Activation of Bacterial Histidine Kinases: Insights into the Kinetics of the cis Autophosphorylation Mechanism

Two-component systems consisting of an input-sensing histidine kinase (HK) and an output-generating response regulator (RR) are one of the key apparatuses utilized by bacteria for adapting to the extracellular milieu. HK autophosphorylation is shown to occur primarily in trans (intermolecular) and more recently shown to occur in cis (intramolecular). Although the catalysis of HK activation remains universal, the reaction scheme for evaluation of the kinetic parameter differs between these designs and cis mode largely remains unexplored. We combined experimental and theoretical approach to unravel two-step mechanism of activation of three cis mode HKs of M. tuberculosis. The new mathematical model yields best-fit parameters to estimate the rates of HK-ATP association and HK autophosphorylation.

phosphoryl group to a specific partner RR protein on a conserved aspartate residue. The changes in the phosphorylation status of RR orchestrate adaptive responses to the environmental cue mostly by altering its DNA binding that lead to change in the expression of downstream genes (1)(2)(3). The TCSs of pathogenic bacteria form attractive drug targets, as the survival of pathogenic bacteria under pressure from host immune responses has been shown to be adversely affected if the functionality of one or more of the TCSs is genetically compromised (4,5). Because the stimuli triggering HKs and the genes modulated by RRs are often unknown (6), studies aim to develop drugs for blocking the autophosphorylation of HKs or the phosphotransfer from HKs to their cognate RRs, which are more general and conserved reactions (7). Accurate biochemical characterization of autophosphorylation and phosphotransfer reactions is therefore likely to be useful in accelerating antibacterial drug development.
Most bacteria contain many distinct TCS proteins, for instance, Mycobacterium tuberculosis contains about 30 TCS proteins, Escherichia coli about 62, and Myxococcus xanthus about 270 (8). In addition to cognate HK¡RR phosphotransfer, in some bacteria such as M. tuberculosis, HKs can "cross phosphorylate" or "cross talk" with noncognate RRs, increasing the number of reactions involving the TCS proteins (9). However, even in the best-studied systems like E. coli, the autophosphorylation kinetics of only 8 of the 31 HKs have been investigated (2,(10)(11)(12)(13)(14). In addition, effective biochemical characterization must define an appropriate reaction scheme for autophosphorylation, which also remains to be investigated for most HKs. Autophosphorylation can happen in cis (intramolecular) or in trans (intermolecular) (3,15). While autophosphorylation in trans was historically reported (16)(17)(18), evidence of cis mode are reported more recently (19,20). The kinetics of autophosphorylation occurring in trans, involving the unphosphorylated protein kinase and ATP as the substrates and the phosphorylated protein kinase as enzyme, has been theoretically analyzed (21) and described using the classical Michaelis-Menten formalism, which may not be optimal on account of the enzyme and product being the same molecule. The assumption of such formalisms that (i) a negligible amount of the product is formed and (ii) excess enzyme is present while evaluating the initial velocities may be disproportionate in HK-limiting conditions owing to low in vitro HK activity reported previously (22)(23)(24). Also this formulation would not be applicable in cis mode, as there the enzyme converts itself into the product. In M. tuberculosis, recent structural studies suggest cis autophosphorylation for the HK PhoR (25). cis autophosphorylation has been argued for the HK PrrB as well: the length and handedness of the loop joining the ␣1 and ␣2 helices in the dimerization and histidine phosphorylation (DHp) domain of PrrB are similar to HKs autophosphorylating in cis in E. coli, such as PhoR and ArcB (26,27). Beyond TCSs, a wide spectrum of protein kinases have also been argued to be autophosphorylated in cis (28). However, a kinetic scheme for analyzing autophosphorylation in cis, handling the aforementioned constraints, is lacking. Our goal in the present study was to develop such a scheme.
We first establish a scheme for elucidating the mechanism of autophosphorylation of the HKs, using MtrB of M. tuberculosis as a model using a variety of techniques and show that it occurs through a cis mechanism. We also report an affinity measurement scheme to establish that our findings are not limited by conformational artifacts associated with mutated proteins, which are necessary for elucidating cis/trans mechanisms. We next utilize a high-throughput assay (HTA) to measure the reaction kinetics using MtrB and extend it to two other HKs, PrrB and PhoR, of M. tuberculosis, argued previously to also autophosphorylate in cis. We then develop a mathematical model to analyze the data and determine the rates of autophosphorylation for them. The rate measurements together with the model unravels a two-step cis autophosphorylation mechanism and allows quantification of the propensity of this reaction across HKs. Our study provides a facile, quantitative template for investigating reactions involving bacterial TCS proteins in a high-throughput manner, which may aid the identification of drug targets (29) and provide insights into the design principles governing TCS signaling and the survival and adaptive responses in bacteria (30)(31)(32).

