Introduction

Substrate inhibition, a common deviation from Michaelis–Menten kinetics, occurs when an enzyme is inhibited by a reaction substrate at high concentrations. This phenomenon is characterized by a velocity curve that initially rises with increasing substrate concentration, reaching a maximum, and then declines, either approaching zero (complete inhibition) or stabilizing at a non-zero level (partial inhibition)1,2. Substrate inhibition is pervasive in enzymatic reactions and serves crucial regulatory functions in various metabolic pathways, influencing the pace and balance of biochemical reactions within cells3,4. Previous studies have shown that approximately 25% of the known enzymes are affected by substrate inhibition1,2,4,5,6,7,8,9,10,11,12 and phosphofructokinase-1 (PFK1, ATP:D-fructose-6-phosphate-1-phosphotransferase, EC 2.7.1.11) is one of them4,10,11,12.

PFK1, a key regulatory enzyme in the glycolytic pathway, plays a crucial role in shaping the biochemical changes in energy metabolism. As a rate-limiting factor in the process, PFK1 irreversibly converts fructose-6-phosphate (F6P) and adenosine triphosphate (ATP) to fructose-1,6-biphosphate (F-1,6-P) and adenosine diphosphate ADP (F6P + ATP → F-1,6-P + ADP)13,14,15,16. Substrate inhibition of PFK1 is a regulatory negative feedback to ensure that resources are not devoted to producing ATP when it is in abundance4.

Enzyme activity is generally measured in terms of the rate of reaction catalyzed by an enzyme expressed in substrate consumed (or product generated) per unit time17. The conventional enzymatic assay involves measuring the initial reaction velocity at various substrate concentrations by coupling the reaction of interest with a chromogenic indicator reaction observable with a spectrophotometer18. The initial velocity is approximated with the slope of the initial linear portion of the reaction curve and the initial substrate concentration is taken as the concentration in effect. PFK1 activity is often expressed in units of fructose 1,6-P (F-1,6-P) generated per minute, which corresponds to the oxidation of twice as many units of NADH observed optically. By using initial velocity as a measure of PFK1 activity, the kinetic and regulatory properties of PFK1 have been extensively studied. Ui (1966) reported that the inhibition of PFK1 by ATP is related to pH19. Bosca et al. (1985) showed that the inhibition of PFK1 by ATP is the basis for the activation effect of fructose-2,6-P2. Fructose-2,6-P2 enhances PFK1 activity by releasing ATP from the regulatory site of PFK1, hence reducing the inhibition effect of ATP20. Dobson et al. (1986) indicated that pH affects PFK1 by affecting the interaction of ATP with the regulatory site of PFK1, hence modifying the inhibition effect of ATP21. Since ATP is both a substrate, which accelerates reaction, and an inhibitor, which impedes reaction; the two effects cannot be easily separated in the conventional initial-velocity measurement.

As an alternative to using only the initial datapoints and substrate concentration, the reaction velocity can be modeled for the entire time course of reaction by accounting for both the substrate and the inhibitor effects. Enzyme activities of pyruvate decarboxylase and alcohol dehydrogenase were estimated by using kinetic models that described the reactions catalyzed by these two enzymes based on Michaelis–Menten and first-order kinetics22. By using a kinetic model for the underlying enzymatic reactions, the substrate inhibition of the reaction catalyzed by cytochrome P450 was found to depend on the concentration of the substrate occupying the inhibitory site6. Furthermore, substrate inhibition of paramecium arginine kinase 3 (AK3) was analyzed by using kinetic reaction models and substrate binding to the catalytic site was found to promote substrate binding to the inhibitory site, hence resulting in stronger substrate inhibition23. In spite of the extensive use of modeling in enzyme studies, surprisingly little has been done on the quantification of PFK1 activity and the degree of substrate inhibition by using a modeling method. Although Yoshino et al. (2015) examined the substrate inhibition of E. coli phosphofructokinase II by ATP using a graphical method, the pH influence was not considered 1. Without a quantitative model for PFK1 activity, Wang et al. (2021, 2022) represented the binding of protons to PFK1 with the Hill equation to account for the pH effect on PFK1 activity, but the effect of ATP as an inhibitor was not included24,25.

