Good-Practice Non-Radioactive Assays of Inorganic Pyrophosphatase Activities

Inorganic pyrophosphatase (PPase) is a ubiquitous enzyme that converts pyrophosphate (PPi) to phosphate and, in this way, controls numerous biosynthetic reactions that produce PPi as a byproduct. PPase activity is generally assayed by measuring the product of the hydrolysis reaction, phosphate. This reaction is reversible, allowing PPi synthesis measurements and making PPase an excellent model enzyme for the study of phosphoanhydride bond formation. Here we summarize our long-time experience in measuring PPase activity and overview three types of the assay that are found most useful for (a) low-substrate continuous monitoring of PPi hydrolysis, (b) continuous and fixed-time measurements of PPi synthesis, and (c) high-throughput procedure for screening purposes. The assays are based on the color reactions between phosphomolybdic acid and triphenylmethane dyes or use a coupled ATP sulfurylase/luciferase enzyme assay. We also provide procedures to estimate initial velocity from the product formation curve and calculate the assay medium’s composition, whose components are involved in multiple equilibria.


Introduction
Inorganic pyrophosphatase (inorganic diphosphatase; EC 3.6.1.1; PPase) is a constitutive, highly specific enzyme that converts pyrophosphate (PP i ), a byproduct and regulator of numerous biosynthetic reactions [1], into a metabolizable phosphate. Ubiquitous soluble PPases dissipate PP i energy as heat, whereas less common membrane-bound PPases use the energy to transport H + or Na + across lipid membranes [2,3]. Both PPase types can also catalyze the reverse reaction of PP i synthesis from P i , and this activity of membrane-bound PPases is also considered physiologically important [4,5]. Soluble PPases are additionally divided into two nonhomologous families-family I, known for nearly a hundred years and found in all kingdoms of life, and family II found in 1998 in prokaryotes [6,7]. Several nonspecific phosphatases also exhibit PPase-like activity [8][9][10].
Although the primary structures of the three groups of specific PPases are completely different, their mechanisms and active sites reveal close similarity, making these enzymes remarkable examples of convergent evolution. All PPases are Mg 2+ -dependent enzymes, but family II PPases additionally require a transition metal ion (Mn 2+ or Co 2+ ) for maximal activity [11]. In total, three to four metal ions per active site are required for PP i conversion [7]. These metal ions, coordinated by numerous protein carboxylates and substrate phosphates, play key roles in catalysis. PP i hydrolysis involves a direct attack of an activated water molecule on a phosphorus atom and a stepwise release of two phosphate molecules. The value of k cat for soluble PPases reaches 10 4 s −1 for PP i hydrolysis and 10 2 s −1 for PP i synthesis [3,4,11]. K m values for most PPases with PP i as substrate lie in the micromolar range.
PPase is generally assayed by measuring phosphate, the product of PP i hydrolysis. Depending on the task, PPase activity is assayed at sub-K m PP i concentrations (in mechanistic studies) or a saturating PP i concentration (in high-throughput screening procedures). The former assay is more demanding concerning sensitivity because of the low K m value, especially with family I and membrane PPases. Sensitivity is also the limiting factor in PP i synthesis studies because the equilibrium PP i , catalog No. FLAAAM, reconstituted with 5 mL of water) inescence. After the signal stabilizes, 2 µL of 10 µM ATP are bration.
Enzyme-bound PPi formation [36] 0 µM) is preequilibrated with Pi and Mg 2+ under appropriate olume and quenched with 10 µL of 5 M trifluoroacetic acid. , precipitated protein is removed by centrifugation, a 10 µL alit is added to 0.2 mL of the PPi assay cocktail (0.2 M Tris-HCl, sulfurylase, 10 µM APS, 6 µL of the luciferase/luciferin solutol, 0.8 mg bovine serum albumin, 30 µM EGTA), and the lud. After the signal stabilizes, 2 µL of 10 µM PPi are added for Determined PPi content refers to the sum of enzyme-bound and medium PPi. The latter is measured similarly but using a 100 times lower enzyme concentration and subtracted.
The third version of the PPi assay accesses the equilibrium of the PPi  2Pi reaction in the active site of PPase. This information is required to estimate the rate constants for individual steps of PPase catalysis [37,38]. This equilibrium is markedly shifted to the left by comparing the equilibrium in solution-up to 20% of enzyme-bound Pi is converted to PPi [37,38]. To assay, the enzyme-bound PPi, the enzyme (20-100 µM) is incubated with Pi and Mg 2+ , inactivated by trifluoroacetic acid to release enzyme-bound PPi into solution, and an aliquot is taken to assay PPi using the coupled ATP sulfurylase/luciferase procedure.
2P i is largely shifted to the right. Here, we summarize our experience in sensitive PPase activity measurements in both directions and present protocols suitable for high-throughput screening and mechanistic studies.

