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In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km?


Taking the reciprocal of both sides of the Michaelis-Menten equation yields the Lineweaver-Burk Equation:

$ dfrac{1}{V} = dfrac{K_m}{V_{max}}dfrac{1}{[S]}+ dfrac{1}{V_{max}} $

Plotting a $ dfrac{1}{V}$ vs. $dfrac{1}{[S]}$ graph, I am told that:

y-int $= dfrac{1}{V_{max}}$ and

x-int $= -dfrac{1}{K_m}$

How are these relationships derived from the lineweaver-burk plot? I can see how the y-intercept can be equal to $dfrac{1}{V_{max}}$ if $dfrac{1}{[S]} = 0$, but I don't see how x-int $= -dfrac{1}{K_m}$ by setting $dfrac{1}{V} = 0$? Can someone demonstrate how these relationships were derived?


Set $ dfrac{1}{V} = 0$ and solve for $dfrac{1}{[S]}$:

$ 0 = dfrac{K_m}{V_{max}}dfrac{1}{[S]}+ dfrac{1}{V_{max}} $

$ -dfrac{1}{V_{max}} = dfrac{K_m}{V_{max}}dfrac{1}{[S]}$

$ -1 = {K_m}dfrac{1}{[S]}$

$ -dfrac{1}{K_m} = dfrac{1}{[S]} = $ x-intercept


10.5: Enzyme Inhibition

Enzymes can be regulated in ways that either promote or reduce their activity. There are many different kinds of molecules that inhibit or promote enzyme function, and various mechanisms exist for doing so. In some cases of enzyme inhibition, for example, an inhibitor molecule is similar enough to a substrate that it can bind to the active site and simply block the substrate from binding. When this happens, the enzyme is inhibited through competitive inhibition , because an inhibitor molecule competes with the substrate for active site binding. On the other hand, in noncompetitive inhibition, an inhibitor molecule binds to the enzyme in a location other than an allosteric site and still manages to block substrate binding to the active site.


Biochemistry : Enzyme Kinetics and Inhibition

The Michaelis constant, , occurs at what value on an enzyme kinetics graph?

Not enough information is given to predict

The Michaelis constant, , is a frequently used value in enzyme kinetics used to essentially describe how much substrate is needed to speed up a reaction. More specifically, it is the substrate concentration required to get a reaction to half of its (the correct answer is ). The itself is given as a specific substrate concentration, and in order to find it, one would draw a straight line across on an enzyme kinetics graph at until reaching the curve. The substrate concentration that is found at that point is the .

Example Question #1 : Vmax And Km

Which of these are representations of ?

III. Y-intercept on a Lineweaver-Burk plot

However, on a Lineweaver-Burk plot, the Y-intercept actually represents . It is the X-intercept that can be derived from. The X-intercept is equivalent to .

Example Question #1 : Vmax And Km

When an enzyme is in the presence of a competitive inhibitor, which of the following will happen to the enzyme?

Competitive inhibitors will block the active site of the enzyme. The presence of the competitive inhibitor increases the amount of substrate required to get the enzyme to half of its maximum velocity. As a result, the of the enzyme will increase. Note that a competitive inhibitor will not affect the maximum velocity of the enzyme.

Example Question #1 : Rate Limiting Step

According to Michaelis-Menton kinetics, what is a characteristic of the rate limiting step in enzyme kinetics?

It dissociates the enzyme-substrate complex into enzyme + substrate

The enzyme-substrate complex is formed

It is dependent on activation energy from catalysis

Does not involve a catalyst

It is dependent on activation energy from catalysis

The enzyme-substrate complex dissociates into enzyme + product. The rate limiting step is providing the activation energy to get to the transition state, which is greatly decreased by an enzyme.

Example Question #2 : Rate Limiting Step

Which of the following best describes the rate-limiting step in a chemical reaction?

It is the step that liberates the most amount of energy in the overall reaction

It is the step that consumes the most amount of energy in the overall reaction

It is the fastest step in the overall reaction

It is always an anabolic reaction

It is the slowest step of the overall reaction

It is the slowest step of the overall reaction

Although chemical reactions are typically displayed in the form of an equation, with reactants on the left and products on the right, these reactions are not a simple one step conversion. Often, there are several individual steps that the reactants go through on their way to becoming products. This is shown by the mechanism for that particular reaction.

Furthermore, when talking about chemical reactions, it is very important to distinguish between two concepts that are sometimes confused with one another. The first concerns the kinetics of the reaction, while the second concerns the thermodynamics.

Chemical kinetics is concerned with time. If a chemical reaction is occurring, kinetics answers the question of how fast the reaction is going. Thermodynamics, on the other hand, is not concerned with time. It doesn't care how fast or how slow a reaction goes. All it cares about is whether a chemical reaction is spontaneous or nonspontaneous. To answer this, thermodynamics considers the energetics of a reaction.

When looking at the answer choices, we can immediately eliminate three of them based on this information. The rate-limiting step of a chemical reaction is not concerned with how much energy is liberated or consumed. Instead, the rate-limiting step is defined as the slowest step out of all the steps that occur for a given chemical reaction. In other words, a reaction can only proceed as fast as its slowest step, just like a chain is only as strong as its weakest link. Further, the rate-limiting step in a reaction may be anabolic or catabolic.

It is important to note, however, that there is one component of energy that does affect the rate of a reaction. This energy is called the activation energy, and it represents how much energy needs to be invested into a reaction in order for that reaction to proceed. The reason why this is distinct from thermodynamics, however, is because thermodynamics cares only about initial and final energy states it doesn't care about how a reaction goes from initial to final, whereas kinetics does. Even though the activation energy for a reaction can change (via enzymes, for instance), this will not affect the initial and final energy levels.


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Heterogeneous Reactions

13.9 Evaluating True Kinetic Parameters

The intrinsic kinetics of zero-order, first-order, and Michaelis–Menten reactions are represented by the parameters k0, k1, and vmax and Km. In general, it cannot be assumed that the values of these parameters will be the same before and after cell or enzyme immobilisation: significant changes can be wrought during the immobilisation process. As an example, Figure 13.20 shows Lineweaver –Burk plots ( Section 12.4.2 ) for free and immobilised β-galactosidase enzyme. According to Eq. (12.43) , the slopes and intercepts of the lines in Figure 13.20 indicate the values of Km/vmax and 1/vmax, respectively. Compared with the results shown for free enzyme, the steeper slopes and higher intercepts obtained for the immobilised enzyme indicate that immobilisation reduces vmax this is a commonly observed result. The value of Km can also be affected [9] .

Figure 13.20 . Lineweaver–Burk plots for free and immobilised β-galactosidase. Enzyme concentrations within the gel are: 0.10 mg ml −1 (●) 0.17 mg ml −1 (□) and 0.50 mg ml −1 (■).

Data from P.S. Bunting and K.J. Laidler, 1972, Kinetic studies on solid-supported β-galactosidase. Biochemistry 11, 4477–4483.

As described in Sections 12.3 and 12.4 Section 12.3 Section 12.4 , kinetic parameters for homogeneous reactions can be determined directly from experimental rate data. However, evaluating the true kinetic parameters of immobilised cells and enzymes is somewhat more difficult. The observed rate of reaction is not the true rate at all points in the catalyst mass transfer processes effectively ‘mask’ the true kinetic behaviour. Accordingly, vmax and Km for immobilised catalysts cannot be estimated using the classical plots described in Section 12.4 . Under the influence of mass transfer, these plots no longer give straight lines over the entire range of substrate concentration [40, 41] .

As illustrated in Figure 13.20 for β-galactosidase, Lineweaver–Burk plots for immobilised enzymes are nonlinear. Often in such plots, however, the deviation from linearity is obscured by the scatter in real experimental data and the distortion of errors due to the Lineweaver–Burk linearisation ( Sections 12.4.2 and 3.3.4 Section 12.4.2 Sections 3.3.4 ). At low substrate concentrations (i.e., large values of 1/s), the Lineweaver–Burk plot appears linear as the reaction exhibits approximate first-order kinetics. By inference then, the apparent linearity of Lineweaver–Burk or similar plots cannot be considered sufficient evidence of the absence of mass transfer restrictions. Even if we mistakenly interpret one of the immobilised enzyme curves in Figure 13.20 as a straight line and evaluate the apparent values of vmax and Km, we can check whether or not we have found the true kinetic parameters by changing the particle radius and substrate concentration over a wide range of values. True enzyme kinetic parameters do not vary with these conditions that affect mass transfer into the catalyst. Therefore, if the slope and/or intercept changes with the Thiele modulus, as indicated in Figure 13.20 , it is soon evident that the system is subject to diffusional limitations. Studies have shown that the effects of diffusion are more pronounced in Eadie–Hofstee plots than in Lineweaver–Burk or Langmuir plots however, all three for immobilised enzymes can be approximated by straight lines over certain intervals.

Although the Lineweaver–Burk plots for immobilised β-galactosidase are nonlinear in Figure 13.20 , we should not conclude that the immobilised enzyme fails to obey Michaelis–Menten kinetics. The kinetic form of reactions is generally maintained after immobilisation of cells and enzymes [9] . The nonlinearity of the Lineweaver–Burk plots is due instead to the effect of mass transfer on the measured reaction rate.

Several methods have been proposed for determining vmax and Km in heterogeneous catalysts [40–43] . The most straightforward approach is experimental: it involves reducing the particle size and catalyst loading and increasing the external liquid velocity to eliminate all mass transfer resistances. The measured rate data can then be analysed for kinetic parameters as if the reaction were homogeneous. However, because it is usually very difficult or impossible to completely remove intraparticle mass transfer effects, procedures involving a series of experiments coupled with theoretical analysis have also been proposed. In these methods, rate data are collected at high and low substrate concentrations using different particle sizes. At high substrate levels, it is assumed that the reaction is zero-order with ηi=1 at low substrate concentrations, first-order kinetics are assumed. These assumptions simplify the analysis but may not always be valid.

When adequate computing facilities are available, true values of vmax and Km can be extracted from diffusion-limited data using iterative calculations based on numerical integration and nonlinear regression [7, 9] . Many iteration loops may be required before convergence to the final parameter values.


In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km? - Biology

Question : 21) Mixed type inhibitors will affect the ________ of an : 2008157

21) Mixed type inhibitors will affect the ________ of an enzymatic reaction.

22) The type of inhibitor that binds to the enzyme (E) but not to the enzyme-substrate (ES) complex is a(n) ________ inhibitor.

23) Which of the following is not true of the enzyme-substrate interaction?

A) Many enzymes are extremely specific regarding a substrate.

B) Many enzymes cannot recognize a stereoisomer of their substrate.

C) Some enzymes accept any of a whole group of substrates.

D) Carboxypeptidase recognizes any of the amino acids from the carboxyl end of a polypeptide.

E) Cells are often able to carry out metabolic activity with only a handful of enzymes.

A) was proposed by Hans Buchner.

B) involves a conformational change in the shape of the enzyme.

C) is also called the lock-and-key model.

D) states that enzyme-substrate interactions are rigid.

E) proposes that very strong covalent bonds are formed upon substrate binding.

25) Why is the Lineweaver—Burk plot important in enzyme kinetics?

A) It is a single-reciprocal plot.

