristic properties. In this framework, the error rate is defined as the ratio of the stationary flux to create the wrong W product to the stationary flux to create the right R product, and the speed is defined as the stationary flux to create the right R product. The proofreading cost is defined as the ratio of the proofreading flux to the product formation flux in the steady state. 21,23,31,37 Thus, in the microscopic picture, the proofreading cost quantifies the fraction of extra NTP molecules hydrolyzed during the futile resetting cycles that do not result in product formation. 31 The proofreading cost is related to the energy dissipation normalized per flux of product formed as discussed in our previous works. [23][24][25]37 Thus, the proofreading cost can be interpreted as an extra energetic cost.

We first investigated why the SARS-CoV RdRp complex has the proofreading ability. To this end, we determined how the error rate and the speed change with the Pol-Exo sliding rate k 2,R . It is clear from Figure 2A that the error rate η monotonically decreases with increasing Pol-Exo sliding rate k 2,R . Thus, the SARS-CoV RdRp complex operates at a lower error rate and removes more wrongly incorporated nucleotides when the Pol-Exo sliding rate is higher. In contrast to the error rate, the speed V depends nonmonotonically on the Pol-Exo sliding rate k 2,R (Figure 2B). For increasing the Pol-Exo sliding rate, the speed first increases and the RdRp complex gets faster until it reaches the maximum, but then the speed starts to decrease due to an increase in the probability of removing correctly incorporated nucleotides from the growing RNA chain. To the left of the maximum, the speed of the RdRp complex decreases as the lower Pol-Exo sliding rate will lead to the complex spending relatively more time in the EW state as is evident by the increased return of the probability P EW (see the Supporting Information Figure S6). Notably, for the estimated parameters, the native system (red dot) is within ∼0.2% of the maximum speed (blue square in Figure 2B).

Given that the Pol-Exo sliding rate constant does not optimize for error rate and further reduction of error rate is possible at a higher sliding rate (see Figure 2A), we asked whether the native error rate η nat = 3.4 × 10 -6 is a tolerable one for the virus given its large genome size. To answer this question, the interplay of the error rate η and the mutation probability p, which is defined as the fraction of virus copies that would have reduced fitness due to critical mutations, was examined. We argue that mutations that either insert a stop codon into a viral protein (nonsense) or that change one amino acid to a significantly different one (severe missense) can have such an effect (see the Supporting Information for exact definitions). The mutation probability p for these mutations was calculated by analyzing the effects of nucleotide substitutions in the SARS-CoV genome analysis (Figure 1B). Assuming that the errors in each step are independent, for each value of the error rate we compute the probability p = 1 -∏ i (1n i η) that at least one such mutation will occur during replication of the whole genome. Here, n i is the number of mutations on the nucleotide and the product is taken over all bases i that may lead to a virus copy with a nonsense or severe missense mutation. As expected, the decrease in the per-step error rate lowers the mutation probability (Figure 2C). We then selected approximate thresholds for the mutation probabilities of the nonsense and severe missense mutations based on the native error rate η nat (red dot). Using this analysis, we determined thresholds of 2% for the probability of nonsense mutations and 5% for the probability of severe missense mutations to postulate a range of error rates that would be tolerable for the virus (gray shaded region in Figure 2C.) One can note that the error rates for the estimated rate of Pol-Exo sliding lie right near the boundary of this range. However, significant reduction or elimination of Pol-Exo sliding and, hence, proofreading would have much higher mutation probabilities as large as ∼0.5 (cyan diamonds for η ∼ 10 -4 ). Therefore, proofreading ensures that the SARS-CoV RdRp complex operates at a tolerable error rate that keeps the mutation probability and, thus, the fraction of nonfunctional virus copies small.

To understand the effect of proofreading on the overall replication rate of the SARS-CoV RdRp complex, we also studied the interplay of the error rate with the effective speed V eff = V(1p)/N, which is defined as the rate of production of functional virus copies (without the above-mentioned deleterious mutations) over the entire SARS-CoV genome for nonsense (dashed-dotted black curve) and severe missense (dashed violet curve) mutations. D. Dependence of the effective speed for the whole SARS-CoV genome V eff on the error rate η for nonsense (dashed-dotted black curve) and severe missense mutations (dashed violet curve). The red dot denotes the native system, the blue square denotes the maximum speed, the orange and green squares denote the maximum effective speed, and the cyan diamonds denote the system at a very low Pol-Exo sliding rate, k 2,R ∼ 10 -4 s -1 .