Continuous Low-Substrate Assay of PP i -Hydrolyzing Activity
The low value of the Michaelis constant of PPases makes it necessary to use highly sensitive assays in kinetic studies conducted at sub-K m substrate concentrations. The assay currently used in our labs combines two approaches to achieve high sensitivity: first, it measures the formation of the intensively colored complex of 12-molibdophosphoric acid with a triphenylmethane dye, methyl green, and, second, it is continuous. The PPase reaction mixture and two-color reagents are continuously pumped off by a peristaltic pump and mixed, and the absorbance of the resulting solution is measured at 620 or 660 nm in a flow photometer. Even though the background absorbance is relatively high, it is automatically subtracted in this continuous assay mode, allowing the precise recording of much smaller changes in absorbance (P i concentration) compared to a fixed-time assay. The principal advantage of the methyl green dye in this application is its low tendency to deposit on the optical cuvette [12].
The phosphate analyzer consists of three main parts: a four-channel peristaltic pump, flow photometer, and paper recorder ( Figure 1A). Alternatively, the photometer's output can be directed to a computer, which calculates initial velocity from the absorbance timecourse. However, we found that the manual procedure, using paper recordings, is less sensitive to signal fluctuations and provides more accurate data. We have used a fourchannel Gilson Minipuls pump equipped with Tygon tubings. Three pump channels are used to deliver a sample, acid/molybdate, and dye/Triton X-305 solutions. Their initial mixing occurs in tubing T-joints and is completed in a mixing device consisting of a glass tube of 1.3 mm inner diameter with 12-15 bubbles of 3.5 mm inner diameter. The fourth pump channel eliminates any air bubbles from the final mixed solution before it is directed to the flow photometer. To prevent excessive air from entering the flow system, the pump is stopped when the sample inlet tubing is transferred between the samples and water. To ensure that the color reaction proceeds to completion, the time intervals between the two mixing events and between the second mixing event and entry to the photometer cuvette should be adjusted to 12 and 32 s, respectively, by using connecting tubings of appropriate diameter and length. The small nonlinearity of the calibration plot at low phosphate content is eliminated by adding a small amount of phosphate to the stock acid/molybdate solution. We have been using ISCO UA-5 and 229 flow photometers. These instruments are no longer produced but can be replaced by any flow photometer operating at 620-660 nm wavelengths and a flow rate of approximately 10 mL/min. Alternatively, one can use a standard spectrophotometer with a flow cuvette, such as a Helma model 178-010-10-40 (10 mm pathlength, 80 µL internal volume).
Three setups of the phosphate analyzer were found useful for standard, high-sensitivity, and low dead-time measurements ( Figure 1A-C). A five-fold increase in sensitivity was achieved in version B by interchanging the sample and molybdate tubings on the pump ( Figure 1B). As the response is linear up to approximately 0.5 absorbance unit, the photometer sensitivity is adjusted to 0.5 absorbance unit per recorder scale in the standard mode (version A) and down to 0.1 unit in the high-sensitivity mode (version B). These settings correspond to approximately 100 and 4 µM P i , respectively, per recorder scale. As each PP i molecule yields 2 molecules of P i , this sensitivity is sufficient for a reliable measurement of initial velocities at PP i concentrations down to 0.5 µM (see Appendix A for further details). PPase concentration in the assay is typically in the pM range when activity is determined at saturating substrate concentrations and optimal pH and temperature values. We have successfully used the phosphate analyzer to determine PPase reaction kinetics over a broad temperature range (reaction mixture temperature 20-60 • C).
is determined at saturating substrate concentrations and optimal pH and temperature values. We have successfully used the phosphate analyzer to determine PPase reaction kinetics over a broad temperature range (reaction mixture temperature 20-60 °C).
The dead-time between withdrawing the sample and mixing it with acid/molybdate is 10 s in the standard mode but can be decreased to 1 s by shifting the first mixing point to the pump inlet ( Figure 1C). In this version, the sample is withdrawn from the reaction vessel at a rate of 1.2 mL/min due to differences in the two lowest pump tubes' flow rates. This setup helps monitor reactions demonstrating nonlinear progress curves because of enzyme activation or inactivation during the reaction. Reagent concentrations were adjusted in versions B and C as indicated in Table 1 to ensure the same optimal final concentrations in the photometer cuvette.  Numbers on the pump refer to flow rate in mL/min (before the slash) and tubing diameter in mm (panel A) or flow rate in mL/min (panels B and C). (D) Actual P i accumulation recordings in setup A with photometer sensitivity of 1 absorbance unit per recorder scale. The calibration data shown at the beginning of the recording was obtained by adding 0-200 µM P i to the reaction buffer. (E) Actual P i accumulation recordings in setup B with photometer sensitivity of 0.1 absorbance unit per recorder scale. The assay mixture of 40 mL volume contained 140 µM PP i , 5 mM MgCl 2 , 50 mM MOPS-KOH, pH 7.2, and 0.03 nM Streptococcus gordonii PPase with a specific activity of 480 s −1 . (F) Actual recordings of P i accumulation in the setup C for rat liver PPase in the presence (a) or absence (b) of slow-binding inhibitor (10 mM fluoride). The arrow marks the moment of enzyme addition. Part of the figure was taken with permission from references [13] (panels A and D) and [14] (panels C and F).
The dead-time between withdrawing the sample and mixing it with acid/molybdate is 10 s in the standard mode but can be decreased to 1 s by shifting the first mixing point to the pump inlet ( Figure 1C). In this version, the sample is withdrawn from the reaction vessel at a rate of 1.2 mL/min due to differences in the two lowest pump tubes' flow rates. This setup helps monitor reactions demonstrating nonlinear progress curves because of enzyme activation or inactivation during the reaction. Reagent concentrations were adjusted in versions B and C as indicated in Table 1 to ensure the same optimal final concentrations in the photometer cuvette. The continuous P i assay is robust and tolerates the presence of many biochemical compounds, including magnesium ions at high concentrations (up to 40 mM). In contrast, Mg 2+ interferes with the P i assay based on 12-molibdophosphoric acid reduction [17]. Nevertheless, it is wise to calibrate the instrument by measuring phosphate standard against background each time when a new component is added to the sample assayed, especially when working in the high-sensitivity mode (version B). Importantly, PP i at high concentrations was found to suppress the P i signal (by 25% for 1.5 mM PP i ) in this mode. This should be taken into consideration in measurements of PPase activity at a variable substrate concentration. The effect is proportional to PP i concentration and likely results from the competitive formation of 12-molibdopyrophosphoric acid, detected by Raman spectroscopy [18].
Other known continuous assays of PPase activity monitor, depending on pH, uptake or release of hydrogen ions during PP i conversion to P i [19,20] or monitor PP i concentration using synthetic PP i sensors [21][22][23][24][25]. These assays are less sensitive and less convenient in that the signal is not proportional to the degree of conversion.
Substances that modulate PPase activity will interfere with any assay. Among them, Tris and other amine buffers were found to dramatically increase the Michaelis constant for family I and membrane PPases [26][27][28]. Accordingly, zwitterionic buffers, such as Tes, Mops, and HEPES, are preferable for work at sub-K m substrate concentrations. Tetramethy-lammonium hydroxide was found to be useful as the second component of the zwitterionic buffers in experiments measuring K + and Na + effects on K + -dependent membrane PPase. However, some batches of this strongly basic compound supplied in glass containers were found to nonspecifically inactivate this enzyme when used as a buffer component instead of KOH.

