Biological Oscillations

Using the Goodwin Oscillator

as a model of negative feedbackJ. Watrous

Biology Department

St. Joseph’s University

July, 2008

The Goodwin oscillator model is used to describe and analyze the behavior of simple negative feedback systems like the one shown above. The model assumes a constant supply of DNA and substrate while the amounts of mRNA, enzyme and product will change with time. As the amount of product increases, it signals a decrease in mRNA production.

Modeling Equations

• Three equations are needed. Both DNA and substrate concentrations are assumed to be at much greater concentrations than mRNA, the enzyme machinery and the product made.

mRNA Production

A Michaelis-Menton type equation is used to describe mRNA production along with a term that accounts for any breakdown of nucleic acid.

dm/dt = ((D/(k+(p^q))) - a*m

Because the product is in the denominator, as it increases, it will reduce any increase in mRNA.

Enzyme Production

• The level of enzymatic activity is a function of two terms: the amount produced from mRNA and the amount broken down.

• de/dt = b*m - c*e

Product produced

• The product produced depends on the substrate available (assumed to be in excess), the reaction is dependent on the enzyme produced and any product breakdown.

• dp/dt = d*e - (n*p/(k+p)

Sample output using Madonna

The output shows a segment of the simulation using a set of parameter values where oscillations can be observed.

Parameter value plots can be constructed to see the role each plays in the simulation and to maximize the amount of product made.

References

Goodwin, B. 1965. Oscillatory behavior in enzymatic control processes. In: Advances in Enzyme Regulation 3: 425.

Murray, J.D. Mathematical Biology. 1989. Springer-Verlag

Biochemical Regulation 2008 obtained from:http://mcb.berkeley.edu/courses/mcb137/exercises/Biochemistry.pdf