Michaelis–Menten kinetics
Michaelis–Menten kinetics
![{\displaystyle [{\ce {P}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c666f77e34c220257d5840b2615c6882aabb89)
![{\displaystyle [{\ce {S}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea89dfff9e9cad8e883f46fc400e35d23e6eb914)
![{\displaystyle v={\frac {\mathrm {d} [{\ce {P}}]}{\mathrm {d} t}}={\frac {V_{\max }{[{\ce {S}}]}}{K_{\mathrm {M} }+[{\ce {S}}]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94a7743232865ec7bb1e0de0d4e3b6bff3b8be4f)
Model
In 1901, French physical chemist Victor Henri found that enzyme reactions were initiated by a bond (more generally, a binding interaction) between the enzyme and the substrate.[2] His work was taken up by German biochemist Leonor Michaelis and Canadian physician Maud Menten, who investigated the kinetics of an enzymatic reaction mechanism, invertase, that catalyzes the hydrolysis of sucrose into glucose and fructose.[3] In 1913, they proposed a mathematical model of the reaction.[4] It involves an enzyme, E, binding to a substrate, S, to form a complex, ES, which in turn releases a product, P, regenerating the original enzyme. This may be represented schematically as
![{\displaystyle {\ce {E{}+S<=>[{\mathit {k_{f}}}][{\mathit {k_{r}}}]ES->[k_{\ce {cat}}]E{}+P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bae00cd4d91917219daf235391295adeb36c84d)
Under certain assumptions – such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by
![{\displaystyle v={\frac {d[{\ce {P}}]}{dt}}=V_{\max }{\frac {[{\ce {S}}]}{K_{\mathrm {M} }+[{\ce {S}}]}}=k_{\mathrm {cat} }[{\ce {E}}]_{0}{\frac {[{\ce {S}}]}{K_{\mathrm {M} }+[{\ce {S}}]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798b38b6f94f0236206d9a34f6a9be342f7bcd02)
![{\displaystyle [{\ce {S}}]\ll K_{M}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70e5ed70da3fe7f5f1ab381c37bf7adff9d995e5)
![{\displaystyle v=k_{\mathrm {cat} }[{\ce {E}}]_{0}{\frac {[{\ce {S}}]}{K_{\mathrm {M} }}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acebe4b4474bd665906cdb0401011917ef508ca7)
![{\displaystyle {\ce {[S]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5909e9989dfe9306325e8dab287928f3c984ee3)
![{\displaystyle [{\ce {S}}]\gg K_{M}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/441b1072d0cd1724042f6a61792adf48b92dfde1)
![{\displaystyle V_{\max }=k_{\ce {cat}}[{\ce {E}}]_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6f14353feb260ab781c8ec562dcb93247e57ee)
![{\displaystyle {\ce {[E]_0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a654ad516b5d2547fd9980bb7590464441d6c38)
The model is used in a variety of biochemical situations other than enzyme-substrate interaction, including antigen-antibody binding, DNA-DNA hybridization, and protein-protein interaction.[7][10] It can be used to characterise a generic biochemical reaction, in the same way that the Langmuir equation can be used to model generic adsorption of biomolecular species.[10] When an empirical equation of this form is applied to microbial growth, it is sometimes called a Monod equation.
Applications
Parameter values vary widely between enzymes:[11]
| Enzyme | (M) | (s−1) | (M−1s−1) |
|---|---|---|---|
| Chymotrypsin | 1.5 × 10−2 | 0.14 | 9.3 |
| Pepsin | 3.0 × 10−4 | 0.50 | 1.7 × 103 |
| T-RNA synthetase | 9.0 × 10−4 | 7.6 | 8.4 × 103 |
| Ribonuclease | 7.9 × 10−3 | 7.9 × 102 | 1.0 × 105 |
| Carbonic anhydrase | 2.6 × 10−2 | 4.0 × 105 | 1.5 × 107 |
| Fumarase | 5.0 × 10−6 | 8.0 × 102 | 1.6 × 108 |
The equation can also be used to describe the relationship between ion channel conductivity and ligand concentration.[17]
Derivation
![{\displaystyle {\begin{aligned}{\frac {d[{\ce {E}}]}{dt}}&=-k_{f}[{\ce {E}}][{\ce {S}}]+k_{r}[{\ce {ES}}]+k_{\mathrm {cat} }[{\ce {ES}}]\\[4pt]{\frac {d[{\ce {S}}]}{dt}}&=-k_{f}[{\ce {E}}][{\ce {S}}]+k_{r}[{\ce {ES}}]\\[4pt]{\frac {d[{\ce {ES}}]}{dt}}&=k_{f}[{\ce {E}}][{\ce {S}}]-k_{r}[{\ce {ES}}]-k_{\mathrm {cat} }[{\ce {ES}}]\\[4pt]{\frac {d[{\ce {P}}]}{dt}}&=k_{\mathrm {cat} }[{\ce {ES}}].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe55cee321a6fc6e858b6e0a1473fff674f7ed7)
![[E] + [ES] = [E]_0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad13d24a63d5ee09ed542b5a281471be77a68b09)
Equilibrium approximation
![{\displaystyle k_{f}[{\ce {E}}][{\ce {S}}]=k_{r}[{\ce {ES}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a027a53d0cef2086d7b096e9ed7a1952d708346b)
From the enzyme conservation law, we obtain[19]
![{\displaystyle [{\ce {E}}]=[{\ce {E}}]_{0}-[{\ce {ES}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82aa61ab7633797beff11f364f755a233e74b1df)
Combining the two expressions above, gives us
![{\displaystyle k_{f}([{\ce {E}}]_{0}-[{\ce {ES}}])[{\ce {S}}]=k_{r}[{\ce {ES}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4895b5689ea10ae2549c696175c3402c30f00c8)
Upon simplification, we get
![{\displaystyle [{\ce {ES}}]={\frac {[{\ce {E}}]_{0}[S]}{K_{d}+[{\ce {S}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097cd93112fd2bce222f6065b551bda9ce86ea80)
![{\displaystyle v={\frac {d[{\ce {P}}]}{dt}}={\frac {V_{\max }{[{\ce {S}}]}}{K_{d}+[{\ce {S}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e39bd00c424f1dc38b1d1c447eec121ab6d33ea)
![{\displaystyle V_{\max }=k_{\mathrm {cat} }[{\ce {E}}]_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9476dfc33c4c323b9becb9fa1e6bbe4fb86845)
Quasi-steady-state approximation
![{\displaystyle k_{f}[{\ce {E}}][{\ce {S}}]=k_{r}[{\ce {ES}}]+k_{\mathrm {cat} }[{\ce {ES}}]=(k_{r}+k_{\mathrm {cat} })[{\ce {ES}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f44026ab5de910033fa61c3b882f915c34c6743f)
![{\displaystyle v={\frac {V_{\max }{[{\ce {S}}]}}{K_{\mathrm {M} }+[{\ce {S}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bde0878e92a154c2c8dbcfbe93221614f545f53)
where
is known as the Michaelis constant.