Sensitive Fixed-Time Assay of PP i -Hydrolyzing Activity
Fixed-time assays measuring the product P i are perhaps most suitable for highthroughput screening of compound libraries for the effects on PPase activity. Most published procedures determine P i as reduced 12-molybdophosphoric acid [29] and perform well with 10 −5 -10 −4 M P i concentrations. Although this sensitivity suffices for most applications, the identification of samples with low PPase activity or assays performed at low PP i concentrations (<10 µM) would require a more sensitive P i assay, such as the assay based on the shift of malachite green spectrum upon binding to 12-molybdophosphoric acid. However, low dye solubility, leading to its precipitation, was a serious drawback of the original malachite green-based procedure [30] and its many later variants.
A solution to this problem was finding that increased acid concentration (2.5 M H 2 SO 4 ) in stock dye solution surprisingly increases dye solubility and stability [15]. This finding allowed the formulation of a single stable color reagent that stops the enzymatic reaction and produces a green-blue color (630 nm), whose intensity is proportional to P i concentration. The color reagent is made daily by adding the nonionic detergent Tween-20 (color stabilizer) to the stock dye/molybdate solution ( Table 1). The signal produced by 2 µM P i (corresponding to 1 µM PP i hydrolyzed) is 0.15 absorbance units in a 1 cm cuvette. The assay can be performed in a 96-well plate ( Figure 2) and quantified with a plate reader.

Sensitive Fixed-Time Assay of PPi-Hydrolyzing Activity
Fixed-time assays measuring the product Pi are perhaps most suitable for highthroughput screening of compound libraries for the effects on PPase activity. Most published procedures determine Pi as reduced 12-molybdophosphoric acid [29] and perform well with 10 -5 -10 -4 M Pi concentrations. Although this sensitivity suffices for most applications, the identification of samples with low PPase activity or assays performed at low PPi concentrations (<10 µ M) would require a more sensitive Pi assay, such as the assay based on the shift of malachite green spectrum upon binding to 12-molybdophosphoric acid. However, low dye solubility, leading to its precipitation, was a serious drawback of the original malachite green-based procedure [30] and its many later variants.
A solution to this problem was finding that increased acid concentration (2.5 M H2SO4) in stock dye solution surprisingly increases dye solubility and stability [15]. This finding allowed the formulation of a single stable color reagent that stops the enzymatic reaction and produces a green-blue color (630 nm), whose intensity is proportional to Pi concentration. The color reagent is made daily by adding the nonionic detergent Tween-20 (color stabilizer) to the stock dye/molybdate solution ( Table 1). The signal produced by 2 µ M Pi (corresponding to 1 µ M PPi hydrolyzed) is 0.15 absorbance units in a 1 cm cuvette. The assay can be performed in a 96-well plate ( Figure 2) and quantified with a plate reader.

Continuous and Fixed-Time Assays of PPi-Synthesizing Activity
Like any catalyst, PPase accelerates the attainment of thermodynamic equilibrium PPi  2Pi in both directions. Although the equilibrium is shifted to the right (∆G 0 = −20-25 kJ/mol), it allows the formation of measurable amounts of PPi under physiological conditions. For instance, the concentration of PPi at equilibrium with 10 mM Pi (pH 7, 25 °C, 1 mM Mg 2+ ) is approximately 1 µ M [31], which is below the detection limits of most analytical methods for PPi determination. However, micromolar sensitivity has been claimed for several assays based on fluorogenic and electrochemical PPi sensors [21][22][23][24][25]. Some of them are commercially available. However, they are of limited use in measurements of initial velocities of PPi synthesis, which require that PPi concentration in the medium does not exceed 10% of the equilibrium concentration to prevent product inhibition.
This problem is elegantly obviated in the coupled-enzyme luminescent procedure of Nyrén and Lundin [32], which later became a core element of "pyrosequencing", a wellestablished DNA sequencing method [33]. In their procedure, PPi is converted to ATP by reaction with adenosine-5′-phosphosulfate (APS), catalyzed by ATP sulfurylase (E.C. 2.7.7.4), and the ATP formed is detected with a firefly luciferase/luciferin system. This method is used in three versions in PPase studies (Table 2 and Figure 3). In the first version, all components of the detection system, including ATP sulfurylase, luciferase, APS, and luciferin, are added to the PPase assay mix, and PPi formation is recorded continu-