B) It illustrates enzyme specificity.

C) It reveals the presence of prosthetic groups in enzymes.

D) It makes it easier to determine V max .

26) The Michaelis constant

A) can be determined using the Lineweaver—Burk plot.

B) is equal to twice the V max .

C) is equal to the substrate concentration at V max /2.

27) Which of the following does accurately describe the Lineweaver—Burk plot?

A) It is a double-reciprocal plot.

B) The y intercept is equal to 1/ V max .

C) The x intercept is -1/ K m .

E) Its slope is the same as the Eadie—Hofstee plot.

28) Which of the following variables is part of the Michaelis—Menten equation?

29) Saturation can be defined as

A) denaturation of an enzyme.

B) the inability to increase reaction velocity beyond a finite upper limit.

C) inhibition of enzyme function by blocking the active site.

D) the substrate concentration at which velocity reaches one-half maximum velocity.


In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km? - Biology

Our current chapter focused on the way in which cells are able to carry out the reactions necessary for life. We began with a discussion of the types of enzymes that you are likely to encounter on Test Day before reviewing thermodynamics and kinetics in relation to enzymes, which are biological catalysts. We went on to discuss the analysis of kinetic data with two different types of graphs, and talked about cooperativity. Because catalysts are generally most active in their native environment, we considered the impact of temperature, pH, and salinity on their activity. All of these are likely to appear on Test Day.

Enzymes need to be regulated we analyzed the basics of feedback mechanisms. We talked about inhibitors of enzymes, which may be reversible or irreversible. The difference between the types of reversible inhibition is a key Test Day concept. Finally, we discussed changes in enzyme activity that may include allosteric activation, covalent modification, or cleavage of inactive zymogens. Let's move on now to discuss the nonenzymatic functions of proteins. You will notice many parallels between the new material and the concepts described in this chapter, like binding affinity. By the end of the next chapter, you'll be ready to face any protein question the MCAT can throw at you!

Concept Summary

Enzymes as Biological Catalysts

·&emspEnzymes are biological catalysts that are unchanged by the reactions they catalyze and are reusable.

·&emspEach enzyme catalyzes a single reaction or type of reaction with high specificity.

o Oxidoreductases catalyze oxidation–reduction reactions that involve the transfer of electrons.

o Transferases move a functional group from one molecule to another molecule.

o Hydrolases catalyze cleavage with the addition of water.

o Lyases catalyze cleavage without the addition of water and without the transfer of electrons. The reverse reaction (synthesis) is often more important biologically.

o Isomerases catalyze the interconversion of isomers, including both constitutional isomers and stereoisomers.

o Ligases are responsible for joining two large biomolecules, often of the same type.

·&emspExergonic reactions release energy &DeltaG is negative.

·&emspEnzymes lower the activation energy necessary for biological reactions.

·&emspEnzymes do not alter the free energy (&DeltaG) or enthalpy (&DeltaH) change that accompanies the reaction nor the final equilibrium position rather, they change the rate (kinetics) at which equilibrium is reached.

Mechanisms of Enzyme Activity

·&emspEnzymes act by stabilizing the transition state, providing a favorable microenvironment, or bonding with the substrate molecules.

·&emspEnzymes have an active site, which is the site of catalysis.

·&emspBinding to the active site is explained by the lock and key theory or the induced fit model.

o The lock and key theory hypothesizes that the enzyme and substrate are exactly complementary.

o The induced fit model hypothesizes that the enzyme and substrate undergo conformational changes to interact fully.

·&emspSome enzymes require metal cation cofactors or small organic coenzymes to be active.

Enzyme Kinetics

·&emspEnzymes experience saturation kinetics: as substrate concentration increases, the reaction rate does as well until a maximum value is reached.

·&emspMichaelis–Menten and Lineweaver–Burk plots represent this relationship as a hyperbola and line, respectively.

·&emspEnzymes can be compared on the basis of their Km and vmax values.

·&emspCooperative enzymes display a sigmoidal curve because of the change in activity with substrate binding.

Effects of Local Conditions on Enzyme Activity

·&emspTemperature and pH affect an enzyme's activity in vivo changes in temperature and pH can result in denaturing of the enzyme and loss of activity due to loss of secondary, tertiary, or, if present, quaternary structure.

·&emspIn vitro, salinity can impact the action of enzymes.

Regulation of Enzyme Activity

·&emspEnzyme pathways are highly regulated and subject to inhibition and activation.

·&emspFeedback inhibition is a regulatory mechanism whereby the catalytic activity of an enzyme is inhibited by the presence of high levels of a product later in the same pathway.

·&emspReversible inhibition is characterized by the ability to replace the inhibitor with a compound of greater affinity or to remove it using mild laboratory treatment.

o Competitive inhibition results when the inhibitor is similar to the substrate and binds at the active site. Competitive inhibition can be overcome by adding more substrate. vmax is unchanged, Km increases.

o Noncompetitive inhibition results when the inhibitor binds with equal affinity to the enzyme and the enzyme–substrate complex. vmax is decreased, Km is unchanged.

o Mixed inhibition results when the inhibitor binds with unequal affinity to the enzyme and the enzyme–substrate complex. vmax is decreased, Km is increased or decreased depending on if the inhibitor has higher affinity for the enzyme or enzyme–substrate complex.

o Uncompetitive inhibition results when the inhibitor binds only with the enzyme–substrate complex. Km and vmax both decrease.

·&emspIrreversible inhibition alters the enzyme in such a way that the active site is unavailable for a prolonged duration or permanently new enzyme molecules must be synthesized for the reaction to occur again.

·&emspRegulatory enzymes can experience activation as well as inhibition.

o Allosteric sites can be occupied by activators which increase either affinity or enzymatic turnover.

o Phosphorylation (covalent modification with phosphate) or glycosylation (covalent modification with carbohydrate) can alter the activity or selectivity of enzymes.

o Zymogens are secreted in an inactive form and are activated by cleavage.

Answers to Concept Checks

1. Catalysts are characterized by two main properties: they reduce the activation energy of a reaction, thus speeding up the reaction, and they are not used up in the course of the reaction. Enzymes improve the environment in which a particular reaction takes place, which lowers its activation energy. They are also regenerated at the end of the reaction to their original form.

2. Enzyme specificity refers to the idea that a given enzyme will only catalyze a given reaction or type of reaction. For example, serine/threonine-specific protein kinases will only place a phosphate group onto the hydroxyl group of a serine or threonine residue.

Addition or synthesis reactions, generally between large molecules often require ATP

Rearrangement of bonds within a compound

Cleavage of a single molecule into two products, or synthesis of small organic molecules

Breaking of a compound into two molecules using the addition of water

Oxidoreductase

Oxidation–reduction reactions (transferring electrons)

Transferase

Movement of a functional group from one molecule to another

4. Enzymes have no effect on the overall thermodynamics of the reaction they have no effect on the &DeltaG or &DeltaH of the reaction, although they do lower the energy of the transition state, thus lowering the activation energy. However, enzymes have a profound effect on the kinetics of a reaction. By lowering activation energy, equilibrium can be achieved faster (although the equilibrium position does not change).

Lock and Key

Induced Fit

·&emspActive site of enzyme fits exactly around substrate

·&emspNo alterations to tertiary or quaternary structure of enzyme

·&emspActive site of enzyme molds itself around substrate only when substrate is present

·&emspTertiary and quaternary structure is modified for enzyme to function

2. Cofactors and coenzymes both act as activators of enzymes. Cofactors tend to be inorganic (minerals), while coenzymes tend to be small organic compounds (vitamins). In both cases, these regulators induce a conformational change in the enzyme that promotes its activity. Tightly bound cofactors or coenzymes that are necessary for enzyme function are termed prosthetic groups.

1. Increasing [S] has different effects, depending on how much substrate is present to begin with. When the substrate concentration is low, an increase in [S] causes a proportional increase in enzyme activity. At high [S], however, when the enzyme is saturated, increasing [S] has no effect on activity because vmax has already been attained.
Increasing [E] will always increase vmax, regardless of the starting concentration of enzyme.

2. Both the Michaelis–Menten and Lineweaver–Burk relationships account for the values of Km and vmax under various conditions. They both provide simple graphical interpretations of these two variables and are derived from the Michaelis–Menten equation. However, the axes of these graphs and visual representation of this information is different between the two. The Michaelis–Menten plot is v vs. [S], which creates a hyperbolic curve for monomeric enzymes. The Lineweaver–Burk plot, on the other hand, is vs. , which creates a straight line.

3. Km is a measure of an enzyme's affinity for its substrate, and is defined as the substrate concentration when an enzyme is functioning at half of its maximal velocity. As Km increases, an enzyme's affinity for its substrate decreases.

4. The x-intercept represents the y-intercept represents .

5. Cooperativity refers to the interactions between subunits in a multisubunit enzyme or protein. The binding of substrate to one subunit induces a change in the other subunits from the T (tense) state to the R (relaxed) state, which encourages binding of substrate to the other subunits. In the reverse direction, the unbinding of substrate from one subunit induces a change from R to T in the remaining subunits, promoting unbinding of substrate from the remaining subunits.

1. As temperature increases, enzyme activity generally increases (doubling approximately every 10°C). Above body temperature, however, enzyme activity quickly drops off as the enzyme denatures.
Enzymes are maximally active within a small pH range outside of this range, activity drops quickly with changes in pH as the ionization of the active site changes and the protein is denatured.
Changes in salinity can disrupt bonds within an enzyme, causing disruption of tertiary and quaternary structure, which leads to loss of enzyme function.

2. Ideal temperature: 37°C = 98.6°F = 310 K

Ideal pH for most enzymes is 7.4 for gastric enzymes, around 2 for pancreatic enzymes, around 8.5.

1. Feedback inhibition refers to the product of an enzymatic pathway turning off enzymes further back in that same pathway. This helps maintain homeostasis: as product levels rise, the pathway creating that product is appropriately downregulated.

2. The four types of inhibitors are: competitive, noncompetitive, mixed, and uncompetitive.

3. Irreversible inhibition refers to the prolonged or permanent inactivation of an enzyme, such that it cannot be easily renatured to gain function.

4. Examples of transient modifications include allosteric activation or inhibition. Examples of covalent modifications include phosphorylation and glycosylation.

5. Zymogens are precursors of an active enzyme. It is critical that certain enzymes (like the digestive enzymes of the pancreas) remain inactive until arriving at their target site.

