# Does the Michaelis-Menten equation take in account the non-enzyme formation of products?

I only recently learned about the Michaelis-Menten equation, since I am not studying biology or anything related. Let's write the equation as $$\frac{d[P]}{dt} = V_{max} \cdot \frac{[S]}{K + [S]}$$ If I'm not mistaken, this equation describes the rate at which products are formed ($\frac{d[P]}{dt}$) as a function of substrate concentration $[S]$ in steady-state. In a translation of Michaelis' paper I read that $V_{max}$ can be written as $k [E]$, where $k$ is a constant and $[E]$ is the total enzyme concentration.

Mathematically one would say that when the total enzyme concentration is zero ($[E] = 0$), the rate at which the product is formed should equal zero. If I recall correctly, in several reactions products are formed without enzymes (Enzymes only enhance the reaction rate?). Is the Michaelis-Menten equation specifically used for situations where $[E]>0$? If so, how does it account for very low enzyme concentrations where product formation from non-enzyme reactions cannot be neglected?

• Just for confirmation; the product formation rate would thus be $V_{0} + \frac{V_{max} \cdot [S]}{K+[S]}$ if blank rates were not subtracted? Is there a different notation/name for the rate of product formation when $V_{0}$ is not subtracted? – SMey1908 Mar 10 '16 at 15:52
• I don't know of any other name, but the equation you give is almost exactly that given by Dixon & Webb, p 461: $v$ = $w$ + ${V_{max}\ s}\over{{K_m + s}}$, where $w$ is the blank rate – user1136 Mar 10 '16 at 16:18