RESULTS
For determination of the autophosphorylation rates of HKs, we first had to establish the experimental conditions where maximal HK autophosphorylation is recorded and then identify the mechanism of their autophosphorylation, i.e., whether it occurs in cis or trans. This was essential, as the reaction steps vary depending on the mechanism and dictate the nature of the kinetic measurements. For our study, we decided to use three functionally characterized HKs of M. tuberculosis, viz., MtrB, PrrB, and PhoR (31), which have been proposed to play a role in regulating M. tuberculosis pathogenicity (33).
Optimal HK autophosphorylation occurs in the presence of Mg 2؉ ions. Given that divalent cations are necessary for HK autophosphorylation (34), we first tested for the preferred divalent cation in the autophosphorylation reaction for each of the selected HKs before kinetic measurements. Autophosphorylation was performed in the presence of 10 mM Ca 2ϩ , Mg 2ϩ , or Mn 2ϩ ions (all chloride salts) for 90 min. Although the presence of Ca 2ϩ and Mn 2ϩ did allow autophosphorylation to occur, Mg 2ϩ ions were the most preferred as they facilitated maximal autophosphorylation (see Fig. S1 in the supplemental material). No autophosphorylation was seen in the absence of cations. We therefore used Mg 2ϩ for all further studies.
MtrB autophosphorylation occurs in cis (intramolecular). As described above, the HKs can undergo phosphorylation in either a cis (intramolecular) or trans (intermolecular) mechanism. PrrB and PhoR have already been proposed to undergo phosphorylation through a cis mode (25,26). We thus used the HK MtrB to identify its phosphorylation mechanism. For this, we first generated a phosphorylation site (His) mutant (in the dimerization and histidine phosphorylation DHp domain [35]), MtrB H305Q , and another mutant defective in ATP binding (in the catalytic ATPase domain), MtrB N419D , both of which by themselves were phosphorylation defective (Fig. 1A). These mutant proteins were coincubated to record the occurrence of trans phosphorylation, if any. The rationale of this experiment is that in a trans model, phosphorylation of the ATPase mutant MtrB N419D can occur as it has a viable phosphoacceptor His site by dimerization with a His site mutant, which is capable of binding ATP. However, we found no phosphorylated MtrB species in this coincubation assay (Fig. 1A). This conspicuous absence of the phosphorylation signal suggested that the mutated proteins are unable to complement, indicative of a cis autophosphorylation mechanism. In this model, the ATP bound to the catalytic domain (ATPase_C) phosphorylates His in the DHp domain of the same molecule; hence, a defect in either the ATP domain or His site will render them phosphorylation defective.
To validate this, we performed a competition experiment by adding increasing amounts of MtrB H305Q in the autophosphorylation reaction of the green fluorescent protein (GFP)-tagged wild-type MtrB protein, as reported in previous studies where the cis/trans mechanism was probed (19,27). In this assay, if trans autophosphorylation is a viable mechanism, the MtrB H305Q mutant should compete with the wild-type MtrB and will show a reduction in the phosphorylation of the wild-type MtrB protein. Here also we failed to record any reduction in MtrB autophosphorylation, which competed with equimolar concentrations and up to 10-fold concentrations of the mutated protein, reinforcing the cis autophosphorylation mechanism (Fig. 1B).
