The hydrolysis of nitrophenyl phosphate (NPP) by alkaline phosphatase enzyme is a first order reaction dependent on the concentration of NPP in solution. A product of this reaction is nitrophenyl anion which has a high molar absorptivity at 410nm. This property makes it relatively easy to observe this reaction as a function of time via a spectrophotometer. By observing the reaction as a function of time it is possible to study the kinetics of this reaction and to determine how the initial reaction rate depends on the initial concentration of NPP. This relationship can be described by the Michaelis-Menton equation which is described in some detail. It is found that, as expected, reaction rate increases with an increase in NPP concentration. The Eadie-Hofstee plot is used to linearize the data and to obtain reasonable approximations for the Vmax and KM parameters found in the Michaelis-Menton equation. A process involving the minimization of a Χ2 value is used to obtain the final values of these two parameters: Vmax = 4.28*10-7 and KM = 3.33*10-4. These parameters produce a qualitatively strong fit for the data obtained and so the Michaelis-Menton equation reasonably accurately describes the relation between initial NPP concentration and reaction rate. It is found, as expected, that the addition of the inhibitor species phosphate decreases the rate of NPA formation. The inhibition constant KI obtained from the apparent KM value of the Michaelis-Menton equation. By averaging the KI value for several concentrations of inhibitor, = (1.79 + 0.25)*10-4 M.

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## Introduction

Alkaline phosphatases are common enzymes found in places such as the mammalian intestine and the bacterium Escherichia coli. They are a family of two-subunit homologous enzymes which catalyze the hydrolysis of a large spectrum of phosphate monoesters to phosphate and neutral molecules. Because they catalyze a large variety of hydrolysis reactions, they are said to be non-specific. This reaction can be expressed as follows where R is any organic compound:

R-OPO32- + H2O R-OH + HPO42- (1)

By removing the phosphate, R becomes a neutral species that may more easily serve to facilitate transport of nutrients across cell membranes or other biological housekeeping processes. Alkaline phosphatases can also be used to remove phosphate groups from DNA molecules which allows for different manipulations of DNA.

It is instructive to examine the kinetics of this reaction in order to more fully understand certain biochemical processes. Alkaline phosphatses are large molecules with a certain number of active sites in which the hydrolysis of smaller molecules is catalyzed. Competition for the active site of these enzymes is one aspect which may affect the kinetics of the reaction. We will examine the hydrolysis of p-nitrophenyl phosphate (NPP) by the E. coli alkaline phosphatase at a pH of 10.6 at room temperature. In particular, the maximum velocity Vmax and the Michaelis constant KM of the reaction are determined and the effect of the product HPO42- as an inhibitor is examined.

## Theory

The initial rate, or the initial velocity, v of enzyme-catalyzed reactions has been observed to follow first order kinetics. That is, the rate of the reaction depends only on the initial substrate concentration. However, this only holds true at low values of substrate concentration and levels off to a maximum velocity V at large concentrations. The general form for the way in which an enzyme catalyzes a reaction is a noncovalent association of the substrate molecule to the enzyme followed by some catalytic steps that result in the product being released from the enzyme.

S + E ES E + P (2)

The enzyme concentration is almost always much smaller than the concentration of the substrate. Due to this, very quickly equilibrium will be reached where, as soon as product is released from the enzyme’s active site, a new substrate molecule will enter the active site. At this production of [ES] is said to be saturated, meaning it has a constant value which does not change with time.

(3)

This is said to be the steady state, or bottleneck, condition where all active enzyme sites are occupied. From this condition and the fact that matter is conserved, one can derive the rate law:

(4)

This is the Michaelis-Menton equation, where the Michaelis constant KM=(k-1 + k2)/k1. This value in a physical context is the substrate concentration midway between zero substrate and the concentration at which the reaction rate has become saturated. Also in the equation Vmaxk2E0 which is said to be the maximum velocity of the reaction. Here, v is the rate of the steady state enzyme reaction.

In order to measure the reaction rates, the concentration of product or substrate must be observed as a function of time. It is expected that the concentration vs. time plot will be initially linear until the substrate is beginning to be used up where it would then level off to a constant value. p-nitrophenyl phosphate is chosen as the substrate because a product of its dephosphorylation, nitrophenolate anion has a high molar absorptivity in the blue wavelengths of visible light. This allows for the use of Beer’s Law to calculate concentration of the product from the absorbance at a specific wavelength of light.