Continuous and Fixed-Time Assays of PP i -Synthesizing Activity
Like any catalyst, PPase accelerates the attainment of thermodynamic equilibrium PP i Molecules 2021, 26, x FOR PEER REVIEW 6 both ATP sulfurylase and PPase need Mg 2+ as a cofactor. The synthesized PPi is conti ously converted into ATP at such a high rate that the steady-state PPi level is not inhibit for PPase and luciferase activities. This is ensured by maintaining high ATP sulfury concentration, such that its doubling does not increase the measured rate of ATP accum lation. In the second assay version, the PPase reaction mix contains ATP sulfurylase/A which continuously converts the formed PPi into ATP, whose concentration is determi at fixed time points with luciferase/luciferin in a separate tube. Importantly, PPase d not hydrolyze ATP in the presence of Mg 2+ . Both assay versions combine the high se tivity of the luciferase ATP determination with the possibility to accumulate detecta amounts of ATP while keeping the steady-state concentration of PPi at a sub-equilibri non-inhibitory level. ATP sulfurylase and luciferase act thus as amplifiers of the PPi-g erated signal and allow the measurement of the initial velocities of PPi synthesis from  [34] Assay mixture contains calculated amounts of potassium phosphate and MgCl2, 0.7 U/mL ATP-sulfurylase, 10 µM APS, 5 µL of luciferin/luciferase solution (Sigma ATP assay mix, catalog no. FLAAM, reconstituted with 5 mL of water), 1 mM dithiothreitol, 0.8 mg/mL bovine serum albumin and 0.1 M MOPS-KOH buffer in a total volume of 0.2 mL. The reaction is initiated by adding PPase, and the time-course of luminescence is followed with a luminometer (e. g., LKB model 1250). After the PPigenerated signal stabilizes, 2 µL of 10 µM ATP are added for internal calibration.
ATP sulfurylase is from BioLabs (UK). Its c centration should be sufficiently high so th its doubling does not change the measured rate of ATP accumulation. The luminescen versus-time dependencies are slightly curv because of the slow inactivation of lucifera in the reaction medium.
PPi synthesis (fixed-time) [35] Assay mixture contains calculated amounts of potassium phosphate and MgCl2, 0.7 U/mL ATP-sulfurylase, 10  Crystalline phosphoric acid (Fluka) is freed from PPi contamination as described in the text.
Enzyme-bound PPi formation [36] PPase (typically 20-100 µM) is preequilibrated with Pi and Mg 2+ under appropriate conditions in a 50 µL volume and quenched with 10 µL of 5 M trifluoroacetic acid. After several minutes, precipitated protein is removed by centrifugation, a 10 µL aliquot of the supernatant is added to 0.2 mL of the PPi assay cocktail (0.2 M Tris-HCl, pH 8.0, 0.4 U/mL ATP sulfurylase, 10 µM APS, 6 µL of the luciferase/luciferin solution, 1 mM dithiothreitol, 0.8 mg bovine serum albumin, 30 µM EGTA), and the luminescence is recorded. After the signal stabilizes, 2 µL of 10 µM PPi are added for internal calibration.
Determined PPi content refers to the sum o enzyme-bound and medium PPi. The latter measured similarly but using a 100 times lower enzyme concentration and subtracte The third version of the PPi assay accesses the equilibrium of the PPi  2Pi react in the active site of PPase. This information is required to estimate the rate constants individual steps of PPase catalysis [37,38]. This equilibrium is markedly shifted to the by comparing the equilibrium in solution-up to 20% of enzyme-bound Pi is converte PPi [37,38]. To assay, the enzyme-bound PPi, the enzyme (20-100 µM) is incubated wit and Mg 2+ , inactivated by trifluoroacetic acid to release enzyme-bound PPi into solut and an aliquot is taken to assay PPi using the coupled ATP sulfurylase/luciferase pro dure.
2P i in both directions. Although the equilibrium is shifted to the right (∆G 0 = −20-25 kJ/mol), it allows the formation of measurable amounts of PP i under physiological conditions. For instance, the concentration of PP i at equilibrium with 10 mM P i (pH 7, 25 • C, 1 mM Mg 2+ ) is approximately 1 µM [31], which is below the detection limits of most analytical methods for PP i determination. However, micromolar sensitivity has been claimed for several assays based on fluorogenic and electrochemical PP i sensors [21][22][23][24][25]. Some of them are commercially available. However, they are of limited use in measurements of initial velocities of PP i synthesis, which require that PP i concentration in the medium does not exceed 10% of the equilibrium concentration to prevent product inhibition.
This problem is elegantly obviated in the coupled-enzyme luminescent procedure of Nyrén and Lundin [32], which later became a core element of "pyrosequencing", a wellestablished DNA sequencing method [33]. In their procedure, PP i is converted to ATP by reaction with adenosine-5 -phosphosulfate (APS), catalyzed by ATP sulfurylase (E.C. 2.7.7.4), and the ATP formed is detected with a firefly luciferase/luciferin system. This method is used in three versions in PPase studies (Table 2 and Figure 3). In the first version, all components of the detection system, including ATP sulfurylase, luciferase, APS, and luciferin, are added to the PPase assay mix, and PP i formation is recorded continuously. This became possible because neither APS nor luciferin affects PPase activity, and both ATP sulfurylase and PPase need Mg 2+ as a cofactor. The synthesized PP i is continuously converted into ATP at such a high rate that the steady-state PP i level is not inhibitory for PPase and luciferase activities. This is ensured by maintaining high ATP sulfurylase concentration, such that its doubling does not increase the measured rate of ATP accumulation. In the second assay version, the PPase reaction mix contains ATP sulfurylase/APS, which continuously converts the formed PP i into ATP, whose concentration is determined at fixed time points with luciferase/luciferin in a separate tube. Importantly, PPase does not hydrolyze ATP in the presence of Mg 2+ . Both assay versions combine the high sensitivity of the luciferase ATP determination with the possibility to accumulate detectable amounts of ATP while keeping the steady-state concentration of PP i at a sub-equilibrium, non-inhibitory level. ATP sulfurylase and luciferase act thus as amplifiers of the PP i -generated signal and allow the measurement of the initial velocities of PP i synthesis from P i .  Determined PP i content refers to the sum of enzyme-bound and medium PP i . The latter is measured similarly but using a 100 times lower enzyme concentration and subtracted.  Several comments on the assay procedure are appropriate. First, phosphoric acid and its salts are often contaminated with PPi, leading to high background luminescence. A low-PPi potassium phosphate can be prepared in the following way: phosphoric acid is diluted to 0.2 M with water, boiled for 3 h, and neutralized with KOH [39]. Second, low but significant background luminescence results from APS being a poor substrate for luciferase. Third, as Pi is added in high concentrations, care should be taken to ensure the assay mixture's desired pH. It is not enough to adjust to this value the pH of the stock Pi solution because the formation of the MgHPO4 complex from H2PO4 -upon mixing stock Pi and Mg 2+ solutions would cause acidification of the medium (by 0.10 pH unit with 50 mM Pi, 5 mM Mg 2+ in 0.1 M MOPS-KOH buffer, pH 7.2). This effect is compensated by adding an appropriate amount of KOH to the assay mixture. An Excel subroutine to calculate this amount, which depends on Pi, Mg 2+ , and H + concentrations and reaction volume, is available (Supplementary Material). Alternatively, one can adjust the pH of the stock Pi solution used at each Mg 2+ concentration to the value (above the desired assay pH) determined in a trial titration.
A different coupled-enzyme assay with similar characteristics has been described [40]. In this assay, PPi is converted to adenosine 5'-triphosphate (ATP) by pyruvate phosphate dikinase in the presence of its substrates, pyruvate phosphate and AMP. The ATP formed is similarly determined by the firefly luciferase reaction. The detection limit for PPi is approximately 10 -8 M. The use of this assay in PPase studies has not yet been reported.