Given that histidine kinases are known to form stable homodimers (19), the above observations could also result due to lack of heterodimer formation in the complementation or competition assay on account of preformed homodimers in the proteins. To address this, we performed microscale thermophoresis studies, wherein rates of association of various MtrB proteins (wild type and mutants) were determined using GFP-tagged MtrB (MtrB-GFP). We first validated that the MtrB-GFP protein is as active as untagged MtrB protein (Fig. S2A) and then subjected it to microscale thermophoresis (MST) analysis using variants of untagged protein for binding analysis. The dissociation constant (K D ) obtained for wild-type MtrB was 1.17 Ϯ 0.21 M, while the K D s were 0.41 Ϯ 0.06 M for MtrB H305Q and 0.77 Ϯ 0.19 M for MtrB N419D (Fig. 1C, D, and E), all very close and slightly lower for mutant proteins. The dissociation constants suggest that the dimerization of MtrB-GFP with its wild-type protein and mutant MtrB H305Q and MtrB N419D proteins occurs at the same affinities and rules out gross structural changes in the mutants that could influence their ability to dimerize, similar to what has been shown by Ashenberg and coworkers using the fluorescence resonance energy transfer (FRET) competition assay (27). Of note, the apparent dissociation constants of dimerization obtained are in the range of that for the HK CheA (K D ϭ 0.4 M) (11) known to autophosphorylate in trans, thus reinforcing the fact that the affinities of homodimers are in the range that allows heterodimer formation.
To further confirm the ability of wild-type MtrB and its mutants to form heterodimers, the wild-type and mutant MtrB proteins were coincubated with the recombinant MtrB-GFP protein and were resolved on a nonreducing 15% SDS-polyacrylamide gel. Here we recorded the presence of all the monomeric and dimeric species as anticipated. Distinct monomers of untagged and GFP-tagged MtrB, homodimers of tagged and GFP-tagged MtrB, along with the heterodimer of GFP-MtrB with untagged MtrB in the Coomassie brilliant blue (CBB)-stained gel as well as in fluorescence imaging were observed, confirming that the wild-type and mutant MtrB proteins do heterodimerize (Fig. 1F). The mutant proteins were also subjected to circular dichroism (CD) spectroscopy analysis, where no gross difference in the secondary structures of MtrB and the mutant proteins MtrB H305Q and MtrB N419D were recorded (Fig. S2B). Overall, our analysis allowed us to establish that MtrB autophosphorylation occurs by a cis mechanism, and using it as the template, we next analyzed the kinetic parameters and developed the mathematical model.
Equivalence of the HTA platform and PAGE/autoradiography. Given that autophosphorylation involves incorporation of the 32 P-labeled phosphoryl moiety from ATP onto an HK molecule, technically any assessment platform wherein labeled HK can be separated from unincorporated ATP can be utilized, including techniques like electrophoresis, chromatography, or filtration (34,(36)(37)(38). Here, we employed a 96-well filter plate platform (referred to as HTA assay henceforth), which allows quantification of acid-labile phosphohistidine in a high-throughput manner. In this assay, the labeled HK is retained on the membrane and unincorporated ATP is washed off using cold washing buffer, which does not affect the phosphohistidine moiety, thereby facilitating quantification of HK phosphorylation. Existing evidence of TCS phosphorylation reactions and information on their kinetics predominantly comes from conventional autoradiography and SDS-PAGE-based assays (2,10,17). Although a 96-well nitrocellulose filterbased plate assay was utilized to study autophosphorylation of the HK CheA (39), its comparative analysis with standard PAGE/autoradiography assay was lacking. Therefore, we first tested the equivalence of the HTA platform by the conventional PAGE/ autoradiography technique. For this, we performed autophosphorylation analysis of MtrB concomitantly on the HTA platform and the PAGE/autoradiography setup. The evolution over time of the normalized concentrations of labeled MtrB (denoted [HK·P H ]; see below) measured using the HTA platform was nearly identical to the normalized intensities measured using PAGE/autoradiography ( Fig. 2A; the corresponding gel and autoradiographs are shown in Fig. S3). Further, the band intensities obtained from autoradiography showed a linear correlation with [HK·P H ] from the HTA platform, yielding a calibration curve of band intensity with respect to the concentration (Fig. 2B). This close agreement gave us confidence that the HTA platform could be used instead of the PAGE/autoradiography technique, which is an indirect technique. To characterize the kinetics of HK autophosphorylation, we utilized the HTA platform for all our subsequent assays.
HK autophosphorylation kinetics. We first measured the concentrations of labeled HK, denoted [HK·P H ], which is equal to [HK~P H ] ϩ [HK-ATP H ] (see mathematical model below), at different time points from the start of the experiment, for fixed initial HK and different initial ATP concentrations (Fig. 3). (Note that we use the subscript H throughout to denote labeled entities and subscript C for unlabeled entities.) To capture initial reaction velocities, we made frequent measurements until 8 min. For any ATP concen- The data present two additional insights into the two-step process. First, the separation of the time scales of these two steps, viz., binding and catalytic activation,    3A) and MtrB (Fig. 3C). The concentrations of HK employed were generally comparable to the total ATP levels employed. The limitation in HK could thus be attributed to a fraction of the employed HK being capable of ATP binding and autophosphorylation. We employed these insights in constructing a mathematical model of the kinetics of cis HK autophosphorylation presented below. Mathematical model. We envision the following reactions following the exposure of HK to ATP: where K e is the equilibrium constant and k p is the phosphorylation rate.
