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. Source: BioCote Interest in applying mathematics to biology Intersectional knowledge
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 system Three 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 Combine three equations to retrieve steady state result (Ozbudak et al.) 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
Phase diagram of the lac system Cells shift between being uninduced and fully induced as parameters 1/ρ and αβ/ρ are varied Saddle node bifurcations Bistable region has hysteretic behavior System response Occurs either hysteretically, or in a graded fashion 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 - Bistable Figure: (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 conditions https://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 Figure: (Ozbudak et al.) ln(y) Values
Distribution of ln(y) Values as Extracellular TMG Levels Increase Theirs(left) vs. Ours(right) T=8 Figure: (Ozbudak et al.) T = 0.5 T = 12 T=1 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!