Although much work has been dedicated to studying PFK1 activity, quantifying the inhibitory effects of ATP and pH on PFK1 requires further investigation. In this study, two methods were used to study the effects of ATP and pH on PFK1 activity, the conventional initial-velocity method and kinetic modeling method. In the modeling method, a kinetic model was developed based on experimental measurements from porcine muscles to describe the underlying enzymatic reactions, and PFK1 activity and inhibitory effects were quantitatively evaluated. In addition, an analysis based on the model was performed to provide insights into the mechanisms involved in the regulation of PFK1 activity.

Material and methods

Muscle samples

Porcine muscle was used as a source of PFK1 in its natural environment. Market-weight pigs (100–125 kg) of similar genetics were harvested at the Utah State University Animal Harvest Facility following standard commercial practices and in accordance with the US Department of Agriculture inspection guidelines. Within 5 min postmortem, the longissimus lumborum (LL) muscle was excised from one side of each carcass, snap-frozen in liquid nitrogen, and stored at − 80 °C until used for the PFK1 assay.

PFK1 assay

PFK1 assay was performed in a 96-well plate by following a modified procedure of England et al. (2014)13. Frozen LL muscle samples were pulverized under liquid nitrogen with a mortar and pestle, and aliquots (∼ 0.1 g) were collected from each sample in 15-ml centrifuge tubes. Samples were homogenized at 1:20 (w/v) in an ice-cold buffer (100 mM K2HPO4, pH 7.4) with a Polytron homogenizer (PT 2500 E, Kinematica AG, Switzerland). Aliquots of the muscle homogenates were then added to a reaction solution containing 3.2 mM MgSO4, 1 mM NADH, 10 mM fructose 6-phosphate, 2 U/ml triosephosphate isomerase, 1 U/ml glycerol-3-phosphate dehydrogenase, and 1 U/ml aldolase.

The buffer pH was adjusted to 5.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9, 8, or 9. These pH values were selected based on the observation that the most pronounced changes in PFK1 activity occurred at pH > 6. Depending on the pH tested, the reaction was buffered with either MES (120 mM, pH 5.3), PIPES (120 mM, pH 6.4–7), HEPES (120 mM, pH 7.1–8), or Tris-base (120 mM, pH 9). For each tested pH, different concentrations of ATP (0.3, 0.625, 1.25, 2.5, 3.75, or 5 mM) were added to initiate the reaction.

Immediately after the addition of ATP, 200-µl aliquots of the reaction mixture were transferred in duplicates to a 96-well microplate, where kinetic measurements were carried out with a spectrophotometer (Epoch 2, BioTek, Winooski, VT, USA). The absorbance resulting from NADH was measured at 340 nm every minute for 7 min at 25 °C and then converted to a concentration of NADH in mM by using a calibration curve. NADH is consumed in proportion with F-1,6-P, the product of PFK1-catalyzed reaction, as further explained in section “Enzymatic reactions in PFK1 assay”. The rate of NADH consumption thus indicates the rate of F-1,6-P production.

Enzymatic reactions in PFK1 assay

The enzymatic reactions occurring in the PFK1 assay include a phosphate transformation reaction catalyzed by PFK1 coupled with an indicator reaction catalyzed by aldolase, triosephosphate isomerase, and glycerophosphate dehydrogenase; i.e.,

$$ {\text{F}}6{\text{P}} + {\text{ ATP}} \to {\text{F}}-1,6-{\text{P }} + {\text{ ADP}} $$
(1)
$$ {\text{F}}-1,6-{\text{P }} + 2{\text{NADH }} + 2{\text{H}}^{ + } \to 2{\text{Glycerol }}-1-{\text{P }} + 2{\text{NAD}}^{ + } $$
(2)

The indicator reaction (Eq. 2) produces a detectable spectrophotometric response because of the oxidation of NADH to NAD+. Equation (2) is in fact a combined form of the following three reactions:

$$ \begin{gathered} {\text{F}}-1,{6} \to {\text{Ga}}-3-{\text{P}} + {\text{ DHAP}} \hfill \\ {\text{Ga}} -3-{\text{P}} \to {\text{DHAP}} \hfill \\ 2{\text{DHAP }} + 2{\text{NADH}} \to 2{\text{Glycerol }}-1-{\text{P }} + 2{\text{NAD}}^{ + } \hfill \\ \end{gathered} $$

When NADH and the enzymes needed are abundant, these reactions are assumed fast and not rate-limiting; therefore, the measured NADH concentration indicates the rate of the PFK1-catalyzed reaction (Eq. 1).