Here, labeled ATP (denoted ATP H ) and unlabeled ATP (denoted ATP C ) rapidly form reversible complexes with the HK, viz., HK-ATP H and HK-ATP C , respectively. Then, ATP in the complexes phosphorylates HK, yielding the phosphorylated analogs, HK~P H and HK~P C . Note that HK dimerization is unimportant, given that autophosphorylation occurs in cis. However, this is not the case when the phosphorylation occurs in trans, and dimerization becomes an essential step and has to be accounted for in determining rates. We neglect dephosphorylation of the HK~P species primarily considered to be ADP driven (14), as we focus on the early stages of the experiment. The evolution of the concentration of the phosphorylated species over time is then given by Note that K e , the equilibrium constant, and k p , the phosphorylation rate, are identical for the labeled and unlabeled forms of ATP. Yet, we track the forms separately because the former form alone can be detected in measurements. To solve the above equations, we invoke the following species conservation relations:

Substituting equation 5 in equation 2 and adding the two equations in equation 2 yields
Substituting equation 7 in equation 5 yields the evolution of [HK] over time: Using equation 8 in the first of equation 2 and solving it with the initial condition that [HK~P H ] ϭ 0 at t ϭ 0, we obtain where we recall that A ϭ k p K e ͓͑ATP H ͔ 0 ϩ ͓ATP C ͔ 0 ͒⁄͑1 ϩ K e ͓͑ATP H ͔ 0 ϩ ͓ATP C ͔ 0 ͒͒.

Equation 10 thus provides an analytical expression for the evolution of [HK·P H ] over time when HK is limiting and [ATP H ] and [ATP C ] remain close to their initial values.
When HK is not limiting, [ATP H ] and [ATP C ] cannot be assumed constant and the differential equations in equation 2 must be integrated along with the constraints in equations 3 and 4 to obtain the corresponding evolution over time. We next fit equation 10 to the mean values of the measured data of HK associated with labeled ATP using the Levenberg-Marquardt algorithm in MatLab with k p , K e , and as adjustable parameters. All the data for a given HK were fit simultaneously. Fits of mathematical model to data and determination of kinetic parameters. The model captured all the trends observed above and provided good fits to the experimentally obtained data of autophosphorylation of PrrB (Fig. 4), PhoR (Fig. S4), and MtrB (Fig. S5). The resulting best-fit parameter estimates along with their 95% confidence intervals are listed in Table 1. We note that for each HK, data of all the ATP levels were fit simultaneously. The same initial HK concentration was used for all the ATP levels in these fits, consistent with the experiments. Experimental uncertainties could introduce variations in the initial HK concentration. Indeed, where the agreement between the best-fit parameter estimates and the data appeared inadequate (such as for MtrB), allowing variation in the initial HK level restored the agreement (Fig. S6). We determined from the fits that of the three HKs studied, PhoR had the highest equilibrium association constant, K e (2.7 mM Ϫ1 ), and hence associated with ATP the most strongly. PhoR was followed by MtrB (K e ϭ 1.7 mM Ϫ1 ) and PrrB (K e ϭ 0.5 mM Ϫ1 ) in the strength of this association. Following association with ATP, PrrB had the highest rate of autophosphorylation (k p ϭ 0.31 min Ϫ1 ) and was followed by MtrB (k p ϭ 0.27 min Ϫ1 ) and PhoR (k p ϭ 0.17 min Ϫ1 ). We also estimated the fraction of the HK proteins that was active and capable of association with ATP and autophosphorylation. We found to be the highest with MtrB (14%), followed by PrrB and PhoR, the lowest at~3%. Note that the overall extent of autophosphorylation is determined by all of these parameter values, viz., K e , k p and , as described by our model (equation 10).
