# The Michaelis–Menten model

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Enzymes are biological molecules that catalyze (speed up) chemical reactions in living cells. They are essential for many processes such as metabolism, digestion, and DNA replication. Enzymes work by binding to specific molecules called substrates and converting them into products.

The rate of an enzyme-catalyzed reaction is often called its **velocity**. It measures how fast the enzyme can transform the substrate into the product. Enzyme velocity can be affected by various factors such as temperature, pH, inhibitors, and the concentration of the enzyme and the substrate.

To measure enzyme velocity, we need to monitor the amount of product formed or substrate consumed over time. This can be done by using various methods such as spectrophotometry, chromatography, or radioisotope labeling. The most common way is to use a spectrophotometer, which measures the absorbance of light by a solution at a specific wavelength. By knowing the relationship between absorbance and concentration, we can calculate the amount of product or substrate in the solution.

However, enzyme velocity is not constant throughout the reaction. It changes depending on how much substrate is available and how much product is accumulated. Therefore, we need to measure the velocity at a specific point in time, usually at the beginning of the reaction when no product is yet present. This is called the **initial velocity** (symbol V0; μmol min-1).

To obtain the initial velocity, we plot the amount of product formed or substrate consumed against time for an enzyme-catalyzed reaction. We then draw a straight line through the linear part of the curve, starting at the zero time-point. The slope of this line is equal to V0.

The following figure shows an example of a plot of product formed against time for an enzyme-catalyzed reaction and how to obtain V0 from it.

**Figure 1:** Plot of product formed against time for an enzyme-catalyzed reaction

The rate of an enzyme-catalyzed reaction depends on several factors, such as the temperature, pH, and the presence of inhibitors or activators. However, two of the most important factors are the concentration of the substrate () and the concentration of the enzyme ().

The substrate is the molecule that binds to the active site of the enzyme and undergoes a chemical transformation to form the product. The enzyme is the biological catalyst that speeds up the reaction by lowering the activation energy required for the substrate to react.

The relationship between and and the reaction rate (v) can be illustrated by using a simple analogy. Imagine that the enzyme is a taxi driver and the substrate is a passenger who wants to go to a destination (the product). The reaction rate is then equivalent to the number of passengers who reach their destination per unit time.

If there are more passengers () than taxi drivers (), then some passengers will have to wait for a taxi to become available. This means that increasing will increase v, but only up to a certain point. Eventually, all the taxi drivers will be busy with passengers, and adding more passengers will not increase v any further. This is called saturation, and it occurs when is much higher than .

If there are more taxi drivers () than passengers (), then some taxi drivers will be idle and looking for passengers. This means that increasing will increase v, but only up to a certain point. Eventually, all the passengers will be picked up by taxi drivers, and adding more taxi drivers will not increase v any further. This is called excess, and it occurs when is much higher than .

The effect of and on v can be mathematically described by using the Michaelis–Menten equation, which will be explained in detail in point 4. For now, we can use a simplified version of the equation:

v = (Vmax * ) / (Km + )

where Vmax is the maximum rate that can be achieved by the enzyme when it is saturated with substrate, and Km is a constant that reflects the affinity of the enzyme for its substrate.

The equation shows that v increases with , but approaches Vmax asymptotically as becomes very large. This means that there is a limit to how fast an enzyme can catalyze a reaction, even if there is plenty of substrate available.

The equation also shows that v increases with Vmax, which is proportional to . This means that adding more enzyme can speed up the reaction, but only if there is enough substrate to bind to it.

The graph below shows how v changes with for different values of Vmax (or ). Note that each curve has a characteristic shape called a hyperbola.

The Michaelis–Menten model is one of the most widely used models to describe the kinetics of enzyme-catalyzed reactions. It was developed independently by two scientists in the early 20th century: Leonor Michaelis and Maud Menten.