Equations to Remember

(2.1) Michaelis–Menten rates:

(2.2) Michaelis–Menten equation:

Shared Concepts

·&emspBiochemistry Chapter 1

o Amino Acids, Peptides, and Proteins

·&emspBiochemistry Chapter 12

o Bioenergetics and Regulation of Metabolism

·&emspBiology Chapter 9

·&emspGeneral Chemistry Chapter 5

·&emspGeneral Chemistry Chapter 7

·&emspGeneral Chemistry Chapter 11

o Oxidation–Reduction Reactions

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In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km? - Biology

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movie update may 2021 May 2021 Movies: Wrath of Man • Here Today • Spiral: From the Book of Saw • A Quiet Place Part II • Cruella • The Human Factor • Profile, movies coming May. The following is a one-stop location for all the movies to look forward to in 2021 and when to expect them. Of course, as we all should be used to by now, dates are subject to change, so be sure check back for shifts in the schedule. Enjoy! recomended site : https://connect.informs.org/network/members/profile?UserKey=6c2c59de-3648-4063-a034-e8cff4bbdf1f https://connect.informs.org/network/members/profile?UserKey=16bb8ec2-a192-43cc-81a2-1ed377c2b649 https://connect.informs.org/network/members/profile?UserKey=1151c99f-2f95-46a8-bdd8-0410d6862cc3 https://connect.informs.org/network/members/profile?UserKey=912d4929-73b6-4c38-a151-45ed3703bd32 https://connect.informs.org/network/members/profile?UserKey=210346ea-1569-42c6-a085-bd4484bd0543 https://connect.informs.org/network/members/profile?UserKey=2b731e2d-6bff-429a-a244-4d4d0f85a618 https://connect.informs.org/network/members/profile?UserKey=f8ada4d3-cfbb-4551-a00d-4231a50cde4d https://connect.informs.org/network/members/profile?UserKey=822e0277-7aac-4646-849f-fcbf0cd667b6 https://connect.informs.org/network/members/profile?UserKey=e0429956-f03a-4e0d-9c38-94dcc92f476b https://connect.informs.org/network/members/profile?UserKey=89fc15a2-63dd-4b3a-bb0a-b362acec59e3 https://connect.informs.org/network/members/profile?UserKey=dcf3dcac-9188-4adb-ada9-f0b63c87fa12 https://connect.informs.org/network/members/profile?UserKey=a4178d4e-3975-4777-a938-b8976bf2c15e https://connect.informs.org/network/members/profile?UserKey=6cb9a3e8-1c92-4730-9337-e5a3d4f56b71 https://connect.informs.org/network/members/profile?UserKey=b4500502-207d-4fcc-82cd-2a5f00b8fddd https://connect.informs.org/network/members/profile?UserKey=2fb2227c-62cc-47de-917d-02db5508293d https://connect.informs.org/network/members/profile?UserKey=e29dc7e2-c81a-4796-af2a-1e2767bed12a https://connect.informs.org/network/members/profile?UserKey=e05b2e64-05b2-4c76-94ee-ca4698755a88 https://connect.informs.org/network/members/profile?UserKey=0ee0e91e-3fc1-4c25-8c27-feafc9f467db https://connect.informs.org/network/members/profile?UserKey=a7718e4e-807a-4d90-ad7f-b883c6bfdcd7 https://connect.informs.org/network/members/profile?UserKey=8a292769-6347-41be-b483-eb64919b8ca8 https://volunteer.alz.org/network/members/profile?UserKey=35cd0e7d-c38f-416c-931a-41224d55f72c https://vrip.unmsm.edu.pe/foro-vrip/profile/hvsdghds/ https://community.afpglobal.org/network/members/profile?UserKey=23a184ef-8bc7-4ef9-8c1e-73c80d467e65 https://community.afpglobal.org/network/members/profile?UserKey=96b868c3-999f-4e78-b7eb-6aafc567093d https://community.afpglobal.org/network/members/profile?UserKey=db7ac908-f911-47a4-bd10-ce0a595890fa https://community.afpglobal.org/network/members/profile?UserKey=7df33d7b-d8b1-46c2-a0bf-29d323c4fd9d https://connect.informs.org/network/members/profile?UserKey=b6baee9e-37a0-455b-9a9f-370f605f777b https://connect.informs.org/network/members/profile?UserKey=d891d4b4-ae89-40f1-9664-d40c95594c1a http://paste.jp/9bc0426d/ https://slexy.org/view/s21WvhFTV5 https://pastebin.com/iKX5BN9C https://pasteio.com/xlb9MOCQr7Cr https://paste.in/ei93EI https://jsfiddle.net/berlinm821/ojt20f7u/2/ https://pastelink.net/2ucpm https://paiza.io/projects/hB6o7e_Q6dMCcjYDZcUiww http://www.mpaste.com/p/sAaFRuS https://telegra.ph/movie-update-may-2021-04-27 https://p.ip.fi/M4fO https://penzu.com/p/c67089c1 https://onlinegdb.com/rJ5Ook8Pu https://justpaste.it/8drs7 https://dpaste.com/CRZMPUWJ4 https://paste.feed-the-beast.com/view/7313cf08 https://paste.ubuntu.com/p/GrYWgDhyxc/ https://paste.firnsy.com/paste/0sMc4O2HcWW https://0paste.com/245320 https://rentry.co/emthk https://friendpaste.com/7Tl1x5y76wYuFsnWDlnA9h https://paste.rs/dPc https://paste.laravel.io/8981a0c9-c46c-49c5-b1e8-d97b434c8204 https://www.mydigoo.com/forums-topicdetail-266611.html https://authors.curseforge.com/paste/ddbe8302 http://network-marketing.ning.com/profiles/blogs/movie-update-may-2021 https://www.unphp.net/decode/2e3e95933979537628df815efc68e2cf/ http://paste4btc.com/0nKEicI6 https://paste.artemix.org/-/9taQ4

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In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km? - Biology

Peter J. Kennelly, PhD & Victor W. Rodwell, PhD

After studying this chapter, you should be able to:

Describe the scope and overall purposes of the study of enzyme kinetics.

Indicate whether &DeltaG, the overall change in free energy for a reaction, is dependent on reaction mechanism.

Indicate whether &DeltaG is a function of the rates of reactions.

Explain the relationship between Keq, concentrations of substrates and products at equilibrium, and the ratio of the rate constants k1/k–1.

Outline how temperature and the concentration of hydrogen ions, enzyme, and substrate affect the rate of an enzyme-catalyzed reaction.

Indicate why laboratory measurement of the rate of an enzyme-catalyzed reaction typically employs initial rate conditions.

Describe the application of linear forms of the Michaelis–Menten equation to the determination of Km and Vmax.

Give one reason why a linear form of the Hill equation is used to evaluate the substrate-binding kinetics exhibited by some multimeric enzymes.

Contrast the effects of an increasing concentration of substrate on the kinetics of simple competitive and noncompetitive inhibition.

Describe the ways in which substrates add to, and products depart from, an enzyme that follows a ping–pong mechanism and do the same for an enzyme that follows a rapid-equilibrium mechanism.

Illustrate the utility of enzyme kinetics in ascertaining the mode of action of drugs.

BIOMEDICAL IMPORTANCE

Enzyme kinetics is the field of biochemistry concerned with the quantitative measurement of the rates of enzyme-catalyzed reactions and the systematic study of factors that affect these rates. Kinetic analysis can reveal the number and order of the individual steps by which enzymes transform substrates into products. Together with site-directed mutagenesis and other techniques that probe the protein structure, kinetic analyses can reveal details of the catalytic mechanism of a given enzyme.

A complete, balanced set of enzyme activities is of fundamental importance for maintaining homeostasis. An understanding of enzyme kinetics thus is important to understanding how physiologic stresses such as anoxia, metabolic acidosis or alkalosis, toxins, and pharmacologic agents affect that balance. The involvement of enzymes in virtually all physiologic processes makes them the targets of choice for drugs that cure or ameliorate human disease. Applied enzyme kinetics represents the principal tool by which scientists identify and characterize therapeutic agents that selectively inhibit the rates of specific enzyme-catalyzed processes. Enzyme kinetics thus plays a central and critical role in drug discovery and comparative pharmacodynamics, as well as in elucidating the mode of action of drugs.

CHEMICAL REACTIONS ARE DESCRIBED USING BALANCED EQUATIONS

A balanced chemical equation lists the initial chemical species (substrates) present and the new chemical species (products) formed for a particular chemical reaction, all in their correct proportions or stoichiometry. For example, balanced equation (1) describes the reaction of one molecule each of substrates A and B to form one molecule each of products P and Q:

The double arrows indicate reversibility, an intrinsic property of all chemical reactions. Thus, for reaction (1), if A and B can form P and Q, then P and Q can also form A and B. Designation of a particular reactant as a “substrate” or “product” is therefore somewhat arbitrary since the products for a reaction written in one direction are the substrates for the reverse reaction. The term “products” is, however, often used to designate the reactants whose formation is thermodynamically favored. Reactions for which thermodynamic factors strongly favor formation of the products to which the arrow points often are represented with a single arrow as if they were “irreversible”:

Unidirectional arrows are also used to describe reactions in living cells where the products of reaction (2) are immediately consumed by a subsequent enzyme-catalyzed reaction. The rapid removal of product P or Q therefore effectively precludes occurrence of the reverse reaction, rendering equation (2) functionally irreversible under physiologic conditions.

CHANGES IN FREE ENERGY DETERMINE THE DIRECTION & EQUILIBRIUM STATE OF CHEMICAL REACTIONS

The Gibbs free energy change &DeltaG (also called either free energy or Gibbs energy) describes both the direction in which a chemical reaction will tend to proceed and the concentrations of reactants and products that will be present at equilibrium. &DeltaG for a chemical reaction equals the sum of the free energies of formation of the reaction products &DeltaGp minus the sum of the free energies of formation of the substrates &DeltaGs. &DeltaG 0 denotes the change in free energy that accompanies transition from the standard state, one-molar concentrations of substrates and products, to equilibrium. A more useful biochemical term is &DeltaG 0 ’, which defines &DeltaG 0 at a standard state of 10 –7 M protons, pH 7.0 (Chapter 11). If the free energy of formation of the products is lower than that of the substrates, the signs of &DeltaG 0 and &DeltaG 0’ will be negative, indicating that the reaction as written is favored in the direction left to right. Such reactions are referred to as spontaneous. The sign and the magnitude of the free energy change determine how far the reaction will proceed. Equation (3) illustrates the relationship between the equilibrium constant Keq and &DeltaG 0 :

where R is the gas constant (1.98 cal/mol°K or 8.31 J/mol°K) and T is the absolute temperature in degrees Kelvin. Keq is equal to the product of the concentrations of the reaction products, each raised to the power of their stoichiometry, divided by the product of the substrates, each raised to the power of their stoichiometry:

For the reaction

&DeltaG 0 may be calculated from equation (3) if the molar concentrations of substrates and products present at equilibrium are known. If &DeltaG 0 is a negative number, Keq will be greater than unity, and the concentration of products at equilibrium will exceed that of the substrates. If &DeltaG 0 is positive, Keq will be less than unity, and the formation of substrates will be favored.

Note that, since &DeltaG 0 is a function exclusively of the initial and final states of the reacting species, it can provide information only about the direction and equilibrium state of the reaction. &DeltaG 0 is independent of the mechanism of the reaction and therefore provides no information concerning rates of reactions. Consequently&mdashand as explained below&mdashalthough a reaction may have a large negative &DeltaG 0 or &DeltaG 0 ’, it may nevertheless take place at a negligible rate.

THE RATES OF REACTIONS ARE DETERMINED BY THEIR ACTIVATION ENERGY

Reactions Proceed via Transition States

The concept of the transition state is fundamental to understanding the chemical and thermodynamic basis of catalysis. Equation (7) depicts a group transfer reaction in which an entering group E displaces a leaving group L, attached initially to R:

The net result of this process is to transfer group R from L to E. Midway through the displacement, the bond between R and L has weakened but has not yet been completely severed, and the new bond between E and R is yet incompletely formed. This transient intermediate&mdashin which neither free substrate nor product exists&mdashis termed the transition state, E···R···L. Dotted lines represent the “partial” bonds that are undergoing formation and rupture. Figure 8–1 provides a more detailed illustration of the transition state intermediate formed during the transfer of a phosphoryl group.

