Enzyme Kinetics - Given Km find substrate concentration at a certain velocity

From what I understand about the Michaelis Menten Model 


* Km defines the amount of substrate required to reach half-saturation.
* 1/2 Vmax corresponds to Km on the x axis
* Generic formula is v = Kcat [ES] [S] / (Km + [S])


**Given Km and the desired velocity in terms of reaction efficiency, can one just plug in Km into the equation and get a relational expression with [S] on one side?** 


**Or do you manipulate the 1/2 Vmax expression?** 




You can solve for [S]. There are two common ways to write the equation, like you have it: 
$$v\_0 = k\_\text{cat} [E\_0] \frac{[S]}{K\_\text{m} + [S]}$$


or after dividing both numerator and denominator by [S]:


$$v\_0 = k\_\text{cat} [E\_0] \frac{1}{\frac{K\_\text{m}}{[S]} + 1}$$


In this form, it is easier to see that [S] occurs once and you can solve for it.


Get [S] out of the denominator of the "big" fraction and isolate the "little" fraction:


$$\frac{K\_\text{m}}{[S]} + 1 = \frac{k\_\text{cat} [E\_0]}{v\_0}$$


$$\frac{K\_\text{m}}{[S]} = \frac{k\_\text{cat} [E\_0]}{v\_0} - 1$$


Finally, take the reciprocal and multiply by $K\_\text{m}$:


$${[S]} = \frac{K\_\text{m}}{\frac{k\_\text{cat} [E\_0]}{v\_0} - 1}$$


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You can also solve for $\mathrm{[S]} $ using Lineweaver-Burk plot (or double reciprocal plot). One way of writing the Michaelis Menten kinetic equation given in [Wikipedia](https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics) is: 
$$v\_{\circ} = V\_\mathrm{Max} \frac {[S]}{K\_\mathrm{M} + [S] } = = k\_\text{cat} \mathrm{[E]\_{\circ}} \frac {[S]}{K\_\mathrm{M} + [S]} $$


If you take the reciprocal of both sides of the equation, you get: 


$$\frac{1}{v\_{\circ}} = \frac{K\_\mathrm{M} + \mathrm{[S]}}{V\_\mathrm{Max} \mathrm{[S]}} = \left(\frac{K\_\mathrm{M}}{V\_\mathrm{Max}}\right)\frac{1}{\mathrm{[S]}} + \frac{1}{V\_\mathrm{Max}}$$


Thus, a plot of $\frac{1}{\mathrm{[S]}}$ vs $\frac{1}{v\_{\circ}}$ is a straight line, which is called Lineweaver-Burk plot:


[![Lineweaver-Burk plot](https://i.stack.imgur.com/RVocG.jpg)](https://i.stack.imgur.com/RVocG.jpg)


Now, if you know the values of $K\_\mathrm{M}$ and $V\_\mathrm{Max}$ for your enzyme and substrate, you can draw a plot upper-hand ($x$-$\text{intercept} = -\frac{1}{K\_\mathrm{M}}$ and $y$-$\text{intercept} = \frac{1}{V\_\mathrm{Max}}$) and determine your substrate concentration ($\mathrm{[S]}$) for each desired velocity ($v\_{\circ}$).


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