A410 = ε410,NPA l cNPA (5)

Here, A is the absorbance at a wavelength of 410nm, l is the light path through the solution, ε is the molar absorptivity of the solution and c is the concentration. Thus, by observing the change in absorbance as a function of time, it is possible to observe the increase in product as a function of time.

Once the reaction rates at each initial substrate concentration are known, it is necessary to determine the parameters Vmax and KM in the Michaelis-Menton equation (Equation 4) in order to fit the data to a mathematical trend. This nonlinear fit is difficult to make without first having reasonable estimates of Vmax and KM. However, by rearranging Equation 4, it is possible to linearize the equation such that reasonable approximations of the parameters can be obtained by performing a simple linear least-squares regression. One such linearization produces the Eadie-Hofstee plot of the form:

v = Vmax – KM (6)

This equation can be used to give reasonable approximations of the two parameters, which can then be optimized to produce the values of Vmax and KM. This optimization is performed by comparing the estimated rate from Equation 4 using the approximate values for the two parameters to the actual rate at different concentrations. A Χ2 is defined to be the sum of the deviations from the actual observations. A computer can be used to minimize this value by varying Vmax and KM where the values for these two parameters at the minimized Χ2 are the best approximations. The error in these values is found by minimizing the Χ2 at several values of Vmax and KM by varying the other parameter. A plot of these points can be used to determine the 95% confidence intervals for these values.

Competitive inhibition is the process by which molecules that resemble the substrate can bind to the active site of the enzyme, establishing a competition between the substrate and this molecule for active sites on the enzyme. HPO4- is a product of the hydrolysis of NPP which acts as an inhibitor to the enzyme reaction. The effect of the addition of phosphate on the rate can be expressed as follows:

(7)

Here, all variables are the same as in Equation 4, where KI is the inhibition constant.

## Experiment

An Ocean Optics USB 400 spectrophotometer is used to take all absorbance readings. All parts of the experiment take place at atmospheric pressure and are open air. In order to establish a basis of comparison, the absorbance at 410 nm is taken for nitrophenolate anion, NPA, the pure product of the enzyme reaction. From this, the molar absorptivity constant can be determined for NPA which can be used to determine the concentration of NPA in solution. Several solutions with a range of concentrations of the substrate nitrophenyl phosphate, NPP, are necessary to observe the effect of substrate concentration on the rate of the enzyme reaction. The concentrations are chosen such that they span the range of the Michaelis-Menton plot. Upon the addition of enzyme to each solution, the concentration of NPA as a function of time is recorded by taking the absorbance reading at 410 nm at each time step. The reaction for each initial NPP concentration is charted for a sufficiently long time such that an accurate least-squares fit can be made for the plot of concentration as a function of time. From this plot, the initial reaction rates for each NPP concentration can be determined and fit to an Eadie-Hofstee plot to determine approximate values for V and KM. To observe the inhibiting effect of phosphate on the enzyme reaction, several solutions with a range of phosphate concentrations all with equal concentrations of NPP must be prepared. Then, upon the addition of enzyme to the solutions, the concentration of NPA as a function of time is observed via the absorbance readings. It is necessary to keep the initial substrate concentration constant in this part so that the variation in the reaction rate can be attributed to only the variation in phosphate concentration.

## Discussion of Results

As expected, increasing the initial substrate concentration served to increase the rate of product formation in the hydrolysis of nitrophenyl phosphate by an alkaline phosphatase enzyme. This can be seen clearly in Figure 1 where the slope of the graph of product (NPP) concentration as a function of time increases as enzyme concentration increases. For the trials with the three highest initial substrate concentrations, the enzyme concentration was half that used in every other run, meaning that the rate of reaction for these trials must be doubled for comparison purposes. This explains why the slopes of these lines are approximately half of what would be expected in Figure 1.

