For an enzyme-catalyzed reaction that follows the Michaelis-Menten kinetics as follows:

$$E + S \overset { { k }_{ 1 } }{ \underset { { k }_{ -1 } }{ \rightleftarrows } } ES \overset { { k }_{ 2 } }{ \rightarrow } E + P$$

The Michaelis-Menten equation is given by:

$$v = \frac { v_{ max }[S] }{ { K }_{ M } + [S] } $$

...where $v$ is the rate or velocity of the reaction, $[S]$ is the substrate concentration, ${v}_{max}$ is the maximum velocity, and ${K}_{M}$ is the Michaelis-Menten constant.

Starting with the rate of product formation:

$$\frac { d[P] }{ dt } = { k }_{ 2 }[ES]$$

Using the steady-state approximation for the concentration of the enzyme-substrate complex, $[ES]$, how do you express the Michaelis-Menten constant, ${K}_{M}$, as a function of the individual rate constants?

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