Saturating Hyperbola Relationship

A verbal description of the relationship

A saturating hyperbola function will increase, with diminishing returns until it asymtoptically approaches a limit. One equation for this is to multiply the maximum limit value by the factor (X/(X+K)).

The equation and the meaning of x, y, and other parameters

Y = m* X/(X+K)

m is the maximum value

K is the half-saturation value for X, when X=K, Y will equal m/2

One or more examples

The growth rate of a plant is a function of the resources available. As resources are depleted, the growth rate decreases. IF resources are replete, the growth rate approaches a maximum value of 10 grams per day per plant. The growth limitation has a 1/2 saturation consant of 3 grams/liter of PO4 in solution. What is the growth rate as a function of grams per liter of PO4 in solution.

The equation would be

growth_rate = max_growth_rate * conc/(conc+K)

growth_rate = 10 * conc/(conc + 3)

 

A graph

Algebraic rules that apply to the use of this equation

One of the algebraic manipulations of this equation that you may see is to represent it as a "double reciprocal" form. In this form, it can can be linearized which makes the estimation of m and K a matter of fitting to a linear equation instead of this curve.

I won't go into this here, but if you have Biology this is called the Lineweaver-Burke plot of the Michaelis-Menten equation.

Characteristic values

For the equation Y = m* X/(X+K)

at X = 0, Y = 0 (and vise verse)

at X = K, Y = m/2 (half the maximal value)

at high X, Y approaches m

Y can't be greater than m