Creatine Kinase Equilibration and ΔGATP over an Extended Range of Physiological Conditions: Implications for Cellular Energetics, Signaling, and Muscle Performance

In this report, we establish a straightforward method for estimating the equilibrium constant for the creatine kinase reaction (CK Keq″) over wide but physiologically and experimentally relevant ranges of pH, Mg2+ and temperature. Our empirical formula for CK Keq″ is based on experimental measurements. It can be used to estimate [ADP] when [ADP] is below the resolution of experimental measurements, a typical situation because [ADP] is on the order of micromolar concentrations in living cells and may be much lower in many in vitro experiments. Accurate prediction of [ADP] is essential for in vivo studies of cellular energetics and metabolism and for in vitro studies of ATP-dependent enzyme function under near-physiological conditions. With [ADP], we were able to obtain improved estimates of ΔGATP, necessitating the reinvestigation of previously reported ADP- and ΔGATP-dependent processes. Application to actomyosin force generation in muscle provides support for the hypothesis that, when [Pi] varies and pH is not altered, the maximum Ca2+-activated isometric force depends on ΔGATP in both living and permeabilized muscle preparations. Further analysis of the pH studies introduces a novel hypothesis around the role of submicromolar ADP in force generation.

Understanding the energetic relationships between ∆G ATP and the enzymes that utilize ATP, as well as the metabolic pathways that generate ATP, requires precise knowledge of the free energy available under both in vivo and experimental conditions [23,24].The essentially irreversible hydrolysis reaction for ATP takes the balanced form [24][25][26][27]. ATP 4− → ADP 3− + HPO 4 2− + H + (1) Because the major cellular ATPases utilize adenine nucleotides complexed with Mg 2+ and produce inorganic phosphate (Pi) and a non-stoichiometrically generated proton (H + ), the ATP hydrolysis reaction in cells can be rewritten more generally as A simple definition of ∆G ATP that implicitly incorporates the nuances of Equations ( 1) and ( 2) can be written as where ∆G 0 ATP is the free energy of ATP hydrolysis under standard conditions of temperature, pressure, and substrate/product concentrations, T is the temperature in • K, and R is the gas constant.In a healthy cell, ∆G ATP is on the order of 100 pN•nm per molecule of ATP, which provides an upper limit to the thermodynamic efficiency of work performed by a cellular ATPase [28].The exact values of ∆G ATP and ∆G 0 ATP , however, can vary significantly in the steady-state, both physiologically and experimentally, as implied by Equations ( 1)- (3).∆G ATP not only varies with changes in [ATP], [ADP], and [Pi] (Equation ( 3)); both ∆G ATP and ∆G 0 ATP are influenced by changes in pH, [Mg 2+ ], and other physicochemical parameters [29].∆G ATP and ∆G 0 ATP can vary dramatically as these parameters (especially Mg 2+ ) are altered, mainly due to formation of non-covalent complexes and the associated binding enthalpies of ions with adenine nucleotides and Pi [23,24].Of the parameters needed to estimate ∆G ATP , measurement of cytoplasmic ADP is particularly challenging.The cytoplasmic concentration of free ADP in healthy cells is typically below the limit of detection for direct measurement in vivo (e.g., by 31 P-NMR) [1,[30][31][32][33].In addition, it is only a small fraction of the total ADP in a cell, which includes protein-bound ADP (e.g., ADP bound to actin in the living cell) that is released during tissue processing to extract ADP for analysis.Further, the collection and extraction process may artifactually increase ADP due to hydrolysis of a fraction of the much greater amounts of ATP [34].
In living cells and in many experiments with skinned muscle fibers [35,36], [ATP], [ADP], and ∆G ATP are buffered by the creatine kinase (CK) or Lohmann reaction.CK catalyzes the reversible transfer of phosphate between phosphocreatine (PCr) and MgADP to resynthesize MgATP: The proton stoichiometric coefficient β (Equation ( 4)) is analogous to the coefficient α for ATP hydrolysis (Equation ( 2)).Because of the large amount of CK in striated muscle and its high activity, the reaction catalyzed by this enzyme is likely to be at or near equilibrium under most conditions.Thus, ∆G ATP in vivo is intimately linked to the CK reaction, and an assumption of near equilibrium can be used to first calculate free cytosolic ADP en route to estimating ∆G ATP .The equilibrium constant K eq for the reaction catalyzed by CK (Equation (4)) is Equation ( 5) is often rewritten in simplified form as an apparent equilibrium constant (K eq ) [29,37]: Equation ( 6) is compatible with most analytical measurements at a given pH, because there is no attempt to distinguish among the various ionic species such as those included in Equation (5).Each sum in Equation ( 6) includes all of the relevant ionic species, including minor species, for example, for ATP: When K eq has been determined, Equation ( 6) can be of utility in metabolic studies for estimating the cytoplasmic concentration of free ADP and, in combination with Equation (3), ∆G ATP .While K eq (Equation ( 6)) is proportional to K eq (Equation ( 5)), the constant of proportionality varies with pH, temperature, free Mg 2+ (reported as pMg = −log [Mg 2+ ], where [Mg 2+ ] is in molar units), etc., which places a severe limitation on how broadly any estimate of K eq [pH, pMg, T] can be applied.Many of the same parameters that affect ∆G ATP and ∆G 0 ATP (Equation ( 3)), including [Mg 2+ ] and pH, along with the temperature and ionic strength (Γ/2), affect K eq (Equation ( 6)).A comprehensive empirical approach to determine [ADP] for calculating ∆G ATP must account for variations of these parameters in living muscle as well as for in vitro studies in order to accurately assess both free energy changes and their physiological consequences.
Central to detailed models of the actomyosin crossbridge cycle is the thermodynamic constraint that ∆G ATP is a primary determinant of steady-state isometric force [38][39][40][41].Furthermore, a quasilinear relation has been identified between ∆G ATP and ATP hydrolysis flux when constant pH is maintained [42,43].Elevated Pi reduces the maximum isometric force in skinned muscle fibers [35,[44][45][46][47][48].The inverse correlation between maximum isometric force and Pi observed in skinned fibers has been confirmed in isolated intact slow-twitch muscle from mice, where lowering Pi resulted in an increase in maximum isometric force [49].Because ∆G ATP varies inversely with Pi (Equation (3)), the observed relationship that the isometric force varies with changes in Pi provides strong support for an energetic constraint on the molecular mechanism of force generation by actomyosin in both skinned fibers and in isolated intact muscles.In accordance with this concept, Karatzaferi et al. [45] reported that the maximum isometric force varies with the change in free energy when [Pi] is varied over several orders of magnitude, leading to the idea that free energy determines isometric force through its influence on actomyosin bond strength.The generality of physiological and experimental circumstances in which it can be directly applied to understand muscle function has not been fully examined.While one could consider varying ∆G ATP through changes in [ATP] and/or [ADP] according to the definition of ∆G ATP (Equation (3)), [ATP] and [ADP] are more challenging to vary in a controlled and independent manner [36,[50][51][52].The multiple influences of pH (Equations ( 2) and (4) plus the involvement of [H + ] in ion-binding equilibria) further contribute to the challenge of obtaining accurate estimates of ∆G ATP , which has prevented rigorous empirical tests of whether cellular ATP-driven processes, molecular motors in particular, can vary either their coupling to or work performed by ∆G ATP , particularly in light of large physiological fluctuations in ∆G ATP [35,[42][43][44][45]47].
In view of the central role of the CK reaction for determining ∆G ATP in many cell types, an important biochemical goal of this study was, first, a quantitative measurement of K eq for the CK reaction (Equation ( 6)) across a broad range of physiological and experimentally relevant pH, [Mg 2+ ], and temperatures while holding Γ/2 constant.With these results, we could use readily determined concentrations of ATP, PCr, and Cr to estimate [ADP] for any combination of pH, [Mg 2+ ], and temperature within the ranges examined.This empirical analysis produced a comprehensive quantitative adjustment of the equilibrium constant across key differences in physiological parameters, permitting direct comparisons of ADP and ∆G ATP among disparate studies.
The relationship between ∆G ATP and mechanical output (e.g., isometric force) could then be examined quantitatively using results from skinned fibers and isolated muscles.
We present both in vitro and in vivo confirmations of the previously reported reciprocal relationship between isometric force and Pi demonstrated in both skinned [35,[44][45][46][47][48] and intact muscles [49] as well as the corresponding relationship between isometric force and ∆G ATP when Pi is varied [45].While the relationship between ∆G ATP and pH is complex, it can be readily predicted using the results of this study.We show here that the relationship between isometric force and ∆G ATP when pH is varied is different from that obtained with Pi when ∆G ATP is modulated by changing pH in either chemically skinned fibers [35] or intact muscle preparations [53].These results suggest that the effects of Pi on actomyosin are directly modulated through free energy changes, while the effects of pH on force may be primarily due to other factors, possibly including [ADP].The methods described here are generally applicable to studies of cellular energetics and mathematical modeling of metabolic flux in striated muscles, including myocardial bioenergetics [54].