## Figure : The plot demonstrates that as substrate concentration increases, so too does the rate of product formation

In order to determine the rate of reaction, a second order trend is fit to the data where the linear term in the equation is taken to be the rate of product formation. A second order least squares regression is used to fit the data because the rate of reaction is not truly constant over the timeframe observed. It is expected that the data will have a very slight negative concavity as, by Equation 2, the substrate concentration is decreasing at the same rate as the increase in NPA, leaving less NPP to form the enzyme-substrate complex which produces NPA. This means that the rate should decrease as a function of time – the second order regression is used to correct for this fact, leaving the linear term to describe the rate of reaction. Figure 2 (attached) displays the approximate second order trends for NPA concentration as a function of time for each initial NPP concentration. These are only approximate trends as these may not be rigorously calculated, and in any case are over a smaller range than that used in subsequent calculations. A more rigorous second order least-squares regression is performed to obtain the reaction rates for each initial substrate concentration. The rates along with the standard deviations for each rate are tabulated in Table 1. As stated previously, it has clearly been shown that as initial substrate concentration is increased, so too does the rate of the reaction. Additionally, the second order linear trend provides a very strong fit to the data. This is evidenced by the fact that the standard deviations of the rates are all 8 or 9 orders of magnitude smaller than the rate, meaning there is very little variance in the data from the trend calculated.

Table 1: Enzyme Reaction Rate as a function of Substrate concentration

[S0]/M

Rate (M/sec)

Rate SD

3.23E-03

3.94E-07

1.25E-15

1.61E-03

3.52E-07

2.50E-16

8.06E-04

2.96E-07

4.54E-16

4.03E-04

2.40E-07

4.77E-16

2.42E-04

1.73E-07

2.73E-16

1.45E-04

1.29E-07

2.58E-17

6.45E-05

7.97E-08

7.56E-17

3.23E-05

4.13E-08

4.78E-17

Once the rates at each initial NPP concentration are known, it is possible to attempt to describe the initial rate of the enzyme reaction as a function of initial substrate concentration. This relation can be described by Equation 4, and so the parameters Vmax and KM must be obtained. As stated previously, it is difficult to perform a nonlinear regression to describe rate as a function of time without having reasonably close guesses for these two parameters. By the process described previously whereby Equation 5 is used to linearize the data, we obtain the Eadie-Hostee plot displayed in Figure 3.

## Figure 3: The Eadie-Hofstee plot linearizes the data such that estimates of Vmax and Km can be obtained

As can be seen in the figure, this method produces a roughly linear plot. By performing a linear least squares fit on the data, we can obtain guesses for the two parameters. Equation 6 makes it apparent that the slope of the plot is -KM and the intercept is Vmax. An analysis of the units bears this out: Vmax has units of M/sec (as does the rate because Vmax is the maximum initial rate of reaction at which the enzyme becomes completely saturated), and KM has units of M (as does substrate concentration because KM is the concentration at which the reaction rate is half that of Vmax). The intercept and slope of the graph also have units of M/sec and M, respectively. From the plot, it is estimated that KM = 2.90*10-4 M and Vmax = 4.10*10-7 M/sec. Using these values for the parameters in the Michaelis-Menton equation (4), a decent fit of the data is obtained as seen in Figure 4. However, it is clear from the plot that the values for both parameters are too low. The plot begins to approach a value which is too low as the plot from the formula falls below the final data point. This suggests that the value of Vmax is too low. Additionally, the plot rises too quickly at low substrate concentrations which would cause KM to occur too early, as does the lower value of Vmax.

## Figure 4: The approximate values of Vmax and Km from the Eadie-Hofstee plot produce only a rough fit of the data

These apparent inaccuracies in the values of Vmax and KM are most likely resulted from the fact that the Eadie-Hostfee plot does not use two separate variables on each axis. However, clearly these values are relatively close to the correct value as the plot roughly fits the data. By the process described previously, a computer can be used to produce more accurate values of Vmax and KM by minimizing the Χ2 value, which is the sum of the squares of the deviations from the data by the value predicted in the formula. Minimizing Χ2 by varying the two parameters gives the values in Table 2.

Table 2: Comparison of Vmax and Km values from Eadie-Hofstee plot and non-linear regression

Eadie-Hofstee

Non-linear Reg.