The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter with N = 29725 bases. Depending on which mutations are postulated to be nonfunctional, different mutation probabilities p lead to different values of V eff . It is expected that this quantity should have a nonmonotonic dependence on the error rate as an increase of speed due to a decrease in Pol-Exo sliding rate competes with the reduction of the fraction of nonmutated viruses produced. The result shown in Figure 2D demonstrates that there is indeed a nonmonotonic dependence of the effective speed on the error rate. The native experimentally observed value of the error rate at η nat = 3.4 × 10 -6 nearly maximizes the effective speed (within ∼1% of the maximum shown by the orange square for nonsense mutations and within ∼4% of the maximum shown by the green square for severe missense mutations). Elimination of the proofreading or a significant increase in the proofreading would lead to a decrease in the effective speed. Therefore, due to the large size of the SARS-CoV genome, the error rate must be small enough to keep the effective speed of the RdRp complex close to its optimal value. Interestingly, the SARS-CoV RdRp complex does not completely optimize its effective speed, and, in fact, the native error rate lies on the non-trade-off branch of the plot in Figure 2D where an increase in the Pol-Exo sliding rate will significantly decrease the error and slightly improve the effective speed. While we could not exclude the possibility that this effect is due to uncertainty in the estimation of the kinetic parameters, we nevertheless asked if other characteristic properties of the system can explain why the SARS-CoV RdRp complex does not completely maximize its effective speed. To this end, the interplay between the effective speed V eff and the proofreading cost C was analyzed. The value of the Pol-Exo sliding rate k 2,R was varied, and the corresponding properties are presented as a parametric plot in Figure 3A. The results indicate that a further increase in effective speed V eff and the corresponding decrease in the error rate would significantly increase the cost. For instance, for the severe missense mutations (blue square), the cost would increase almost by ∼1 order of magnitude relative to the cost of the native system (red dot). This nearly 1 order of magnitude increase in the cost is much larger than the corresponding increase in the effective speed of only ∼4%. Thus, it is possible that the SARS-CoV RdRp complex cannot increase its effective speed to the maximum V eff because the proofreading cost C would increase dramatically.

To test the robustness of our conclusions with respect to the estimated uncertainties in the kinetic parameters of the SARS-CoV RdRp complex, we performed a sensitivity analysis. To this end, we randomly perturbed the values of the kinetic parameters k ±i,R from their estimated values stated in the Table S1. Perturbed values of the right nucleotide parameters were sampled from uniform distributions with the respective ranges of [0.5k ±i,R , 2k ±i,R ] and [3k ±i,R , 6k ±i,R ] (200 from each) while keeping all of the discrimination factors f i fixed. The first range was chosen because we would like to examine the robustness of our conclusions for smaller variations (i.e., perturbations) of the kinetic parameters. We chose the second range to include the range of the polymerization rate constants at the physiological temperature (i.e., 600-700 s -1 ) proposed by Shannon et al. 32 For each parameter set, we analyzed the effect of the perturbations in the kinetic parameters on the error rate η, speed V, and proofreading cost C of the SARS-CoV RdRp complex. First, we tested whether or not the native error rate would still be far from the minimum error rate but much smaller than the error rate with no proofreading. Second, we tested if the speed would still be nearly optimized. Last, we tested if the proofreading cost at the optimal speed would still be much larger than the native cost. The results of the analysis (see the Supporting Information, Figures S4 andS5, for more details) show that our conclusions about the error rate, speed, and perhaps to a lesser extent the cost are robust in both ranges.

The results discussed so far investigated the interplay among the error rate (mutation probability), speed, and proofreading cost of the SARS-CoV RdRp complex as we varied the rate of Pol-Exo sliding. This analysis shows that the value of this rate constant estimated from the experimentally determined error rate nearly optimizes the speed while keeping the error rate tolerable. Somewhat surprisingly, in contrast to other studied enzymatic systems, 21,23,24,37 our analysis points to the importance of minimizing the proofreading cost rather than maximization of the speed. To understand how these results would hold when other kinetic parameters are varied, we next performed a more global analysis of the trade-offs.

