Enzymes are used in cheese-making to degrade proteins in milk, changing their solubility, and causing the proteins to precipitate. Many industrial processes ranging from fruit juice production to paper production to biofuel production utilize enzymes.
Contributors Introduction The Michaelis-Menten model 1 is the one of the simplest and best-known approaches to enzyme kinetics. It takes the form of an equation relating reaction velocity to substrate concentration for a system where a substrate S binds reversibly to an enzyme E Michaeliswordmeaning pdf form an enzyme-substrate complex ES, which then reacts irreversibly to generate a product P and to regenerate the free enzyme E.
This system can be represented schematically as follows: Haldane in 2and a version of it follows: For the scheme previously described, kon is the bimolecular association rate constant of enzyme-substrate binding; koff is the unimolecular rate constant of the ES complex dissociating to regenerate free enzyme and substrate; and kcat is the unimolecular rate constant of the ES complex dissociating to give free enzyme and product P.
Note that kon has units of concentration-1time-1, and koff and kcathave units of time Once these rate constants have been defined, we can write equations for the rates of change of all the chemical species in the system: In most systems, the ES concentration will rapidly approach a steady-state — that is, after an initial burst phase, its concentration will not change appreciably until a significant amount of substrate has been consumed.
This is also the reason that well-designed experiments measure reaction velocity only in regimes where product formation is linear with Michaeliswordmeaning pdf.
As long as we limit ourselves to studying initial reaction velocities, we can assume that [ES] is constant: We know that the free enzyme concentration [E] is equal to the total enzyme concentration [ET] minus [ES].
Making these substitutions gives us: We also define KM in terms of the rate constants as follows: This second assumption is the free ligand approximation, and is valid as long the total enzyme concentration is well below the KM of the system.
The larger the kcat is relative to koff, the greater the difference between KD and KM. Briggs and Haldane made no assumptions about the relative values of koff and kcat, and so Michaelis-Menten kinetics are a special case of Briggs-Haldane kinetics.
The Briggs-Haldane approach frees us from the last of these three. Remember that this approximation states that the free substrate concentration the [S] in the Michaelis-Menten equation is close to the total substrate concentration in the system — which is, in fact, the true independent variable in most experimental set-ups.
For such unusual cases, we derive a kinetic equation from the scheme above without resorting to the free ligand approximation.
The rates of change of the various species are given by four differential equations: As the KM becomes larger than [ET], the curve described by the quadratic equation approaches the hyperbola described by the Michaelis-Menten equation.
Differences in affinity between very tight-binding substrates are reflected in the sharpness of the inflection, meaning that KM values recovered from fits to experimental data are disproportionately sensitive to error in the data points surrounding the inflection point.
As a rule of thumb, the quadratic equation should be used in preference to the Michaelis-Menten equation whenever the KM is less than five-fold larger than [ET].
Derivations of this type that do not use the free ligand approximation grow rapidly more complicated for more complex kinetic systems. For anything more complicated than a one-site model, it is better to use kinetic simulations as detailed in part II of this guide.
Building on the derivation of Michaelis-Menten kinetics, we now turn to enzymes with multiple substrate-binding sites. Sequential Models Early work in this regard was carried out by Adair and Pauling, operating under the rapid equilibrium approximation.
Remember that this assumption states that all substrate binding and dissociation steps happen much more rapidly than catalytically productive steps. InKing and Altman showed how to solve any kinetic system without resorting to this approximation, but this is unnecessarily complicated for our purposes.
Even if the rapid equilibrium approximation does not hold for our system, the forms of the equations we derive will not change — instead, the significance of the various KM values will be different. We begin with the simplest model of multiple binding: Here, an enzyme E can bind a single molecule of substrate S to form a singly-occupied complex ES with equilibrium dissociation constant KD1.
ES can either react irreversibly to form product P with rate kcat1, or can bind a second substrate molecule S to form a doubly-occupied complex ESS with dissociation constant KD2.
In turn, ESS can irreversibly form P with rate kcat2. This is represented schematically below: In order to calculate the reaction velocity for this system, we need to know the relative concentrations of the active species ES and ESS. We will describe a straightforward derivation of these quantities, and then a shortcut that allows one to quickly solve more complicated kinetic systems.
From the definitions of the two dissociation constants, we have:Word Equations for Enzyme Kinetics Model # of free S molecules at time t +1 = # of free S molecules at time t # of free S molecules binding to free E – # of ES complexes splitting into E + S + Write similar word equations for the other 3 variables.
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