Pos Error

Neg Error

Vmax (M/sec)

4.10E-07

4.28E-07

.32E-07

.28E-07

Km/M

2.90E-04

3.33E-04

.47E-04

.58E-04

As expected, the values of both parameters have increased, where Vmax = 4.28*10-7 and where KM = 3.33*10-4. The positive and negative errors for these two parameters are obtained by the process described above. At different values of each parameter, the Χ2 value is minimized while varying only the other variable. The plots of this process are appended. The confidence interval is taken to be 4.28*Χ2; that is the values at which these graphs cross this value corresponds to the lower and upper limits of each parameter. This error is also recorded in Table 2. Then by using Equation 4, one can use the multiplicative formula for error to determine error bars for the data points. These errors are tabulated in Table 3 for each initial substrate concentration.

Table 3: Enzyme Reaction Rate as a function of Substrate concentration with Errors from Km and Vmax

[S0]/M

Rate (M/sec)

pos error

neg error

3.23E-03

3.94E-07

6.29E-08

7.33E-08

1.61E-03

3.52E-07

5.62E-08

6.55E-08

8.06E-04

2.96E-07

4.73E-08

5.51E-08

4.03E-04

2.40E-07

3.83E-08

4.47E-08

2.42E-04

1.73E-07

2.76E-08

3.22E-08

1.45E-04

1.29E-07

2.06E-08

2.40E-08

6.45E-05

7.97E-08

1.27E-08

1.48E-08

3.23E-05

4.13E-08

6.60E-09

7.68E-09

Using these corrected values for Vmax and KM, a more accurate formula to describe the data is obtained. Figure 5 clearly shows that the plot using these new values provides a much better fit to the data than does the plot of the approximate values for the parameters. All of the issues seen in the first plot are no longer present, and the formula clearly falls well within the error bars on each data point. Despite lacking a quantitative means of expressing the quality of the fit, it is possibly to qualitatively state by looking at the plot that the formula with the determined parameter values closely matches the experimental data.

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## Figure 5: The optimized values of Vmax and Km from the estimates given by the Eadie-Hofstee plot produce a strong fit

Finally, having confidence in the ability for the Michaelis-Menton equation to describe the effect of initial concentration on the rate of reaction, it is possible to study the effect of the addition of an inhibitor species on the rate of the reaction. Keeping the initial concentration of substrate fixed, the concentration of inhibitor in solution was varied. As would be expected, the addition of inhibitor species greatly reduced the rate of product formation. This is shown in Table 4.

Table 4: Determination of mean inhibition costant Ki

[I]/M

Rate (M/sec)

Km app/M

Ki/M

0

2.40e-7

3.33e-4

0

3.23E-03

2.34E-08

6.84E-03

1.62E-04

6.45E-03

1.69E-08

9.80E-03

2.27E-04

9.68E-03

7.61E-09

2.14E-02

1.47E-04

[S]/M

Avg Ki/M

1.79E-04

4.03E-04

SE

.25E-04

Also observed is that the addition of more inhibitor species results in an increasingly reduced rate of product reaction. This makes sense because as more inhibitor species exists in solution, there are more molecules that can bind to the enzymes’ active sites, thereby inhibiting the formation of the enzyme-substrate that is necessary for product formation. As a result of the rate decreasing as the inhibitor concentration is increased, the apparent KM value greatly increases. This makes sense because the substrate would have to be significantly more concentrated to effectively compete and reach the concentration at which the reaction rate is half that of Vmax. From Equation 7, it is possible to calculate the value of KI from the inhibitor concentration, the rate and Vmax. Averaging the values of KI for each inhibitor concentration gives a = (1.79 + 0.25)*10-4 M.

## Conclusion

The alkaline enzyme-catalyzed hydrolysis of nitrophenyl phosphate is a first order reaction that depends on the initial concentration of substrate in solution when the concentration of enzyme is kept constant. A second order least-squares regression gives the initial rate of reaction as a function of time, where the rate of product formation increases as the initial concentration of substrate increases. The Eadie-Hosfstee plot was used to approximate the parameters Vmax and KM that are necessary to describe the relation between substrate concentration and rate of reaction. Optimizing these parameters through the minimization of a X2 value gives Vmax = 4.28*10-7 and KM = 3.33*10-4. The error in these parameters can be determined by each parameter separately to minimize the X2 value at different values for each parameter. These values for the parameters in the Michaelis-Menton equation produce a qualitatively strong fit for the data. The addition of an inhibitor species served to decrease the rate of product formation where the increase in inhibitor concentration results in a decrease in rate. Using the Michaelis-Menton