Transcript Biological Oscillations

Biological Oscillations
Using the Goodwin Oscillator
as a model of negative feedback
J. 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. SpringerVerlag
Biochemical Regulation 2008 obtained from:
http://mcb.berkeley.edu/courses/mcb137/exercises
/Biochemistry.pdf