FIGURE 8–1 Formation of a transition state intermediate during a simple chemical reaction, Shown are three stages of a chemical reaction in which a phosphoryl group is transferred from leaving group L to entering group E. Top: entering group E (A) approaches the other reactant, L-phosphate (B). Notice how the three oxygen atoms linked by the triangular lines and the phosphorus atom of the phosphoryl group form a pyramid. Center: as E approaches L-phosphate, the new bond between E and the phosphate group begins to form (dotted line) as that linking L to the phosphate group weakens. These partially formed bonds are indicated by dotted lines. Bottom: formation of the new product, E-phosphate (P), is now complete as the leaving group L (Q) exits. Notice how the geometry of the phosphoryl group differs between the transition state and the substrate or product. Notice how the phosphorus and three oxygen atoms that occupy the four corners of a pyramid in the substrate and product become coplanar, as emphasized by the triangle, in the transition state.

Reaction (7) can be thought of as consisting of two “partial reactions,” the first corresponding to the formation (F) and the second to the subsequent decay (D) of the transition state intermediate. As for all reactions, characteristic changes in free energy, &DeltaGF and &DeltaGD are associated with each partial reaction:

For the overall reaction (10), &DeltaG is the sum of &DeltaGF and &DeltaGD. As for any equation of two terms, it is not possible to infer from &DeltaG either the sign or the magnitude of &DeltaGF or &DeltaGD.

Many reactions involve multiple transition states, each with an associated change in free energy. For these reactions, the overall &DeltaG represents the sum of all of the free energy changes associated with the formation and decay of all of the transition states. Therefore, it is not possible to infer from the overall &DeltaG the number or type of transition states through which the reaction proceeds. Stated another way, overall thermodynamics tells us nothing about kinetics.

&Delta GF Defines the Activation Energy

Regardless of the sign or magnitude of &DeltaG, &DeltaGF for the overwhelming majority of chemical reactions has a positive sign. The formation of transition state intermediates therefore requires surmounting energy barriers. For this reason, &DeltaGF for reaching a transition state is often termed the activation energy, Eact. The ease&mdashand hence the frequency&mdashwith which this barrier is overcome is inversely related to Eact. The thermodynamic parameters that determine how fast a reaction proceeds thus are the &DeltaGF values for formation of the transition states through which the reaction proceeds. For a simple reaction, where &prop means “proportionate to,”

The activation energy for the reaction proceeding in the opposite direction to that drawn is equal to –&DeltaGD.

NUMEROUS FACTORS AFFECT THE REACTION RATE

The kinetic theory&mdashalso called the collision theory&mdashof chemical kinetics states that for two molecules to react they (1) must approach within bond-forming distance of one another, or “collide,” and (2) must possess sufficient kinetic energy to overcome the energy barrier for reaching the transition state. It therefore follows that anything that increases the frequency or energy of collision between substrates will increase the rate of the reaction in which they participate.

Temperature

Raising the temperature increases the kinetic energy of molecules. As illustrated in Figure 8–2, the total number of molecules whose kinetic energy exceeds the energy barrier Eact (vertical bar) for formation of products increases from low (A) through intermediate (B) to high (C) temperatures. Increasing the kinetic energy of molecules also increases their rapidity of motion and therefore the frequency with which they collide. This combination of more frequent and more highly energetic, and hence productive, collisions increases the reaction rate.

FIGURE 8–2 The energy barrier for chemical reactions. (See text for discussion.)

Reactant Concentration

The frequency with which molecules collide is directly proportionate to their concentrations. For two different molecules A and B, the frequency with which they collide will double if the concentration of either A or B is doubled. If the concentrations of both A and B are doubled, the probability of collision will increase fourfold.

For a chemical reaction proceeding at constant temperature that involves one molecule each of A and B,

the number of the molecules that possess kinetic energy sufficient to overcome the activation energy barrier will be a constant. The number of collisions with sufficient energy to produce product P therefore will be directly proportionate to the number of collisions between A and B, and thus to their molar concentrations, denoted by the square brackets:

Similarly, for the reaction represented by

which can also be written as

The corresponding rate expression is

For the general case, when n molecules of A react with m molecules of B,

Replacing the proportionality sign with an equals sign by introducing a rate constant k characteristic of the reaction under study gives equations (20) and (21), in which the subscripts 1 and –1 refer to the forward and reverse reactions, respectively:

The sum of the molar ratios of the reactants defines the kinetic order of the reaction. Consider reaction (5). The stoichiometric coefficient for the sole reactant, A, is 2. Therefore, the rate of production of P is proportional to the square of [A] and the reaction is said to be second order with respect to reactant A. In this instance, the overall reaction is also second order. Therefore, k1 is referred to as a second-order rate constant.

Reaction (12) describes a simple second-order reaction between two different reactants, A and B. The stoichiometric coefficient for each reactant is 1. Therefore, while the overall order of the reaction is 2, it is said to be first order with respect to A and first order with respect to B. In the laboratory, the kinetic order of a reaction with respect to a particular reactant, referred to as the variable reactant or substrate, can be determined by maintaining the concentration of the other reactants at a constant, or fixed, concentration in large excess over the variable reactant. Under these pseudo-first-order conditions, the concentration of the fixed reactant(s) remains virtually constant. Thus, the rate of reaction will depend exclusively on the concentration of the variable reactant, sometimes also called the limiting reactant. The concepts of reaction order and pseudo-first-order conditions apply not only to simple chemical reactions but also to enzyme-catalyzed reactions.

Keq Is a Ratio of Rate Constants

While all chemical reactions are to some extent reversible, at equilibrium the overall concentrations of reactants and products remain constant. At equilibrium, the rate of conversion of substrates to products therefore equals the rate at which products are converted to substrates:

The ratio of k1 to k–1 is termed the equilibrium constant, Keq. The following important properties of a system at equilibrium must be kept in mind.

1. The equilibrium constant is a ratio of the reaction rate constants (not the reaction rates).

2. At equilibrium, the reaction rates (not the rate constants) of the forward and back reactions are equal.

3. Equilibrium is a dynamic state. Although there is no net change in the concentration of substrates or products, individual substrate and product molecules are continually being interconverted.

4. The numeric value of the equilibrium constant Keq can be calculated either from the concentrations of substrates and products at equilibrium or from the ratio k1/k–1.

THE KINETICS OF ENZYMATIC CATALYSIS

Enzymes Lower the Activation Energy Barrier for a Reaction

All enzymes accelerate reaction rates by lowering &DeltaGF for the formation of transition states. However, they may differ in the way this is achieved. Where the mechanism or the sequence of chemical steps at the active site is essentially equivalent to those for the same reaction proceeding in the absence of a catalyst, the environment of the active site lowers &DeltaGF by stabilizing the transition state intermediates. To put it another way, the enzyme can be envisioned as binding to the transition state intermediate (Figure 8–1) more tightly than it does to either substrates or products. As discussed in Chapter 7, stabilization can involve (1) acid–base groups suitably positioned to transfer protons to or from the developing transition state intermediate, (2) suitably positioned charged groups or metal ions that stabilize developing charges, or (3) the imposition of steric strain on substrates so that their geometry approaches that of the transition state. HIV protease (see Figure 7–6) illustrates catalysis by an enzyme that lowers the activation barrier by stabilizing a transition state intermediate.

Catalysis by enzymes that proceeds via a unique reaction mechanism typically occurs when the transition state intermediate forms a covalent bond with the enzyme (covalent catalysis). The catalytic mechanism of the serine protease chy-motrypsin (see Figure 7–7) illustrates how an enzyme utilizes covalent catalysis to provide a unique reaction pathway.

ENZYMES DO NOT AFFECTKeq

While enzymes undergo transient modifications during the process of catalysis, they always emerge unchanged at the completion of the reaction. The presence of an enzyme therefore has no effect on &DeltaG 0 for the overallreaction, which is a function solely of the initial and final states of the reactants. Equation (25) shows the relationship between the equilibrium constant for a reaction and the standard free energy change for that reaction:

This principle is perhaps most readily illustrated by including the presence of the enzyme (Enz) in the calculation of the equilibrium constant for an enzyme-catalyzed reaction:

Since the enzyme on both sides of the double arrows is present in equal quantity and identical form, the expression for the equilibrium constant,

reduces to one identical to that for the reaction in the absence of the enzyme:

Enzymes therefore have no effect on Keq.

MULTIPLE FACTORS AFFECT THE RATES OF ENZYME-CATALYZED REACTIONS

Temperature

Raising the temperature increases the rate of both uncatalyzed and enzyme-catalyzed reactions by increasing the kinetic energy and the collision frequency of the reacting molecules. However, heat energy can also increase the kinetic energy of the enzyme to a point that exceeds the energy barrier for disrupting the noncovalent interactions that maintain its three-dimensional structure. The polypeptide chain then begins to unfold, or denature, with an accompanying loss of the catalytic activity. The temperature range over which an enzyme maintains a stable, catalytically competent conformation depends upon&mdashand typically moderately exceeds&mdashthe normal temperature of the cells in which it resides. Enzymes from humans generally exhibit stability at temperatures up to 45–55°C. By contrast, enzymes from the thermophilic microorganisms that reside in volcanic hot springs or undersea hydrothermal vents may be stable at temperatures up to or even above 100°C.

The temperature coefficient (Q10) is the factor by which the rate of a biologic process increases for a 10°C increase in temperature. For the temperatures over which enzymes are stable, the rates of most biological processes typically double for a 10°C rise in temperature . Changes in the rates of enzyme-catalyzed reactions that accompany a rise or fall in body temperature constitute a prominent survival feature for “cold-blooded” life forms such as lizards or fish, whose body temperatures are dictated by the external environment. However, for mammals and other homeothermic organisms, changes in enzyme reaction rates with temperature assume physiologic importance only in circumstances such as fever or hypothermia.

Hydrogen Ion Concentration

The rate of almost all enzyme-catalyzed reactions exhibits a significant dependence on hydrogen ion concentration. Most intracellular enzymes exhibit optimal activity at pH values between 5 and 9. The relationship of activity to hydrogen ion concentration (Figure 8–3) reflects the balance between enzyme denaturation at high or low pH and effects on the charged state of the enzyme, the substrates, or both. For enzymes whose mechanism involves acid–base catalysis, the residues involved must be in the appropriate state of protonation for the reaction to proceed. The binding and recognition of substrate molecules with dissociable groups also typically involves the formation of salt bridges with the enzyme. The most common charged groups are carboxylate groups (negative) and protonated amines (positive). Gain or loss of critical charged groups adversely affects substrate binding and thus will retard or abolish catalysis.

FIGURE 8–3 Effect of pH on enzyme activity. Consider, for example, a negatively charged enzyme (E – ) that binds a positively charged substrate (SH+). Shown is the proportion (%) of SH+ [] and of E – [///] as a function of pH. Only in the cross-hatched area do both the enzyme and the substrate bear an appropriate charge.