Ionic Equilibria in the Assay Media
Magnesium, the essential cofactor of PPases, has a dual role in catalysis-it activates the enzyme and substrates by forming complexes with both. Because the enzyme can sense only the actual substrate species in the medium, but not total substrate concentration, the magnesium complexes of the substrates PPi and Pi are named "true substrates." A meaningful analysis of enzyme kinetics with the Michaelis-Menten equation should be The third version of the PP i assay accesses the equilibrium of the PP i amounts of ATP while keeping the steady-state concentration of PPi at a sub-equilibrium, non-inhibitory level. ATP sulfurylase and luciferase act thus as amplifiers of the PPi-generated signal and allow the measurement of the initial velocities of PPi synthesis from Pi.

Assay Type and Procedure
Comments PPi synthesis (continuous) [34] Assay mixture contains calculated amounts of potassium phosphate and MgCl2, 0.7 U/mL ATP-sulfurylase, 10 µM APS, 5 µL of luciferin/luciferase solution (Sigma ATP assay mix, catalog no. FLAAM, reconstituted with 5 mL of water), 1 mM dithiothreitol, 0.8 mg/mL bovine serum albumin and 0.1 M MOPS-KOH buffer in a total volume of 0.2 mL. The reaction is initiated by adding PPase, and the time-course of luminescence is followed with a luminometer (e. g., LKB model 1250). After the PPigenerated signal stabilizes, 2 µL of 10 µM ATP are added for internal calibration.
ATP sulfurylase is from BioLabs (UK). Its concentration should be sufficiently high so that its doubling does not change the measured rate of ATP accumulation. The luminescenceversus-time dependencies are slightly curved because of the slow inactivation of luciferase in the reaction medium.
PPi synthesis (fixed-time) [35] Assay mixture contains calculated amounts of potassium phosphate and MgCl2, 0.7 U/mL ATP-sulfurylase, 10  Crystalline phosphoric acid (Fluka) is freed from PPi contamination as described in the text.

Enzyme-bound PPi formation [36]
PPase (typically 20-100 µM) is preequilibrated with Pi and Mg 2+ under appropriate conditions in a 50 µL volume and quenched with 10 µL of 5 M trifluoroacetic acid. After several minutes, precipitated protein is removed by centrifugation, a 10 µL aliquot of the supernatant is added to 0.2 mL of the PPi assay cocktail (0.2 M Tris-HCl, pH 8.0, 0.4 U/mL ATP sulfurylase, 10 µM APS, 6 µL of the luciferase/luciferin solution, 1 mM dithiothreitol, 0.8 mg bovine serum albumin, 30 µM EGTA), and the luminescence is recorded. After the signal stabilizes, 2 µL of 10 µM PPi are added for internal calibration.
Determined PPi content refers to the sum of enzyme-bound and medium PPi. The latter is measured similarly but using a 100 times lower enzyme concentration and subtracted.
The third version of the PPi assay accesses the equilibrium of the PPi  2Pi reaction in the active site of PPase. This information is required to estimate the rate constants for individual steps of PPase catalysis [37,38]. This equilibrium is markedly shifted to the left by comparing the equilibrium in solution-up to 20% of enzyme-bound Pi is converted to PPi [37,38]. To assay, the enzyme-bound PPi, the enzyme (20-100 µM) is incubated with Pi and Mg 2+ , inactivated by trifluoroacetic acid to release enzyme-bound PPi into solution, and an aliquot is taken to assay PPi using the coupled ATP sulfurylase/luciferase procedure.
2P i reaction in the active site of PPase. This information is required to estimate the rate constants for individual steps of PPase catalysis [37,38]. This equilibrium is markedly shifted to the left by comparing the equilibrium in solution-up to 20% of enzyme-bound P i is converted to PP i [37,38]. To assay, the enzyme-bound PP i , the enzyme (20-100 µM) is incubated with P i and Mg 2+ , inactivated by trifluoroacetic acid to release enzyme-bound PP i into solution, and an aliquot is taken to assay PP i using the coupled ATP sulfurylase/luciferase procedure.
Several comments on the assay procedure are appropriate. First, phosphoric acid and its salts are often contaminated with PP i , leading to high background luminescence. A low-PP i potassium phosphate can be prepared in the following way: phosphoric acid is diluted to 0.2 M with water, boiled for 3 h, and neutralized with KOH [39]. Second, low but significant background luminescence results from APS being a poor substrate for luciferase. Third, as P i is added in high concentrations, care should be taken to ensure the assay mixture's desired pH. It is not enough to adjust to this value the pH of the stock P i solution because the formation of the MgHPO 4 complex from H 2 PO 4 − upon mixing stock P i and Mg 2+ solutions would cause acidification of the medium (by 0.10 pH unit with 50 mM P i , 5 mM Mg 2+ in 0.1 M MOPS-KOH buffer, pH 7.2). This effect is compensated by adding an appropriate amount of KOH to the assay mixture. An Excel subroutine to calculate this amount, which depends on P i , Mg 2+ , and H + concentrations and reaction volume, is available (Supplementary Material). Alternatively, one can adjust the pH of the stock P i solution used at each Mg 2+ concentration to the value (above the desired assay pH) determined in a trial titration.
A different coupled-enzyme assay with similar characteristics has been described [40]. In this assay, PP i is converted to adenosine 5'-triphosphate (ATP) by pyruvate phosphate dikinase in the presence of its substrates, pyruvate phosphate and AMP. The ATP formed is similarly determined by the firefly luciferase reaction. The detection limit for PP i is approximately 10 −8 M. The use of this assay in PPase studies has not yet been reported.