31 P-NMR Analysis of Solutions
Solutions that mimic the intracellular environment (Section 4.1) were first analyzed by 31 P-NMR.Figure 1 shows a representative series of 31 P-NMR spectra obtained over the entire pH range (five discrete pH values: pH 6.0, 6.5, 7.0, 7.5, and 8.0), pMg 3.0, 30 • C, and 50 mM Cr added. 31P-NMR was used to validate significant aspects of solution composition and to demonstrate that equilibrium was achieved following addition of CK and prior to termination of the reaction for further analyses.
Int. J. Mol.Sci.2023, 24, 13244 5 is the calculated fraction of Pi that is in the second (of th protonation states of Pi (Figure 2D-F and Equation ( 9)).In Equation ( 9), pKa is the neg log of the acid dissociation constant at 20 °C, dpKa/dT is the change in pKa per °C, a is a scaling factor.All calculated values of [H2PO4 -]/Σ[Pi] at pMg 2 (points in Figure were simultaneously fitted to Equation ( 9); the regression predictions are shown as in Figure 2D and the fit parameter estimates are provided in Table 1.This was repe twice more for pMg 3 (Figure 2E and Table 1) and pMg 4 (Figure 2F and Table 1).evident in Figure 2, the data (left panels) and calculations (right panels) follow sim trends, although the fit parameters (Table 1) indicate a slightly greater variation of with [Mg 2+ ] and temperature than predicted.1.An external standard (H3PO4, chemical shift δ = 0) was placed in a small capillary and cen in the coil's sensitive volume.Spectra were acquired on a Varian 600 MHz spectrometer at the p phorus frequency (242 MHz) using an 8000 Hz sweep width.Data are the sum of 1024 trans collected with a 1.0 s recycle delay and a π/2 pulse width and 1024 complex points and zeroonce to a total of 2048 data points and exponentially filtered prior to the Fourier transform.that Pi chemical shift (δPi) moves from right to left along the δ-axis with increasing pH (botto top), corresponding to deprotonation of H2PO4 -with pKa around neutral pH.Additionally, not the γATP and βATP peaks become sharper with increasing pH (bottom-to-top) and that peak ting is evident under most conditions for the γATP and αATP peaks, as well as for the βADP αADP peaks where they are detectable.− with pK a around neutral pH.Additionally, note that the γATP and βATP peaks become sharper with increasing pH (bottom-to-top) and that peak splitting is evident under most conditions for the γATP and αATP peaks, as well as for the βADP and αADP peaks where they are detectable. 31P-NMR spectra obtained at equilibrium (Figure 1) allowed determination of a pK a for H + binding by Pi within the pH range 6-8 along with how that pK a is affected by temperature and pMg (Figure 2).Such information is useful for calibration of pH i in living tissue by evaluating the chemical shift of Pi relative to PCr (note that the chemical shift of PCr relative to an external standard of H 3 PO 4 is −2.54 ppm, and is essentially constant over the physiological pH range).These relationships were quantified as adapted from Kost [55]: where δ Pi is the 31 P-NMR chemical shift difference between Pi at a given pH and the external standard (see spectra in Figure 1).The temperature dependencies of the extreme acid chemical shift δ A (T), the extreme basic chemical shift δ B (T), and the difference between them were consistent with, and were assumed to be the same as those described by Kost [55].
The variable ∆ in Equation ( 8) was necessary to allow for a small offset in chemical shift between the current dataset and the values presented in Kost [55]. Figure 2A-C shows 31 P-NMR chemical shift titrations for Pi (δ Pi ) at 10  2A), pMg 3.0 (Figure 2B), and pMg 4.0 (Figure 2C).All data at each pMg were simultaneously fitted to Equation (8); the resulting fits are shown in Figure 2A-C and the parameter estimates are provided in Table 1.
The coefficients from the chemical shift data (Figure 2A-C and Table 1) were corroborated with calculations from ion binding equilibria that were used for calculating the solution composition (Section 4.1), fitted to the following relationship: where Σ[Pi] is the sum over all relevant ionic forms of Pi, and has the same form as Equation (7) describing Σ[ATP] (Section 4.1): Thus, [H 2 PO 4 − ]/Σ[Pi] is the calculated fraction of Pi that is in the second (of three) protonation states of Pi (Figure 2D-F and Equation ( 9)).In Equation ( 9), pK a is the negative log of the acid dissociation constant at 20  2D) were simultaneously fitted to Equation (9); the regression predictions are shown as lines in Figure 2D and the fit parameter estimates are provided in Table 1.This was repeated twice more for pMg 3 (Figure 2E and Table 1) and pMg 4 (Figure 2F and Table 1).As is evident in Figure 2, the data (left panels) and calculations (right panels) follow similar trends, although the fit parameters (Table 1) indicate a slightly greater variation of pK a with [Mg 2+ ] and temperature than predicted.In all panels, blue is 10 °C, green is 20 °C, yellow is 30 °C, and red is 40 °C.In panels (A-C), all data in each panel (single pMg) were simultaneously fitted to Equation (8) by nonlinear least squares regression, while in panels (D-F) all values in each panel (single pMg) were simultaneously fitted to Equation ( 9) by nonlinear least squares regression; regression parameter estimates are provided in Table 1.Note the qualitative correspondence between the left and right panels.
Table 1.Regression coefficients for pH titrations of Pi from 31 P-NMR experiments and calculated predictions at three different free magnesium concentrations.Values are nonlinear least squares regression parameter estimates ± SE for the curves shown in Figure 2. Rows labeled NMR are parameter estimates for 31 P-NMR chemical shift data fitted to Equation (8) (Figure 2A-C).Rows labeled "calc" are parameter estimates for regression of Pi titration calculated predictions from binding equilibria as described in Section 4.1 fitted to Equation (9) (Figure 2D-F  In all panels, blue is 10 After equilibrium was demonstrated by 31 P-NMR (Figure 1), the CK reaction was terminated by denaturing the enzyme, followed by HPLC analysis of metabolites (Section 4.3); SDS was chosen as the denaturant (Section 4.2) to minimize spontaneous hydrolysis of analytes that occurs with some other methods of stopping enzyme-catalyzed reactions.Representative chromatograms for two conditions in the large matrix of solutions (Section 4.1) are shown in Figure 3: pMg 4.0, pH 8, and 10 • C (Figure 3A,C) and pMg 2.0, pH 8, and 40 • C (Figure 3B,D).These two conditions represent low or high concentrations, respectively, for both ADP (anion exchange chromatography in Figure 3A,B) and Cr (cation exchange chromatography in Figure 3C,D).2A-C).Rows labeled "calc" are parameter estimates for regression of Pi titration calculated predictions from binding equilibria as described in Section 4.1 fitted to Equation (9) (Figure 2D-F After equilibrium was demonstrated by 31 P-NMR (Figure 1), the CK reaction was terminated by denaturing the enzyme, followed by HPLC analysis of metabolites (Section 4.3); SDS was chosen as the denaturant (Section 4.2) to minimize spontaneous hydrolysis of analytes that occurs with some other methods of stopping enzyme-catalyzed reactions.Representative chromatograms for two conditions in the large matrix of solutions (Section 4.1) are shown in Figure 3: pMg 4.0, pH 8, and 10 °C (Figure 3A,C) and pMg 2.0, pH 8, and 40 °C (Figure 3B,D).These two conditions represent low or high concentrations, respectively, for both ADP (anion exchange chromatography in Figure 3A,B) and Cr (cation exchange chromatography in Figure 3C,D).Both samples initially contained 1 mg/mL rabbit CK.At equilibrium as determined by monitoring the reactions using 31 P-NMR spectroscopy (Figure 1), each reaction was stopped with addition of SDS at 2% per mg CK (Section 4.2).For both HPLC methods, detection was by optical absorbance at 210 nm and peak areas were quantified against calibration curves determined using known standards.Numerical scales on vertical (optical absorbance) axes correspond to detector output in millivolts.Note that scales for the vertical (optical absorbance) axes are the same for panels (A,B) and for panels (C,D).
To optimize estimation of apparent equilibrium constants using regression analysis on HPLC data, we reformulated Equation (6) (Keq′) to incorporate pH, i.e., to bring it closer to Keq (Equation ( 5)).While Keq″ (Equation (11)) includes pH, it retains compatibility with  C, pMg 2 and pH 8.Both samples initially contained 1 mg/mL rabbit CK.At equilibrium as determined by monitoring the reactions using 31 P-NMR spectroscopy (Figure 1), each reaction was stopped with addition of SDS at 2% per mg CK (Section 4.2).For both HPLC methods, detection was by optical absorbance at 210 nm and peak areas were quantified against calibration curves determined using known standards.Numerical scales on vertical (optical absorbance) axes correspond to detector output in millivolts.Note that scales for the vertical (optical absorbance) axes are the same for panels (A,B) and for panels (C,D).
To optimize estimation of apparent equilibrium constants using regression analysis on HPLC data, we reformulated Equation ( 6) (K eq ) to incorporate pH, i.e., to bring it closer to K eq (Equation ( 5)).While K eq (Equation ( 11)) includes pH, it retains compatibility with the analytical measurements of metabolites where the individual species are not distinguished experimentally: Rearrangement of Equation ( 11) yields a form that is suitable for nonlinear regression analysis on the aggregate of the HPLC data obtained for the family of solutions (all pH values) at a single pMg and temperature: However, in order to obtain the desired estimates of K eq using Equation ( 12) it was necessary to evaluate β, which is the net proton stoichiometric coefficient for hydrolysis of PCr to Cr (Equations ( 5), (11), and ( 12)).Therefore, we estimated β by calculating values for each of the experimental conditions according to the approach employed for constructing solutions (Section 4.1).For each combination of temperature and pMg, the calculated β was <1.0 at pH 6 and β → 1 as pH increased from 6 to 8 (Figure 4), which is in general agreement with previous estimates [56,57].Of all of the conditions examined in this study (Section 4.1), the two conditions included in Figure 4 illustrate the smallest (10 • C, pMg 4, dashed line) and largest (30 • C, pMg 3, solid line) variations in β calculated over the pH range of 6-8.
Rearrangement of Equation ( 11) yields a form that is suitable for nonlinear regression analysis on the aggregate of the HPLC data obtained for the family of solutions (all pH values) at a single pMg and temperature: However, in order to obtain the desired estimates of Keq″ using Equation ( 12) it was necessary to evaluate β, which is the net proton stoichiometric coefficient for hydrolysis of PCr to Cr (Equations ( 5), (11), and ( 12)).Therefore, we estimated β by calculating values for each of the experimental conditions according to the approach employed for constructing solutions (Section 4.1).For each combination of temperature and pMg, the calculated β was <1.0 at pH 6 and β → 1 as pH increased from 6 to 8 (Figure 4), which is in general agreement with previous estimates [56,57].Of all of the conditions examined in this study (Section 4.1), the two conditions included in Figure 4 illustrate the smallest (10 °C, pMg 4, dashed line) and largest (30 °C, pMg 3, solid line) variations in β calculated over the pH range of 6-8.
At a given temperature, chromatographic analyses of pMg and pH showed that ADP increased as the amount of Cr added increased and that ADP increased as pH increased at a given temperature, pMg, and Cr (Figure 5).Simultaneously fitting all of the data at 30 °C and pMg 3 for all pH values to Equation ( 12), the regression estimate of Keq″ was 1.075 × 10 9 M −1 ± 0.036 × 10 9 (Figure 5).For the purposes of this study, it was sufficient to obtain a regression estimate of Keq″ at each of the combinations of pMg and temperature (twelve total combinations yielding twelve estimates of Keq″ [pMg, T]), as the major goal of these measurements was to estimate [ADP] in experiments where it is difficult to measure [ADP] directly, thereby enabling calculation of ΔGATP).HPLC data for all twelve combinations of temperature and pMg were independently fitted to Equation ( 12 4)).pH dependence of β was predicted according to the ion binding equilibria (Section 4.1) for all conditions of this study.The smallest range of predicted values as a function of pH was obtained at pMg 4.0 and 10 °C (dashed line), while the largest range of predicted values as a function of pH was obtained at pMg At a given temperature, chromatographic analyses of pMg and pH showed that ADP increased as the amount of Cr added increased and that ADP increased as pH increased at a given temperature, pMg, and Cr (Figure 5).Simultaneously fitting all of the data at 30 • C and pMg 3 for all pH values to Equation ( 12), the regression estimate of K eq was 1.075 × 10 9 M −1 ± 0.036 × 10 9 (Figure 5).For the purposes of this study, it was sufficient to obtain a regression estimate of K eq at each of the combinations of pMg and temperature (twelve total combinations yielding twelve estimates of K eq [pMg, T]), as the major goal of these measurements was to estimate [ADP] in experiments where it is difficult to measure [ADP] directly, thereby enabling calculation of ∆G ATP ).HPLC data for all twelve combinations of temperature and pMg were independently fitted to Equation ( 12) (Appendix A) to obtain a matrix of estimates of K eq [pMg, T].   12) using values of β, as illustrated in Figure 4. Colors correspond to pH (blue, pH 8; cyan, pH 7.5; green, pH 7; orange, pH 6.5; red, pH 6).Comparable analyses were performed for each combination of temperature and pMg, resulting in a total of 12 plots (Figure A1) and regression estimates of Keq″ [pMg, T] (Figure 6).