ASSAYS OF ENZYME-CATALYZED REACTIONS TYPICALLY MEASURE THE INITIAL VELOCITY

Most measurements of the rates of enzyme-catalyzed reactions employ relatively short time periods, conditions that approximate initial rate conditions. Under these conditions, only traces of product accumulate, rendering the rate of the reverse reaction negligible. The initial velocity (vi) of the reaction thus is essentially that of the rate of the forward reaction. Assays of enzyme activity almost always employ a large (10 3 –10 7 ) molar excess of substrate over enzyme. Under these conditions, v 1 . is proportionate to the concentration of enzyme. Measuring the initial velocity therefore permits one to estimate the quantity of enzyme present in a biologic sample.

SUBSTRATE CONCENTRATION AFFECTS THE REACTION RATE

In what follows, enzyme reactions are treated as if they had only a single substrate and a single product. For enzymes with multiple substrates, the principles discussed below apply with equal validity. Moreover, by employing pseudo-first-order conditions (see above), scientists can study the dependence of reaction rate upon an individual reactant through the appropriate choice of fixed and variable substrates. In other words, under pseudo-first-order conditions the behavior of a multisubstrate enzyme will imitate one having a single substrate. In this instance, however, the observed rate constant will be a function of the rate constant k1 for the reaction as well as the concentration of the fixed substrate(s).

For a typical enzyme, as substrate concentration is increased, vi. increases until it reaches a maximum value Vmax (Figure 8–4). When further increases in substrate concentration do not further increase vi, the enzyme is said to be “saturated” with the substrate. Note that the shape of the curve that relates activity to substrate concentration (Figure 8–4) is hyperbolic. At any given instant, only substrate molecules that are combined with the enzyme as an enzyme-substrate (ES) complex can be transformed into a product. Since the equilibrium constant for the formation of the enzyme-substrate complex is not infinitely large, only a fraction of the enzyme may be present as an ES complex even when the substrate is present in excess (points A and B of Figure 8–5). At points A or B, increasing or decreasing [S] therefore will increase or decrease the number of ES complexes with a corresponding change in vi. At point C (Figure 8–5), however, essentially all the enzyme is present as the ES complex. Since no free enzyme remains available for forming ES, further increases in [S] cannot increase the rate of the reaction. Under these saturating conditions, vi. depends solely on&mdashand thus is limited by&mdashthe rapidity with which product dissociates from the enzyme so that it may combine with more substrate.

FIGURE 8–4 Effect of substrate concentration on the initial velocity of an enzyme-catalyzed reaction.

FIGURE 8–5 Representation of an enzyme in the presence of a concentration of substrate that is below Km (A), at a concentration equal to Km (B), and at a concentration well above Km(C). Points A, B, and C correspond to those points in Figure 8–4.

THE MICHAELIS–MENTEN & HILL EQUATIONS MODEL THE EFFECTS OF SUBSTRATE CONCENTRATION

The Michaelis–Menten Equation

The Michaelis–Menten equation (29) illustrates in mathematical terms the relationship between initial reaction velocity vi and substrate concentration [S], shown graphically in Figure 8–4:

The Michaelis constant Km is the substrate concentration at which vi is half the maximal velocity (Vmax/2) attainable at a particular concentration of the enzyme. Km thus has the dimensions of substrate concentration. The dependence of initial reaction velocity on [S] and Km may be illustrated by evaluating the Michaelis–Menten equation under three conditions.

1. When [S] is much less than Km (point A in Figures 8-4 and 8-5), the term Km + [S] is essentially equal to Km. Replacing Km + [S] with Km reduces equation (29) to

where ≈ means “approximately equal to.” Since Vmax and Km are both constants, their ratio is a constant. In other words, when [S] is considerably below Km, vi is proportionate to k[S]. The initial reaction velocity therefore is directly proportional to [S].

2. When [S] is much greater than Km (point C in Figures 8–4 and 8–5), the term Km + [S] is essentially equal to [S]. Replacing Km + [S] with [S] reduces equation (29) to

Thus, when [S] greatly exceeds Km, the reaction velocity is maximal (Vmax) and unaffected by further increases in the substrate concentration.

3. When (point B in Figures 8–4 and 8–5):

Equation (32) states that when [S] equals Km, the initial velocity is half-maximal. Equation (32) also reveals that Km is&mdashand may be determined experimentally from&mdashthe substrate concentration at which the initial velocity is half-maximal.

A Linear Form of the Michaelis–Menten Equation Is Used to Determine Km & Vmax

The direct measurement of the numeric value of Vmax, and therefore the calculation of Km, often requires impractically high concentrations of substrate to achieve saturating conditions. A linear form of the Michaelis–Menten equation circumvents this difficulty and permits Vmax and Km to be extrapolated from initial velocity data obtained at less than saturating concentrations of the substrate. Start with equation (29),

Equation (35) is the equation for a straight line, , where and . A plot of 1/vi. as y as a function of 1/[S] as x therefore gives a straight line whose y intercept is 1/Vmax and whose slope is Km/Vmax. Such a plot is called a double reciprocal or Lineweaver–Burk plot (Figure 8–6). Setting the y term of equation (36) equal to zero and solving for x reveals that the x intercept is –1/Km:

FIGURE 8–6 Double-reciprocal or Lineweaver–Burk plot of 1/v1 versus 1/[S] used to evaluate Km and Vmax.

Km is thus most readily calculated from the negative x intercept.

The greatest virtue of the Lineweaver–Burk plot resides in the facility with which it can be used to determine the kinetic mechanisms of enzyme inhibitors (see below). However, in using a double-reciprocal plot to determine kinetic constants it is important to avoid the introduction of bias through the clustering of data at low values of 1/[S]. To avoid this bias, prepare a solution of substrate whose dilution into an assay will produce the maximum desired concentration of the substrate. Now use the same volume of solutions prepared by diluting the stock solution by factors of 1:2, 1:3, 1:4, 1:5, etc. The data will then fall on the 1/[S] axis at intervals of 1, 2, 3, 4, 5, etc. Alternatively, a single-reciprocal plot such as the Eadie–Hofstee (vi versus vi/[S]) or Hanes–Woolf ([S]/vi versus [S]) plot can be used to minimize clustering.

The Catalytic Constant, kcat

Several parameters may be used to compare the relative activity of different enzymes or of different preparations of the same enzyme. The activity of impure enzyme preparations typically is expressed as a specific activity (Vmaxdivided by the protein concentration). For a homogeneous enzyme, one may calculate its turnover number (Vmax divided by the moles of enzyme present). But if the number of active sites present is known, the catalytic activity of a homogeneous enzyme is best expressed as its catalytic constant, kcat (Vmax divided by the number of active sites, St):

Since the units of concentration cancel out, the units of kcat are reciprocal time.

By what measure should the efficiency of different enzymes, different substrates for a given enzyme, and the efficiency with which an enzyme catalyzes a reaction in the forward and reverse directions be quantified and compared? While the maximum capacity of a given enzyme to convert substrate to product is important, the benefits of a high kcat can only be realized if Km is sufficiently low. Thus, catalytic efficiency of enzymes is best expressed in terms of the ratio of these two kinetic constants, kcat/Km.

For certain enzymes, once substrate binds to the active site, it is converted to product and released so rapidly as to render these events effectively instantaneous. For these exceptionally efficient catalysts, the rate-limiting step in catalysis is the formation of the ES complex. Such enzymes are said to be diffusion-limited, or catalytically perfect, since the fastest possible rate of catalysis is determined by the rate at which molecules move or diffuse through the solution. Examples of enzymes for which kcat/Km approaches the diffusion limit of 10 8 –10 9 M –1 s –1 include triosephosphate isomerase, carbonic anhydrase, acetylcholinesterase, and adenosine deaminase.

In living cells, the assembly of enzymes that catalyze successive reactions into multimeric complexes can circumvent the limitations imposed by diffusion. The geometric relationships of the enzymes in these complexes are such that the substrates and products do not diffuse into the bulk solution until the last step in the sequence of catalytic steps is complete. Fatty acid synthetase extends this concept one step further by covalently attaching the growing substrate fatty acid chain to a biotin tether that rotates from active site to active site within the complex until synthesis of a palmitic acid molecule is complete (Chapter 23).

Km May Approximate a Binding Constant

The affinity of an enzyme for its substrate is the inverse of the dissociation constant Kd for dissociation of the enzyme-substrate complex ES:

Stated another way, the smaller the tendency of the enzyme and its substrate to dissociate, the greater the affinity of the enzyme for its substrate. While the Michaelis constant Km often approximates the dissociation constant Kd, this is by no means always the case. For a typical enzyme-catalyzed reaction:

The value of [S] that gives is

When , then

Hence, 1/Km only approximates 1/Kd under conditions where the association and dissociation of the ES complex are rapid relative to catalysis. For the many enzyme-catalyzed reactions for which k–1 + k2 is not approximately equal to k–1, 1/Km will underestimate 1 /Kd.

The Hill Equation Describes the Behavior of Enzymes That Exhibit Cooperative Binding of Substrate

While most enzymes display the simple saturation kinetics depicted in Figure 8–4 and are adequately described by the Michaelis–Menten expression, some enzymes bind their substrates in a cooperative fashion analogous to the binding of oxygen by hemoglobin (Chapter 6). Cooperative behavior is an exclusive property of multimeric enzymes that bind substrate at multiple sites.

For enzymes that display positive cooperativity in binding the substrate, the shape of the curve that relates changes in vi to changes in [S] is sigmoidal (Figure 8–7). Neither the Michaelis–Menten expression nor its derived plots can be used to evaluate cooperative kinetics. Enzymologists therefore employ a graphic representation of the Hill equation originally derived to describe the cooperative binding of O2 by hemoglobin. Equation (44) represents the Hill equation arranged in a form that predicts a straight line, where k’ is a complex constant:

FIGURE 8–7 Representation of sigmoid substrate saturation kinetics.

Equation (44) states that when [S] is low relative to k’, the initial reaction velocity increases as the nth power of [S].

A graph of log vi/(Vmaxvi) versus log[S] gives a straight line (Figure 8–8), where the slope of the line n is the Hill coefficient, an empirical parameter whose value is a function of the number, kind, and strength of the interactions of the multiple substrate-binding sites on the enzyme. When , all binding sites behave independently and simple Michaelis–Menten kinetic behavior is observed. If n is greater than 1, the enzyme is said to exhibit positive cooperativity. Binding of substrate to one site then enhances the affinity of the remaining sites to bind additional substrate. The greater the value for n, the higher the degree of cooperativity and the more markedly sigmoidal will be the plot of vi. versus [S]. A perpendicular dropped from the point where the y term log vi/(Vmaxvi) is zero intersects the x-axis at a substrate concentration termed S50, the substrate concentration that results in half-maximal velocity. S50 thus is analogous to the P50 for oxygen binding to hemoglobin (Chapter 6).

FIGURE 8–8 A graphical representation of a linear form of the Hill equation is used to evaluate S50, the substrate concentration that produces half-maximal velocity, and the degree of cooperativity n.

