Multistability in the lactose utilization network of E. coli

Lauren Nakonechny, Katherine Smith, Michael Volk, Robert Wallace

Mentor: J. Ruby Abrams

Motivation

● Understanding biological switches in the context of multistability in mathematical systems

● Recreating, verifying, and expanding upon mathematical results from “Multistability in the lactose utilization network of Escherichia coli” by Ozbudak et al.○ Interest in applying mathematics to

biology○ Intersectional knowledgeSource: BioCote

Multistability of (general) systems● Multiple internal states in response to

single set of external outputs● Biological “switches”● Positive feedback loops typically

responsible for multistability● Phase diagrams

○ Quantitatively modelling parameters of biological systems; measuring internal states as external parameters vary

○ Determine requirements for switch within a system

○ Predict behavior of systems Source: BioNinja

Biological background● Escherichia coli

○ Bacteria in mammalian gut○ Metabolize glucose as preferred

source of carbon○ In absence of glucose, will

metabolize lactose

● lac operon○ Gene segment in E. coli DNA○ Responsible for expression of

enzymes used to break down lactose into simpler sugars

○ Selectively turned on and off; strictly regulated

Source: University of Arizona Dept. of Chemistry & Biochemistry

Key players of the lac operon● Repressor protein (LacI): prevents

operon expression○ Always bound to operator○ Blocks transcription of gene○ Removed in presence of lactose

● CAP-cAMP complex: facilitates efficient lactose metabolism

○ Cyclic AMP (cAMP) binds to catabolite activator protein (CAP)

○ Assists with attachment of RNA polymerase

○ High levels of cAMP in presence of glucose

Source: Khan Academy

How does it all work together?Situation Repressor? CAP-cAMP complex? Operon Expression

High glucose, no lactose Bound to operator Low cAMP; not attached None

Glucose and lactose Released from operator Low cAMP; not attached Some; inefficient

No glucose, high lactose Released from operator High cAMP; attached to promoter

High

No glucose, no lactose Bound to operator High cAMP; attached to promoter

None

Biological methods● Vary 2 external inputs: extracellular

concentrations of glucose and TMG in 1000 E.coli cells

○ TMG = non-metabolizable lactose analog○ Measure levels of fluorescent reporter proteins:

■ GFP - measures operon expression■ HcRed - measures CRP-cAMP levels

● KEY:○ Red arrow - activation○ Red blunt end - inhibition○ Black arrow - protein creation○ Dotted arrow - uptake across cell membrane

Figure: (Ozbudak et al.)

Modeling the lac systemThree equations (Ozbudak et al.): Parameters:

● R = active concentration of LacI● RT = total concentration of LacI● x = intracellular concentration of TMG● x0 = half-saturation of TMG● n = Hill coefficient● y = concentration of LacY● τx, τy = time constants● α = maximum growth of LacY● β = measure of TMG uptake per LacY

Modeling the lac system

● α - lac expression level obtained if every repressor molecule were inactive (maximum induction)

● ρ (repression factor) - ratio of maximal to basal (read: every repressor molecule is active) activity

● β (transport rate) - TMG uptake rate per LacY molecule

Combine three equations to retrieve steady state result (Ozbudak et al.)

Phase diagram of the lac system● Cells shift between being uninduced and fully

induced as parameters 1/ρ and αβ/ρ are varied○ System response○ Occurs either hysteretically, or in a graded fashion

● Saddle node bifurcations● Bistable region has hysteretic behavior

○ Cusp at ρ=9● Beyond cusp, response occurs in graded fashion

○ Expression levels of individual cells move continuously between values

Figure: (Ozbudak et al.)

Our work● Exploration of the parametric

equations that describe the boundary of the bistable region

● Verifying plots based off of dynamic equations

● Developing general Matlab framework

Figure: (Ozbudak et al.)

Our work● Red Dot - Inflection point● Blue Line: Trajectory as μ is

changed

Midterm future work● Generate data and model

trajectories with differential equations

● Arrows indicate initial conditions

● Red : TMG > 30 μM to turn on initially uninduced cells

● Blue : TMG < 3 μM to turn off initially induced cells

● Proves hysteresis Grey Region - BistableFigure: (Ozbudak et al.)

Sigmoidal switching● Given a specified amount of time and

a given parameter set y,x will into the switch “on/off” position - solving the ODE using ODE45 in Matlab

● Plot (dy/dt) cubic function to find the fixed points

● Fixed points will provide an indication of how switching occurs

Linearization - Jacobian● Linearize the system using

the Jacobian ● Evaluate the determinant

and the trace● Possible fixed points are

saddles and stable nodes

Figure: (Ozbudak et al.)

1 stable fixed point● Red circle indicates stable node

PPlane

3 stable fixed points● Red circles indicate stable nodes● Black circle indicates saddle

PPlane

1 stable fixed point● Red circle indicates stable node

PPlane

ODE Matlab Simulation● Adjust parameter T● Provide a set of initial conditions - black

dots ● Run system for a given time and track final

yf values - red dots● Background hue depicts the vector field -

final values are expected to be found in the lighter regions

Figure: (Ozbudak et al.)

ODE Matlab Simulation● Compare yf values with

experimental data● Switching occurs around T = 10

Figure: (Ozbudak et al.)

ODE Matlab Simulation● Once T has increased from T= 2

to T = 30 the processing of lactose has completely switched into the “on” position - meaning lactose is now being processed by the cell

Figure: (Ozbudak et al.)

Simulation comparison to experimental data● Turning “on” occurs at lower

values of T for the system of ODEs

● Turning “on/off” is dependent on the selection of initial conditions - hysteresis

Figure: (Ozbudak et al.)

Time evolution of initial conditionshttps://www.youtube.com/watch?v=V-MgNAJtNJw&feature=youtu.be

https://www.youtube.com/watch?v=kL_k8FZYmoQ&feature=youtu.be

https://www.youtube.com/watch?v=AHUsa-4BrcQ&feature=youtu.be

Qualitative comparison to Shraiman’s data

ln(Y) ValuesFigure: (Ozbudak et al.)

Figure: (Ozbudak et al.)

Distribution of ln(Y) Values as Extracellular TMG Levels Increase

Theirs(left) vs. Ours(right)T = 0.5 T = 1

T = 8 T = 12 T = 20

Qualitative comparison to experimental data

Figure: (Ozbudak et al.)

Sources & Acknowledgments● Ozbudak, Ertugrul M., Thattai, Mukund, Lim, Han N., Shraiman, Boris I., & van

Oudenaarden, Alexander. Multistability in the lactose utilization network of Escherichia coli. Nature. 427, 737-740 (2004).

● Hansen, L. H., Knudsen, S. & Sorenson, S. J. The effect of the lacY gene on the induction of IPTG inducible promoters, studied in Escherichia coli and Pseudomonas fluorescens. Curr. Microbiol. 36, 341-347 (1998).

● Yagil, G. & Yagil, E. On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys. J. 11, 11-27 (1971).

A huge THANK YOU to our mentor, J. Ruby Abrams, and to Dr. Ildar Gabitov for their huge contributions to this project!