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Author: Subject: Enzymes, myoglobin and rectangular hyperbolas
Carbon8
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Enzymes, myoglobin and rectangular hyperbolas

When you plot the velocity of a typical enzymatic reaction (ie. one that displays Michaelis-Menten kinetics) as a function of the substrate concentration, you get a curve that is steep and linear at the beginning of the reaction. But as the substrate concentration increases and the enzyme binding sites become more and more occupied, the reaction velocity asymptotically approaches a maximum value (denoted Vmax), which occurs when the binding sites are saturated.

The equation for this curve can be written as: v = Vmax [S]/(Km + [S]), where [S] is the substrate concentration, Km is the Michaelis-Menten constant, and v is the velocity of the reaction.

Similarly, when you plot the fractional saturation curve of myoglobin as a function of the partial pressure of oxygen, you get a homologous curve, whose equation can be written as: Y(O2) = pO2/(Kd + pO2), where Y(O2) is the fraction of myoglobin molecules binding O2, Kd is the dissociation constant for myoglobin and O2, and pO2 is the partial pressure of O2 in the system.

In both of the above cases the equation can be simplified to the form: y = x/(x + constant). And if you plot a few examples of this kind of equation at a site like Desmos, you can produce a curve that looks like the enzyme and myoglobin curves (ie. steep at the beginning, but which flatten out as the enzyme and myoglobin binding sites become saturated with substrate and O2, respectively).

https://www.desmos.com/calculator

All of the biochemistry textbooks I have consulted (including Lehninger, Voet and Stryer) describes these kinds of curves as "rectangular hyperbolas." The typical formula for a rectangular hyperbola is: x^2 - y^2 = constant.

http://mathworld.wolfram.com/RectangularHyperbola.html
https://en.wikipedia.org/wiki/Hyperbola#Rectangular_hyperbol...

My question is this: why do these textbooks call an equation like y = x/(x + constant) a rectangular hyperbola?

I just don't see the connection.

P.S. I have attached a two-page PDF excerpt regarding myoglobin from a biochemistry textbook as an example.

Attachment: myoglobin excerpt.pdf (409kB)
Metacelsus
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It is a rectangular hyperbola, it's just rotated (and also normally only a small part of the hyperbola is plotted). See: https://en.wikipedia.org/wiki/Rotation_of_axes#Rotation_of_c...

When multiplied out the equation becomes: xy + (constant)*y - x = 0
This conforms to the general equation of a conic section, namely a hyperbola.

[Edited on 7-13-2018 by Metacelsus]

As below, so above.
Carbon8
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Thanks Metacelsus, your answer has prompted me to revisit some math from many years ago.

 Sciencemadness Discussion Board » Special topics » Biochemistry » Enzymes, myoglobin and rectangular hyperbolas Select A Forum Fundamentals   » Chemistry in General   » Organic Chemistry   » Reagents and Apparatus Acquisition   » Beginnings   » Miscellaneous   » The Wiki Special topics   » Technochemistry   » Energetic Materials   » Biochemistry   » Radiochemistry   » Computational Models and Techniques   » Prepublication Non-chemistry   » Forum Matters   » Legal and Societal Issues   » Detritus