Biotecnología y biomedicina Investigación

agosto 31, 2020

Tiempo de lectura: 8 minutos

Introduction: What is the Michaelis-Menten equation?

If you were to type “Michaelis-Menten equation?” in your search bar, you would probably end up with a result similar to this:

$v_{0}=\frac{v_{max}[S]}{[S]+K{m}}$

For someone who is slightly familiar with chemical notation, you will soon realise that this equation involves substrate concentration, $[S]$ , and some velocities, $v_{0}$ and $v_{max}$ . However, what may have caught you off guard is $K{m}$ , which stands for the Michaelis-Menten constant. This equation is generally used when discussing enzyme kinetics and it is extremely useful for scientists to determine a variety of parameters related to enzymes.

In today’s article, I will explain how to reach this equation, using a basis of fundamental assumptions and experimental observations, and illustrate its application in the fascinating world of Biochemistry.

Rate of reaction

We must bear in mind that the Michaelis-Menten equation is only a model of what we believe happens in our bodies based on experimental data. Enzymes, the proteins responsible for catalysing a variety of reactions within our cells, must first bind to the substrate to form the enzyme-substrate complex. Later on, if the enzyme has been activated or is found in the appropriate conditions, it will form the product. At this point, the product is released and the enzyme can catalyse a new reaction. This process can be summarised as follows:

$E+S \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} ES \underset{}{\stackrel{k_2}{\rightleftharpoons}} E+P$

Now that we know what the process entails, we can begin to study its velocity. Firstly, we should learn the general rate of reaction equation (how much product or how much substrate is consumed over time). For example, $A+B\rightarrow C$ , the rate of reaction would be:

$v=k\cdot [A]^{m}[B]^{n}$

Above all, what we must take away from this equation is that the rate of reaction is some constant, k, times the concentration of the substrates.

Secondly, we must determine a key feature when discussing rates of reaction: the rate-limiting step, the slowest step in the reaction. Therefore, to write the equation for the rate of reaction,we must include all of the substrates involved until that point. For enzyme-catalysed reactions, we will assume this is the second step. Now we can determine the rates of reaction for the first reaction in this article:

Rate of formation of ES complex: $v_1=k_1 \cdot [E][S]$

Rate of dissociation of ES complex: $v_2=k_{-1} \cdot [ES] + k_2 \cdot [ES]$

Note that for this last rate, we have added two expressions. Both of them involve the dissociation of the enzyme-substrate complex, but one is the reverse of the first step and the second is the formation of product.

Reaching the Michaelis-Menten equation

This may all seem simple, yet, there are still a few issues we must sort out before we can reach the Michaelis-Menten equation.

We have to assume that our enzyme is functioning at a steady rate, in a kind of equilibrium. Thus, the rate of formation of ES should be equal to the rate of dissociation of ES.

$v_1=k_1 \cdot [E][S]=[ES] \cdot (k_{-1}+k_2) = v_2$

Similarly:

$\frac{[E][S]}{[ES]} = \frac{k_{-1}+k_2}{k_1}= K_m$

Finally, we have discovered what the Michaelis-Menten constant is.

Finding comparable variables

Unfortunately, although it looks great on paper, the variables in the equation we reached are difficult to measure in the lab. How can we possibly measure the concentration of enzyme-substrate complex? This is why we need to slightly adapt the formula to obtain another one that can actually be used. To do so, let’s have a look at some facts we already know.

We know that the total concentration of enzyme will be equal to the concentration of free enzymes plus the concentration of enzyme forming the enzyme-substrate complex. That is to say, $[E_T]=[E]+[ES]$

In addition, we also know what the initial velocity $v_0$ and the $v_{max}$ are. You can see these two concepts in the graph below. I would recommend that you study it carefully, as I will be using it from now on.

Figure 8.11. Michaelis-Menten Kinetics. — Graph plotting reaction velocity against substrate concentration

$v_0=k_2 \cdot [ES]$

This is because the formation of product depends on the concentration of enzyme-substrate complex. As a result, the initial velocity of reaction is simply this concentration times the constant, $k_2$ .

$v_{max}=k_2 \cdot [E_T]$

Again, this is observed clearly in the graph. When all of the enzymes are occupied by the substrate, there are no more available. In other words, the rate of reaction cannot increase any further since the enzymes are saturated.

Taking all of this into account, we can attempt to simplify the Michaelis-Menten equation. The only step left is substituting these expressions into the equation.

$K_m=(\frac{[E_T]}{ES}-1)[S]$

$\frac{v_{max}}{v_0} = \frac{k_2[E_T]}{k_2[ES]}=\frac{[E_T]}{[ES]}$

And if we put it all together by substituting the second equation into the first one:

$K_m=(\frac{v_{max}}{v_0}-1) \cdot [S]$

$v_0=\frac{v_{max}[S]}{K_m+[S]}$

That is how you derive the Michaelis-Menten equation.