Dependence of Keq″ for the Creatine Kinase Reaction on Mg 2+ and Temperature
Nonlinear regression parameter estimates of CK Keq″ (Equation ( 12)), obtained as shown in Figures 5 and A1, are shown in Figure 6 as a 3D surface plot.Multiple linear regression was performed to obtain a simple predictive equation for Keq″ [pMg, T] over the entire matrix of conditions employed: where T is the temperature in °C and the three regression parameter estimates (in parentheses) are provided ± SE regression (multiple R 2 = 0.855).Predictions from the multiple regression (Equation ( 13)) are shown in Figure 6 connected by thick blue lines.The empirical relationship in Equation ( 13) can be used in combination with Equation ( 12) to obtain estimates of cytoplasmic ADP levels over the broad physiologically and experimentally relevant range of pH 6-8, pMg 2-4, and T = 10-40 °C.This result is useful on its own for experiments on living cells using results that are typically measured in bioenergetic experiments, such as 31 P-NMR spectroscopy in combination with chemical analysis.12) using values of β, as illustrated in Figure 4. Colors correspond to pH (blue, pH 8; cyan, pH 7.5; green, pH 7; orange, pH 6.5; red, pH 6).Comparable analyses were performed for each combination of temperature and pMg, resulting in a total of 12 plots (Figure A1) and regression estimates of K eq [pMg, T] (Figure 6).

Dependence of K eq for the Creatine Kinase Reaction on Mg 2+ and Temperature
Nonlinear regression parameter estimates of CK K eq (Equation ( 12)), obtained as shown in Figures 5 and A1, are shown in Figure 6 as a 3D surface plot.Multiple linear regression was performed to obtain a simple predictive equation for K eq [pMg, T] over the entire matrix of conditions employed: where T is the temperature in • C and the three regression parameter estimates (in parentheses) are provided ± SE regression (multiple R 2 = 0.855).Predictions from the multiple regression (Equation ( 13)) are shown in Figure 6 connected by thick blue lines.The empirical relationship in Equation ( 13) can be used in combination with Equation ( 12) to obtain estimates of cytoplasmic ADP levels over the broad physiologically and experimentally relevant range of pH 6-8, pMg 2-4, and T = 10-40 • C.This result is useful on its own for experiments on living cells using results that are typically measured in bioenergetic experiments, such as 31 P-NMR spectroscopy in combination with chemical analysis.A1).Error bars are the SE regression for each estimate of Keq″.The results of a multiple linear least squares regression on all of the Keq″ data (Equation ( 13)) are shown by thick blue lines.This simple relationship allows for prediction of [ADP] (which is not always directly measurable, i.e., as illustrated in Figures 1 and 3) over a wide range of conditions, ranging from those relevant to intact tissue in vivo to experiments with permeabilized muscle fibers.

Estimation of ΔGATP
To determine ΔGATP for evaluation of mechanical measurements under various biochemical conditions, we evaluated the following relationship, which is a more complete description of Equation ( 3) that takes into account pH and ion binding equilibria (Section 2.1): where fATP, fADP, and fPi are: All of the ratios in Equation (15) vary with [H + ], [Mg 2+ ], and temperature, and can be calculated from the equilibrium binding constants as described for the solution calculations (Section 2.1).Points are regression estimates of K eq (Equations ( 10) and ( 11)) obtained as illustrated (Figures 5 and A1).Error bars are the SE regression for each estimate of K eq .The results of a multiple linear least squares regression on all of the K eq data (Equation ( 13)) are shown by thick blue lines.This simple relationship allows for prediction of [ADP] (which is not always directly measurable, i.e., as illustrated in Figures 1 and 3) over a wide range of conditions, ranging from those relevant to intact tissue in vivo to experiments with permeabilized muscle fibers.

Estimation of ∆G ATP
To determine ∆G ATP for evaluation of mechanical measurements under various biochemical conditions, we evaluated the following relationship, which is a more complete description of Equation ( 3) that takes into account pH and ion binding equilibria (Section 2.1): where f ATP , f ADP , and f Pi are: All of the ratios in Equation (15) vary with [H + ], [Mg 2+ ], and temperature, and can be calculated from the equilibrium binding constants as described for the solution calculations (Section 2.1).Equation ( 14) can be expanded to a form that is more useful for calculating the individual contributions of each component: (16) The first three terms in Equation ( 16) comprise ∆G 0 ATP (Equation ( 3)), which together account for the pH and Γ/2 dependencies of ∆G ATP as well as part of the temperature dependence.K ATP was set to 9.91 × 10 −7 M 2 , meaning that ∆G 0 ATP was −32 kJ mol −1 at pH 7, pMg 3, and 37  (16).For example, ∆G ATP varies with pH because of the proton concentration term (RT ln[H + ]) and because pH affects the ratios f ATP , f ADP , and f Pi (Equation ( 15)).In addition, there is an influence of pH on K eq that markedly alters [ADP] at given levels of [ATP], [PCr], and [Cr] (Figures 5 and A1).

Influence of [Pi] on ∆G ATP and Muscle Force
Within living skeletal muscle, the cytoplasmic Pi concentration can vary over a wide range [37,58].In permeabilized muscle, logarithmic increases in [Pi] depress the maximum Ca 2+ -activated isometric force [45,59,60].Taken together, these observations suggest that within rather wide and physiologically relevant limits the force likely varies linearly with ∆G ATP (Equation ( 3)), at least with respect to variations in Pi concentration.To quantitatively examine this possibility, we first examined tetanic force of isolated soleus muscle from mice in the presence and absence of pyruvate in the bathing medium (Figure 7A).In the presence of pyruvate, the intracellular Pi of slow muscle is reduced from approximately 6 mM (control) to 1 mM or less, as determined by 31 P-NMR spectroscopy [49].Force for isolated soleus muscle was normalized to the control condition.Fast muscle results are not included because the resting Pi is much lower (~1 mM or less) compared with slow muscle [49,58], and we could not discern through our methods whether addition of pyruvate reduced intracellular Pi substantially enough to influence isometric force.
In order to allow variation of [Pi] beyond what is possible in vivo, we measured the maximum steady-state isometric force of single skinned fibers from rabbit soleus (Figure 7B, red squares) and psoas (Figure 7B, open squares) muscle when [Pi] in the bathing solution was varied between 0.1 and 36 mM.This range was the maximum extent of variation that could be achieved without exceeding the ionic strength constraint (Section 4.4.1) on the upper end of the Pi concentration range or adding a Pi "mop" [45,59,61] to extend the lower end.We verified that sufficient activating Ca 2+ was present to achieve maximum force despite the decrease in Ca 2+ -sensitivity (rightward shift of the force-pCa relation) observed at elevated Pi levels in both psoas (Figure A2A and Table A1) and soleus (Figure A2B and Table A1) muscle fibers.Our observation of decreased Ca 2+ -sensitivity with elevated Pi is consistent with previous observations by others [62][63][64][65][66].Each data point in Appendix C was normalized to bracketing control measurements at 0.1 mM Pi.The force for each skinned psoas fiber (Figure 7B) was renormalized to the regression estimate of force at 1 mM Pi, a concentration that is comparable to what is found in living fast muscle fibers [58].Force for skinned soleus fibers (Figure 7B) was renormalized to the regression estimate of force at 6 mM Pi for consistency with the isolated soleus muscle data (Figure 7A).
For all three muscle preparations, the maximum isometric force decreased with increasing Pi (Figures 7, A2 and A3A,B), which corresponds to the force decreasing as ∆G ATP became less negative (Figure 7).∆G ATP was calculated for each experimental condition according to Equation (16), assuming that the CK reaction (Equation ( 4)) was at equilibrium in the muscle preparations.The slopes for the three relationships between force and ∆G ATP were linear, and were similar for intact and skinned soleus preparations over the experimental ranges examined.Note that the skinned fiber and isolated muscle datasets from soleus muscles are offset on the horizontal axes in Figure 7 because the skinned fiber conditions (primarily the levels of Cr and ADP and the temperature) were not designed to exactly match the conditions in living muscle cytoplasm.experimental ranges examined.Note that the skinned fiber and isolated muscle datasets from soleus muscles are offset on the horizontal axes in Figure 7 because the skinned fiber conditions (primarily the levels of Cr and ADP and the temperature) were not designed to exactly match the conditions in living muscle cytoplasm.[Pi] was varied in intact muscles (panel (A)) by altering the substrate supplied extracellularly [49], and the force and biochemical data are from that study; [Pi] in the presence of pyruvate was set to 1 mM in the plot, with the upper limit determined [49].[Pi] for skinned fiber experiments (panel (B)) was 0.1-36 mM.The psoas force at the first Pi concentration in panel (B) was previously published in Chase and Kushmerick [35], and the psoas force at the second Pi concentration in panel (B) was previously published in Chase and Kushmerick [36]; all other data in panel (B) are previously unpublished.Measurements on skinned fibers in panel (B) were from N = 28 psoas fibers from ten rabbits and N = 19 soleus fibers from six rabbits.As described in the text, the force for the psoas fibers in panel (B) was normalized to 1 mM Pi, while the soleus force was normalized to 6 mM to reflect the higher basal levels of Pi in slow muscle [58].The points in panel (A) are mean values, while the points in panel (B) are individual measurements.The regression slopes are −0.0248 in panel (A), −0.0306 (regression SE 0.0029) for soleus data in panel (B), and −0.0406 (regression SE 0.0038) for psoas data in panel (B); note that the slopes are not significantly different for soleus data in panels (A,B), while the slope for skinned psoas fibers is significantly steeper than that for skinned soleus fibers.
The [Pi]-dependence of isometric force for permeabilized fibers from rabbit psoas (Figure A3A) and soleus (Figure A3B) muscles illustrates that slow fibers are more sensitive to Pi in the sense that the force declines to a greater extent at lower concentrations of Pi.Considering this observation in the context of physiological levels of intracellular Pi, where fast muscle has much lower levels of cytoplasmic Pi at rest [58], the data in Figure 7 and Figure A3A,B indicate that the maximum isometric force of fast muscle should be higher than that of slow muscle in healthy living muscles.Control experiments where sulfate concentration was varied at a constant baseline of 0.1 mM Pi showed that [SO4 = ] caused only a small decline in isometric force in permeabilized fibers from both fast (Figure A3C) and slow (Figure A3D) muscles relative to that observed over the same concentration range of Pi (Figure A3A,B).Thus, the inhibitory effects of Pi on isometric force are not due to nonspecific effects of multivalent anions, and the variation of force with ΔGATP when [Pi] is varied (Figure 7) can be directly attributed to the contribution of [Pi] to ΔGATP.
Our examination of the relationship between unloaded shortening velocity (VUS) and Pi (Figure A4) confirmed prior studies that showed Pi to have little or no effect on the rate [Pi] was varied in intact muscles (panel (A)) by altering the substrate supplied extracellularly [49], and the force and biochemical data are from that study; [Pi] in the presence of pyruvate was set to 1 mM in the plot, with the upper limit determined [49].[Pi] for skinned fiber experiments (panel (B)) was 0.1-36 mM.The psoas force at the first Pi concentration in panel (B) was previously published in Chase and Kushmerick [35], and the psoas force at the second Pi concentration in panel (B) was previously published in Chase and Kushmerick [36]; all other data in panel (B) are previously unpublished.Measurements on skinned fibers in panel (B) were from N = 28 psoas fibers from ten rabbits and N = 19 soleus fibers from six rabbits.As described in the text, the force for the psoas fibers in panel (B) was normalized to 1 mM Pi, while the soleus force was normalized to 6 mM to reflect the higher basal levels of Pi in slow muscle [58].The points in panel (A) are mean values, while the points in panel (B) are individual measurements.The regression slopes are −0.0248 in panel (A), −0.0306 (regression SE 0.0029) for soleus data in panel (B), and −0.0406 (regression SE 0.0038) for psoas data in panel (B); note that the slopes are not significantly different for soleus data in panels (A,B), while the slope for skinned psoas fibers is significantly steeper than that for skinned soleus fibers.
The [Pi]-dependence of isometric force for permeabilized fibers from rabbit psoas (Figure A3A) and soleus (Figure A3B) muscles illustrates that slow fibers are more sensitive to Pi in the sense that the force declines to a greater extent at lower concentrations of Pi.Considering this observation in the context of physiological levels of intracellular Pi, where fast muscle has much lower levels of cytoplasmic Pi at rest [58], the data in Figures 7 and A3A,B indicate that the maximum isometric force of fast muscle should be higher than that of slow muscle in healthy living muscles.Control experiments where sulfate concentration was varied at a constant baseline of 0.1 mM Pi showed that [SO 4 = ] caused only a small decline in isometric force in permeabilized fibers from both fast (Figure A3C) and slow (Figure A3D) muscles relative to that observed over the same concentration range of Pi (Figure A3A,B).Thus, the inhibitory effects of Pi on isometric force are not due to nonspecific effects of multivalent anions, and the variation of force with ∆G ATP when [Pi] is varied (Figure 7) can be directly attributed to the contribution of [Pi] to ∆G ATP .
Our examination of the relationship between unloaded shortening velocity (V US ) and Pi (Figure A4) confirmed prior studies that showed Pi to have little or no effect on the rate limiting step for unloaded shortening [47,48,52].This means that, in contrast to isometric force (Figures 7 and A3A,B), ∆G ATP does not directly influence V US in skeletal muscle.