Ionic Equilibria in the Assay Media
Magnesium, the essential cofactor of PPases, has a dual role in catalysis-it activates the enzyme and substrates by forming complexes with both. Because the enzyme can sense only the actual substrate species in the medium, but not total substrate concentration, the magnesium complexes of the substrates PP i and P i are named "true substrates." A meaningful analysis of enzyme kinetics with the Michaelis-Menten equation should be done in terms of these true substrates. Furthermore, the substrate-concentration dependencies of enzyme activity should be measured at a fixed free Mg 2+ ion concentration because it, not the total magnesium concentration, determines the fraction of the metal-complexed enzyme and, hence, its activity.
Pyrophosphoric acid undergoes a four-step dissociation, of which only the last two steps with the pK a3 of 6.38 and pK a4 of 9. done in terms of these true substrates. Furthermore, the substrate-concentration dependencies of enzyme activity should be measured at a fixed free Mg 2+ ion concentration because it, not the total magnesium concentration, determines the fraction of the metal-complexed enzyme and, hence, its activity. Pyrophosphoric acid undergoes a four-step dissociation, of which only the last two steps with the pKa3 of 6.38 and pKa4 of 9.11 should be considered in enzyme studies. Magnesium forms four complexes with three PPi species: MgP2O7 2-, Mg2P2O7, MgHP2O7 -, and MgH2P2O7 (Scheme 1). Furthermore, alkali metal ions, often present in the assay medium, form additional complexes: MP2O7 3-, M2P2O7 2-, and MHP2O7 2-(M = K + or Na + ). At fixed pH and alkali metal ion concentrations, the complexation in this system can be described by simple Scheme 2, in which each species is the sum of all differently protonated individual species of the same magnesium:PPi stoichiometry. Scheme 1. Complex formation between PPi, H + , Mg 2+ , K + , and Na + ions under physiological conditions. The number above or beside arrows refers to the minus logarithm of the respective dissociation constant [41].

Scheme 2.
A simplified description of complexation between PPi and Mg 2+ at fixed pH and alkali metal ion concentrations.
The values of Kmpp and Km2pp can be calculated at each pH value and alkali metal concentration from the dissociation constants of the individual PPi complexes with H + , Mg 2+ , K + , and Na + (tabulated in a previous publication [41]). Knowing the values of Kmpp and Km2pp allows calculating the total magnesium and PPi concentrations needed to maintain the required concentrations of MgPPi or Mg2PPi complex and free Mg 2+ . This algorithm was embedded in an Excel subroutine that calculates the composition of the assay medium at 25 °C and chosen pH and alkali metal concentrations (Supplementary Material).
The choice between MgPPi and Mg2PPi as the true substrate is largely arbitrary because the ratio of Mg2PPi and MgPPi concentrations is constant at a fixed Mg 2+ concentration, and switching from one complex to the other would only change the apparent value of the Michaelis constant. In most reported studies, MgPPi has been assumed to be the true substrate for PPases of families I and II and Mg2PPi to be such for membrane PPases. However, one should keep in mind that steady-state kinetic studies can define only stoichiometry of the statistically significant enzyme species, but not the ways of their for-Scheme 1. Complex formation between PP i , H + , Mg 2+ , K + and Na + ions under physiological conditions. The number above or beside arrows refers to the minus logarithm of the respective dissociation constant [41]. done in terms of these true substrates. Furthermore, the substrate-concentration dependencies of enzyme activity should be measured at a fixed free Mg 2+ ion concentration because it, not the total magnesium concentration, determines the fraction of the metal-complexed enzyme and, hence, its activity. Pyrophosphoric acid undergoes a four-step dissociation, of which only the last two steps with the pKa3 of 6.38 and pKa4 of 9.11 should be considered in enzyme studies. Magnesium forms four complexes with three PPi species: MgP2O7 2-, Mg2P2O7, MgHP2O7 -, and MgH2P2O7 (Scheme 1). Furthermore, alkali metal ions, often present in the assay medium, form additional complexes: MP2O7 3-, M2P2O7 2-, and MHP2O7 2-(M = K + or Na + ). At fixed pH and alkali metal ion concentrations, the complexation in this system can be described by simple Scheme 2, in which each species is the sum of all differently protonated individual species of the same magnesium:PPi stoichiometry. Scheme 1. Complex formation between PPi, H + , Mg 2+ , K + , and Na + ions under physiological conditions. The number above or beside arrows refers to the minus logarithm of the respective dissociation constant [41].

