The powerpoint presentation for the in class "oral report" of the final project in Dr. Jeff Hasty's SP15 Modeling and Computation in Bioengineering course at UCSD by Joaquin Reyna, Lanie Happ, Rohit Mande and I.

1. 2D System of Lac
Operon Dynamics:
mRNA and Lactose
Joaquin Reyna
Lanie Happ
Rohit Mande
Derek Bever

2. Lac Operon Background
 “Adaptive enzymes” coined by André Lwoff from observing yeast (1940)
 Classical lac operon model developed by Francois Jacob and Jacques
Monod using E. Coli (1949 – 1950s)
 First genetic regulatory mechanism to be fully understood/documented
 The lac operon has become the foremost model/example of prokaryotic
gene regulation.

3. OFF
state
ON
state
Lac Operon Activity

4. Our Simplified Model
Assumption: Glucose concentration is low and the effect of cAMP is removed
M′ = basal transcription rate + mRNA induction via lactose - degradation of mRNA
L′ = uptake of lactose by permease - dilution of lactose - breakdown of lactose by
β-galactosidase

5. M′ = a + (b * L5)/(c + L5) - d * M
L′ = e * M - f * L - (g * M * L)/(h + L)
Our Simplified Model
mRNA Parameters Lactose Parameters
a - basal transcription rate
b - maximal transcription rate
c - transcription capacity as a
result of lactose activation
d - mRNA degradation rate
constant
e - rate constant of lactose influx as
a result of mRNA (= to permease)
f - lactose degradation rate
constant
g - maximal β-galactosidase
degradation rate
h - β-galactosidase activity
capacity as a result of lactose
activation

6. M’ = 0.05 + (L5 / (1 + L5)) - M
L’ = M - 0.2L - (ML / (2 + L))
Bistabilty in the Lac Operon
A
B
C
Fixed points:
A: (1.0388, 2.3717) =
nodal sink
Lac operon is ON
B: (0.18585, 0.69071) =
saddle point
C: (0.050605, 0.22721) =
nodal sink
Lac operon is OFF

7. The Change in mRNA and
Lactose over Time
mRNA vs. timeLactose vs. time

8. Changing the Dynamics by
Varying the Concentration of
External Lactose (e)
e << 1
(e = 0.3)
e = 1
e >> 1
(e = 3)

9. Lac Operon Dynamics using
Parameters from Literature1
Parameter Description Value
b
maximum transcription
initiation rate
~0.18 min-1
d
degradation rate of mRNA
in E. coli
~0.46 mRNA/min-1
e
maximum rate of permease
activity (lactose into cell)
~6.0X104 min-1
g
maximum rate of β-
galactosidase activity
(breakdown of lactose)
~3.8x104 min-1
1Santillan, M. “Bistable Behavior in a model of the lac Operon in Escherichia coli with Variable Growth Rate.”
Biophys Journal 2008 March 15. 94(6): 2065-2081

10. Fixed Point:.
A: (0.5, 55003.4485)
= nodal sink
Limitation:
Simple model does not
seem to exhibit
bistable behavior using
experimentally
determined
parameters.
mRNA is not the cap
for lactose. In reality it’s
β-galactosidase
production.
Lac Operon Dynamics using Parameters
from Literature
A

11. Further Model Development
 Include a glucose variable
 Include a protein variable
 Include a cell growth
variable
New Research Ideas
 Understand the effect of
multiple operator binding
sites
 Research the effect of
different lac operon alleles
on dynamics
Future Directions

12.  Santillán, M. and MC Mackey. “Quantitative approaches to
the study of bistability in the lac operon of Escherichia coli.” J R
Source Interface 5 (2008): S29-39
 Santillán, M. “Bistable Behavior in a model of the lac Operon in
Escherichia coli with Variable Growth Rate.” Biophys Journal
2008 March 15. 94(6): 2065-2081
 Yildirim, N. et. al. “Dynamics and bistability in a reduced model
of the lac operon.” Chaos 14 (2004): 279-92
 Díaz-Hernández O, Santillán M. Bistable Behavior of the Lac
Operon in E. Coli When Induced with a Mixture of Lactose and
TMG. Frontiers in Physiology. 2010;1:22.
doi:10.3389/fphys.2010.00022.
 Müller-Hill, Benno. The Lac Operon. Berlin; New York: Walter de
Gruyter, 1996. Print.
References

13. QQuestions
???