Validation of parameter estimates. To validate the above parameter estimates, we performed independent experiments using the HTA platform, where the autophosphorylation reactions were performed for longer durations (up to 3 h for PhoR and up to 6 h for PrrB and MtrB) and using different initial HK and ATP concentrations. The reaction kinetics were then predicted using the best-fit parameter estimates above in equation 10. The resulting predictions were in close agreement with experimental data for all three HKs (Fig. 5). For instance, with PrrB, the measured [HK·P H ] was~1 Ϯ 0.11 nM at 320 min, and the corresponding model prediction with the parameter values in Table 1 was 0.93 nM (Fig. 5A). Similarly, the measured PhoR·P H was~0.56 Ϯ 0.06 nM at 160 min, while the corresponding model prediction using parameter values in Table 1 was 0.53 nM (Fig. 5B), and with MtrB, the measured [HK·P H ] was~1.7 Ϯ 0.08 nM at 320 min, and the corresponding model prediction with the parameter values in Table 1 Table 1. The data are shown as means Ϯ SD (n ϭ 3). a Best-fit parameter values (95% confidence intervals) obtained from fits of our model to data (Fig. 4 and Fig.  S4 and S5 in the supplemental material). was 1.35 nM (Fig. 5C). The model underpredicts the data for MtrB but captures it using ϭ 0.17 (dashed line) which is within the 95% confidence intervals on in Table 1.
To further validate the two-step mechanism captured by the model, we performed independent experiments of the autophosphorylation kinetics with the phosphorylationdefective histidine mutant MtrB H305Q which is proficient in ATP binding, with various concentrations of unlabeled ATP and a fixed concentration of labeled ATP ([ATP H ] 21 nM) using the HTA platform. In this experiment, it is anticipated that only the first step of the reaction, i.e., ATP binding, will be captured at all time points, and no contribution of the second step, i.e., phosphorylation, will be recorded and that is what was observed (Fig. 6). The data were fit using our model (equation 10) with k p ϭ 0, given the absence of phosphorylation, and with the best-fit parameter estimates of K e above.
was used as an adjustable parameter. The model provides good fits to the data at all concentrations of unlabeled ATP yielding ϭ 9%, which is within the 95% confidence interval on of MtrB (Table 1). We also ruled out nonspecific ATP binding by using a mutant defective in ATP binding, MtrB N419D , which is also phosphorylation defective (Fig. S7). This close agreement between model predictions and independent experimental measurements provides a successful test of the two-step mechanism as well as of the parameters estimated using the HTA platform and our mathematical model.
Overall, our findings reveal a two-step activation model for the bacterial HKs, which involves their association with ATP, an extremely rapid step, followed by a slower cis autophosphorylation reaction step.

DISCUSSION
On account of the prevailing view of the mechanism of autophosphorylation of HKs being the trans mode, rate measurements and modeling of the cis mode of phosphorylation have not been extensively explored. We present here experimental evidence to demonstrate the existence of the cis mode of autophosphorylation along with a combined theoretical and experimental analysis of the kinetics of autophosphorylation of three HKs of M. tuberculosis implicated in regulating its virulence. We found that cis autophosphorylation of the HKs was orchestrated via a two-step mechanism: rapid, reversible association of HK with ATP, followed by the phosphorylation of the HK by the associated ATP (Fig. 7), which has not been reported previously but at the same time was anticipated based on the structure-function properties of the HKs (15). We developed a mathematical model based on this mechanism that captured experimental data quantitatively. Experiments with mutant proteins provided a direct evidence of this two-step cis mechanism. Using fits of our model to the data, we estimated the rates of the autophosphorylation reaction, which allowed characterization of the HKs in terms of their relative propensity to be activated.
We employed a high-throughput assay (HTA) platform for our measurements, which yielded readouts in close agreement with classical PAGE/autoradiography but using 10-fold-lower labeled ATP and~20-fold-lower data processing time and where reactions can be run in parallel in 96 reactions per plate. Our model presents a single analytical expression, akin to the classical Michaelis-Menten formalism, that allows facile data analysis. A predictive rule for the mode of autophosphorylation (cis or trans) has been established structurally by studying the determinants of the loop handedness of the HKs (27). Our combined experimental and mathematical analysis may be readily scaled up to the large number of HKs that still remain poorly studied and may be autophosphorylated in cis.