Conventional initial-velocity method

In the conventional initial-velocity method, the reaction rate during the initial phase is measured without differentiating the dual effects of substrate. In this work, the slope of the initial linear portion of the measured NADH (the indicator) concentrations over time was calculated to yield the initial F-1,6-P generation velocity as a measure of PFK1 activity. Linearity was assessed visually and the first 2–5 measured values were used to compute the slope. The initial ATP concentration was taken as the concentration in effect, as conventionally done.

Kinetic modeling method

In the kinetic modeling method, the enzymatic reactions are modeled as a kinetic process based on the measurements made over time (as opposed to only the initial points in the conventional method). The product production velocity, v, is represented as a function of the substrate-inhibitor concentration [ATP] and hydrogen ion concentration [H+], both of which vary continuously during the reactions, as explained further below.

Kinetic model structure development

When reaction (2) is not limiting, the concentrations of ATP and NADH vary proportionally; i.e.,

$$\frac{d[ATP]}{dt}=\frac{1}{2}\frac{d[NADH]}{dt}=-v$$
(3)

where \(v\) represents the reaction velocity of both reactions; [ATP] and [NADH] are the concentrations of ATP and NADH, respectively; and t is time.

The reaction catalyzed by PFK1 (Eq. 1) does not obey the classical Michaelis–Menten kinetics because of substrate inhibition even at constant pH. PFK1 is an allosteric enzyme which is characterized by the existence of both catalytic and allosteric sites10,26,27. While ATP is a substrate for the reaction, it also inhibits PFK1 activity; in other words, ATP affects reaction velocity in two ways: as a substrate, which binds to the catalytic site, and as an inhibitor, which binds to the allosteric inhibitory site of PFK1. For convenience, we refer to the effect of substrate on reaction velocity without enzyme inhibition as the “substrate effect” and the reduction in reaction velocity by an inhibitor from the uninhibited case as the “inhibitor effect.” In addition, the PFK1-catalyzed reaction is highly sensitive to hydrogen ions.

Because the affinity of the catalytic site to ATP is much greater than that of the inhibitory site (as shown later in section “Further analysis of PFK1 activity based on the model”), we simplify the reactions by assuming that ATP binds to the catalytic site before binding to the inhibitory site of PFK1. Moreover, F6P was at a much higher concentration in the assay and thus is assumed not rate-limiting. The reactions accounting for the effects of substrate inhibition and hydrogen ions on PFK1 are then as represented in Fig. 1.

Fig. 1
figure 1

Reactions accounting for the effects of substrate inhibition and hydrogen ions on PFK1. E, S and P denote enzyme, substrate, and product, which are PFK1, ATP, and F-1,6-P, respectively; E·S and S·E·S are, respectively, enzyme–substrate complexes formed through substrate binding to the catalytic site and to both the catalytic site and inhibitory site of the enzyme. KE1, KE2, KES1, KES2, KSES1, KSES2 are the dissociation constants for EH+, E, EH+S, ES, SEH+S, SES , respectively; Ks and Ki are the dissociation constants for S and E and that for E·S and S, respectively; Kcat1 and Kcat2 are the catalytic rates of product generation from the E·S and S·E·S complexes, respectively.

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Based on steady-state and rapid equilibrium assumptions, a reaction velocity equation can be derived from the reactions presented in Fig. 1 as shown below:

$$v={K}_{cat1}[E \cdot S]+{K}_{cat2}[S \cdot E\cdot S]=\frac{\left(\frac{{v}_{{\max}}^{E \cdot S}}{{f}_{2}}[S]+\frac{{v}_{max}^{S \cdot E\cdot S}}{{f}_{2}{K}_{i}}\frac{{f}_{3}}{{f}_{2}}{[S]}^{2}\right)}{\left({K}_{s}\frac{{f}_{1}}{{f}_{2}}+[S]+\frac{1}{{K}_{i}}\frac{{f}_{3}}{{f}_{2}}{[S]}^{2}\right)}$$
(4)

with

$${f}_{1}=\frac{\left[{H}^{+}\right]}{{K}_{E1}}+1+\frac{{K}_{E2}}{\left[{H}^{+}\right]}$$
(5)
$${f}_{2}=\frac{\left[{H}^{+}\right]}{{K}_{ES1}}+1+\frac{{K}_{ES2}}{\left[{H}^{+}\right]}$$
(6)
$${f}_{3}=\frac{\left[{H}^{+}\right]}{{K}_{SES1}}+1+\frac{{K}_{SES2}}{\left[{H}^{+}\right]}$$
(7)

where [S], [H+], [E·S] and [S·E·S] are the concentrations of ATP, hydrogen ions, E·S and S·E·S complexes, respectively; \({v}_{max}^{E \cdot S} \;\text{ and } \;{v}_{max}^{S \cdot E\cdot S}\) are the maximum velocities for the catalyzed reactions of E·S and S·E·S, respectively.

