2D System of Lac Operon Dynamics: mRNA and Lactose

Joaquin ReynaLanie Happ

Rohit MandeDerek Bever

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.

OFF stat

e

ON stat

e

Lac Operon Activity

Our Simplified Model

Assumption: Glucose concentration is low and the effect of cAMP is removedM′ = basal transcription rate + mRNA induction via lactose - degradation of mRNA

L′ = uptake of lactose by permease - dilution of lactose - breakdown of lactose byβ-galactosidase

M′ = a + (b * L5)/(c + L5) - d * ML′ = e * M - f * L - (g * M * L)/(h

+ L)

Our Simplified Model

mRNA Parameters

Lactose Parametersa - basal transcription rate b - maximal transcription ratec - transcription capacity as a result of lactose activationd - mRNA degradation rate constant

e - rate constant of lactose influx as a result of mRNA (= to permease)f - lactose degradation rate constantg -  maximal β-galactosidase degradation rateh - β-galactosidase activity capacity as a result of lactose activation

M’ = 0.05 + (L5 / (1 + L5)) - ML’ = M - 0.2L - (ML / (2 + L))

Bistabilty in the Lac Operon

A

B

C

Fixed points:A: (1.0388, 2.3717) = nodal sinkLac operon is ON

B: (0.18585, 0.69071) = saddle point

C: (0.050605, 0.22721) = nodal sinkLac operon is OFF

The Change in mRNA and Lactose over Time

mRNA vs. time

Lactose vs. time

Changing the Dynamics by Varying the Concentration of External Lactose (e)

e << 1(e = 0.3)e = 1e >> 1(e = 3)

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

gmaximum 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

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

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

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

QQuestions???