Previous studies have reported trans autophosphorylation of several HKs (16,18), in which dimerization precedes the reaction (10,40). Although the autophosphorylation mechanism is conserved regardless of directionality as the same product, i.e., phosphorylated HK, which is necessary for the phosphorylation of a response regulator, the structural differences in the helix loops of the DHp domain dictating the directionality emphasize the necessity for a dimerization event for HK trans autophosphorylation (27). The bimolecular autocatalytic event involving ATP and HK as the substrates and phosphorylated HK within a dimer as the enzyme follows a random bi-bi mechanism which has been described using the classical Michaelis-Menten equations (21). Kinetics of HK autophosphorylation has been studied by employing such reaction schemes yielding kinetic parameters for ATP and HK as the substrate of HKs namely, CheA (11,41,42), NtrC (23), VanS (22), FixL (43), and other HKs from various organisms like Shewanella oneidensis (gene identifier [ID] or accession no. 1169886), and Legionella pneumophila (gene ID 19831845) (36). However, our experiments and many reported recently (19,20,25,27,44) indicate that some HKs undergo autophosphorylation by the cis mode, that the cis autophosphorylation is expected to be dimerization independent, and that the monomer can be considered a kinase-competent state of the HK (3) and can be modeled without requiring the dimerization process in the reaction steps. However, we recorded the presence of dimers even for HK, which undergoes autophosphorylation in cis, which was evident in the MST and other interaction experiments. The significance of this dimerization process is not clear at this time and does not conform with the models generated with trans autophosphorylation process. Further, the low fraction of HK that is active in vitro makes HK limiting even at the initial time points in our experiments and those reported previously (22)(23)(24). The Michaelis-Menten equations therefore could not be readily applied to such data. It is important to note here that the small active fraction of HKs that we obtained ( of~3 to 14%) is similar to those obtained for many other HKs, such as VanS of Enterococcus faecium ( of~10%) (22,24) and the HK NtrC of Escherichia coli ( of~30%) (23), attributed to intrinsically low basal level of functional conformers, due to use of only the C-terminal domain of the HKs for activity assays. Dephosphorylation of HK~P may also underlie the low percentage activity of HK, but this appears negligible in our experiments given the close agreement between our predictions without adjustable parameters and experimental measurements of HK autophosphorylation over extended durations (Fig. 5). This consonance between our estimates of the active HK fraction and those reported previously, together with the validation of our model predictions using independent experiments ( Fig.  5 and 6) gives us confidence in our approach and design of the rate equations.
Given that evidence of cis autophosphorylation of HKs is growing (19,25,44), our experimental workflow and mathematical analysis are expected to provide insights into the underlying reaction scheme and allow robust estimation of kinetic parameters of the rate. We expect that the kinetic parameters will provide inputs to theoreticians interested in analyzing the behavior of TCS systems and their implications for bacterial survival and adaptation. Simultaneously, the parameters would aid the comparative evaluation of drug targets and potentially accelerate antibacterial drug development. Step 1 indicates the presence of rapid and reversible association of ATP with HK.
Step 2 is the slow autophosphorylation of HK in response to stimulation by ligand, which is the rate-limiting step. P, phosphate group.
(PerkinElmer, USA) in the 32 P channel using scintillation fluid (0.5% [wt/vol] 2,5-diphenyl oxazole and 0.005% [wt/vol] 1,4-bis-4-methyl-5-phenyl-2-benzene oxazoyl benzene in toluene). A negative-control reaction having the same components but lacking the HK was also processed on the high-throughput assay (HTA) platform in the same way to determine the retention of radiolabel alone on the membrane. At the same time, an internal control (same as the negative control) was directly used for scintillation counting to determine the exact disintegrations per minute (dpm) of the ␥-32 P-labeled ATP used per reaction. Experiments were also performed with different amounts of unlabeled ATP added initially (along with 1 Ci of ␥-32 P-labeled ATP) with fixed amounts of HK, and a time course experiment with fixed amounts of ATP and HK was also performed. Each experiment was performed in triplicate.
Data fitting using mathematical model. We fit equation 10 of our model (see below) to the mean values of the data of the concentrations of HK associated with labeled ATP measured as a function of time, following the start of the experiment. Data for various initial concentrations of unlabeled ATP for any given HK were fit simultaneously. k p , K e , and were used as adjustable parameters. The fitting was done using the Levenberg-Marquardt algorithm in MatLab using the inbuilt function NLINFIT. The fitting also yielded 95% confidence intervals on the parameters as well as the best-fit predictions. The goodness of fit was estimated by computing R 2 values. The fit parameters were also tested by comparing model predictions with independent experiments with different initial concentrations of the reactants and run for extended durations.