Equations (3)–(7) form a kinetic model structure with adjustable parameters and are used to analyze the kinetic characteristics of the enzymatic reactions in the PFK1 assay.

Parameter estimation and model validation

The model parameters in Eqs. (3)–(7) were adjusted by using the Levenberg–Marquardt algorithm to achieve optimal fit to experimental measurements28. The least-squares optimization algorithm was implemented in MATLAB (Version 2018, Mathworks, Natick, MA, USA).

The relative standard error of predictions (RESP) and the coefficient of determination (R2) were calculated to assess the degree of agreement between model predictions and experimental observations. The RESP and R2 were calculated as:

$$RESP=\frac{1}{n}\sqrt{\frac{\sum_{i=1}^{N}{\left({x}_{i}^{*}-{x}_{i}\right)}^{2}}{{\sum }_{i=1}^{N}{{x}_{i}^{*}}^{2}}}$$
(8)
$${R}^{2}=1-\frac{{\sum }_{i=1}^{N}{({x}_{i}^{*}-{x}_{i})}^{2}}{{\sum }_{i=1}^{N}{\left({x}_{i}^{*}-\frac{{x}_{i}^{*}}{N}\right)}^{2}}$$
(9)

where \({x}_{i}^{*}\) and \({x}_{i}\) are the ith measurement and predicted data points, respectively; N is the number of data points in n experimental runs with n being the number of runs.

Results and discussion

Conventional initial-velocity method

The initial reaction velocity at a given initial ATP concentration and pH was obtained as the slope of the initial linear part of [F-1,6-P] as a function of time and is plotted against pH in Fig. 2. As noted previously, the velocity of substrate (ATP or F6P) consumption or product (F-1,6-P) generation is proportional in magnitude to (double) the consumption rate of the measured NADH indicator. A trendline is fitted to the data in Fig. 2 to show the general pattern of variations.

Fig. 2
figure 2

Scatterplot of measured initial velocities along with fitted trendlines for initial ATP concentrations of 0.3, 0.625, 1.25, 2.5, 3.75, and 5 mM at varied pH levels. The datapoints are averages of two animals with two replicate assays each. Because of the mixed effects of ATP as substrate and inhibitor, the ATP effects are not obvious and consistent from the measurement.

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Several observations may be made from Fig. 2. Changes in pH considerably influence the PFK1 activity indicated by the measured initial velocity. The variations follow roughly skewed bell-shaped curves at various initial ATP concentrations. The reaction starts a higher pH when ATP concentration increases. The reaction does not proceed when assayed at low pH in the presence of high ATP concentrations. Increasing the initial ATP concentration has a marked influence on the initial reaction velocity and this effect is more pronounced around pH 8, about which the initial reaction velocities peak. From the measurements, the peak velocity increased as ATP concentration increased from 0.3 to 1.25 mM but then decreased as the ATP concentration further increased from 1.25 to 5 mM, indicating a complexity of the ATP effects on the initial velocity. Moreover, since ATP promotes the reaction as a substrate and impedes it as an inhibitor at the same time, its effect on the measured initial velocity is not pronounced and clear.

It has been reported that the degree of inhibition of a reaction can be expressed by the ratio of the velocity in the presence of an inhibitor to that in the absence of the inhibitor29. In the absence of the inhibitor, the dependence of reaction velocity on the substrate concentration can be described by the Michaelis–Menten rate equation \(v=\frac{{v}_{max}[S]}{{K}_{m}+[S]}\), where vmax is the maximum reaction velocity, Km is the Michaelis–Menten constant, and [S] is the substrate concentration. However, the parameter values (vmax and Km) cannot be determined unless there is a way, experimental or theoretical, to separate or isolate the substrate effect from the inhibitor effect. To suppress the substrate effect for observation of the inhibitor effect, the enzymatic reaction may be assumed to follow the first-order kinetics represented by \(v=\frac{d[P]}{dt}=k\left[S\right]\), with [P] and [S] being product and substrate concentrations, respectively; and k being a constant. The initial reaction velocity would be proportional to the substrate concentration. As a result, the initial velocity may be divided by the initial ATP concentration as an approximate way to suppress the substrate effect. Figure 3 shows the initial velocities divided by the initial ATP concentrations plotted against pH at various initial ATP concentrations with trendlines.

