Lecture 24, Outline. Michaelis-Menten kineticsInterpretations and uses of the Michaelis-Menten equationEnzyme inhibitors: types and kinetics. Enzyme Kinetics Equation. Michaelis-Menten Equation. Initial Velocity (vo) and [S] . The concentration of substrate [S] present will greatly influence the rate of product formation, termed the velocity (v) of a reaction. Studying the effects of [S] on the velocity of a reaction is complicated by the reversibility of enzyme reactions, e.g. conversion 31977
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Interpretations and uses of the Michaelis-Menten equation
Enzyme inhibitors: types and kinetics
Enzyme Kinetics Equation
Initial Velocity (vo) and [S]
The concentration of substrate [S] present will greatly influence the rate of product formation, termed the velocity (v) of a reaction. Studying the effects of [S] on the velocity of a reaction is complicated by the reversibility of enzyme reactions, e.g. conversion of product back to substrate. To overcome this problem, the use of initial velocity (vo) measurements are used. At the start of a reaction, [S] is in large excess of [P], thus the initial velocity of the reaction will be dependent on substrate concentration
Initial Velocity (vo) and [S] (cont)
When initial velocity is plotted against [S], a hyperbolic curve results, where Vmax represents the maximum reaction velocity. At this point in the reaction, if [S] >> E, all available enzyme is "saturated" with bound substrate, meaning only the ES complex is present.
Substrate Saturation of an Enzyme
A. Low [S] B. 50% [S] or Km C. High, saturating [S]
Steady State Assumption
The M-M equation was derived in part by making several assumptions. An important one was: the concentration of substrate must be much greater than the enzyme concentration. In the situation where [S] >> [E] and at initial velocity rates, it is assumed that the changes in the concentration of the intermediate ES complex are very small over time (vo). This condition is termed a steady-state rate, and is referred to as steady-state kinetics. Therefore, it follows thatthe rate of ES formation will be equal to the rate ES breakdown.
Michaelis-Menten Equation Derivation
Rate of ES formation = k1([ET] - [ES])[S] (where [ET] is total concentration of enzyme E and k-2 is considered neglible)
Rate of ES breakdown to product = k-1[ES] + k2[ES]
Michaelis-Menten Equation Derivation (cont)
Thus for the steady state assumption:
k1([ET] - [ES])[S] = k-1[ES] + k2[ES]
This equation is the basis for the final Michaelis-Menten following algebraic rearrangement and substitution of Km and Vmax terms.
Meaning of Km
An important relationship that can be derived from the Michaelis-Menten equation is the following: If vo is set equal to 1/2 Vmax, then the relation Vmax /2 = Vmax[S]/Km + [S] can be simplied to Km + [S] = 2[S], orKm = [S].This means that at one half of the maximal velocity, the substrate concentration at this velocity will be equal to the Km. This relationship has been shown experimentally to be valid for many enzymes much more complex in regards to the number of substrates and catalytic steps than the simple single substrate model used to derive it.
Meaning of Km (cont)
The significance of Km will change based on the different rate constants and which step is the slowest (also called the rate-limiting step). In the simplest assumption, the rate of ES breakdown to product (k2) is the rate-determining step of the reaction, so k-1 >> k2 and Km = k-1/k1. This relation is also called a dissociation constant for the ES complex and can be used as a relative measure of the affinity of a substrate for an enzyme (identical to Kd). However if k2 >> k-1 or k2 and k-1 are similar, then Km remains more complex and cannot be used as a measure of substrate affinity.
Uses of Km
Experimentally, Km is a useful parameter for characterizing the number and/or types of substrates that a particular enzyme will utilize (an example will be discussed). It is also useful for comparing similar enzymes from different tissues or different organisms. Also, it is the Km of the rate-limiting enzyme in many of the biochemical metabolic pathways that determines the amount of product and overall regulation of a given pathway. Clinically, Km comparisons are useful for evaluating the effects mutations have on protein function for some inherited genetic diseases.
Meaning of Vmax
The values of Vmax will vary widely for different enzymes and can be used as an indicator of an enzymes catalytic efficiency. It does not find much clinical use.
There are some enzymes that have been shown to have the following reaction sequence:
In this situation, the formation of product is dependent on the breakdown of an enzyme-product complex, and is thus the rate-limiting step defined by k3.
Derivation of kcat
A more general term has been defined, termed kcat, to describe enzymes in which there are multiple catalytic steps and possible multiple rate-limiting steps. The Michaelis-Menten equation can be substituted with kcat
Definition and Use of kcat
The constant, kcat (units of sec-1), is also called the turnover number because under saturating substrate conditions, it represents the number of substrate molecules converted to product in a given unit of time on a single enzyme molecule. In practice, kcat values (not Vmax) are most often used for comparing the catalytic efficiencies of related enzyme classes or among different mutant forms of an enzyme.
Two Substrate Reactions
Many enzyme reactions involve two or more substrates. Though the Michaelis-Menten equation was derived from a single substrate to product reaction, it still can be used successfully for more complex reactions (by using kcat).
Two Substrate Reactions (cont)
In random order reactions, the two substrates do not bind to the enzyme in any given order; it does not matter which binds first or second.
In ordered reactions, the substrates bind in a defined sequence, S1 first and S2 second.