Influence of pH on ∆G ATP , [ADP], and Muscle Force
In view of the strong relationship between the maximum Ca 2+ -activated isometric force and ∆G ATP when [Pi] is varied (Figure 7), we extended the investigation by re-examining previously published measurements made with isolated perfused cat muscles [67] and skinned fibers from rabbit psoas and soleus muscles [35] when the pH surrounding the myofilaments was varied.In the study by Harkema et al. [67], intracellular acidification of biceps and soleus muscles was achieved by perfusion with a hypercapnic perfusate, and pH was determined by 31 P-NMR in a manner comparable to that shown in Figure 1; force was then normalized to the normocapnic condition.In the study on single skinned fibers from rabbit muscles, the maximum steady-state isometric force was measured using psoas and soleus fibers when the pH of the bathing solution was varied between pH 6 and 8 [35]; the force for each skinned fiber was normalized to the pH 7.1 condition, which is comparable to that in living fast and slow muscles.
To examine the dependence of the isometric force on free energy when pH was varied, ∆G ATP was calculated for each experimental condition (Equation ( 16)), assuming that the CK reaction was at equilibrium.Force declined and ∆G ATP became more negative with decreased pH in both living and permeabilized muscles (Figure 8A).The slopes for the skinned fiber relationships between force and ∆G ATP were nonlinear and the slopes were positive for all preparations, opposite to what was observed when Pi was varied (Figure 7).In both intact and skinned muscles, the slope for fast fiber types was steeper than that for slow fiber types (Figure 8A).We conclude that the variation in ∆G ATP with pH (Figure 8A), in contrast to Pi (Figure 7), was not due to any direct influence of pH on ∆G ATP .

Influence of pH on ΔGATP, [ADP], and Muscle Force
In view of the strong relationship between the maximum Ca 2+ -activated isometric force and ΔGATP when [Pi] is varied (Figure 7), we extended the investigation by re-examining previously published measurements made with isolated perfused cat muscles [67] and skinned fibers from rabbit psoas and soleus muscles [35] when the pH surrounding the myofilaments was varied.In the study by Harkema et al. [67], intracellular acidification of biceps and soleus muscles was achieved by perfusion with a hypercapnic perfusate, and pH was determined by 31 P-NMR in a manner comparable to that shown in Figure 1; force was then normalized to the normocapnic condition.In the study on single skinned fibers from rabbit muscles, the maximum steady-state isometric force was measured using psoas and soleus fibers when the pH of the bathing solution was varied between pH 6 and 8 [35]; the force for each skinned fiber was normalized to the pH 7.1 condition, which is comparable to that in living fast and slow muscles.
To examine the dependence of the isometric force on free energy when pH was varied, ΔGATP was calculated for each experimental condition (Equation ( 16)), assuming that the CK reaction was at equilibrium.Force declined and ΔGATP became more negative with decreased pH in both living and permeabilized muscles (Figure 8A).The slopes for the skinned fiber relationships between force and ΔGATP were nonlinear and the slopes were positive for all preparations, opposite to what was observed when Pi was varied (Figure 7).In both intact and skinned muscles, the slope for fast fiber types was steeper than that for slow fiber types (Figure 8A).We conclude that the variation in ΔGATP with pH (Figure 8A), in contrast to Pi (Figure 7), was not due to any direct influence of pH on ΔGATP.[67] and (panels (A,B)) single skinned fibers from rabbit psoas (open squares fit with solid gray lines) and soleus (red filled squares fit with dashed red lines) muscles [35].Hypercapnic (acidic cytoplasmic pH 6.48-6.6)force data from cat muscles were normalized to the normocapnic condition (higher force; cytoplasmic pH 7.09-7.11)for each muscle type.Skinned fiber force data were normalized to that at pH 7.1 for the same fiber.Lower forces were associated with acidic (lower) pH and higher forces were associated with basic (higher) pH [35,67].The points represent the average ± SD.ΔGATP was estimated according to Equation ( 16) and [ADP] was estimated according to Equation ( 12) using values of Keq″ from  [67] and (panels (A,B)) single skinned fibers from rabbit psoas (open squares fit with solid gray lines) and soleus (red filled squares fit with dashed red lines) muscles [35].Hypercapnic (acidic cytoplasmic pH 6.48-6.6)force data from cat muscles were normalized to the normocapnic condition (higher force; cytoplasmic pH 7.09-7.11)for each muscle type.Skinned fiber force data were normalized to that at pH 7.1 for the same fiber.
Lower forces were associated with acidic (lower) pH and higher forces were associated with basic (higher) pH [35,67].The points represent the average ± SD. ∆G ATP was estimated according to Equation ( 16) and [ADP] was estimated according to Equation ( 12) using values of K eq from Equation (13).In panel (A) smooth curves were drawn through the points, while in panel (B) the lines represent nonlinear regression fits to Equation (17) (regression parameter estimates for K m are provided in the text).Note that in panel (A) the slope is steeper for fast muscles than for slow muscles when comparing within a species.
∆G ATP varies with pH in part because of a direct contribution of [H + ] in Equation (16) and in part because it influences the terms in Equation ( 16) containing f ATP , f ADP , and f Pi (Equation ( 15)) as well as [ADP] (Equation ( 12) and Figure 5).In particular, [ADP] at acidic pH is reduced to very low levels, on the order of 10 nM at pH 6.0 in the experiments described here (Equation ( 12) and Figure 5) and lower than what was attained in our prior experiments examining the effects of ADP on skeletal muscle contractility [36].Therefore, we examined the relationship between the steady-state isometric force and [ADP] (Figure 8B).The isometric force data for both fast and slow fiber types were described by a saturable binding relation (Equation ( 17)), with the affinity constants (K m ) estimated by nonlinear least squares regression (±SE) of 24.0 ± 5.3 nM for psoas fibers and 9.0 ± 0.9 nM for soleus fibers (Figure 8B).These values are consistent with the lack of effect of higher concentrations of ADP on isometric force at pH 7.1 reported in Chase and Kushmerick [36], and would be slightly lower if we considered only the proportion of ADP in the Mg 2+bound form (MgADP).However, it seems likely that protons modulate force by additional mechanisms beyond altering [ADP].

Discussion
The main results of this study are three-fold.First, we established a comprehensive formalism relating the apparent equilibrium constant (K eq ) for the creatine kinase reaction to broad changes in critical components, specifically Mg 2+ and temperature, that differ within and between experimental preparations and protocols.In circumstances where the CK reaction is at equilibrium, these factors, along with pH, influence two parameters, namely, cytosolic ADP and ∆G ATP , in a predictable manner-even though they typically cannot be measured directly-when [ATP], [Pi], [PCr], and [Cr] have been determined.Second, these results were applied to calculate ADP and ∆G ATP for experimental conditions in which the biochemical conditions were known and mechanical measurements could be made in skeletal muscle preparations.Third, we found that there appears to be marked variability in the contractile efficiency of force generation by skeletal muscle with changes in energetic conditions due to altered [Pi] or pH.

Estimation of K eq for the Creatine Kinase Reaction and Cytoplasmic Free ADP
The results of the first portions of our analysis allow quantitative estimation of intracellular pH from 31 P-NMR spectra (Figure 2 and Table 1) and K eq for the CK reaction (Figure 6 and Equation ( 13)) and ADP (Figure 5 and Equation ( 12)) from a combination of chemical and 31 P-NMR assays over a considerably wider range of physiological and biochemically relevant conditions than previously examined experimentally.
Our results on the use of δ Pi from 31 P-NMR spectra to estimate pH i (Figure 2, Equation ( 8), and Table 1) are in good agreement with the approach of Kost [55] over a similar range of temperatures, and extend the analysis over a wider range of Mg 2+ concentrations, a value that can be determined experimentally [30,[68][69][70].Data from these titration curves (Figure 2) are quite useful for in vivo 31 P-NMR studies at and beyond 37 • C.These curves were produced without incidental modulation of the solution, as occurs, e.g., during traditional titrations that incrementally add acid or base, thereby changing Γ/2.Thus, we avoided any influence on the chemical shift endpoints for Pi (the extreme acid δ A (T) and basic δ B (T) chemical shifts) or the pK a of Equation ( 8) [55].
The concept of determining K eq for the CK reaction en route to estimation of cytoplasmic [ADP] is well established [29], although its applicability has previously been limited to narrow ranges of conditions (note that K eq defined in Equation ( 6) applies to a specific pH, in contrast to K eq defined in Equation ( 11)).K eq for the CK reaction reported by Lawson and Veech [29] for physiologic conditions of 37 • C and pH 7.0 has been widely used, often with adjustments necessary for experimental temperature and/or pH.Lawson and Veech [29] evaluated the dependence of K eq on [Mg 2+ ] over a wider range of [Mg 2+ ] concentrations than reported here, though at a constant pH 7.They varied pH (pH ~7-8) over a limited range of [Mg 2+ ]; however, utilizing these broader ranges of conditions to calculate ADP typically requires estimation, interpolation, and in many instances, extrapolation.
The effect of temperature (5-38 • C) on the observed K eq for the creatine kinase reaction at pH ~7 has been reported from empirical studies [71], showing that CK K eq increases as temperatures decreases.This is in agreement with the data in Figure 6 and the corresponding negative regression coefficient for the temperature term in Equation ( 13).Further theoretical work extrapolated values for CK K eq as a function of both temperature and ionic strength [72].Golding et al. [73] calculated that at 38 • C, K eq increases when pH or pMg decrease.The former agrees with the expectations from Equations ( 6) and (10), and the latter is consistent with the data in Figure 6 and the corresponding, negative regression coefficient for the pMg term in Equation (13).However, Golding et al. [73] did not include binding constants for important cations known to be present in the cytosol, including K + and Ca 2+ , presumably because of their impact on proton binding coefficients, making extrapolation to these extremes difficult to interpret [74].
Considering the dependence on interpolation and extrapolation from experimental measurements to obtain an estimate of CK K eq along with the potential for wide-ranging estimates of [ADP], we decided that a comprehensive strategy was necessary to generate a comprehensive set of empirically derived values for CK K eq (ultimately K eq ).This strategy involved the construction of a matrix of model solutions utilizing binding constants and enthalpic terms for metabolite binding of all important ions present within the cytosol, sensitive analytic methods to determine metabolite contents for calculation of the equilibrium values for each condition, and a statistical approach to derive coefficients for proton stoichiometry over the entire data matrix.The calculations that went into constructing solutions such as those used in our experiments are well-established and have been received considerable attention and research effort [35,44,64,[74][75][76][77].As is evident in these references, a primary focus is to use these solutions to mimic major (though not all) specific aspects of the intracellular milieu in experiments on permeabilized muscle.
The results in Figure 6, along with the regression results in Equation ( 13), allow for reliable estimation of K eq across the pH range of 6.0-8.0,pMg range of 2.0-4.0, and temperature range of 10-40 • C. From this, [ADP] can be estimated using Equation (12) for given conditions of [ATP], [PCr], [Cr], [Mg 2+ ], pH, and temperature when CK is present with sufficient activity to achieve equilibrium.This appears to be the best approach for obtaining estimates of [ADP] under physiological conditions, and would be useful for studies on striated [37,57,58] and smooth [78] muscles.A FRET biosensor for ADP has been developed [79]; however, it cannot be expressed in vivo because its synthesis includes covalent modification of the protein component with rhodoamine fluorescent labels.Perhaps a FRET biosensor for ATP that can be expressed in cells [80] could be altered to discriminate physiologically relevant levels of ADP in the presence of the much higher levels of ATP found in healthy cells.