Interesting conclusions

Analysing the equations we just reached, we find other interesting conclusions. If an enzyme has a high $K_m$ , we know it will have low substrate affinity because $[E][S]>[ES]$ . On the contrary, if an enzyme has a low $K_m$ , we know it will have high substrate affinity since $[E][S]<[ES]$ .

Next, going back to the graph, we can appreciate that when $[S]=K_m$ , $v_0= \frac{v_{max}}{2}$ . Simply put, when the substrate concentration equals the enzyme’s $K_m$ , the reaction rate is exactly half of its maximum rate.

I find it truly amazing that such a simple equation can provide us with so much useful information. In the next few paragraphs, we will explore some more applications of this equation, but in the meantime, if you want to learn more about the processes of the cell, read my previous article about protein trafficking (in Spanish).

Applications of the Michaelis-Menten equation

If we’re being picky, the Michaelis-Menten equation does not have that many applications, but other formulas derived from it do.

Through a series of steps, we can express the reciprocal of the Michaelis-Menten equation. It even has a name: the Lineweaver-Burk plot. This is extremely useful to show if the kinetics of an enzyme are being changed by inhibitors.

$\frac{1}{v_{0}}= \frac{K_m}{v_{max}} \cdot \frac{1}{[S]}+ \frac{1}{v_{max}}$

Different times of inhibition shown using the Lineweaver-Burk plot

Inhibition can affect enzymes two ways: either their allosteric sites are occupied by the inhibitor, thus preventing them from catalysing reactions or the inhibitor competes with the substrate, meaning there is less desired product formed overall.

As you can see in the graphs, in the presence of a competitive inhibitor, the maximum velocity of the enzyme is unaffected, but it requires a higher substrate concentration. The reason for this is that the enzyme cannot distinguish between the inhibitor and its true substrate. The more substrate, the less likely the enzyme will form a complex with the inhibitor.

An uncompetitive inhibitor is slightly different. Both the $K_m$ and the $v_{max}$ decrease. This is because it binds to the enzyme-substrate complex as a whole. You can read more about these special types of inhibitors here.

In the presence of a noncompetitive inhibitor, the Michaelis-Menten constant stays the same. Namely, the concentration of substrate needed to reach half of the maximum velocity, remains unchanged. What is affected by this type of inhibitor is the maximum velocity of the reaction, that is reduced considerably since the inhibitor blocks the allosteric site of the enzyme and changes its shape.

There is still another form of inhibitors called mixed inhibitors, that show a mix of properties between these mentioned previously, but we will omit them in this article for simplicity.

Quick summary

We have learnt about the Michaelis-Menten equation, a useful tool that can be used to determine enzyme kinetics. I have shown you how to reach it using some simple observations from graphs and general enzyme knowledge. Lastly, we have seen the Lineweaver-Burk plot, which inevitably led to the discussion about inhibitors.

I think it is worthy to mention that the two scientists who first discovered this equation were Leonor Michaelis and Maud Menten, a biochemist and a physician, respectively. The point I would like to make with this is that, although it is a relatively simple equation, it required the efforts of two highly qualified individuals to find it. Sometimes, the most obvious things in science have to be worked out in teams. Thanks to this collaboration the discipline of Biochemistry has been able to evolve, especially in reference to kinetics.

I hope you have enjoyed the article and are looking forward to the next one!

Bibliography

Berg, J., Tymoczko, J., & Stryer, L. (2002). The Michaelis-Menten Model Accounts for the Kinetic Properties of Many Enzymes. Retrieved 23 August 2020, from https://www.ncbi.nlm.nih.gov/books/NBK22430/
Laidler, K. (2020). Chemical kinetics – Some kinetic principles. Retrieved 23 August 2020, from https://www.britannica.com/science/chemical-kinetics/Some-kinetic-principles
Le, H., Algaze, S., & Tan, E. (2020). Michaelis-Menten Kinetics. Retrieved 23 August 2020, from https://chem.libretexts.org/Bookshelves/Biological_Chemistry/Supplemental_Modules_(Biological_Chemistry)/Enzymes/Enzymatic_Kinetics/Michaelis-Menten_Kinetics
Lineweaver–Burk plot. (2020). Retrieved 23 August 2020, from https://en.wikipedia.org/wiki/Lineweaver–Burk_plot
Cover image: Odra Noel

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Lilly Pubillones

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Trinity College Hartford
Biochemistry

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What is the Michaelis-Menten equation?