Fig. 3
figure 3

Measured initial velocities divided by initial ATP concentrations vs. pH with trendlines for varied ATP concentrations (0.3, 0.625, 1.25, 2.5, 3.75, and 5 mM). The datapoints are averages of two animals with two replicate assays each. The division partially remove the substrate effect of ATP and thus makes its inhibition effect more pronounced.

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As shown in Fig. 3, division by the initial ATP concentration as an approximate way to suppress the substrate effect substantially changed the initial velocity curves. At a given pH, the initial reaction velocity, as an indicator of PFK1 activity, decreases as ATP concentration increases from 0.3 to 5 mM, suggesting that the degree of PFK1 inhibition is enhanced by increasing ATP concentration. Moreover, based on the trendlines, the peak velocity (indicating the optimum pH for PFK1 activity) appears to shift slightly to the acidic side as ATP concentration increases.

While the conventional initial-velocity method is a convenient way to observe or estimate enzyme activity, it involves several difficulties and inaccuracies. First, the initial substrate/inhibitor (ATP) concentration is used as the concentration in effect, but the substrate is constantly being consumed at different rates. Therefore, when a measurement is made, the substrate concentration does not only differ from the initial value used for an experiment but also differs by a different amount across experiments because the reaction velocity varies (see Fig. 4 and section “Parameter estimation and model validation” for more explanations). As a result, the inhibitor concentration in effect is uncertain. Second, it is difficult to measure the initial velocity consistently across experiments. Even if the first measurement could be made at precisely the same time after initiation of the reactions, which is a challenge practically, the reactions are at different stages because the velocity differs. As shown in Fig. 3, the scatter in the measured initial velocity increased as ATP concentration decreased because measurement inconsistencies were greater when the reaction velocity was higher. Third, unless there is a way to nullify the substrate or the inhibitor effect, the two effects cannot be separated in the measured initial velocities. While division by the initial substrate (ATP) concentration may help suppress the substrate effect, the assumption of substrate effect following simple first-order kinetics is inconsistent with the purpose of the experiments, which is to analyze the enzyme behavior in an enzymatic reaction (thus not of first order). Therefore, a kinetic model can be used to advantage to avoid these difficulties.

Fig. 4
figure 4

Example plots comparing model predictions against experimental observations for pH 7. Symbols represent experimental data while the solid lines represent the model predictions. The datapoints are averages of two animals with two replicate assays each. NADH was the measured output of the assay and thus used as the modeled output, but F-1,6-P or ATP concentration can be easily calculated from that of NADH. To minimize uncertainties, the first measured NADH values, which varied somewhat, were used as the initial condition for the model (see text for more explanations).

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Kinetic modeling method

Parameter estimation and model validation

Comparisons between model predictions and experimental measurements for an example pH are shown in Fig. 4 and the optimized parameters for the kinetic model are listed in Table 1.

Table 1 Model parameter values used for model predictions shown in Fig. 4.
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The degree of agreement between model predictions and measurements was assessed in terms of the relative standard error of prediction (RESP) and the determination coefficient (R2) value, which were 0.1025 ± 0.0131 and 0.9398 ± 0.0907, respectively, over all experiment runs, indicating that the model structure could closely represent the kinetic variations in experimental measurements.

Because the model describes the variables as functions of time, a time-variant rather than an assumed constant ATP concentration is in effect as an inhibitor. Moreover, the model does not need to rely on the time point of reaction initiation as time zero and the initial concentrations as initial conditions. Because it is practically challenging to start a reaction and make the first measurement at precisely the same times for all experiments, the time of reaction initiation is an unreliable time origin. Fortunately, the first measurement can be used as the initial condition for the model to reduce this uncertainty. Because the rates of substrate consumption and NADH consumption are related through Eq. (3), the first substrate concentration can be determined as:

$$\left[ S \right]_{1} = \left[ S \right]_{0} - \frac{1}{2}\left( {[NADH]_{0} - \left[ {NADH} \right]_{1} } \right) $$
(10)

where [S]1 is the first measured substrate (ATP or F6P) concentration to be used as the initial condition; [S]0 and [NADH]0 are the substrate and the NADH concentrations, respectively, used to start the reaction; and [NADH]1 is the first measured NADH concentration.