These two reactions share a common feature termed a ternary complex, formed between E, ES1, ES2 and ES1S2. In this situation, no product is formed before both substrates bind to form ES1S2.
Two Substrate Reactions (cont)
Another possibility is that no ternary complex is formed and the first substrate S1 is converted to product P1 before S2 binds. These types of reactions are termed ping-pong or double displacement reactions.
Km and kcat Example:HSV-1 Thymidine Kinase
A phosphorylation kinase reaction: T (thymidine) + ATP is converted to TMP (thymidine monophosphate) + ADP
In herpesvirus infected cells, this viral encoded TK phosphorylates the antiviral drug acyclovir; this is the pharmacological basis of most herpesvirus treatments
In the last 10 years, this approach has been applied to cancer gene therapies with HSV-TK and ganciclovir
Thymidine Kinetic Constants for HSV-1 Thymidine Kinase (ONLY AN EXAMPLE!!)
Ganciclovir Kinetic Constants for HSV-1 Thymidine Kinase (ONLY AN EXAMPLE!)
Lineweaver-Burk (double reciprocal plot)
If the reciprocal (1/X) of the Michaelis-Menten equation is done, after algebraic simplification the following equation results:
This relation is written in the format of the equation for a straight line, y = mx + b, where y = 1/vo, m (slope) = Km/Vmax, x = 1/[S] and the y-intercept, b = 1/Vmax. When this relation is plotted,the result is a straight line graph
Lineweaver-Burk (double reciprocal plot) (cont)
Uses of double reciprocal plot
The x intercept value is equal to -1/Km. The biggest advantage to using the double reciprocal plot is a more accurate determination of Vmax, and hence Km. It is also useful in characterizing the effects of enzyme inhibitors and distinguishing between different enzyme mechanisms.
Enzyme Inhibitor Types
Inhibitors of enzymes are generally molecules which resemble or mimic a particular enzymes substrate(s). Therefore, it is not surprising that many therapeutic drugs are some type of enzyme inhibitor. The modes and types of inhibitors have been classified by their kinetic activities and sites of actions. These include Reversible Competitive Inhibitors, Reversible Non-Competitive Inhibitors, and Irreversible Inhibitors
Definition of Ki
For reversible inhibitors, a term Ki can be determined.
For competitive inhibitors, the following relation can be used: Km + I = Km (1 + [I] / Ki ) ; (where Km + I is the determined Km in the presence of [I]).
Determining the Ki for other inhibitor types is related but much more complex and not within the scope of this lecture or course
Uses of Ki
Ki values are used to characterize and compare the effectiveness of inhibitors relative to Km. This parameter is especially useful and important in evaluating the potential therapeutic value of inhibitors (drugs) of a given enzyme reaction. For example, Ki values are used for comparison of the different types of HIV protease inhibitors.In general, the lower the Ki value, the tighter the binding, and hence the more effective an inhibitor is.
Vmax - No change
Km INCREASES - indicates a direct interaction
of the inhibitor in the active site
Reversible Competitive Inhibition
Competitive inhibitors compete with the substrate for binding at the active site (as E + I). In the double reciprocal plot for a competitive inhibitor acting at the substrate site for the following reasons, notice with increasing concentration of inhibitor, the Vmax does not change; however, the Km of the substrate is increased. This also reflects the reversible nature of the inhibitor; there is always some concentration of substrate which can displace the inhibitor.
Vmax DECREASES - inhibitor affects rate of reaction
by binding to site other than substrate active-site
Km - No change
Reversible Non-Competitive Inhibition
Non-competitive inhibitors combine with both the enzyme (E + I) and the enzyme-substrate (EI + S) complex. The inhibitor binds to a site other that the substrate site, and is thus independent of the presence or absence of substrate. This action results in a conformational change in the protein that affects a catalytic step and hence decreases or eliminates enzyme activity (formation of P). Notice in the reciprocal plot, a non-competitive inhibitor does not affect the binding of the substrate (Km), but it does result in a decrease in Vmax. This can be explained by the fact that since inhibitor bound to an enzyme inactivates it, the more EI formed will lower [ES] and thus lower the overall rate of the reaction Vmax.
Irreversible inhibitors generally result in the destruction or modification of an essential amino acid required for enzyme activity. Frequently, this is due to some type of covalent link between enzyme and inhibitor. These types of inhibitors range from fairly simple, broadly reacting chemical modifying reagents (like iodoacetamide that reacts with cysteines) to complex inhibitors that interact specifically and irreversibly with active site amino acids. (termed suicide inhibitors). These inhibitors are designed to mimic the natural substrate in recognition and binding to an enzyme active site. Upon binding and some catalytic modification, a highly reactive inhibitor product is formed that binds irreversibly and inactivates the enzyme. Use of suicide inhibitors have proven to be very clinically effective
REMEMBER - The types of enzyme inhibitors described have been for relatively simple, single substrate-product reactions that obey Michaelis-Menten kinetics. However, not all enzyme inhibitors will necessarily be one type of inhibitor. Especially for some multi-substrate reactions, a particular inhibitor can be competitive for one substrate and non-competitive with a second or third substrate. Also, suicide inhibitors by design are generally competitive inhibitors of a substrate, and therefore must first bind in the active site.