First, following the approach introduced in Yu et al., 24 we asked if the observed proofreading cost is at or near its global minimum value for a given error rate. To this end, we generated the scatter plot in Figure 3B from variation of all the rate constants with global parameter sampling under thermodynamic constraints, i.e., fixed discrimination factors (selectivities) f i = k i,W /k i,R for each elementary reaction and fixed chemical potential differences Δμ cycle around each biochemical cycle (see the Supporting Information for details). The interplay between the proofreading cost C and the error rate η shows a trade-off between the two properties with the cost-error bound that gives the theoretical minimum cost for a given error rate. Along this boundary, a decrease in the error rate leads to an unavoidable increase in the proofreading cost. On one hand, without the proofreading the cost might vanish (magenta dot for C = 0 from a corresponding Michaelis-Menten system with no futile cycles where k 2,R = k 3,R = 0), but this leads to an ∼30 fold increase in the error. Thus, these results further confirm that the dissipation is unavoidable for the SARS-CoV RdRp complex to achieve the tolerable error rate. On the other hand, further minimizing the error would lead to a significant increase in the cost of ∼10-fold (blue square) as we saw with the single parameter variation in Figure 3A. Consequently, due to the global trade-off between the cost and the error rate, the SARS-CoV RdRp complex must keep its proofreading cost small enough to maintain a tolerable error rate. Importantly, as one can see from the inset of Figure 3B, the native system lies very close to the trade-off bound (within ∼6%). Thus, the RdRp complex nearly minimizes the proofreading cost for the observed error rate, and our analysis further confirms the importance of considering the energy dissipation for understanding the functionality of the SARS-CoV RdRp complex.

To understand the underlying reason why the cost of the native system does not lie exactly on the cost-error bound, the interplay between the cost C and the effective speed V eff for a fixed error rate was analyzed. To determine if changes in other kinetic parameters can lead to further cost minimization for the same error, we varied the polymerization rate k 1,R = k p,R and the Pol-Exo sliding rate k 2,R so that the error rate was fixed at its native value and computed the proofreading cost C and effective speed V eff . The result in Figure 3C shows the nonmonotonic curve for the cost versus the effective speed. The native SARS-CoV RdRp complex (red dot) sits on a nontrade-off branch of the cost-speed curve; i.e., a decrease in the costs will result in a decrease in the speed. The SARS-CoV RdRp complex could further decrease the cost (red dot) by almost ∼6% to reach the minimum cost (turquoise square). However, the RdRp complex is prevented from completely optimizing the cost because the effective speed decreases by ∼4-fold. Therefore, the SARS-CoV RdRp complex adopts an economic strategy to keep the proofreading cost small (i.e., close to the theoretical minimum cost) while maintaining a high speed to replicate its genome.

The results of our study on the interplay of the characteristic properties from single and global parameter variation appear to indicate that the proofreading cost is highly prioritized by the SARS-CoV RdRp complex. To gain further insight into the importance of the proofreading cost relative to the speed and the error rate as criteria for optimization, we followed the protocol developed in ref 23 and analyzed how far each characteristic property is from its optimal value that respectively minimizes the cost C, maximizes the speed V, and minimizes the error rate η. To this end, each kinetic parameter was varied over 6 orders of magnitude around the "native" value of k 0 ([10 -3 k 0 , 10 3 k 0 ]), and the following function was considered, i k j j j j j y

Here, k i denotes the rate constant varied with index i, and k max/min is the constant at which the speed, or error rate max/min ) has an optimal value (maximum or minimum). The larger this value of d k,i , the closer the native value is to the optimal value (see the Supporting Information for more details). The results shown in Figure S12 indicate that at least three of the kinetic parameters k 1,R = k p,R , k -1,R , and k 2,R are optimized for the proofreading cost. The other two parameters, k -2,R and k 3,R , are optimized for the maximal speed with the error close in second. Given the difference in trends for different steps, we opted against averaging these quantities as we previously did in ref 23. For comparison, the analysis was repeated also for T7 DNA polymerase that uses the same kinetic scheme. The results are qualitatively similar with one exception, k 2,R , where the speed is optimized instead of the proofreading cost. Taken all together, our results show that the proofreading cost is one of the most important features of the SARS-CoV RdRp complex, perhaps along with the speed. In comparison with other biological systems (e.g., T7 DNA polymerase, E. coli ribosome, and isoluecine tRNA synthetase), 21,23,24,35 the relative importance of cost versus speed may be even larger for the SARS-CoV RdRp complex. This finding raises a question on the physiological meaning of the cost optimization from the evolutionary perspective. To better understand the prioritization of the proofreading cost in the SARS-CoV RdRp complex, we propose the following biochemical arguments to explain it. By keeping the proofreading cost small, the SARS-CoV RdRp complex limits the cytoplasmic accumulation of monophosphate nucleotides that are produced as a result of the proofreading steps during the futile cycles. It is well established that accumulation of these molecules, e.g., AMP molecules created as byproducts of ATP hydrolysis, can trigger innate immune defense mechanisms or cell autophagy. 38 In particular, activation of AMP-activated protein kinase (AMPK) in response to an increased cellular AMP/ATP ratio has been linked to cellular defenses against pathogens. [39][40][41] Notably, the ability of SARS-CoV-2 to interfere with AMPK and thereby reduce autophagy has been reported. 42 Thus, keeping the proofreading cost minimal can be critical for the success of the SARS-CoV RdRp complex to replicate the viral genome in the infected host cell without alerting the cell defense mechanisms. At the same time, it is possible that other molecular mechanisms might force the SARS-CoV RdRp complex to optimize the proofreading cost. It will be important to test this idea in more detail using both experimental and theoretical approaches.