As a result of using the first measurement as the initial condition, the plots in Fig. 4 start from varied initial points that are different from the concentration (1 mM) used to start the reactions.

The same could be done in the initial-velocity method. After such a correction, however, the inhibitor (ATP) concentrations deviate by varied amounts from the designed values (0.3, 0.625 etc.), making data analysis and presentation more difficult. More significantly, the correction does not address the fact that ATP concentration changes continuously and any single constant value used is not the true concentration in effect.

Kinetic modeling method to analyze the effects of ATP and pH on PFK1 activity

The reaction velocities in the presence of inhibition were calculated by using Eq. (4) and the parameter values listed in Table 1 and plotted against pH at various ATP concentrations in Fig. 5a. As shown in Fig. 5a, a change in pH substantially influences the PFK1 activity and thus the reaction velocity, and a set of bell-like curves are obtained at various ATP concentrations. The optimum pH for PFK1 activity (corresponding to the peak reaction velocity) is shifted to the alkaline side with increasing ATP concentration. The effect of ATP on PFK1 activity is more obvious over the pH range from 8 to 10. Moreover, increasing the ATP concentration has a marked influence on the maximum PFK1 activity. The maximum PFK1 activity initially increases as ATP concentration increases from 0.3 to 1.25 mM and then decreases as the ATP concentration further increases from 1.25 to 5 mM. This trend of the maximum PFK1 activity as a function of ATP concentration is consistent with the result obtained by using the conventional initial-velocity method as shown in Fig. 2.

Fig. 5
figure 5

Reaction velocities in the presence (a) and absence (b) of substrate inhibition.

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To simulate reaction velocities in the absence of inhibition, f1, f2, and f3 in Eq. (4) can be set to 1 to eliminate the pH effect and \({v}_{max}^{S \cdot E\cdot S}\) replaced with \({v}_{max}^{E \cdot S}\) to eliminate the substrate inhibitor effect on production generation from S·E·S to give,

$${v}_{mm}=\frac{\left({v}_{{\max}}^{E \cdot S} [\text{S}]+\frac{{v}_{max}^{E \cdot S}}{{K}_{i}}{[S]}^{2}\right)}{\left({K}_{s}+[S]+\frac{1}{{K}_{i}}{[S]}^{2}\right)}$$
(11)

Equation (11) represents the substrate effect on reaction velocity without inhibition and is plotted against ATP concentration in Fig. 5b. The plot is analogous to the Michaelis–Menten saturation curve, which exhibits a hyperbolic increase with substrate increase. At low ATP concentrations (lower than 1 mM), the velocity increases almost linearly with the substrate concentration, while at high substrate concentrations (greater than 7 mM), the velocity exhibits saturation near 350 μmol F-1,6-P min−1 mg−1 of muscle.

The degree of reaction velocity reduction or PFK1 activity inhibition may be shown as the ratio of the velocity in the presence of inhibition to that in the absence of the inhibition and the ratio curves are plotted against pH at various ATP concentrations in Fig. 0.6 in two different formats: Fig. 6a as a family of curves while Fig. 6b as a contour plot.

Fig. 6
figure 6

A family of curves (a) and contour plot (b) of PFK1 activity shown by the reaction velocity with inhibition as a percentage of reaction velocity without inhibition against pH at various ATP concentrations.

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As shown in Fig. 6, the optimum pH for PFK1 activity (peak velocity) shifts to the alkaline side with increasing ATP concentrations. Furthermore, ATP concentration has a marked influence on the PFK1 activity, and the effect of ATP on PFK1 activity is pronounced over the pH range from 7 to 9. As can be seen in Fig. 6, at a given pH, PFK1 activity continually decreases as ATP concentration increases from 0.3 to 5 mM, suggesting that the degree of inhibition of PFK1 is enhanced by increasing ATP concentration. This is in agreement with the result obtained by using the conventional initial-velocity method as shown in Fig. 3. However, in contrast to the conventional initial-velocity method, which gives only a qualitative picture, the kinetic modeling method is useful in obtaining the degree of inhibition based on reaction velocity. As shown in Fig. 6, for example, at the pH optimum, about 55% of PFK1 activity is inhibited at ATP concentration of 0.3 mM, while approximately 80% of PFK1 activity is inhibited when ATP concentration is 5 mM.