KINETIC ANALYSIS DISTINGUISHES COMPETITIVE FROM NONCOMPETITIVE INHIBITION

Inhibitors of the catalytic activities of enzymes provide both pharmacologic agents and research tools for the study of the mechanism of enzyme action. The strength of the interaction between an inhibitor and an enzyme depends on forces important in protein structure and ligand binding (hydrogen bonds, electrostatic interactions, hydrophobic interactions, and van der Waals forces see Chapter 5). Inhibitors can be classified on the basis of their site of action on the enzyme, on whether they chemically modify the enzyme, or on the kinetic parameters they influence. Compounds that mimic the transition state of an enzyme-catalyzed reaction (transition state analogs) or that take advantage of the catalytic machinery of an enzyme (mechanism-based inhibitors) can be particularly potent inhibitors. Kinetically, we distinguish two classes of inhibitors based upon whether raising the substrate concentration does or does not overcome the inhibition.

Competitive Inhibitors Typically Resemble Substrates

The effects of competitive inhibitors can be overcome by raising the concentration of substrate. Most frequently, in competitive inhibition the inhibitor (I) binds to the substrate-binding portion of the active site thereby blocking access by the substrate. The structures of most classic competitive inhibitors therefore tend to resemble the structures of a substrate, and thus are termed substrate analogs. Inhibition of the enzyme succinate dehydrogenase by malonate illustrates competitive inhibition by a substrate analog. Succinate dehydrogenase catalyzes the removal of one hydrogen atom from each of the two-methyl-ene carbons of succinate (Figure 8–9). Both succinate and its structural analog malonate ( – OOC&mdashCH2&mdashCOO – ) can bind to the active site of succinate dehydrogenase, forming an ES or an EI complex, respectively. However, since malonate contains only one methylene carbon, it cannot undergo dehydrogenation.

FIGURE 8–9 The succinate dehydrogenase reaction.

The formation and dissociation of the EI complex is a dynamic process described by

for which the equilibrium constant Ki is

In effect, a competitive inhibitor acts by decreasing the number of free enzyme molecules available to bind substrate, ie, to form ES, and thus eventually to form product, as described below.

A competitive inhibitor and substrate exert reciprocal effects on the concentration of the EI and ES complexes. Since the formation of ES complexes removes free enzyme available to combine with the inhibitor, increasing [S] decreases the concentration of the EI complex and raises the reaction velocity. The extent to which [S] must be increased to completely overcome the inhibition depends upon the concentration of the inhibitor present, its affinity for the enzyme, Ki, and the affinity, Km, of the enzyme for its substrate.

Double-Reciprocal Plots Facilitate the Evaluation of Inhibitors

Double-reciprocal plots distinguish between competitive and noncompetitive inhibitors and simplify evaluation of inhibition constants. vi is determined at several substrate concentrations both in the presence and in the absence of the inhibitor. For classic competitive inhibition, the lines that connect the experimental data points converge at the y-axis (Figure 8–10). Since the y intercept is equal to 1/Vmax, this pattern indicates that when 1/[S] approaches 0, viis independent of the presence of inhibitor. Note, however, that the intercept on the x-axis does vary with inhibitor concentration&mdashand that since –1/Km is smaller than 1/Km, Km (the “apparent Km”) becomes larger in the presence of increasing concentrations of the inhibitor. Thus, a competitive inhibitor has no effect on Vmax but raises Km, the apparent Km for the substrate. For a simple competitive inhibition, the intercept on the x-axis is

FIGURE 8–10 Lineweaver–Burk plot of simple competitive inhibition. Note the complete relief of inhibition at high [S] (ie, low 1/[S]).

Once Km has been determined in the absence of inhibitor, Ki can be calculated from equation (47). Ki values are used to compare different inhibitors of the same enzyme. The lower the value for Ki, the more effective the inhibitor. For example, the statin drugs that act as competitive inhibitors of HMG-CoA reductase (Chapter 26) have Ki values several orders of magnitude lower than the Km for the substrate HMG-CoA.

Simple Noncompetitive Inhibitors Lower Vmax But Do Not Affect Km

In strict noncompetitive inhibition, binding of the inhibitor does not affect binding of the substrate. Formation of both EI and EIS complexes is therefore possible. However, while the enzyme-inhibitor complex can still bind the substrate, its efficiency at transforming substrate to product, reflected by Vmax, is decreased. Noncompetitive inhibitors bind enzymes at sites distinct from the substrate-binding site and generally bear little or no structural resemblance to the substrate.

For simple noncompetitive inhibition, E and EI possess identical affinity for the substrate, and the EIS complex generates product at a negligible rate (Figure 8–11). More complex noncompetitive inhibition occurs when binding of the inhibitor does affect the apparent affinity of the enzyme for the substrate, causing the lines to intercept in either the third or fourth quadrants of a double-reciprocal plot (not shown). While certain inhibitors exhibit characteristics of a mixture of competitive and noncompetitive inhibition, the evaluation of these inhibitors exceeds the scope of this chapter.

FIGURE 8–11 Lineweaver–Burk plot for simple noncompetitive inhibition.

A Dixon plot is sometimes employed as an alternative to the Lineweaver–Burk plot for determining inhibition constants. The initial velocity (vi) is measured at several concentrations of inhibitor, but at a fixed concentration of the substrate (S). For a simple competitive or noncompetitive inhibitor, a plot of 1/vi versus inhibitor concentration [I] yields a straight line. The experiment is repeated at different fixed concentrations of the substrate. The resulting set of lines intersects to the left of the y-axis. For competitive inhibition, a perpendicular dropped to the x-axis from the point of intersection of the lines gives –Ki (Figure 8–12, top). For noncompetitive inhibition the intercept on the x-axis is –Ki (Figure 8–12, bottom). Pharmaceutical publications frequently employ Dixon plots to illustrate the comparative potency of competitive inhibitors.

FIGURE 8–12 Applications of Dixon plots. Top: competitive inhibition, estimation of Ki. Bottom: noncompetitive inhibition, estimation of Ki.

A less rigorous alternative to Ki as a measure of inhibitory potency is the concentration of inhibitor that produces 50% inhibition, IC50. Unlike the equilibrium dissociation constant Ki, the numeric value of IC50 varies as a function of the specific circumstances of substrate concentration, etc, under which it is determined.

Tightly Bound Inhibitors

Some inhibitors bind to enzymes with such high affinity, M, that the concentration of inhibitor required to measure Ki falls below the concentration of enzyme typically present in an assay. Under these circumstances, a significant fraction of the total inhibitor may be present as an EI complex. If so, this violates the assumption, implicit in classical steady-state kinetics, that the concentration of free inhibitor is independent of the concentration of enzyme. The kinetic analysis of these tightly bound inhibitors requires specialized kinetic equations that incorporate the concentration of enzyme to estimate Ki or IC50 and to distinguish competitive from noncompetitive tightly bound inhibitors.

Irreversible Inhibitors “Poison” Enzymes

In the above examples, the inhibitors form a dissociable, dynamic complex with the enzyme. Fully active enzyme can therefore be recovered simply by removing the inhibitor from the surrounding medium. However, a variety of other inhibitors act irreversibly by chemically modifying the enzyme. These modifications generally involve making or breaking covalent bonds with aminoacyl residues essential for substrate binding, catalysis, or maintenance of the enzyme’s functional conformation. Since these covalent changes are relatively stable, an enzyme that has been “poisoned” by an irreversible inhibitor such as a heavy metal atom or an acylating reagent remains inhibited even after the removal of the remaining inhibitor from the surrounding medium.

Mechanism-Based Inhibition

“Mechanism-based” or “suicide” inhibitors are specialized substrate analogs that contain a chemical group that can be transformed by the catalytic machinery of the target enzyme. After binding to the active site, catalysis by the enzyme generates a highly reactive group that forms a covalent bond to, and blocks the function of a catalytically essential residue. The specificity and persistence of suicide inhibitors, which are both enzyme-specific and unreactive outside the confines of the enzyme’s active site, render them promising leads for the development of enzyme-specific drugs. The kinetic analysis of suicide inhibitors lies beyond the scope of this chapter. Neither the Lineweaver–Burk nor the Dixon approach is applicable since suicide inhibitors violate a key boundary condition common to both approaches, namely that the activity of the enzyme does not decrease during the course of the assay.

MOST ENZYME-CATALYZED REACTIONS INVOLVE TWO OR MORE SUBSTRATES

While several enzymes have a single substrate, many others have two&mdashand sometimes more&mdashsubstrates and products. The fundamental principles discussed above, while illustrated for single-substrate enzymes, apply also to multisubstrate enzymes. The mathematical expressions used to evaluate multisubstrate reactions are, however, complex. While a detailed analysis of the full range of multisubstrate reactions exceeds the scope of this chapter, some common types of kinetic behavior for two-substrate, two-product reactions (termed “BiBi” reactions) are considered below.

Sequential or Single-Displacement Reactions

In sequential reactions, both substrates must combine with the enzyme to form a ternary complex before catalysis can proceed (Figure 8–13, top). Sequential reactions are sometimes referred to as single-displacement reactions because the group undergoing transfer is usually passed directly, in a single step, from one substrate to the other. Sequential Bi–Bi reactions can be further distinguished on the basis of whether the two substrates add in a random or in a compulsory order. For random-order reactions, either substrate A or substrate B may combine first with the enzyme to form an EA or an EB complex (Figure 8–13, center). For compulsory-order reactions, A must first combine with E before B can combine with the EA complex. One explanation for why some enzymes employ compulsory-order mechanisms can be found in Koshland’s induced fit hypothesis: the addition of A induces a conformational change in the enzyme that aligns residues that recognize and bind B.

FIGURE 8–13 Representations of three classes of Bi–Bi reaction mechanisms. Horizontal lines represent the enzyme. Arrows indicate the addition of substrates and departure of products. Top: an ordered Bi–Bi reaction, characteristic of many NAD(P)H-dependent oxidoreductases. Center: a random Bi–Bi reaction, characteristic of many kinases and some dehydrogenases. Bottom: a ping–pong reaction, characteristic of aminotransferases and serine proteases.

Ping–Pong Reactions

The term “ping–pong” applies to mechanisms in which one or more products are released from the enzyme before all the substrates have been added. Ping–pong reactions involve covalent catalysis and a transient, modified form of the enzyme (see Figure 7–4). Ping–pong Bi–Bi reactions are often referred to as double displacement reactions. The group undergoing transfer is first displaced from substrate A by the enzyme to form product P and a modified form of the enzyme (F). The subsequent group transfer from F to the second substrate B, forming product Q and regenerating E, constitutes the second displacement (Figure 8–13, bottom).

Most Bi–Bi Reactions Conform to Michaelis–Menten Kinetics

Most Bi–Bi reactions conform to a somewhat more complex form of Michaelis–Menten kinetics in which Vmax refers to the reaction rate attained when both substrates are present at saturating levels. Each substrate has its own characteristic Km value, which corresponds to the concentration that yields half-maximal velocity when the second substrate is present at saturating levels. As for single-substrate reactions, double-reciprocal plots can be used to determine Vmax and Km. vi is measured as a function of the concentration of one substrate (the variable substrate) while the concentration of the other substrate (the fixed substrate) is maintained constant. If the lines obtained for several fixed-substrate concentrations are plotted on the same graph, it is possible to distinguish a ping–pong mechanism, which yields parallel lines (Figure 8–14), from a sequential mechanism, which yields a pattern of intersecting lines (not shown).

