Michaelis-Menten Equation - Interactive Graph
When studying biochemical and physiological processes, it is often necessary to measure the rate at which a given reaction or process proceeds to completion. For example, if we examine the rate of biochemical reactions catalyzed by enzymes, or the rate of carrier-mediated transport of molecules across biological membranes, we commonly find that at low enzyme or transporter substrate concentrations, the reaction rate increases almost in a linear fashion with increasing substrate concentration. However, as the substrate concentration is increased to higher and higher levels, the reaction rate no longer increases in proportion to the increase in substrate concentration. Thus, at higher substrate concentrations, the reaction rate no longer increases in a linear manner. Rather, increases in the substrate concentration lead to progressively smaller and smaller increases in the reaction rate. In fact, at very high substrate concentrations, the rate begins to asymptote to a steady-state level, and additional increases in the substrate concentration do not lead to an increase in the reaction rate (see Fig. 1). This type of relationship is referred to as hyperbolic and demonstrates saturation of the enzyme or transporter at high substrate concentrations. Saturation is caused by the fact that there is a fixed number of enzyme or transporter molecules, each with a fixed number of substrate binding sites. At high substrate concentrations, all of the binding sites have substrate bound and each enzyme or transporter molecule is working as fast as its intrinsic rate to catalyze the reaction (for enzymes) or transport the substrate across the membrane (for transporters).

Figure 1. A plot of the reation velocity as a function of the substrate concentration as described by the Michaelis-Menten equation.
When examined at different substrate concentrations, the rate of a reaction catalyzed by an enzyme, or the rate of carrier-mediated transport across a biological membrane, follows a hyperbolic relationship. At low substrate concentrations, the rate increases almost in a linear fashion with the substrate concentration. At very high substrate concentrations, the rate exhibits saturation, where additional increases in the substrate concentration no longer increase the reaction velocity. This type of saturation kinetics is adequately described by the Michaelis-Menten equation. Vmax is the maximum reaction velocity. Km is the Michaelis constant and is the substrate concentration that gives rise to 50% Vmax.
The Michaelis-Menten equation (see below) is commonly used to study the kinetics of reaction catalysis by enzymes as well as the kinetics of transport by transporters. Typically, the rate of reaction (or reaction velocity) is experimentally measured at several substrate concentration values. The range of substrate concentrations is chosen such that very low reaction rates as well as saturating rates are measured. A plot of the reaction rate versus the substrate concentration reveals two important kinetic parameters: Vmax and Km (see Fig. 1). Vmax is the maximum reaction rate that is observed at saturating substrate concentrations. Vmax is a function of the intrinsic rate of the enzyme or transporter as well as a function of the total number of enzyme/transporter molecules that give rise to the measured rate. Km is referred to as the Michaelis constant and is the substrate concentration at which the reaction rate is exactly half of Vmax. Km is inversely related to the apparent affinity of the enzyme/transporter for its substrate. Therefore, a low numerical value of Km refers to a very high affinity of interaction between the protein and its substrate. This is because it takes a very small amount (i.e., low concentration) of the substrate to reach 50% of the saturating concentration. Conversely, a high numerical value of Km is indicative of a low affinity of the enzyme/transporter for its substrate. This is because it takes a large amount (i.e., high concentration) of the substrate to reach 50% of the saturating concentration. Thus, Km is a very useful parameter by which the affinity of the protein for various substrates can be compared.
It is important to emphasize that the kinetics of transport for many transport proteins exhibit features that are very similar to those of enzymes. Similar to enzymes, transporters show specificity with respect to the substrate transported and, in addition, the rate of substrate transport across a biological membrane exhibits saturation at high substrate concentrations. Therefore, the kinetics of many transport processes can be studied by using Michaelis-Menten kinetics. The Michaelis-Menten equation can adequately describe the dependence of transport rate on the substrate concentration for facilitative transporters, secondary active transporters (cotransporters and exchangers), and primary active transporters (i.e., pumps).
If the protein under study has more than one (i.e., two or more) substrate binding sites, and if there is cooperativity with respect to substrate binding to the protein, a plot of the reaction rate as function of the substrate concentration is no longer hyperbolic and may assume a sigmoidal shape. In this case, the Michaelis-Menten equation is no longer the appropriate equation to use for studying the rate of reaction as a function of the substrate concentration. Instead, the Hill equation is the appropriate equation to use. Indeed, the Michaelis-Menten equation is a special case of the Hill equation where the protein under study has only one substrate binding site.
Michaelis-Menten equation

• V is the reaction velocity (rate of reaction progression per unit time) and may be expressed in many different forms such as mmol/s, mol/min, etc.
• Vmax is the maximum velocity of the reaction. It has the same units as the reaction velocity (V). It is the highest reaction rate that can be achieved at saturating substrate concentrations.
• [S] is the substrate concentration. It can be expressed in many different forms such as pM, nM, μM, mM, M, ng/mL, %, etc.
• Km is the Michaelis constant. It is the substrate concentration that gives rise to a reaction velocity that is 50% of Vmax. Km has the same units as the substrate concentration. Km provides useful information about the "apparent affinity" of the protein under study (enzyme, transporter, etc.) for the substrate. Affinity can be thought of as how tightly the substrate binds to the enzyme or transporter protein. The lower the numerical value of Km, the higher the apparent affinity for the substrate (i.e., it takes a lower substrate concentration to reach 50% saturation). Conversely, the higher the numerical value of Km, the lower the apparent affinity for the substrate (i.e., it takes a higher substrate concentration to reach 50% saturation). Some investigators refer to Km simply as Khalf or K0.5.
• The Michaelis-Menten equation represents a special case of the Hill equation, where the Hill coefficient has been set to one.
Michaelis-Menten equation - Interactive graph
The interactive graph provided below allows for a good understanding of the Michaelis-Menten equation, how the reaction velocity changes as a function of the substrate concentration, and how changes in Vmax and Km alter the shape of the graph.
Enter appropriate numerical values for the Maximum velocity (Vmax) and Michaelis constant (Km) in the cells below. Then use the Add Plot button to generate a plot of reaction velocity versus the substrate concentration based on the Michaelis-Menten equation presented above. An appropriate substrate concentration range will be used automatically. An unlimited number of plots may be added, but only the first twenty plots are assigned unique colors. Optionally, the units may also be entered in the cells available. Any unit label entered will be added to the title for the corresponding axis.
Depending on user preference, the position of Vmax on the y-axis (i.e., Reaction Velocity axis), the position of Km on the x-axis (i.e., Substrate Concentration axis), legends for all plots, and grid lines may be shown on the graph.
After generating a plot, use the top, left gray arrowheads to adjust the y-axis (i.e., Reaction Velocity) scale, and the bottom, right gray arrowheads to adjust the x-axis (i.e., Substrate Concentration) scale. When the Auto Scale feature is checked, the y-axis maximum is set to be equal to the highest Vmax plotted, and the x-axis maximum is set to be equal to five times the highest Km plotted. The auto scale feature is applied automatically for the very first plot generated.
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 Value Unit Show on Plot Other Options Maximum velocity (Vmax) Vmax Km Legend Grid lines Auto scale Michaelis constant (Km)

Posted: Monday, September 1, 2014
Last updated: Monday, December 8, 2014