Scheme 2.
A simplified description of complexation between PPi and Mg 2+ at fixed pH and alkali metal ion concentrations.
The values of Kmpp and Km2pp can be calculated at each pH value and alkali metal concentration from the dissociation constants of the individual PPi complexes with H + , Mg 2+ , K + , and Na + (tabulated in a previous publication [41]). Knowing the values of Kmpp and Km2pp allows calculating the total magnesium and PPi concentrations needed to maintain the required concentrations of MgPPi or Mg2PPi complex and free Mg 2+ . This algorithm was embedded in an Excel subroutine that calculates the composition of the assay medium at 25 °C and chosen pH and alkali metal concentrations (Supplementary Material).
The choice between MgPPi and Mg2PPi as the true substrate is largely arbitrary because the ratio of Mg2PPi and MgPPi concentrations is constant at a fixed Mg 2+ concentration, and switching from one complex to the other would only change the apparent value of the Michaelis constant. In most reported studies, MgPPi has been assumed to be the true substrate for PPases of families I and II and Mg2PPi to be such for membrane PPases. However, one should keep in mind that steady-state kinetic studies can define only stoichiometry of the statistically significant enzyme species, but not the ways of their formation because schemes assuming different true substrates will lead to the same rate law The values of K mpp and K m2pp can be calculated at each pH value and alkali metal concentration from the dissociation constants of the individual PP i complexes with H + , Mg 2+ , K + and Na + (tabulated in a previous publication [41]). Knowing the values of K mpp and K m2pp allows calculating the total magnesium and PP i concentrations needed to maintain the required concentrations of MgPP i or Mg 2 PP i complex and free Mg 2+ . This algorithm was embedded in an Excel subroutine that calculates the composition of the assay medium at 25 • C and chosen pH and alkali metal concentrations (Supplementary Material).
The choice between MgPP i and Mg 2 PP i as the true substrate is largely arbitrary because the ratio of Mg 2 PP i and MgPP i concentrations is constant at a fixed Mg 2+ concentration, and switching from one complex to the other would only change the apparent value of the Michaelis constant. In most reported studies, MgPP i has been assumed to be the true substrate for PPases of families I and II and Mg 2 PP i to be such for membrane PPases. However, one should keep in mind that steady-state kinetic studies can define only stoichiometry of the statistically significant enzyme species, but not the ways of their formation because schemes assuming different true substrates will lead to the same rate law and are, therefore, indistinguishable. These considerations are especially important to remember when working with the kinetic schemes considering both substrate and Mg 2+ binding to the enzyme. P i is represented by a mixture of H 2 PO 4 − and HPO 4 2− forms in the pH range 4-10, and only the latter species forms 1:1 complexes with Mg 2+ , K + , and Na + (Scheme 3). The MgP i complex is generally assumed to be the true substrate in the PPase-catalyzed PP i synthesis. The total concentrations of Mg 2+ and P i needed to maintain the required concentrations of the MgP i complex and free Mg 2+ ions are calculated using the apparent dissociation constant for MgP i . The latter is obtained similarly at each pH value and alkali metal concentration from the dissociation constants of the individual P i complexes with H + , Mg 2+ , K + and Na + [42][43][44][45]. This algorithm was also embedded in an Excel subroutine (Supplementary Material), which additionally calculates the amount of the alkali needed to compensate for the acidification of the assay medium because of the selective complexation of the HPO 4 2− species by Mg 2+ . and are, therefore, indistinguishable. These considerations are especially important to remember when working with the kinetic schemes considering both substrate and Mg 2+ binding to the enzyme. Pi is represented by a mixture of H2PO4 -and HPO4 2-forms in the pH range 4-10, and only the latter species forms 1:1 complexes with Mg 2+ , K + , and Na + (Scheme 3). The MgPi complex is generally assumed to be the true substrate in the PPase-catalyzed PPi synthesis. The total concentrations of Mg 2+ and Pi needed to maintain the required concentrations of the MgPi complex and free Mg 2+ ions are calculated using the apparent dissociation constant for MgPi. The latter is obtained similarly at each pH value and alkali metal concentration from the dissociation constants of the individual Pi complexes with H + , Mg 2+ , K + , and Na + [42][43][44][45]. This algorithm was also embedded in an Excel subroutine (Supplementary Material), which additionally calculates the amount of the alkali needed to compensate for the acidification of the assay medium because of the selective complexation of the HPO4 2-species by Mg 2+ . Scheme 3. Complex formation between Pi, H + , Mg 2+ , K + , and Na + ions under physiological conditions. The number above or beside arrows refers to the minus logarithm of the respective dissociation constant [42][43][44][45].

Conflicts of Interest:
The authors declare no conflicts of interest.

Estimation of Initial Velocities from Product Formation Curves
If substrate (PPi) concentration remains saturating during the recording time, resulting in a linear time-course, the initial velocity is obtained straightforwardly. However, if substrate consumption exceeds 20%, drawing the tangent to the product accumulation curve by eye may result in a large error in the initial velocity, more commonly underestimating [46] (pp. 80-82). There are two ways to increase the accuracy of such analysis. One can derive the explicit equation or system of equations that describe the integral reaction kinetics and fit it to the product formation curve. A simpler alternative is to use a semiempirical procedure to manually draw the zero-time tangent to the curve. The procedure described below determines the second point, though, which the tangent must pass, in addition to the zero-time point. The required initial information includes the instrument signal (absorbance increase in recorder paper divisions) upon complete substrate conversion (Pinf). This parameter is calculated from the expected Pi concentration upon complete Scheme 3. Complex formation between P i , H + , Mg 2+ , K + , and Na + ions under physiological conditions. The number above or beside arrows refers to the minus logarithm of the respective dissociation constant [42][43][44][45].

Conflicts of Interest:
The authors declare no conflict of interest.

