A&S 320: Mathematical Modeling in Biology
David Murrugarra
Department of Mathematics,
University of Kentucky
http://www.ms.uky.edu/~dmu228/AS320/
Spring 2016
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
1 / 10
E. coli and the lac operon
The concept of operon was first described by Jacob and Monod in the
early 1960s. They shared the Nobel Prize in Physiology or Medicine
1965.
Jacques Monod.
David Murrugarra (University of Kentucky)
Francois Jacob.
A&S 320: Section 1.2-1.3
Spring 2016
2 / 10
E. coli and the lac operon
In general, an operon is a segment of DNA that includes a promoter,
an operator and one or more structural genes.
Two flavors of operon:
Inducible: the default status is OFF. In the presence of an inducer
binding to the promoter, the operon is induced, or turned ON.
Classic example: the lac operon
Repressible: the default status is ON. In the presence of a
repressor binding to the promoter, the operon is repressed or
turned OFF. Classic example: the trp operon
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
3 / 10
The lac operon – basic biology
E. coli prefer glucose, but they will use lactose as an energy
source if there is no glucose.
The lac operon is the cell machinery that transports external
lactose into the cell, forcing various proteins to be made so that
the lactose can be utilized by the cell and bringing in more lactose.
When the lactose is all used up, the operon shuts itself OFF.
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
4 / 10
No lactose means the operon is OFF
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
5 / 10
Lactose present means the operon is ON.
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
6 / 10
A Boolean Network Model of the Lac Operon
Wiring diagram
Nodes:
E: the LacZ polypeptide.
M: the mRNA.
L: internal lactose.
Parameters:
Le : external lactose.
Ge : external glucose.
A positive influence is indicated by an arrow. A negative
influence is depicted by a circle. Self-regulating links are
not pictured.
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
7 / 10
A Boolean Network Model of the Lac Operon
Regulatory Rules
E(t + 1) = M(t)
M(t + 1) = Ge ∧ (L(t) ∨ Le (t))
L(t + 1) = Ge ∧ ((E(t) ∧ Le (t)) ∨ (L(t) ∧ E(t)))
Parameters:
Le : external lactose.
Ge : external glucose.
Wiring diagram
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
8 / 10
Dynamics of the Model: State Space
The order of the variables is (M, E, L). A. State space for
Le = Ge = 0. B. State space for Le = 0 and Ge = 1.
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
9 / 10
Dynamics of the Model
State space for Le = 1 and Ge = 1
State space for Le = 1 and Ge = 0
David Murrugarra (University of Kentucky)
A&S 320: Section 1.2-1.3
Spring 2016
10 / 10