Further analysis of PFK1 activity based on the model

As shown in Table 1, the parameter Ki, which represents the dissociation constant of a substrate molecule binding to the inhibitory site to form S·E·S, is approximately fourfold of Ks, which denotes the dissociation constant of a substrate molecule binding to the catalytic site to form E·S. This indicates that the affinity of the substrate binding to the catalytic site is about four times that to the inhibitory site. \({v}_{max}^{E \cdot S}\) is approximately five times \({v}_{max}^{S \cdot E\cdot S}\), suggesting that when a substrate binds to the inhibitory site of the enzyme to form S·E·S, the catalytic activity of the enzyme is reduced by a factor of 5.

During the reactions, the enzyme is present in three forms: (1) Unsaturated or free form (E), which makes no direct contribution to product generation, (2) In complex E·S, which is not substrate-inhibited and produces product at one rate, and (3) In complex S·E·S, which is substrate-inhibited and produces a product as a lower rate. The overall observed velocity (\(v\)) of the reaction catalyzed by PFK1 thus depends on the proportions of these three enzyme populations. [E·S], [S·E·S] and [E] as proportions of the total enzyme concentration [ET] = [E] + [E·S] + [S·E·S] can be calculated as \(\frac{[\text{E}\cdot \text{S}]}{[{\text{E}}_{\text{T}}]}=\frac{1}{\left(\frac{{K}_{s}}{\left[S\right]}\frac{{f}_{1}}{{f}_{2}}+1+\frac{\left[S\right]}{{K}_{i}}\frac{{f}_{3}}{{f}_{2}}\right)}\), \(\frac{\left[\text{S}\cdot \text{E}\cdot \text{S}\right]}{[{\text{E}}_{\text{T}}]}=\frac{\left[\text{E}\cdot \text{S}\right]\left[\text{S}\right]}{{K}_{i}\left(\frac{{f}_{2}}{{f}_{3}}\right)}\), and \(\frac{\left[E\right]}{[{\text{E}}_{\text{T}}]}=\frac{\left(\frac{{f}_{1}}{{f}_{2}}\right)\left[\text{E}\cdot \text{S}\right]{K}_{s}}{\left[\text{S}\right]}\), respectively, and are plotted against ATP concentration in Fig. 7.

Fig. 7
figure 7

Proportions of [E·S], [S·E·S] and [E] against ATP concentration.

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As shown in Fig. 7, the fraction of [E·S] initially increases with [ATP] and reaches a maximum of 5% at an ATP concentration of 0.12 mM and then drops as the ATP concentration increases. At the same time, [S·E·S] increases in a sigmoid pattern and its fraction reaches 90% near a concentration of 0.6 mM of ATP. This shows that when [ATP] is elevated, the inhibited form (S·E·S) quickly becomes the dominant form. While the affinity of ATP to the catalytic site of PFK1 is higher than that to the inhibitory site, the velocity of product generation via the S·E·S pathway (i.e., consumption of S·E·S) is much slower than that via the E·S pathway as shown below, leading to accumulation of the inhibited form when ATP concentration increases.

The velocities of product generation by the E·S and S·E·S pathways can be computed as.

\({v}_{ES}=\frac{\left(\frac{{v}_{max}^{E \cdot S}}{{f}_{2}}\right)}{\left(\frac{{K}_{s}}{\left[ATP\right]}\frac{{f}_{1}}{{f}_{2}}+1+\frac{\left[ATP\right]}{{K}_{i}}\frac{{f}_{3}}{{f}_{2}}\right)}\) and \({v}_{SES}=\frac{\left(\frac{{v}_{max}^{S \cdot E\cdot S}\left[S\right]}{{f}_{2}{K}_{i}}\frac{{f}_{3}}{{f}_{2}}\right)}{\left(\frac{{K}_{s}}{\left[S\right]}\frac{{f}_{1}}{{f}_{2}}+1+\frac{\left[S\right]}{{K}_{i}}\frac{{f}_{3}}{{f}_{2}}\right)}\) and are plotted against ATP concentration in Fig. 8a. In addition, the percent contributions of E·S and S·E·S pathways to product generation (\(\frac{{v}_{ES}}{{v}_{ES}+{v}_{SES}}\) and \(\frac{{v}_{SES}}{{v}_{ES}+{v}_{SES}}\)) are plotted against ATP concentration in Fig. 8b.