Leonor Michaelis (1875-1949) was a German biochemist who studied the effects of pH, temperature, and inhibitors on enzyme activity. He also pioneered the use of spectrophotometry to measure enzyme reactions. He collaborated with Maud Menten in 1913 at the University of Berlin to study the kinetics of invertase, an enzyme that catalyzes the hydrolysis of sucrose.

Maud Menten (1879-1960) was a Canadian physician and biochemist who made significant contributions to enzymology, histochemistry, and immunology. She was one of the first women to earn a medical degree in Canada and to pursue a career in biomedical research. She worked with Michaelis in Berlin and later with other prominent scientists in the United States and Canada.

Michaelis and Menten proposed a simple mathematical model to explain the relationship between the rate of an enzyme-catalyzed reaction and the concentration of the substrate. Their model was based on the assumption that an enzyme binds reversibly to its substrate to form an enzyme-substrate complex, which then either dissociates back to the enzyme and substrate or converts to the enzyme and product. They derived an equation that relates the initial velocity of the reaction to the maximum velocity, the substrate concentration, and a constant called the Michaelis constant.

The Michaelis–Menten model was a breakthrough in enzymology because it provided a quantitative framework to analyze enzyme kinetics and compare different enzymes. It also helped to elucidate some of the factors that affect enzyme activity, such as substrate affinity, enzyme concentration, and inhibitors. The model is still widely used today, although it has been modified and extended to account for more complex situations.

The Michaelis–Menten equation describes how the reaction velocity (v) depends on the substrate concentration () and the enzyme characteristics. The equation is:

v = ^{Vmax}/_{Km + }

where Vmax is the maximum velocity that the enzyme can achieve when it is fully saturated with substrate, and Km is the Michaelis constant, which represents the substrate concentration at which the reaction velocity is half of Vmax.

The equation can be derived from the following assumptions:

- The enzyme (E) binds to the substrate (S) reversibly to form an enzyme-substrate complex (ES).
- The ES complex can either dissociate back to E and S, or proceed to form the product (P) and release E.
- The rate of each step is proportional to the concentration of the reactants, and can be expressed by a rate constant (k1, k2, and k3).
- The concentration of the enzyme () is much lower than the concentration of the substrate (), so that remains constant throughout the reaction.
- The concentration of the ES complex () reaches a steady state, meaning that its rate of formation is equal to its rate of consumption.

Based on these assumptions, we can write the following equations for each step:

E + S ⇌ ES → E + P

d[ES]/dt = k1[ES] - k2[ES] - k3[ES] = 0

d[P]/dt = k3[ES] = v

By rearranging and solving for [ES], we get:

[ES] = k1[E][S] / (k2 + k3)

By substituting this into the equation for v, we get:

v = k3[k1[E][S] / (k2 + k3)]

We can simplify this equation by introducing two new terms: Vmax and Km. Vmax is defined as the maximum velocity that can be achieved when is equal to , meaning that all the enzyme molecules are bound to substrate. This occurs when is very high and much greater than Km. In this case, we can write:

Vmax = k3

Km is defined as the substrate concentration at which v is half of Vmax. This occurs when is equal to Km. In this case, we can write:

Vmax/2 = k3[k1[E][S] / (k2 + k3)]

By rearranging and solving for Km, we get:

Km = (k2 + k3) / k1

By substituting these terms into the equation for v, we get the final form of the Michaelis–Menten equation:

v = Vmax / (Km + )

The equation shows that v increases as increases, but approaches a limit of Vmax as becomes very large. The value of Km indicates how sensitive v is to changes in . A low Km means that v reaches half of Vmax at a low , implying that the enzyme has a high affinity for the substrate. A high Km means that v reaches half of Vmax at a high , implying that the enzyme has a low affinity for the substrate. The ratio of Vmax to Km is often used as a measure of enzyme efficiency. A high ratio means that the enzyme can achieve a high velocity at a low substrate concentration. A low ratio means that the enzyme requires a high substrate concentration to achieve a high velocity.