FIGURE 8–14 Lineweaver–Burk plot for a two-substrate ping–pong reaction. An increase in concentration of one substrate (S1) while that of the other substrate (S2) is maintained constant for changes both the x and yintercepts, but not the slope.

Product inhibition studies are used to complement kinetic analyses and to distinguish between ordered and random Bi–Bi reactions. For example, in a random-order Bi–Bi reaction, each product will act as a competitive inhibitor in the absence of its coproducts regardless of which substrate is designated the variable substrate. However, for a sequential mechanism (Figure 8–13, top), only product Q will give the pattern indicative of competitive inhibition when A is the variable substrate, while only product P will produce this pattern with B as the variable substrate. The other combinations of product inhibitor and variable substrate will produce forms of complex noncompetitive inhibition.

KNOWLEDGE OF ENZYME KINETICS, MECHANISM, AND INHIBITION AIDS DRUG DEVELOPMENT

Many Drugs Act as Enzyme Inhibitors

The goal of pharmacology is to identify agents that can

1. Destroy or impair the growth, invasiveness, or development of invading pathogens.

2. Stimulate endogenous defense mechanisms.

3. Halt or impede aberrant molecular processes triggered by genetic, environmental, or biologic stimuli with minimal perturbation of the host’s normal cellular functions.

By virtue of their diverse physiologic roles and high degree of substrate selectivity, enzymes constitute natural targets for the development of pharmacologic agents that are both potent and specific. Statin drugs, for example, lower cholesterol production by inhibiting 3-hydroxy-3-methylglutaryl coenzyme A reductase (Chapter 26), while emtricitabine and tenofovir disoproxil fumarate block replication of the human immunodeficiency virus by inhibiting the viral reverse transcriptase (Chapter 34). Pharmacologic treatment of hypertension often includes the administration of an inhibitor of angiotensin-converting enzyme, thus lowering the level of angiotensin II, a vasoconstrictor (Chapter 42).

Enzyme Kinetics Defines Appropriate Screening Conditions

Enzyme kinetics plays a crucial role in drug discovery. Knowledge of the kinetic behavior of the enzyme of interest is necessary, first and foremost, to select appropriate assay conditions for detecting the presence of an inhibitor. The concentration of substrate, for example, must be adjusted such that sufficient product is generated to permit facile detection of the enzyme’s activity without being so high that it masks the presence of an inhibitor. Second, enzyme kinetics provides the means for quantifying and comparing the potency of different inhibitors and defining their mode of action. Noncompetitive inhibitors are particularly desirable, because&mdashby contrast to competitive inhibitors&mdashtheir effects can never be completely overcome by increases in substrate concentration.

Most Drugs Are Metabolized In Vivo

Drug development often involves more than the kinetic evaluation of the interaction of inhibitors with the target enzyme. Drugs may be acted upon by enzymes present in the patient or pathogen, a process termed drug metabolism.For example, penicillin and other &beta-lactam antibiotics block cell wall synthesis in bacteria by irreversibly poisoning the enzyme alanyl alanine carboxypeptidase-transpeptidase. Many bacteria, however, produce &beta-lactamases that hydrolyze the critical &beta-lactam function in penicillin and related drugs. One strategy for overcoming the resulting antibiotic resistance is to simultaneously administer a &beta-lactamase inhibitor with a &beta-lactam antibiotic.

Metabolic transformation is sometimes required to convert an inactive drug precursor, or prodrug, into its biologically active form (Chapter 53). 2&prime-Deoxy-5-fluorouridylic acid, a potent inhibitor of thymidylate synthase, a common target of cancer chemotherapy, is produced from 5-fluorouracil via a series of enzymatic transformations catalyzed by a phospho-ribosyl transferase and the enzymes of the deoxyribonucleo-side salvage pathway (Chapter 33). The effective design and administration of prodrugs requires knowledge of the kinetics and mechanisms of the enzymes responsible for transforming them into their biologically active forms.

The study of enzyme kinetics&mdashthe factors that affect the rates of enzyme-catalyzed reactions&mdashreveals the individual steps by which enzymes transform substrates into products.

&DeltaG, the overall change in free energy for a reaction, is independent of reaction mechanism and provides no information concerning rates of reactions.

Keq, a ratio of reaction rate constants, may be calculated from the concentrations of substrates and products at equilibrium or from the ratio k1/k–1. Enzymes do not affect Keq.

Reactions proceed via transition states for which &DeltaGF is the activation energy. Temperature, hydrogen ion concentration, enzyme concentration, substrate concentration, and inhibitors all affect the rates of enzyme-catalyzed reactions.

Measurement of the rate of an enzyme-catalyzed reaction generally employs initial rate conditions, for which the virtual absence of product precludes the reverse reaction.

Linear forms of the Michaelis–Menten equation simplify determination of Km and Vmax.

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A value of n greater than 1 indicates positive cooperativity.

The effects of simple competitive inhibitors, which typically resemble substrates, are overcome by raising the concentration of the substrate. Simple noncompetitive inhibitors lower Vmax but do not affect Km.

For simple competitive and noncompetitive inhibitors, the inhibitory constant Ki is equal to the equilibrium dissociation constant for the relevant enzyme-inhibitor complex. A simpler and less rigorous term for evaluating the effectiveness of an inhibitor is IC50, the concentration of inhibitor that produces 50% inhibition under the particular circumstances of the experiment.

Substrates may add in a random order (either substrate may combine first with the enzyme) or in a compulsory order (substrate A must bind before substrate B).

In ping–pong reactions, one or more products are released from the enzyme before all the substrates have been added.

Applied enzyme kinetics facilitates the identification and characterization of drugs that selectively inhibit specific enzymes. Enzyme kinetics thus plays a central and critical role in drug discovery, in comparative pharmacodynamics, and in determining the mode of action of drugs.

Cook PF, Cleland WW: Enzyme Kinetics and Mechanism. Garland Science, 2007.

Copeland RA: Evaluation of Enzyme Inhibitors in Drug Discovery. John Wiley & Sons, 2005.

Cornish-Bowden A: Fundamentals of Enzyme Kinetics. Portland Press Ltd, 2004.

Dixon M: The determination of enzyme inhibitor constants. Biochem J 195355:170.

Dixon M: The graphical determination of Km and Ki. Biochem J 1972129:197.

Fersht A: Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding. Freeman, 1999.

Fraser CM, Rappuoli R: Application of microbial genomic science to advanced therapeutics. Annu Rev Med 200556:459.

Henderson PJF: A linear equation that describes the steady-state kinetics of enzymes and subcellular particles interacting with tightly bound inhibitors. Biochem J 1972127:321.

Schramm, VL: Enzymatic transition-state theory and transition-state analogue design. J Biol Chem 2007282:28297.

Schultz AR: Enzyme Kinetics: From Diastase to Multi-enzyme Systems. Cambridge University Press, 1994.

Segel IH: Enzyme Kinetics. Wiley Interscience, 1975.

Wlodawer A: Rational approach to AIDS drug design through structural biology. Annu Rev Med 200253:595.

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Results and discussion

Initial rate measurements reveal that hydrogen peroxide is a reversible noncompetitive inhibitor of the CsOxOx catalyzed oxidation of oxalate

Product inhibition is a special case of inhibition in which the inhibitor is also a product of the enzyme catalyzed reaction. Typically kinetic studies of enzymatic activity are done under conditions at which the amount of product present during the reaction is effectively zero. This removes the effects of the reverse reaction forming reactants as well as those effects observed by the binding of product to enzyme thus simplifying the velocity equation dramatically, and, in the case of a simple unireactant system, producing the well-known Michaelis-Menten equation (Eq 1). Introducing non-zero initial concentrations of product reintroduces the product dependent terms of the velocity equation and often allows one to differentiate between two similar potential kinetic models. The Michaelis-Menton and Lineweaver-Burk plots for varying concentrations of hydrogen peroxide are shown in Fig 1A and 1B, respectively. The percent of the uninhibited specific activity and KM values for the CsOxOx catalyzed oxidation of oxalate in the presence and absence of hydrogen peroxide measured by MIMS is presented in Table 1. Initial concentrations of hydrogen peroxide reduce the apparent maximum velocity of catalysis with only mild perturbations (a 60% increase) of the observed KM app values. These data suggest that hydrogen peroxide behaves as a noncompetitive inhibitor of CsOxOx. Noncompetitive inhibitors may bind to the enzyme form that the substrate binds and a different enzyme species [44, 45]. In the classic noncompetitive case, the equilibrium constants for the binding of inhibitor with enzyme alone (KI) and inhibitor with another enzyme species (αKI, described further below) are equal. When the two equilibrium constants are different, traditional inhibition nomenclature becomes ambiguous with this case often defined as a “mixed-type” inhibitor and not a noncompetitive inhibitor. However, Cleland states that since there is no a priori reason that the equilibrium constants should be equal, mixed inhibition should be referred to as noncompetitive inhibition [44]. By this definition, a noncompetitive inhibitor will reduce the apparent maximal velocity, but can reduce, increase or have no effect on the apparent KM value. All models consistent with this observation have hydrogen peroxide reversibly binding both CsOxOx alone and another CsOxOx complex.

Michaelis-Menten (A) and Lineweaver-Burk (B) plots of the initial rates of oxalate oxidation by CsOxOx demonstrating the effects of varying initial hydrogen peroxide concentrations: black, no H2O2 red, 2 mM H2O2 blue, 4 mM H2O2 green, 10 mM H2O2.

To assign KI app and αKI app values for the noncompetitive inhibition observed in Fig 1A and 1B, experiments at substrate concentrations resulting in Lineweaver-Burk plots of equidistance and equal weight points were carried out in a larger range of hydrogen peroxide concentrations. The resulting plots of reciprocal initial oxalate oxidase activity versus reciprocal oxalate concentration for CsOxOx are shown in Fig 2A. Fig 2B shows the secondary plots of the slopes and intercepts of the reciprocal plot linear best fit lines versus hydrogen peroxide concentration. The αKI app and KI app values are estimated to be 7.91 ± 1.12 and 2.84 ± 1.06 from the x-intercepts secondary plots [44]. The slope and intercept secondary plots are linear (R 2 values greater than 0.99) which suggests that the hydrogen peroxide inhibition of CsOxOx is complete (no product is released from the inhibitor and the different enzyme species) and not partial (up to 40 mM hydrogen peroxide). From these data, α was estimated to be 2.79 ± 1.12. A way to mathematically check this value is to use the regression lines from the reciprocal plots (Fig 2B) [44]. The lines intersect at the point -0.207±0.011 on the x-axis and 0.195±0.042 on the y-axis, which corresponds to an α value of 2.25 ± 0.59. The values for the α constant determined using two different the methods are in good agreement.

(A) Lineweaver-Burk plots of initial oxalate oxidase activity versus oxalate concentration for CsOxOx in varying H2O2 concentrations: circles, no H2O2 squares, 10 mM H2O2 diamonds, 20 mM H2O2 triangles, 40 mM H2O2. (B) Secondary plots of the slopes and intercepts from of the reciprocal plot linear best fit lines (Fig 2A) versus hydrogen peroxide concentration circles, slopes squares, intercepts.