Estimation of ∆G ATP
The results described in the previous section greatly expand the range of physiological and experimental conditions for which ∆G ATP can be more easily and reliably estimated based on direct measurements of parameters that are part of many experimental routines.Longstanding studies of Alberty and co-workers, as well as others, have provided calculations for estimating ∆G ATP under a wide variety of conditions [23,25,[81][82][83][84][85][86].To apply the results of these studies to living tissues, however, requires knowledge of cytoplasmic [ADP] in addition to [ATP], [Pi], pH, [Mg 2+ ], etc.Thus, this extensive body of valuable work on its own is not sufficient to estimate cellular ∆G ATP .
We estimated ∆G ATP to be −58.9kJ mol −1 for mouse soleus (slow) muscle, with glucose as substrate at 25 • C (Figure 7A), and to be −57.3kJ mol −1 for cat soleus (slow) muscle and −64.6 kJ mol −1 for cat biceps (fast) muscle with normocapnic perfusate at 37 • C (Figure 8A).The difference between the slow and fast muscle types stems largely from the higher levels of Pi in slow muscles at rest, though there is a contribution from slightly lower ATP levels in slow muscles as well [49,58,67].Perhaps surprisingly, differences in PCR and Cr contribute little to the fiber type difference in ∆G ATP ; the resulting ADP levels due to the CK reaction are not very different (15.8 µM for mouse soleus, 27.6 µM for cat soleus, and 16.3 µM for cat biceps).The implications of these results impact the precise calculation of free [ADP] to values that in certain circumstances may be lower than previously calculated, necessitating the reinvestigation of previously reported ADP-dependent processes.
∆G ATP values from our skinned fiber experiments were substantially more negative than those from intact muscles of the same fiber type (Figures 7 and 8A).We estimated ∆G ATP to be −71.4 to −71.5 kJ mol −1 for permeabilized fibers from rabbit psoas (fast) muscle at 1 mM Pi and 12 • C (psoas controls in Figures 7B and 8A, respectively).∆G ATP for permeabilized soleus fibers would be exactly the same for the same solution conditions (e.g., −71.4 kJ mol −1 for soleus control in Figure 8A), although ∆G ATP was less negative for the soleus control in Figure 7B because force normalization in the [Pi] experiments accounted for the higher basal [Pi] in that fiber type [49,58,67].A substantial contributor to the more negative values of ∆G ATP for permeabilized muscles is the much lower [ADP] (~2 orders of magnitude) due to lesser amounts of Cr (also ~2 orders of magnitude, per Equation ( 12)) present in the control conditions for skinned fibers (Figure 8B) [36].

Implications for Actomyosin Interactions and the Physiology of Skeletal Muscle
The results of this study allowed us to make initial steps toward quantifying the relationship between the maximum Ca 2+ -activated isometric force and available energy over a wide range of conditions in situations where all of the relevant parameters can be controlled and/or measured.ATP plays two roles in the actomyosin crossbridge cycle: binding of MgATP to a nucleotide-free (rigor) crossbridge results in rapid dissociation of the myosin head from the thin filament; then, ATP hydrolysis by the myosin head (Equation ( 2)) provides the energy for the mechanical power stroke in the next crossbridge cycle.∆G ATP provides the ultimate limit for work performed by actomyosin [38,40,45,87].The dependence of actomyosin function on ∆G ATP is mechanistically important for understanding the energetics of actomyosin's ATPase cycle and for assessing physiological changes during hypoxia and muscle fatigue, where there are substantial alterations in cytosolic metabolite concentrations and cellular energy status.
Our data in Figures 7 and A3A,B are consistent with other studies on skinned muscle fibers in which the maximal steady-state isometric force varies logarithmically with [Pi] over a wide concentration range, including the physiological range of [Pi] [44,[46][47][48]52,60,64,65,[88][89][90].However, force for fast skeletal muscle fibers plateaus below ~100 µM [45,59,91], a concentration range that we did not explore (Figures 7 and A3A,B).Thus, within the limits of our experimental measurements, the data in Figure 7 are consistent with the ∆G ATP limiting force when [Pi] is varied (the second to last term on the right side of Equation ( 16)) according to the description of Pate et al. [92].
Our [Pi] data, when plotted on a linear scale (Figure A3A,B), are in apparent agreement with others suggesting that slow fibers are more sensitive to Pi compared with fast fibers at low Pi concentrations, but are less sensitive to Pi at high concentrations [48,88,93].The latter is consistent with the difference in slopes when force is plotted against ∆G ATP when [Pi] is varied (Figure 7B), which effectively corresponds to plotting on a logarithmic axis for [Pi].The slopes in Figure 7B are significant because of their relation to the energetics of actomyosin interactions [45,59,92].Interestingly, the slope obtained with intact soleus from mouse appears to be similar to that obtained with skinned fibers from rabbit soleus (Figure 7).The slope for intact muscle, however, was not as well defined as that for skinned fibers due to the greater difficulty in controlling cytosolic Pi in living muscle.Only two Pi concentrations were achieved for intact mouse soleus (Figure 7A), and the leftmost point (low [Pi] in the presence of pyruvate) is an upper limit for [Pi] because the limit of detection by 31 P-NMR is ~1 mM.Thus, the slope for intact muscle could be less steep.Regardless, the relationship between isometric force, [Pi], and ∆G ATP applies whether mechanical events leading to force generation occur prior to or after release of Pi from the myosin head [65,88,89,[94][95][96][97].
In contrast to Pi, the mechanism of force inhibition by pH is difficult to predict because protons can participate in the crossbridge cycle in multiple ways.During the ATPase cycle, proton release (stoichiometric coefficient α in Equation ( 2)) occurs simultaneously with Pi release because the affinity for H + of the phosphate moiety changes when it is cleaved from the terminal γ position within the ATP molecule.Thus, it is reasonable to assume that isometric force should vary with [H + ] in a manner analogous to that observed with [Pi].While the data of Nosek, Fender and Godt [46] are consistent with this hypothesis, subsequent measurements have suggested that this is not the case [35,60,98].The present study provides further evidence that the effects of pH are distinct from those of Pi (Figures 7 and 8).
When interpreted in terms of ∆G ATP , the effects of pH on isometric force observed in living muscle vary in a consistent manner with observations in permeabilized muscle regardless of fiber type, i.e., all of the relationships in Figure 8A have a positive slope.In addition, the slopes for fast muscles in Figure 8A are steeper compared with those for slow muscles.Studies on the direct effects of altered pH on muscle force generation indicate that force inhibition by H + is lower at physiological temperatures than at the lower temperatures that have often been used for experiments on reduced systems [53,98], which likely explains all or part of the differences in slopes between skinned fiber data obtained at cooler temperatures than data for intact muscle (Figure 8A).At 37 • C, acidification of isolated muscles did not affect the energetic cost of contraction [67].The inhibition of isometric force by Pi (e.g., Figure A3A,B) is similarly reduced as temperature is increased close to physiological levels [45,48,62].
The positive slopes of the relationships between isometric force and ∆G ATP when pH was varied (Figure 8A) are opposite to what was observed when [Pi] was varied (Figure 7).Significantly, the positive slopes in Figure 8A are opposite both to expectations and what is energetically possible [45,59,92].Therefore, we conclude that it is not ∆G ATP per se that determines isometric force production but the free energy associated with a specific step or steps associated with Pi release in the actomyosin ATPase cycle.
In search of an explanation of the effects of pH on force, we took advantage of the fact that our results for determining CK K eq over a wide range of conditions (Figure 6 and Equation ( 13)) allowed us to determine [ADP] for the pH solutions.The results (Figure 8B) appear consistent with high-affinity (nM) binding of MgADP to the myosin head.However, additional experiments are required to distinguish the effects of pH from those due to nucleotides.

Solution Composition for Biochemical Analyses
All chemicals and enzymes were of the highest degree of purity available and were obtained from Sigma Chemical Co.(St.Louis, MO, USA).Solutions for biochemical analyses were designed to mimic cytosolic composition over a broad range of physiological conditions and temperatures, and were constructed using known binding constants for each species (Table A2) [76,99].Solution composition (in mM) was 145 Na + , 6.5 K + , 2.5 EGTA (pCa 9), 8 MgATP, 30 PCr, and 1 Pi.No ADP was added.pMg (−log [Mg 2+ ], where [Mg 2+ ] is in molar units) was either 2, 3, or 4, as estimates of intracellular [Mg 2+ ] i are typically within that range [30,100].The pH range of 6-8 was examined because it is the most physiologically and experimentally relevant range [101][102][103]; pH buffer was 50 mM MES at pH 6 and 6.5, 50 mM MOPS at pH 7, or 50 mM TES at pH 7.5 and 8.0 to achieve optimal buffering based on the buffer pK a s.In all solutions for biochemical analyses we used Γ/2 = 0.25 M, with the ionic balance adjusted using acetate as the anion and Tris as the cation.The solutions were titrated to their final pH at each experimental temperature (10, 20, 30, or 40 • C); thus, there were 60 combinations of pH, pMg, and temperature, covering the range of physiologically relevant conditions as well as those most commonly used in biochemical experiments.To determine K eq , CK was added at ~75 units/mL.Under each of the 60 combinations, either no Cr was added or 0.5, 5, or 50 mM Cr was added immediately prior to addition of CK, leading to four discrete values of Cr.This allowed us to manipulate ATP/ADP at each condition (240 solutions in total).
Solution composition was calculated using a program that utilized the National Institute of Standards and Technology (NIST) Critically Selected Stability Constants of Metal Complexes Database [76,99,104].The desired [H + ] was calculated with correction for Γ/2 and temperature using the following equation from Khoo [105]: where T is the desired temperature ( • C).The first protonation of Cr and the equivalent protonation of PCr were not included in calculations because K a > 10 14 for both, meaning that protonation is essentially complete over the entire 6-8 pH range.