The Michaelis–Menten equation describes how the reaction velocity (v) depends on the substrate concentration () and two other parameters: the maximum velocity (Vmax) and the Michaelis constant (Km).

Vmax is the theoretical maximum rate of the reaction when all the enzyme molecules are bound to the substrate. It is proportional to the enzyme concentration and the rate constant k3.

Km is a measure of how effectively the enzyme binds to the substrate. It is equal to the substrate concentration at which the reaction velocity is half of Vmax. It is inversely proportional to the affinity of the enzyme for the substrate.

The Michaelis–Menten equation can be written as:

v = Vmax / (Km + )

This equation shows that:

- When is much lower than Km, the reaction velocity is directly proportional to . This means that the enzyme is not saturated with substrate and the rate of product formation depends on how much substrate is available.
- When is much higher than Km, the reaction velocity approaches Vmax. This means that the enzyme is saturated with substrate and the rate of product formation depends on how fast the enzyme can convert the substrate to product.
- When is equal to Km, the reaction velocity is half of Vmax. This means that half of the enzyme molecules are bound to substrate and half are free.

The value of Km can vary widely depending on the enzyme and the substrate. A low Km indicates a high affinity of the enzyme for the substrate, meaning that it can achieve a high reaction velocity even at low substrate concentrations. A high Km indicates a low affinity of the enzyme for the substrate, meaning that it requires a high substrate concentration to reach a high reaction velocity.

The value of Km can also be affected by factors such as temperature, pH, inhibitors, and activators that influence the enzyme activity or stability. Therefore, Km can provide useful information about how an enzyme behaves under different conditions.

One of the advantages of the Michaelis–Menten model is that it can be easily visualized by plotting the reaction velocity (v) against the substrate concentration (). This allows us to see how the enzyme kinetics are affected by different parameters, such as the maximum velocity (Vmax), the Michaelis constant (Km), and the enzyme concentration ().

The most common graphical representation of the Michaelis–Menten model is the **hyperbolic curve**, which shows that v increases with until it reaches a plateau at Vmax. The shape of the curve reflects the saturation of the enzyme with substrate at high . The Km value can be estimated from the curve as the that gives half of Vmax. The hyperbolic curve can be expressed by the following equation:

v = Vmax / (Km + )

The hyperbolic curve can be transformed into a **linear plot** by taking reciprocals of both sides of the equation. This gives the **Lineweaver–Burk plot**, which shows a straight line with slope Km/Vmax, intercept 1/Vmax on the y-axis, and intercept -1/Km on the x-axis. The Lineweaver–Burk plot can be useful for determining Vmax and Km more accurately than from the hyperbolic curve, as well as for analyzing enzyme inhibition. However, it also has some drawbacks, such as amplifying experimental errors and distorting the data at low . The Lineweaver–Burk plot can be expressed by the following equation:

1/v = Km/Vmax + 1/Vmax

Another way to transform the hyperbolic curve into a linear plot is by rearranging the equation to express v/ as a function of v. This gives the **Eadie–Hofstee plot**, which shows a straight line with slope -Km/Vmax, intercept Vmax/Km on the y-axis, and no intercept on the x-axis. The Eadie–Hofstee plot has some advantages over the Lineweaver–Burk plot, such as being less sensitive to experimental errors and showing deviations from Michaelis–Menten kinetics more clearly. However, it also has some disadvantages, such as being less intuitive and requiring more data points. The Eadie–Hofstee plot can be expressed by the following equation:

v/ = -Km/Vmaxv + Vmax/Km

The graphical representation of the Michaelis–Menten model can help us understand how enzymes work and how they are regulated by different factors. By plotting v against , we can see how fast an enzyme can catalyze a reaction and how much substrate it needs to reach its maximum efficiency. By transforming the hyperbolic curve into a linear plot, we can estimate Vmax and Km more precisely and compare different enzymes or inhibitors. The graphical representation of the Michaelis–Menten model is therefore a powerful tool for studying enzyme kinetics.

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