Reversible noncompetitive inhibition and irreversible inhibition may be distinguished by plotting Vmax app data as a function of [E]t, where [E]t represents the total units of enzyme activity added to the assay [41]. These data for the hydrogen peroxide inhibition of the CsOxOx catalyzed oxidation of oxalate are shown in Fig 3. The rate of product formation is equal to the product of the enzyme concentration and a kinetic factor which is a function of all substrate, product activator, and inhibitor concentrations. The reaction rate of an enzyme catalyzed reaction is, therefore, a linear function of the enzyme concentration with a y-intercept of 0 and a slope of f([S], [P], [A], [I]) [42]. A non-zero x-intercept implies that the concentration of enzyme has been changed with the difference of the x-intercept from zero defining the concentration of active enzyme removed or added. Reversible inhibitors by definition do not permanently inactivate the enzyme and therefore can only affect the slope of the line but not its x-intercept. In Fig 3 the curves with and without initial concentrations of hydrogen peroxide present do not appear to have different x-intercepts. The x-intercept for the plot without inhibitor present was 7 nM ± 8 nM and the x-intercept for the plot with inhibitor present was 8 nM ± 9 nM. These x-intercepts from both linear fits are not significantly different from each other and they are not significantly different from zero (one standard deviation from the x-intercept for both lines includes zero) suggesting that CsOxOx is not inactivated upon addition of hydrogen peroxide within the time frame of the kinetic assay. Since the curve with hydrogen peroxide present has a smaller slope and goes through the approximate origin, these data are consistent with hydrogen peroxide being a reversible noncompetitive inhibitor of the CsOxOx catalyzed oxidation of oxalate. In the case of an irreversible inhibitor, the slope of the line with inhibitor present would have the same slope and intersect the x-axis at a position equal to the amount of enzyme that is irreversibly inactivated [41, 42]. It is important to note that these data (Fig 3) show that the concentration of irreversibly inactivated enzyme is negligible under initial rate conditions.

Each point represents a Vmax app determination at five concentrations of oxalate (0.2, 0.5, 1.0, 5.0, and 10.0 mM).

Turnover-generated hydrogen peroxide leads to inactivation of CsOxOx

HPLC chromatograms monitoring the amounts of hydrogen peroxide and oxalate at discrete time points are shown in Fig 4A (no catalase present) and 4B (catalase present). Hydrogen peroxide, oxalate and succinate standards eluted at 6.3 minutes, 6.9 minutes, and 12.4 minutes, respectively and informed the peak assignments shown. In the absence of catalase the amount of oxalate remaining after 24 hours was approximately 85% of the initial amount (10 mM) and the appearance of an accumulation of hydrogen peroxide was visible in the chromatogram. The proximity of the hydrogen peroxide peak and the oxalate peak confounded precise integration of their respective areas. After 24 hours, an aliquot of the sample without catalase had no detectable activity in the MIMS assay suggesting that the CsOxOx present had become inactivated by the build-up of turnover-generated hydrogen peroxide. Further, the addition of catalase to the inactivated enzyme does not return any active CsOxOx from the observed inactivation. These results suggest that hydrogen peroxide is having at least two separate effects on CsOxOx. First, hydrogen peroxide behaves as a reversible noncompetitive inhibitor of the initial reaction velocity as described above. Separately, hydrogen peroxide also appears to be involved in the slow onset inactivation of the CsOxOx enzyme observable minutes to hours after initial enzyme turnover.

An aliquot of the incubation reaction without (A) or with (B) recombinant bovine catalase was analyzed at initiation, 20 minutes, 1 hour, 2 hours, and 24 hours.

In order to observe the effects of turnover on CsOxOx, a plot of reaction velocities after 0 and 2 hours of turnover is shown in Fig 5. Similar to the data in Fig 3, the slope of the turnover “treated” (two hours of turnover in the presence of 10 mM oxalate) initial rate curve is smaller than that of the “untreated” (no prior turnover) control. This suggests that a perturbation of the kinetic function is at least partially due to the build-up of the hydrogen peroxide product during the two hours of turnover. In Fig 5, however, the x-intercept of the turnover treated curve is different from the control curve. This observation is consistent with inactivation of CsOxOx over the two hour time period [41, 42]. The inactivation of the enzyme is only seen when the enzyme is pre-incubated with oxalate. Pre-incubation in buffer or buffer and hydrogen peroxide showed no inactivation. Specifically, CsOxOx preincubated with 10 mM hydrogen peroxide for two hours then assayed at final concentrations of 1 mM and 10 mM hydrogen peroxide resulted in measured initial rates that were commensurate with those expected from the inhibition studies described above (91.4% and 44.3% of the uninhibited specific activity, respectively). Enzyme pre-incubated with 10 mM potassium oxalate possessed less than 20 percent of the original enzyme activity after two hours (data not shown) and no activity was detectable after 24 hours. Table 2 shows the results of MIMS measurements of samples with no prior turnover, turnover in the absence and in the presence of 1.0 mg/mL bovine catalase as described in the Materials and Methods section. The enzyme was reasonably stable under the conditions of the experiment with 73 percent of original enzyme activity remaining after incubation at 25°C for 24 hours. When catalase is present, however, approximately 80 percent of the enzyme activity remains after 24 hours. These data suggest that hydrogen peroxide or a derivative there of has a key role in the turnover-dependent inactivation of CsOxOx.

It is not immediately apparent how the presence of turnover-generated hydrogen peroxide inactivates CsOxOx. One proposed mechanism for barley oxalate oxidase catalysis identifies the fraction of enzyme containing Mn 3+ as the active form of the enzyme [28]. In this proposal, turnover is initiated after oxalate binds to the metal center and an electron is transferred from the oxalate ligand to a Mn 2+ ion. The loss of an electron facilitates the decarboxylation of the oxalate ligand leaving a manganese bound carbon dioxide radical anion which is proposed to react with diatomic oxygen to produce carbon dioxide and a hydroperoxyl radical. Spectroscopic studies consistent with the persistence of the hydroperoxyl radical species being critical to enzyme turnover support this proposal. Whitaker et al have hypothesized that the hydroperoxyl radical directly reoxidizes the Mn 2+ ion, reactivating the enzyme to the Mn 3+ form for further turnover. Loss of the hydroperoxyl radical would thus prevent the reoxidation of the metal center rendering it inactive [28]. A similar mechanism of inactivation of OxDC was put forth to describe the EPR observation of a spin trapped carbon dioxide radical species suggesting a “leaky” active site from which the carbon dioxide radical can dissociate and enter bulk solution. Again, it was proposed that in the absence of an electron sink at the active site, the inactivation of the enzyme would occur after carbon dioxide radical dissociation [33]. It is important to note that the Whitaker et al mechanism and observations for monocupin barley oxalate oxidase may not be accurate for CsOxOx. One significant difference is that the barley enzyme is inactivated under anaerobic condition in the presence of oxalate and remains so even after the reintroduction of oxygen. In contrast, CsOxOx, while inactive under anaerobic conditions in the presence of oxalate, regains activity upon reintroduction of oxygen (data not shown). Another significant difference is that the presence of catalase accelerated enzyme inactivation of the barley enzyme while catalase protects against inactivation in CsOxOx. These observations are not consistent with a mechanism for CsOxOx utilizing a hydroperoxyl radical alone for the regeneration of catalytically competent enzyme, and supports the role of the Mn(II) in the reductive activation of diatomic oxygen. Our results suggest a completely different method of inactivation. We observe that enzyme inactivation occurs during CsOxOx turnover and only in the presence of hydrogen peroxide. This suggests that hydrogen peroxide is reacting irreversibly with a turnover intermediate rendering the enzyme inactive reminiscent of the hydrogen peroxide inactivation of catalase [46].

MIMS measurements reveal approximately 1.3 moles of CO2 is produced per one mole of O2 consumed. When this ratio is plotted as a function of oxalate (from 0.5 to 10 mM) in the reaction mixture (Figure C in S1 File), the result is essentially a horizontal line indicating that the ratio is independent of initial oxalate concentration. Furthermore, the ratio of CO2 produced per one mole of O2 consumed is independent of hydrogen peroxide concentration (in both initial rate measurements and in the case of turnover induced inactivation). Despite the application of numerous assay techniques and much interest in the stoichiometry of this reaction, this is to our knowledge the first report of the observed stoichiometric ratio of moles of CO2 produced per mole of O2 consumed of oxalate oxidase. It has been reported that for sorghum OxOx, the consumption of oxalate was directly related to the formation of hydrogen peroxide [6]. The application of Warburg manometric techniques to peroxisomal preparations from the leaves of the spinach beet enzyme were not able to establish the stoichiometry due to the small amount of oxygen consumed and carbon dioxide formed [10]. Manometric measurements of rates of the enzyme from the parasitic fungus Tilletia controversa were confounded by impure enzyme preparations [47]. Liquid scintillation procedures were used to relate 14 CO2 formation to 14 C-oxalate disappearance but the consumption of oxygen was not measured in those experiments [48]. Interestingly, oxalate decarboxylase from Bacillus subtilis (BsOxDC) has been mutated (SENS161-4DASN) to carry out the oxidation reaction and a ratio of 1.3 moles of hydrogen peroxide produced to 1 mole of the one electron oxidation of the dye ABTS has been reported [27]. Recent MIMS measurements of this mutant BsOxDC assumed a ratio of 2 moles of CO2 is produced per one mole of O2 consumed at low oxalate concentrations [29]. This assumption contributed to the conclusion that only 9% of the total number of active sites of the SENS161-4DASN mutant BsOxDC carry out the oxidation reaction with the remaining sites nonoxidatively decarboxylating oxalate.

Previously proposed mechanistic schemes all assume a 2 to one molar relationship between CO2 production and O2 consumption. That a 1.3 to one ratio is observed in all conditions tested raises the possibility that an additional oxidation may be taking place. One possible explanation could reside in the fate of the carbon dioxide radical anion formed after the first decarboxylation. As mentioned above, the spin trapping of this species in both CsOxOx and OxDC [25, 32–34], suggests that it leaks out the active site. This radical may react with dissolved oxygen from bulk solution to form a superoxide radical. Superoxide radicals have been trapped in the case of the OxDC catalyzed redox neutral decarboxylation of oxalate [33]. Since superoxide radicals (pKa of 4.88) exist primarily as hydroperoxyl radicals under our experimental conditions, escape of this species from the active site and subsequent oxidation of exogenous species, would result in the observed increased consumption of oxygen [28].

Circular Dichroism (CD) studies of CsOxOx show no global structural changes in the presence of hydrogen peroxide

CD experiments were performed to monitor the secondary structure as a function of hydrogen peroxide concentration and temperature. Fig 6A shows that the CD spectrum for CsOxOx is unperturbed by hydrogen peroxide concentrations up to 20 mM. These data indicate that hydrogen peroxide does not affect the overall global protein structure of the enzyme. In contrast, the effect of temperature on the CD spectrum of CsOxOx is shown in Fig 6B. Spectra taken at higher temperatures exhibit a deeper minimum and a shift toward the near UV region. Typically, examples in the literature of the effect of temperature on the CD spectrum of proteins show a lesser degree of molar ellipticity upon increasing temperature [49]. There are, however, examples where the CD spectra display a greater degree of molar ellipticity as does CsOxOx [50].


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