Nuclear Magnetic Resonance Spectroscopy
Phosphorus NMR ( 31 P-NMR) spectroscopy was performed on two high-field spectrometers.For equilibration studies, the spectrometer was a 7T GN 300 (Bruker Instruments, Billerica, MA, USA) using a 10 mm broadband commercial NMR probe tuned to the phosphorus frequency (121 MHz).A subset of experiments was performed at higher field strengths using a Varian 600 MHz Anova spectrometer (Varian, Palo Alto, CA) at the phosphorus frequency (242 MHz).A representative series of 31 P-NMR spectra obtained at 600 MHz over the entire pH range 6-8, pMg 3.0, 30 • C, and 50 mM added Cr is shown in Figure 1.Magnetic field homogeneity was shimmed (usually less than 0.07 PPM) on the available proton signal prior to the start of the experiment.Data were acquired at 300 MHz with a π/2 pulse width (18 µs at 90 W), 15 s delay, and 4K data points.Transformed data were the sum of 64 acquisitions that were apodized with a 3 Hz exponential filter prior to Fourier transformation.All experiments were referenced to an external standard of dilute phosphoric acid (δ = 0 PPM) that was placed in a small glass capillary and positioned concentrically in the center of the NMR tube.
To confirm equilibration of the CK reaction in each solution, serially acquired 31 P-NMR data from each solution were obtained at 300 MHz.The temperature was maintained (±1 • C) by a blanket of dry N 2 gas that was first passed through a set of copper coils immersed in either a water bath or an acetone-dry ice bath, depending on the desired temperature.The solution temperature was measured using a thermocouple immersed in the solution while placed within the probe.Spectra were acquired while avoiding saturation of spectral resonances (recycle time 5*T1s), then an aliquot was removed for later analysis.At this point, the phosphorylation reaction was initiated by the addition of CK (~75 units/mL) to each tube and the tube was returned to the spectrometer for further acquisitions in order to follow the approach to equilibration.Serial spectra were acquired during the time course to equilibration and were halted when the change in peak area for PCr differed by less than 5% from the previous acquisition.At equilibration, the sample was removed from the magnet, kept at constant temperature by rapid immersion in a water bath, and CK was denatured by the addition of 2% SDS per mg of CK as previously described [32].Samples were then frozen at −70 • C for later chemical analysis by anion and cation HPLC.
Analysis of spectral areas and chemical shift positions was performed on summed data processed with a 3 Hz line broadening and zero-filled one time prior to Fourier transformation.Spectra were analyzed for peak positions and integral areas using commercially available software (Bruker Instruments, Billerica, MA, USA).To obtain pH titration curves for Pi at each pMg and temperature (Equation ( 7)), the chemical shift of Pi (δ Pi ) relative to the external standard was obtained from NMR spectra as adapted from Kost [55] with more comprehensive consideration of the ionic interactions.

HPLC Analysis
Chromatographic analysis was performed on stable SDS-treated samples as previously described using a Waters Millennium HPLC system (Waters Corp, Milford, MA, USA) [106].In brief, nucleotides and PCr content were analyzed using a Vydac 303NT405 NTP anion exchange column (Vydac, Hesperia, CA, USA) with a phosphate gradient from 50 mM (pH 4.5) to 400 mM (pH 2.7) linearly applied over 20 min (Figure 3A,B).Creatine was determined by cation exchange chromatography using a Waters amino acid column under isocratic conditions with 25 mM sodium phosphate (pH 7.8) (Figure 3C,D).Detection of all analytes was by absorbance at 210 nm and quantification was from calibration curves determined using known standards.

Muscle Mechanics
In order to examine the relationship between muscle mechanical parameters and ∆G ATP , we utilized data from previously published studies on muscle mechanics of permeabilized fibers [35,36] and mechanics along with biochemical analyses of isolated muscle tissues [49,67].All protocols for harvesting muscle tissue from animals were in accordance with the policies and standards of the National Institutes of Health/National Research Council Guide for the Care and Use of Laboratory Animals.Muscle tissues were obtained according to protocols approved by the Institutional Animal Care and Use Committee (IACUC) as described in the original publications and at Michigan State University and Florida State University (approved protocol 0118).

Single Permeabilized Muscle Fiber Studies
Single chemically permeabilized ("skinned") muscle fibers were dissected from rabbit psoas or soleus muscles and prepared for mechanical experimentation using published methods [35,36,53,104,107,108].Single fiber segments (length ~2 mm) were isolated in a cold bath (4 • C) of 50% glycerol-relaxing solution and the fiber ends were chemically cross-linked by localized microapplication of chemical fixative (5% glutaraldehyde plus 1 mg/mL fluorescein for visualization) to generate 'artificial tendons' that minimize end compliance.The fixed ends were wrapped in Al foil 'T clips' (KEM-MIL, Hayward, CA, USA) and the T-clips were placed on hooks on a motor and force transducer (using silicone adhesive for stabilization) mounted on the modified stage of a Leitz Diavert (Wetzlar, Germany) inverted microscope for mechanical measurements.
Activating (pCa < 5) and relaxing (pCa > 8) solutions for fiber mechanics experiments were prepared as described in [35,36].The composition of the control solution (in mM) was 5 MgATP, 1 Pi, 4 EGTA, 15 PCr, 100 monovalent cations (sum of K + plus Na + ), 3 Mg 2+ (pMg 2.52), 50 MOPS, and 1 mg/mL CK.Control pH was 7.1 and was adjusted at 12 • C (the experimental temperature).When pH was varied, the pH buffer was varied as well to maintain buffering capacity, as follows: MES at pH 6.0 and 6.5; MOPS at pH 7.1 (control) and 7.3; MOPS, HEPES, or TES at pH 7.5; and EPPS at pH 8.0.[Ca 2+ ] was adjusted by adding appropriate amounts of Ca(acetate) 2 ; EDTA was substituted for EGTA when it was more appropriate as the Ca 2+ buffer, taking into consideration Mg 2+ binding by EDTA.
[Pi] was varied from 0.1 mM (no added Pi) to 36 mM.Γ/2 = 0.16 M in all solutions for permeabilized fiber mechanics, with the ionic balance adjusted using Tris as the cation and acetate as the anion.
Experimental control, data acquisition, and data analysis were accomplished using custom software described previously [35,36,53,104,107,108].The stability of the fiber structure and mechanical properties during activation were maintained by transient shortening of the fibers every 5 s at a rate that was at least as fast as the maximum shortening velocity, which reduced the force to zero; periodic unloading was followed by rapid re-stretching to the initial isometric length (L 0 ).The initial sarcomere length (L s ) was set to 2.6 µm in relaxing conditions.Following a brief initial control activation, fibers were returned to relaxing solution, L s was adjusted if necessary, and L 0 , fiber diameter, and passive force were measured.Maximum isometric force was determined first in control conditions, then in an experimental condition (varied pH or [Pi]), then in control conditions.Normalized force was calculated as the experimental force divided by the average of the two bracketing controls.The similarity of force during first and last activations indicated that the fiber structure and function were stable under the examined conditions.
V US was measured at maximum Ca 2+ -activation using the slack test [109] adapted as described previously in [35,36].Normalized force and normalized V US were calculated as the experimental value divided by the average of the two bracketing controls.

Studies on Intact Muscles from Mice
Isolated mouse muscle experiments were conducted as described in [49,53,110,111], with modifications.Both soleus (SOL) muscles were ligated at the proximal and distal tendons with 5.0 silk sutures, removed from the hindlimbs, and immediately placed in organ baths.Muscles were incubated in modified mouse Ringer's solution (in mM: 117 NaCl, 4.6 KCl, 25 NaHCO 3 , 2.5 CaCl 2 , 1.16 MgSO 4 , and 11 glucose) containing 10 mg/L gentamycin and equilibrated with 95% O 2 /5% CO 2 .The pH was 7.4 at 37 • C. Superfusate temperature was measured in a subset of experiments using a K-type thermocouple (Omega Engineering, Stamford, CT, USA) adjacent to the muscle and maintained at 37 ± 0.2 • C by circulating water through a glass-jacketed organ bath (Radnoti Glass Technology, Inc., Monrovia, CA, USA).
Isolated SOL muscles were mounted for mechanical measurements by tying one end of the muscle ligature to a stationary hook and the other end to an isometric force transducer fitted on a micrometer.Muscles were aligned with the axis of the transducer and the length was adjusted to optimal resting length (L o ) using the length-tension relationship.Electrical stimulation was delivered via two Pt plate electrodes adjacent to the muscle and generated using a Grass S88 Stimulator (Grass Instruments, Quincy, MA, USA).The pulse duration was 0.2 ms and stimulation trains for tetanic force were delivered at fusion frequency (~70 Hz) for 0.5-1.2s.Force was recorded using an ADC model AT MIO16E (National Instruments, Austin, TX, USA) controlled by commercially available software (LabScribeNI, iWorx, Dover, NH, USA).Analysis of mechanical transients was performed using a custom algorithm for physiological data developed in this laboratory using the MATLAB programming environment (MathWorks, Natick, MA, USA) [112].

Statistical Analysis
Nonlinear regression analyses were initially performed using SigmaPlot version 8.0 (SPSS Inc., Richmond, CA, USA) and validated using R version 4.0.5 or later.Regression parameter estimates are provided ± the standard error (SE) of the regression.
Table A1.Parameter estimates from nonlinear regressions of force-pCa data at 0.1 or 20 mM Pi from single permeabilized fibers from rabbit psoas (Figure A2A) and soleus (Figure A2B) muscles.The parameter estimates correspond to the four lines in Figure A2 when each of the four datasets was fitted to the Hill Equation (Equation (A2)).All values for pCa 50  For comparison with prior studies, we plotted the maximum isometric force versus Pi data for permeabilized psoas (Figure A3A; N = 28 fibers) and soleus (Figure A3B; N = 19 fibers) muscle fibers.These data suggest that the maximum isometric force of soleus fibers is more affected by Pi at lower concentrations compared with psoas fibers, with an apparent binding affinity (K m ) of 2.4 ± 0.7 mM versus 17.5 ± 5.2 mM, respectively, when the data were fitted to Equation (A3): The dependent variable in Equation (A3) is the normalized isometric force (Force normalized ([Pi])), which was obtained by first subtracting the passive force (the small amount of force measured at pCa 8) from all force measurements from the same fiber and second by dividing the remaining active force by the active force at saturating Ca 2+ and 0.1 mM Pi for the same fiber.Thus, the normalized force was 1.0 for the data point at 0.1 mM Pi.Note that Equation (A3), which includes an asymptotic force to which the isometric force declines at very high levels of Pi (F(∞)), represents a model that is not entirely consistent with the alternate model implied in Figure 7.
Combined with the decrease in Ca 2+ sensitivity with increased Pi (Figure A2), the decrease in maximal isometric force in both fast and slow fiber types (the right side of each panel in Figures A2 and A3A,B) indicates that substantially more Ca 2+ would be required to achieve the same force levels in the presence of elevated Pi, while the loss of force due to the rightward shifts of Ca 2+ sensitivity (Figure A2) could potentially be overcome only at the low end of the submaximal range of Ca 2+ concentrations due to the loss of maximal isometric force (Figure A3A,B).
To test for nonspecific effects of multivalent anions, we examined the effects of sulfate ([SO 4 = ]) on the maximum isometric force of permeabilized fibers from rabbit psoas (Figure A3C; N = 7 fibers) and soleus (Figure A3D; N = 7 fibers) muscles.All measurements were made in the presence of 0.1 mM Pi, and no added sulfate was the control condition used for normalizing force, as described above.Comparing the small effects of sulfate (Figure A3C,D) with the effects of Pi (Figure A3A,B) on the isometric force of permeabilized muscle fibers suggests that the effects of Pi are specific for that anion.
In addition, we examined the effects of [Pi] on the velocity of unloaded shortening (V US ) as measured using the slack test [109] (Section 4.4.1).The data in Figure A4 indicate that there is a small inhibitory effect of Pi in both fiber types, with the effect being slightly larger in soleus fibers (Figure A4B).The effect of Pi on V US (Figure A4A) is clearly smaller than its effect on the isometric force (Figure A3A), consistent with the expectation that ∆G ATP is not limiting for unloaded shortening.In addition, we examined the effects of [Pi] on the velocity of unloaded shortening (VUS) as measured using the slack test [109] (Section 4.4.1).The data in Figure A4 indicate that there is a small inhibitory effect of Pi in both fiber types, with the effect being slightly larger in soleus fibers (Figure A4B).The effect of Pi on VUS (Figure A4A) is clearly smaller than its effect on the isometric force (Figure A3A), consistent with the expectation that ΔGATP is not limiting for unloaded shortening.In addition, we examined the effects of [Pi] on the velocity of unloaded shortening (VUS) as measured using the slack test [109] (Section 4.4.1).The data in Figure A4 indicate that there is a small inhibitory effect of Pi in both fiber types, with the effect being slightly larger in soleus fibers (Figure A4B).The effect of Pi on VUS (Figure A4A) is clearly smaller than its effect on the isometric force (Figure A3A), consistent with the expectation that ΔGATP is not limiting for unloaded shortening.Each point represents the normalized value of V US (normalized to V US obtained at 0.1 mM Pi in the same fiber, comparable to normalization of force data as described in Figures A2 and A3) at maximum Ca 2+ -activation.Measurements are from N = 18 psoas fibers from eight rabbits in panel (A) and N = 9 soleus fibers from six rabbits in panel (B).The control values of V US were 3.3 ± 0.8 FL s −1 (mean ± SD) for psoas fibers and 0.7 ± 0.2 FL s −1 (mean ± SD) for soleus fibers.The lines are linear least squares regression fits constrained to pass through V US = 1 at 0.1 mM Pi.