Assumptions and limitations
The first step in the derivation applies the law of mass action, which is reliant on free diffusion. However, in the environment of a living cell where there is a high concentration of proteins, the cytoplasm often behaves more like a gel than a liquid, limiting molecular movements and altering reaction rates.[21] Although the law of mass action can be valid in heterogeneous environments,[22] it is more appropriate to model the cytoplasm as a fractal, in order to capture its limited-mobility kinetics.[23]
![{\displaystyle \varepsilon _{m}={\frac {\ce {[E]_{0}}}{[{\ce {S}}]_{0}+K_{\ce {M}}}}\ll 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/438c880796d0bff343d10a74392b3dba9e7c1fd8)
It is also important to remember that, while irreversibility is a necessary simplification in order to yield a tractable analytic solution, in the general case product formation is not in fact irreversible. The enzyme reaction is more correctly described as
![{\displaystyle {\ce {E{}+S<=>[{\mathit {k_{f_{1}}}}][{\mathit {k_{r_{1}}}}]ES<=>[{\mathit {k_{f_{2}}}}][{\mathit {k_{r_{2}}}}]E{}+P.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b661b61bc9cfd6e6dd3eb133e7cfd63d6deafb6b)
In general, the assumption of irreversibility is a good one in situations where one of the below is true:
- The concentration of substrate(s) is very much larger than the concentration of products:
![{\displaystyle {\ce {[S]\gg [P].}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6505b0b596821fdbe306e0357a8a25fcb0246f1a)
This is true under standard in vitro assay conditions, and is true for many in vivo biological reactions, particularly where the product is continually removed by a subsequent reaction.
- The energy released in the reaction is very large, that is
In situations where neither of these two conditions hold (that is, the reaction is low energy and a substantial pool of product(s) exists), the Michaelis–Menten equation breaks down, and more complex modelling approaches explicitly taking the forward and reverse reactions into account must be taken to understand the enzyme biology.
Determination of constants
![[S]](https://wikimedia.org/api/rest_v1/media/math/render/svg/292bbb82029aa583c5d2ac5fa1d7e4fedf537d8b)
In 1997 Santiago Schnell and Claudio Mendoza suggested a closed form solution for the time course kinetics analysis of the Michaelis–Menten kinetics based on the solution of the Lambert W function.[28]
Namely,
![{\displaystyle {\frac {[{\ce {S}}]}{K_{\mathrm {M} }}}=W(F(t))\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9a2b7a3ffec425635e862d91323b89d18d5051)
where W is the Lambert W function and
![{\displaystyle F(t)={\frac {[{\ce {S}}]_{0}}{K_{\mathrm {M} }}}\exp \!\left({\frac {[{\ce {S}}]_{0}}{K_{\mathrm {M} }}}-{\frac {V_{\max }}{K_{\mathrm {M} }}}\,t\right)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b838ca40a839b85dd58c1a5590925d5a0d5583ec)
Role of substrate unbinding
The Michaelis-Menten equation has been used to predict the rate of product formation in enzymatic reactions for more than a century. Specifically, it states that the rate of an enzymatic reaction will increase as substrate concentration increases, and that increased unbinding of enzyme-substrate complexes will decrease the reaction rate. While the first prediction is well established, the second is more elusive. Mathematical analysis of the effect of enzyme-substrate unbinding on enzymatic reactions at the single-molecule level has shown that unbinding of an enzyme from a substrate can reduce the rate of product formation under some conditions, but may also have the opposite effect. As substrate concentrations increase, a tipping point can be reached where an increase in the unbinding rate results in an increase, rather than a decrease, of the reaction rate. The results indicate that enzymatic reactions can behave in ways that violate the classical Michaelis-Menten equation, and that the role of unbinding in enzymatic catalysis still remains to be determined experimentally.[31]
See also
Enzyme kinetics
Functional response
Lineweaver–Burk plot
Reaction progress kinetic analysis
Steady state (chemistry)
Hill equation (biochemistry)