Multistability in the lactose utilization network of E. coli Lauren Nakonechny, Katherine Smith, Michael Volk, Robert Wallace Mentor: J. Ruby Abrams
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!