Appendix D
Table A2.Summary of the binding constants used to construct model solutions and calculations of proton stoichiometric coefficients, showing the metal ion-binding constants used for calculating the solutions (Section 4.1) and H + stoichiometric constants α (Equation ( 2)) and β (Equation (4) and Figure 4).

Equilibrium
log K eq log K eq log K eq log K eq Reference(s) 10

Figure 1 .
Figure 1. 31 P-NMR spectroscopy of model solutions containing inorganic phosphate (Pi), phos creatine (PCr), ATP (left-to-right, the peaks correspond to the γ, α, and β phosphate resonan ADP (only visible at pH 7.5 and 8.0; left-to-right, the peaks correspond to the β and α phosp resonances) in the presence of CK and with 50 mM added Cr at pMg 3.0, 30 °C, and pH 6.0, 6.5 7.5, or 8.0 (bottom-to-top stacked plot).Details of solution composition were as described in Se 4.1.An external standard (H3PO4, chemical shift δ = 0) was placed in a small capillary and cen in the coil's sensitive volume.Spectra were acquired on a Varian 600 MHz spectrometer at the p phorus frequency (242 MHz) using an 8000 Hz sweep width.Data are the sum of 1024 trans collected with a 1.0 s recycle delay and a π/2 pulse width and 1024 complex points and zeroonce to a total of 2048 data points and exponentially filtered prior to the Fourier transform.that Pi chemical shift (δPi) moves from right to left along the δ-axis with increasing pH (botto top), corresponding to deprotonation of H2PO4 -with pKa around neutral pH.Additionally, not the γATP and βATP peaks become sharper with increasing pH (bottom-to-top) and that peak ting is evident under most conditions for the γATP and αATP peaks, as well as for the βADP αADP peaks where they are detectable.

Figure 1 .
Figure 1. 31 P-NMR spectroscopy of model solutions containing inorganic phosphate (Pi), phosphocreatine (PCr), ATP (left-to-right, the peaks correspond to the γ, α, and β phosphate resonances), ADP (only visible at pH 7.5 and 8.0; left-to-right, the peaks correspond to the β and α phosphate resonances) in the presence of CK and with 50 mM added Cr at pMg 3.0, 30 • C, and pH 6.0, 6.5, 7.0, 7.5, or 8.0 (bottom-to-top stacked plot).Details of solution composition were as described in Section 4.1.An external standard (H 3 PO 4 , chemical shift δ = 0) was placed in a small capillary and centered in the coil's sensitive volume.Spectra were acquired on a Varian 600 MHz spectrometer at the phosphorus frequency (242 MHz) using an 8000 Hz sweep width.Data are the sum of 1024 transients collected with a 1.0 s recycle delay and a π/2 pulse width and 1024 complex points and zero-filled once to a total of 2048 data points and exponentially filtered prior to the Fourier transform.Note that Pi

Figure 2 .
Figure 2. (A-C) Experimentally determined 31 P-NMR chemical shift of Pi (δPi) titrated between pH 6-8 and (D-F) calculated pH-dependence of [H2PO4 − ]/Σ[Pi].(A,D) pMg 2; (B,E) pMg 3; (C,F) pMg 4.In all panels, blue is 10 °C, green is 20 °C, yellow is 30 °C, and red is 40 °C.In panels (A-C), all data in each panel (single pMg) were simultaneously fitted to Equation (8) by nonlinear least squares regression, while in panels (D-F) all values in each panel (single pMg) were simultaneously fitted to Equation (9) by nonlinear least squares regression; regression parameter estimates are provided in Table1.Note the qualitative correspondence between the left and right panels.

Figure 3 .
Figure 3. HPLC quantitation of (A,B) PCr, ADP, and ATP by anion exchange chromatography and (C,D) Cr by cation exchange chromatography (Section 4.3) in solutions designed to mimic the cytosol under different metabolic conditions (Section 4.1).Representative sample #226 (panels (A,C); low ADP; no added Cr) was held at 10 °C, pMg 4 and pH 8. Representative sample #80 (panels (B,D); high ADP; 50 mM added Cr) was held at 40 °C, pMg 2 and pH 8.Both samples initially contained 1 mg/mL rabbit CK.At equilibrium as determined by monitoring the reactions using 31 P-NMR spectroscopy (Figure1), each reaction was stopped with addition of SDS at 2% per mg CK (Section 4.2).For both HPLC methods, detection was by optical absorbance at 210 nm and peak areas were quantified against calibration curves determined using known standards.Numerical scales on vertical (optical absorbance) axes correspond to detector output in millivolts.Note that scales for the vertical (optical absorbance) axes are the same for panels (A,B) and for panels (C,D).

Figure 3 .
Figure 3. HPLC quantitation of (A,B) PCr, ADP, and ATP by anion exchange chromatography and (C,D) Cr by cation exchange chromatography (Section 4.3) in solutions designed to mimic the cytosol under different metabolic conditions (Section 4.1).Representative sample #226 (panels (A,C); low ADP; no added Cr) was held at 10 • C, pMg 4 and pH 8. Representative sample #80 (panels (B,D); high ADP; 50 mM added Cr) was held at 40 • C, pMg 2 and pH 8.Both samples initially contained 1 mg/mL rabbit CK.At equilibrium as determined by monitoring the reactions using 31 P-NMR spectroscopy (Figure1), each reaction was stopped with addition of SDS at 2% per mg CK (Section 4.2).For both HPLC methods, detection was by optical absorbance at 210 nm and peak areas were quantified against calibration curves determined using known standards.Numerical scales on vertical (optical absorbance) axes correspond to detector output in millivolts.Note that scales for the vertical (optical absorbance) axes are the same for panels (A,B) and for panels (C,D).
) (Appendix A) to obtain a matrix of estimates of Keq″ [pMg, T].

Figure 4 .
Figure 4. Calculated pH dependence and range of the stoichiometric coefficient of proton consumption (β) by ADP rephosphorylation via transfer of Pi from PCr (Equation (4)).pH dependence of β was predicted according to the ion binding equilibria (Section 4.1) for all conditions of this study.The smallest range of predicted values as a function of pH was obtained at pMg 4.0 and 10 °C (dashed line), while the largest range of predicted values as a function of pH was obtained at pMg

Figure 4 .
Figure 4. Calculated pH dependence and range of the stoichiometric coefficient of proton consumption (β) by ADP rephosphorylation via transfer of Pi from PCr (Equation (4)).pH dependence of β was predicted according to the ion binding equilibria (Section 4.1) for all conditions of this study.The smallest range of predicted values as a function of pH was obtained at pMg 4.0 and 10 • C (dashed line), while the largest range of predicted values as a function of pH was obtained at pMg 3.0 and 30 • C (solid line); all other predicted values of β fell within the range between the two lines shown.
Int. J. Mol.Sci.2023, 24, 13244 9 of 31 3.0 and 30 °C (solid line); all other predicted values of β fell within the range between the two lines shown.

Figure 5 .
Figure 5. Nonlinear least squares regression estimation of Keq″ for model solutions at pMg 3.0 and 30 °C.Total concentrations of ADP, ATP, Cr, and PCr were measured by HPLC (Figure 3), although only ADP and Cr are plotted here because they vary most widely by the amount of Cr added.All data (all Cr and pH values) were used to simultaneously obtain a single estimate of Keq″ [pMg 3.0, 30 °C] by nonlinear least squares regression fitting of the data to Equation (12) using values of β, as illustrated in Figure4.Colors correspond to pH (blue, pH 8; cyan, pH 7.5; green, pH 7; orange, pH 6.5; red, pH 6).Comparable analyses were performed for each combination of temperature and pMg, resulting in a total of 12 plots (FigureA1) and regression estimates of Keq″ [pMg, T] (Figure6).

Figure 5 .
Figure 5. Nonlinear least squares regression estimation of K eq for model solutions at pMg 3.0 and 30 • C. Total concentrations of ADP, ATP, Cr, and PCr were measured by HPLC (Figure 3), although only ADP and Cr are plotted here because they vary most widely by the amount of Cr added.All data (all Cr and pH values) were used to simultaneously obtain a single estimate of K eq [pMg 3.0, 30 • C] by nonlinear least squares regression fitting of the data to Equation (12) using values of β, as illustrated in Figure4.Colors correspond to pH (blue, pH 8; cyan, pH 7.5; green, pH 7; orange, pH 6.5; red, pH 6).Comparable analyses were performed for each combination of temperature and pMg, resulting in a total of 12 plots (FigureA1) and regression estimates of K eq [pMg, T] (Figure6).

Figure 6 .
Figure 6.Creatine kinase Keq″ as a function of free Mg 2+ (pMg) and temperature at Γ/2 = 0.25 M. Points are regression estimates of Keq″ (Equations (10) and (11)) obtained as illustrated (Figures 5 andA1).Error bars are the SE regression for each estimate of Keq″.The results of a multiple linear least squares regression on all of the Keq″ data (Equation (13)) are shown by thick blue lines.This simple relationship allows for prediction of [ADP] (which is not always directly measurable, i.e., as illustrated in Figures1 and 3) over a wide range of conditions, ranging from those relevant to intact tissue in vivo to experiments with permeabilized muscle fibers.