Estimation of Initial Velocities from Product Formation Curves
If substrate (PP i ) concentration remains saturating during the recording time, resulting in a linear time-course, the initial velocity is obtained straightforwardly. However, if substrate consumption exceeds 20%, drawing the tangent to the product accumulation curve by eye may result in a large error in the initial velocity, more commonly underestimating [46] (pp. 80-82). There are two ways to increase the accuracy of such analysis. One can derive the explicit equation or system of equations that describe the integral reaction kinetics and fit it to the product formation curve. A simpler alternative is to use a semi-empirical procedure to manually draw the zero-time tangent to the curve. The procedure described below determines the second point, though, which the tangent must pass, in addition to the zero-time point. The required initial information includes the instrument signal (absorbance increase in recorder paper divisions) upon complete substrate conversion (P inf ). This parameter is calculated from the expected P i concentration upon complete substrate conversion (twice the initial PP i concentration) and instrument sensitivity. One will also need an approximate K m value in terms of total PP i .
In a reaction with substrate exhaustion, the product accumulation curve will deviate from the tangent at the zero-time point, corresponding to initial velocity ( Figure A1). The difference between the two curves (∆P) increases with time, and its knowledge at a single value of P suffices to draw the zero-time tangent. The smaller the correction (∆P), the more accurate is the initial velocity estimate. Obviously, the ∆P value will decrease with increasing P inf and decreasing P and K m values. The equation for ∆P is obtained by combining Equation A1 and Equation A2, where V is the maximal velocity, t is time, K m is the Michaelis constant, and P inf is the initial substrate concentration. Equation A1 describes product (P 1 ) accumulation in time in a zero-order reaction proceeding with the rate equal to the enzyme-catalyzed reaction's initial velocity. Equation A2 describes the true integral kinetics of the enzyme-catalyzed reaction [46] (p. 78). At any time point t, P 1 = P + ∆P, and, accordingly, Equation A3 is valid. After its rearrangement, one obtains Equation A4, which links ∆P to P inf , P, and K m values.
Molecules 2021, 26, x FOR PEER REVIEW 10 of 13 substrate conversion (twice the initial PPi concentration) and instrument sensitivity. One will also need an approximate Km value in terms of total PPi. In a reaction with substrate exhaustion, the product accumulation curve will deviate from the tangent at the zero-time point, corresponding to initial velocity ( Figure A1). The difference between the two curves (∆P) increases with time, and its knowledge at a single value of P suffices to draw the zero-time tangent. The smaller the correction (∆P), the more accurate is the initial velocity estimate. Obviously, the ∆P value will decrease with increasing Pinf and decreasing P and Km values. The equation for ∆P is obtained by combining Equation A1 and Equation A2, where V is the maximal velocity, t is time, Km is the Michaelis constant, and Pinf is the initial substrate concentration. Equation A1 describes product (P1) accumulation in time in a zero-order reaction proceeding with the rate equal to the enzyme-catalyzed reaction's initial velocity. Equation A2 describes the true integral kinetics of the enzyme-catalyzed reaction [46] (p. 78). At any time point t, P1 = P + ∆P, and, accordingly, Equation A3 is valid. After its rearrangement, one obtains Equation A4, which links ∆P to Pinf, P, and Km values. The dependencies of ∆P on Pinf calculated at several fixed P and Km values are presented in Figure A2. These graphs are used to determine ∆P at given Pinf and P values and draw the tangent for any reaction with substrate depletion. The top left panel applies to a first-order reaction. As expected, the ∆P value decreases, and the accuracy of the initial velocity estimate increases accordingly with increasing Pinf and decreasing P. Optimally, P should be chosen between 20 and 50% of Pinf, also keeping in mind that the signal-tonoise ratio becomes unfavorable at low P values. Although this treatment involves visual interpolation and is approximate, it decreases the error in the estimated initial velocity to less than 10%, which is sufficient for most uses. The simplicity of the procedure compared to that of Cornish-Bowden [47] and other known ones [46] (pp. 80-85) has made it a useful aid in the kinetic studies of PPase. Although we have been mainly using this correction The dependencies of ∆P on P inf calculated at several fixed P and K m values are presented in Figure A2. These graphs are used to determine ∆P at given P inf and P values and draw the tangent for any reaction with substrate depletion. The top left panel applies to a first-order reaction. As expected, the ∆P value decreases, and the accuracy of the initial velocity estimate increases accordingly with increasing P inf and decreasing P. Optimally, P should be chosen between 20 and 50% of P inf , also keeping in mind that the signal-tonoise ratio becomes unfavorable at low P values. Although this treatment involves visual interpolation and is approximate, it decreases the error in the estimated initial velocity to less than 10%, which is sufficient for most uses. The simplicity of the procedure compared to that of Cornish-Bowden [47] and other known ones [46] (pp. 80-85) has made it a useful aid in the kinetic studies of PPase. Although we have been mainly using this correction procedure in the continuous assays, it is also applicable to and is especially important for the fixed-time assay carried out at non-saturating substrate concentration.
Molecules 2021, 26, x FOR PEER REVIEW 11 of 13 procedure in the continuous assays, it is also applicable to and is especially important for the fixed-time assay carried out at non-saturating substrate concentration. Figure A2. The dependence of ∆P in Figure A1 on Pinf at seven fixed values of P (shown on one panel and kept same on the other panels) and six values of Km (shown on the panels) for an enzyme-catalyzed reaction. For simplicity, the values of ∆P, Pinf, P, and Km are shown in terms of the same arbitrary unit, for instance, recorder paper division. These graphs were created for an enzyme that obeys simple Michaelis-Menten kinetics with the product to substrate ratio of one. To use these graphs for PPase, which produces two Pi molecules from one PPi molecule, one must divide by two the measured values of P and Pinf if they are expressed in terms of Pi. Note that axis scaling is different on the two bottom panels. Figure A2. The dependence of ∆P in Figure A1 on P inf at seven fixed values of P (shown on one panel and kept same on the other panels) and six values of K m (shown on the panels) for an enzyme-catalyzed reaction. For simplicity, the values of ∆P, P inf , P and K m are shown in terms of the same arbitrary unit, for instance, recorder paper division. These graphs were created for an enzyme that obeys simple Michaelis-Menten kinetics with the product to substrate ratio of one. To use these graphs for PPase, which produces two P i molecules from one PP i molecule, one must divide by two the measured values of P and P inf if they are expressed in terms of P i . Note that axis scaling is different on the two bottom panels.