Fig. 8
figure 8

Velocities (a) and percent contributions (b) of E·S and S·E·S pathways to product generation.

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As can be seen in Fig. 8a, the velocity of product generation through the E·S pathway initially goes up and reaches a maximum of 8 μmol F-1,6-P min−1 per mg of muscle at approximately 0.15 mM of ATP concentration and then decreases as the ATP concentration increases, while the velocity of product generation through the S·E·S pathway increases rapidly in a sigmoid way and reaches a maximum of 35 μmol F-1,6-P/min per mg of muscle near a concentration of 1.2 mM of ATP.

Figure 8b shows that the E·S pathway is likely the dominant pathway to generate product when ATP concentration is lower than 0.12 mM, while afterwards, the S·E·S pathway prevails, resulting in slower overall velocity because the S·E·S pathway is much slower in product generation.

The conventional initial-velocity method is simple and commonly used, and the initial velocity is widely accepted as an indicator of enzyme activity in most enzymatic assays. The method, however, is not an effective means of quantifying enzyme activity under substrate inhibition because of the difficulties discussed earlier, the most significant of which is that the substrate effect and inhibitor effect cannot be separated in the measured initial velocity, making it difficult to quantify the specific impact of the substrate on enzyme kinetics30. To mitigate the interference of substrate effects on inhibitor effects, a common approach is to normalize the initial velocity by the initial substrate concentration under the assumption of first-order kinetics31,32. While this adjustment can reduce substrate interference to some extent, it does not fully account for complex kinetic behaviors, such as non-competitive or mixed inhibition, where the assumption of first-order kinetics may not hold true33,34.

The kinetic modeling method reveals enzyme activity by using a kinetic model representing the underlying enzymatic reactions. The primary advantage of the method is that it allows for quantitative separation of the substrate effect and inhibitor effect so that the latter can be quantified and modeled. The model parameter values can provide insights into different aspects of the underlying reactions such as affinity between reactants, and the model can be used to analyze the relative contributions of different reaction pathways and different forms of an enzyme, as illustrated herein, which are crucial for understanding enzyme regulation and designing targeted inhibitors35,36,37. Moreover, the kinetic modeling approach does not rely on precise timing of initial measurements, as it can accommodate variations in reaction initiation times by using the first experimental data point as the initial condition. This flexibility makes the modeling method robust for analyzing enzymatic reactions under various experimental conditions and across different reaction time frames.

The initial-velocity method showed quantitatively similar trends and effects of ATP and pH as the modeling method, indicating a level of agreement between the two methods. The former is simple and can be used to gain a general picture of enzyme activity as affected by pH and a substrate inhibitor. It can also be used to determine an approximate optimal range of pH for an enzyme. However, while the initial-velocity method provides a rapid overview of enzyme behavior, the kinetic modeling method excels in quantifying the degree of inhibition and elucidating the underlying kinetic mechanisms with higher accuracy and detail.

In summary, while the initial-velocity method remains valuable for its simplicity and quick assessment of enzyme activity, especially in preliminary screenings, the kinetic modeling method stands out for its ability to rigorously quantify and model enzyme kinetics under complex conditions, such as substrate inhibition. Combining these methods can provide complementary insights into enzyme function and regulation, thereby advancing our understanding of enzymatic processes in biological system.

Conclusions

A kinetic modeling method is presented for evaluating phosphofructokinase (PFK1) activity, an important factor in energy metabolism, and quantifying the inhibitory effects of ATP and pH. In comparison to the conventional initial-velocity method, which gives a qualitative picture of the inhibition of PFK1 activity under varied conditions, the modelling method can separate the substrate effect from the inhibitor effect of ATP and thus quantify the inhibitor effect. The use of a kinetic model to evaluate enzyme activity and quantify the degree of inhibition can be extended to other enzymes if the underlying reaction mechanisms are known. Kinetic analysis of PFK1 based on the model indicated that the affinity of the substrate binding to the catalytic sites is fourfold of that to the inhibitory site. Increasing ATP concentration leads to a reduced overall reaction velocity as a result of increasing proportion of ATP binding to both the catalytic and the inhibitory sites of PFK1.