Figure 6 .
Figure 6.Creatine kinase K eq as a function of free Mg 2+ (pMg) and temperature at Γ/2 = 0.25 M.Points are regression estimates of K eq (Equations (10) and (11)) obtained as illustrated (Figures5 and A1).Error bars are the SE regression for each estimate of K eq .The results of a multiple linear least squares regression on all of the K eq data (Equation (13)) are shown by thick blue lines.This simple relationship allows for prediction of [ADP] (which is not always directly measurable, i.e., as illustrated in Figures1 and 3) over a wide range of conditions, ranging from those relevant to intact tissue in vivo to experiments with permeabilized muscle fibers.

Figure 7 .
Figure 7. Variation of maximum Ca 2+ -activated force with ΔGATP when ΔGATP was altered by varying [Pi] in (A) intact mouse soleus muscles (red filled triangles with blue borders and with dashed red line) or (B) single skinned fibers from rabbit psoas (open squares with solid gray line) and soleus (red filled squares with dashed red line) muscles.[Pi]was varied in intact muscles (panel (A)) by altering the substrate supplied extracellularly[49], and the force and biochemical data are from that study; [Pi] in the presence of pyruvate was set to 1 mM in the plot, with the upper limit determined[49].[Pi] for skinned fiber experiments (panel (B)) was 0.1-36 mM.The psoas force at the first Pi concentration in panel (B) was previously published in Chase and Kushmerick[35], and the psoas force at the second Pi concentration in panel (B) was previously published in Chase and Kushmerick[36]; all other data in panel (B) are previously unpublished.Measurements on skinned fibers in panel (B) were from N = 28 psoas fibers from ten rabbits and N = 19 soleus fibers from six rabbits.As described in the text, the force for the psoas fibers in panel (B) was normalized to 1 mM Pi, while the soleus force was normalized to 6 mM to reflect the higher basal levels of Pi in slow muscle[58].The points in panel (A) are mean values, while the points in panel (B) are individual measurements.The regression slopes are −0.0248 in panel (A), −0.0306 (regression SE 0.0029) for soleus data in panel (B), and −0.0406 (regression SE 0.0038) for psoas data in panel (B); note that the slopes are not significantly different for soleus data in panels (A,B), while the slope for skinned psoas fibers is significantly steeper than that for skinned soleus fibers.

Figure 7 .
Figure 7. Variation of maximum Ca 2+ -activated force with ∆G ATP when ∆G ATP was altered by varying [Pi] in (A) intact mouse soleus muscles (red filled triangles with blue borders and with dashed red line) or (B) single skinned fibers from rabbit psoas (open squares with solid gray line) and soleus (red filled squares with dashed red line) muscles.[Pi]was varied in intact muscles (panel (A)) by altering the substrate supplied extracellularly[49], and the force and biochemical data are from that study; [Pi] in the presence of pyruvate was set to 1 mM in the plot, with the upper limit determined[49].[Pi] for skinned fiber experiments (panel (B)) was 0.1-36 mM.The psoas force at the first Pi concentration in panel (B) was previously published in Chase and Kushmerick[35], and the psoas force at the second Pi concentration in panel (B) was previously published in Chase and Kushmerick[36]; all other data in panel (B) are previously unpublished.Measurements on skinned fibers in panel (B) were from N = 28 psoas fibers from ten rabbits and N = 19 soleus fibers from six rabbits.As described in the text, the force for the psoas fibers in panel (B) was normalized to 1 mM Pi, while the soleus force was normalized to 6 mM to reflect the higher basal levels of Pi in slow muscle[58].The points in panel (A) are mean values, while the points in panel (B) are individual measurements.The regression slopes are −0.0248 in panel (A), −0.0306 (regression SE 0.0029) for soleus data in panel (B), and −0.0406 (regression SE 0.0038) for psoas data in panel (B); note that the slopes are not significantly different for soleus data in panels (A,B), while the slope for skinned psoas fibers is significantly steeper than that for skinned soleus fibers.

Figure 8 .
Figure 8. Maximum Ca 2+ -activated isometric force variation with (A) ΔGATP and (B) [ADP] when pH was altered.We reanalyzed previously published data from (panel (A)) isolated perfused biceps (open circles with blue error bars connected by solid gray line) and soleus (red filled circles with blue error bars connected by red dashed line) muscles from cat[67] and (panels (A,B)) single skinned fibers from rabbit psoas (open squares fit with solid gray lines) and soleus (red filled squares fit with dashed red lines) muscles[35].Hypercapnic (acidic cytoplasmic pH 6.48-6.6)force data from cat muscles were normalized to the normocapnic condition (higher force; cytoplasmic pH 7.09-7.11)for each muscle type.Skinned fiber force data were normalized to that at pH 7.1 for the same fiber.Lower forces were associated with acidic (lower) pH and higher forces were associated with basic (higher) pH[35,67].The points represent the average ± SD.ΔGATP was estimated according to Equation (16) and [ADP] was estimated according to Equation (12) using values of Keq″ from

Figure 8 .
Figure 8. Maximum Ca 2+ -activated isometric force variation with (A) ∆G ATP and (B) [ADP] when pH was altered.We reanalyzed previously published data from (panel (A)) isolated perfused biceps (open circles with blue error bars connected by solid gray line) and soleus (red filled circles with blue error bars connected by red dashed line) muscles from cat[67] and (panels (A,B)) single skinned fibers from rabbit psoas (open squares fit with solid gray lines) and soleus (red filled squares fit with dashed red lines) muscles[35].Hypercapnic (acidic cytoplasmic pH 6.48-6.6)force data from cat muscles were normalized to the normocapnic condition (higher force; cytoplasmic pH 7.09-7.11)for each muscle type.Skinned fiber force data were normalized to that at pH 7.1 for the same fiber.

Figure A2 .
Figure A2.Isometric force-pCa relationships for single permeabilized fibers from (A) rabbit psoas and (B) rabbit soleus muscles (Section 4.4.1) at 0.1 mM Pi (open squares and solid line in panel (A) ; red squares and dashed line in panel (B)) or 20 mM Pi (gray squares and dotted line in panel (A) ; dark red squares and dotted line in panel (B)).Measurements are from N = 4 psoas fibers from two rabbits in panel (A) and N = 3 soleus fibers from two rabbits in panel (A).Not shown are data points at pCa ≥ 8 that corresponded to normalized force = 0, as the absolute value (passive force) for each fiber was subtracted from all force values before plotting.Parameter estimates for nonlinear regressions on the Hill Equation (Equation (A2)) are provided in TableA1.

Figure A2 .
Figure A2.Isometric force-pCa relationships for single permeabilized fibers from (A) rabbit psoas and (B) rabbit soleus muscles (Section 4.4.1) at 0.1 mM Pi (open squares and solid line in panel (A); red squares and dashed line in panel (B)) or 20 mM Pi (gray squares and dotted line in panel (A); dark red squares and dotted line in panel (B)).Measurements are from N = 4 psoas fibers from two rabbits in panel (A) and N = 3 soleus fibers from two rabbits in panel (A).Not shown are data points at pCa ≥ 8 that corresponded to normalized force = 0, as the absolute value (passive force) for each fiber was subtracted from all force values before plotting.Parameter estimates for nonlinear regressions on the Hill Equation (Equation (A2)) are provided in TableA1.

Figure A3 .
Figure A3.Dependence of isometric force on (A,B) [Pi] and (C,D) [sulfate] for single permeabilized fibers from (A,C) rabbit psoas and (B,D) rabbit soleus muscles (Section 4.4.1).Each point represents the normalized maximum force (normalized as described for the data in Figure A2) for one muscle fiber at one concentration of Pi (A,B) or sulfate (C,D).Measurements are from N = 28 psoas fibers from ten rabbits in panel (A), N = 19 soleus fibers from six rabbits in panel (B), N = 7 psoas fibers from one rabbit in panel (C), and N = 7 soleus fibers from one rabbit in panel (D).The lines in panels (A,B) were fitted to Equation (A3) using nonlinear least squares regression, while the lines in panels (C,D) were fitted using linear least squares regression constrained to pass through normalized force = 1.0 at [sulfate] = 0.

Figure A4 .
Figure A4.Dependence of the velocity of unloaded shortening (VUS) on [Pi] for single permeabilized fibers from (A) rabbit psoas (open squares) and (B) rabbit soleus muscles (red squares) (Section

Figure A3 .
Figure A3.Dependence of isometric force on (A,B) [Pi] and (C,D) [sulfate] for single permeabilized fibers from (A,C) rabbit psoas and (B,D) rabbit soleus muscles (Section 4.4.1).Each point represents the normalized maximum force (normalized as described for the data in Figure A2) for one muscle fiber at one concentration of Pi (A,B) or sulfate (C,D).Measurements are from N = 28 psoas fibers from ten rabbits in panel (A), N = 19 soleus fibers from six rabbits in panel (B), N = 7 psoas fibers from one rabbit in panel (C), and N = 7 soleus fibers from one rabbit in panel (D).The lines in panels (A,B) were fitted to Equation (A3) using nonlinear least squares regression, while the lines in panels (C,D) were fitted using linear least squares regression constrained to pass through normalized force = 1.0 at [sulfate] = 0.

Figure A4 .
Figure A4.Dependence of the velocity of unloaded shortening (VUS) on [Pi] for single permeabilized fibers from (A) rabbit psoas (open squares) and (B) rabbit soleus muscles (red squares) (Section Figure A4.Dependence of the velocity of unloaded shortening (V US ) on [Pi] for single permeabilized fibers from (A) rabbit psoas (open squares) and (B) rabbit soleus muscles (red squares) (Section 4.4.1).Each point represents the normalized value of V US (normalized to V US obtained at 0.1 mM Pi in the same fiber, comparable to normalization of force data as described in FiguresA2 and A3) at maximum Ca 2+ -activation.Measurements are from N = 18 psoas fibers from eight rabbits in panel (A) and N = 9 soleus fibers from six rabbits in panel (B).The control values of V US were 3.3 ± 0.8 FL s −1 (mean ± SD) for psoas fibers and 0.7 ± 0.2 FL s −1 (mean ± SD) for soleus fibers.The lines are linear least squares regression fits constrained to pass through V US = 1 at 0.1 mM Pi.
• C, dpK a /dT is the change in pK a per • C, and A is a scaling factor.All calculated values of [H 2 PO 4 − ]/Σ[Pi] at pMg 2 (points in Figure ).

green is 20 • C, yellow is 30 • C, and red is 40 • C. In panels (A-C), all data in each panel (single pMg) were simultaneously fitted to Equation (8) by nonlinear least squares regression, while in panels (D-F) all values in each panel (single pMg) were simultaneously fitted to Equation (9) by nonlinear least squares regression; regression parameter estimates are provided inTable 1 .
Note the qualitative correspondence between the left and right panels.

Table 1 .
Regression coefficients for pH titrations of Pi from 31 P-NMR experiments and calculated predictions at three different free magnesium concentrations.Values are nonlinear least squares regression parameter estimates ± SE for the curves shown in Figure2.Rows labeled NMR are parameter estimates for 31 P-NMR chemical shift data fitted to Equation (8) (Figure ).
• C [29].Values for [ADP] were calculated from Equation (12) using K eq from Equation (13) with the values of [ATP], [PCr], [Cr], and [H + ] measured in each solution.Note that pH, [Mg 2+ ], and temperature each affect ∆G ATP nonlinearly in Equation and n Hill are provided ± SE regression, as are values for F max at 20 mM Pi.