In summary, our analysis of the interplay among error rate, speed, and proofreading cost for the SARS-CoV RdRp complex has revealed some important and unexpected results on the functionality of this complicated multiprotein structure. We have shown that the SARS-CoV RdRp complex operates at a tolerable error rate that keeps the fraction of nonfunctional virus copies low. Furthermore, the tolerable error rate allows the SARS-CoV RdRp complex to nearly optimize its effective speed when certain types of mutations (i.e., nonsense and severe missense) are considered. Consequently, proofreading is necessary for the SARS-CoV RdRp complex to keep the rate of production of functional virus copies nearly maximal, but the The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter rate would decrease if the Pol-Exo sliding rate is too high. Furthermore, the SARS-CoV RdRp complex cannot reach the maximum effective speed for the production of functional virus copies because the proofreading cost would increase by about 1 order of magnitude. Moreover, the error rate for the SARS-CoV RdRp complex cannot be further decreased for the same reason. Given that the proofreading cost prevents the SARS-CoV RdRp complex from further reducing its error rate and completely optimizing its effective speed, it would be easy to assume that this quantity is completely minimized. However, the SARS-CoV RdRp complex is prevented from completely minimizing the proofreading cost to the theoretical minimum cost due to a slow down in the effective speed. Finally, our quantitative study on ranking the proofreading cost, speed, and error rate of the SARS-CoV RdRp complex places the proofreading cost as one of the most important characteristic properties along with the speed. Our results in this work have uncovered some intriguing features on the trade-offs and optimization of the characteristic properties of the SARS-CoV RdRp complex. Specifically, we now understand that the proofreading ability conferred by the nsp14 ExoN domain 43 is necessary for SARS-CoV to replicate its unusually large genome at nearly optimal speed while, at the same time, keeping the fraction of nonfunctional virus copies low. This finding directly contrasts with other RNA viruses that do not have the proofreading ability but instead rely heavily on high mutation rates and diversity of their small genomes to continue infecting their hosts. 1,9 Furthermore, the proofreading cost itself is nearly optimized for the RdRp complex, and the importance of the cost may enable SARS-CoV to avoid triggering defense pathways in the host cell. 38 However, our current knowledge on the functionality of SARS-CoV and its associated proofreading mechanism is still far from complete. We note that another advantage of proofreading in the SARS-CoV RdRp complex is that the error-correction ability renders the virus resistant to medical treatments with certain types of drugs (i.e., nucleotide analogues that mimic naturally occurring nucleotides) such as Ribavirin. 7,9 Therefore, it is important to encourage the development and testing of viable drugs against SARS-CoV such as nucleotide analogues like Remdesivir and others that can strategically target both the nsp12 RdRp and the nsp14 ExoN domains of the SARS-CoV RdRp complex. 9,[44][45][46][47] ■ ASSOCIATED CONTENT * sı Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.1c00190.

Derivation and description of the SARS-CoV RdRp complex KPR model and the underlying mathematical formalism; definitions of the error rate η, speed V, and proofreading cost C; qualitative and quantitative sensitivity analysis on the robustness of our conclusions for the error rate η, speed V, and proofreading cost C due to variation of all the kinetic parameters k i,R ; interplay of the stationary probabilities for each biochemical state in the KPR scheme with the Pol-Exo sliding rate k 2,R ; complete description and mathematical details of our bioinformatics analysis; effective speed V eff as a function of the error rate η using the complex formalism and the simple power law for nonsense mutations and severe missense mutations; speed V and effective speed V eff for two different definitions of the speed; interplay of the effective speed V eff with the proofreading cost C; histograms of the effective speed V eff distributions at a fixed error rate η nat from the variation of both the polymerization k 1,R = k p,R and the Pol-Exo sliding k 2,R rate constants; and mathematical and graphical description of the quantitative methods for ranking the importance of the characteristic properties for each kinetic parameter k i,R individually (PDF)