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Network Topology as a Driver of Bistability in the lac Operon
Brandilyn Stigler, Alan Veliz-Cuba
TL;DR
The paper asks whether lac-operon bistability requires detailed quantitative functions or can arise from network topology and interaction signs. It constructs and analyzes a Boolean network including glucose control, then reduces it to lac mRNA and lactose; both models show the reported ON/OFF dynamics, supporting topology as the key determinant within this model.
Problem
Prior lac-operon models are numerous and mostly continuous, while the existing discrete model does not predict bistability.
Method
The authors construct a Boolean network including catabolite repression and inducer exclusion, analyze its state-space dynamics, and reduce it by deleting vertices while preserving topological features.
Results
The full model has ON/OFF switching and, with stochasticity, bistability; the reduced lac mRNA–lactose model exhibits the same dynamics.
Takeaways & Limitations
The results support the hypothesis that network topology and interaction signs are key to the lac operon’s dynamical properties.
Takeaways & Limitations
A future multi-state framework may be needed for a more refined qualitative description incorporating features such as multiple promoter and operator regions.
Abstract
from arXiv · showhide
The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. We include the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits bistability. Further we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions.
1. Introduction
The lac operon motivates discrete modeling because existing formulations are mostly mathematical, while network topology and interaction signs may capture gene-network dynamics. The proposed Boolean model includes glucose control mechanisms and supports bistability, while reduction identifies a lac-gene/lactose core with matching dynamics.
- Modeling motivation: Most lac-operon models use complex mathematical formulations, especially differential-equation frameworks, whereas discrete models offer global state-space analysis.Boolean networks encode topology and interaction type through Boolean expressions and permit computation of the entire state space.
- Modeling motivation: The proposed Boolean model includes catabolite repression and inducer exclusion, the two main glucose control mechanisms.These mechanisms address glucose regulation of lac-operon induction.
- Modeling motivation: The model predicts bistability, with the lac operon occupying either an ON induced state or an OFF uninduced state.The paper reports only two steady states corresponding to these alternatives.
- Prior work and contribution: The paper extends prior discrete lac-operon modeling by addressing bistability and incorporating both glucose-control mechanisms.The authors describe the earlier Setty model as the first discrete model but note that it does not predict bistability.
- Modeling motivation: Reducing the network to lac genes and lactose preserves the full model’s dynamics, identifying these components as the core of the operon’s qualitative behavior.This reduction tests whether topology and interaction type, rather than detailed quantitative functions, determine the dynamics.
2. Model
The paper models lac-operon regulation with a Boolean network incorporating glucose control and analyzes its topology, steady states, and stochastic bistability. The model predicts glucose- and lactose-dependent ON/OFF behavior and reproduces bistable hysteresis under stochastic inducer uptake.
- Boolean Network: The Boolean network represents lac-operon mRNAs, proteins, and sugars as binary variables governed by logical update functions.The model includes catabolite repression and inducer exclusion, with extracellular lactose and glucose treated as parameters.
- Network Topology: Network topology reveals positive feedback loops through M → P → L → A → R → M and M → B → A → R → M, with no negative feedback loops.Edges encode regulatory dependencies and interaction signs in the wiring diagram.
- Dynamics: The model has a single OFF steady state when glucose is present or both extracellular glucose and lactose are absent, and an ON steady state when lactose is present without glucose.The four parameter combinations yield one steady state each, with the ON state occurring for (Le, Ge) = (1, 0).
- Bistability: Stochastic inducer uptake produces a bistable region in which induced and uninduced cells coexist and can switch states as inducer levels decrease or increase.The experiments use populations of cells, vary inducer levels, and evaluate the population after 10 time units.
- Bistability: The resulting in silico hysteresis experiments show the same qualitative pattern as earlier bistability experiments, whereas the corresponding behavior was absent without stochasticity.This identifies stochasticity in inducer uptake as necessary for the observed bistable behavior in model H.
3. Reduced Model
The Boolean network H is systematically reduced by simplifying functions, deleting nonfunctional edges, and removing vertices while inheriting their functionality. The resulting model h retains the M–L positive feedback loop and the original bistable ON/OFF dynamics, supporting a topological core centered on mRNA and lactose.
- Reduction method: The reduction method first simplifies Boolean functions and removes edges absent from the resulting expressions, then deletes eligible vertices while preserving their functionality through inherited regulation.Step (1) has priority over Step (2), which applies only to vertices without self-loops.
- Reduction steps: Deleting R replaces its incoming and outgoing edges with positive edges from Al and A to M, preserving the signs of the corresponding paths.This is the first illustrated application of step 2 to model H.
- Reduction steps: After deleting P, B, C, R, Al, and Ll, the reduced wiring diagram is described by HM = ¬Ge ∧(A ∨L ∨Le), HA = L ∧M, and HL = ¬Ge ∧M ∧Le.These functions retain regulation by glucose, external lactose, mRNA, and lactose in the reduced network.
- Reduced model: Deleting A and simplifying with L∧M ∨L = L produces model h, whose variables are M, L, Le, and Ge, with Le and Ge as parameters.The final Boolean functions are hM = ¬Ge ∧(Le ∨L) and hL = ¬Ge ∧Le ∧M.
- Reduced model: The reduction preserves the paths from Ge and Le to M and the positive feedback loop involving M and L.The reduced model therefore retains the key interaction structure identified in the full network.
- Dynamics and bistability: The reduced model has two steady states corresponding to ON and OFF, preserves the original dynamics, and retains a bistable region in experiments.It has one positive feedback loop rather than the several loops in H, suggesting that loop existence, not loop number, underlies the steady-state behavior; stochasticity and M–L interaction are implicated in bistability.
4. Discussion
The paper presents a Boolean lac-operon model that includes glucose control and reproduces ON/OFF switching and stochastic bistability. A reduced model identifies lac mRNA and lactose as the core while supporting a topology-based explanation of the dynamics.
- The Boolean model includes catabolite repression and inducer exclusion, and predicts bistability when stochasticity is included.
- The model reproduces the lac operon’s ON/OFF switching dynamics.
- The reduced model shows that lac mRNA and lactose form the core of the lac operon and exhibit the same dynamics.
- The findings support the claim that dynamical properties depend on network topology and interaction signs rather than quantitative component functions.
- A possible extension is a multi-state framework for a more refined qualitative description, including multiple promoter and operator regions.
Supporting Information
The supporting information specifies Boolean rules for model construction, including a repressor- and CAP-dependent mRNA rule and alternative glucose-control models.
- When repressor is high, mRNA production is low; when repressor is low and CAP is high, mRNA production is high.The rule is expressed as HM = ¬R ∧ C.
- Model H includes catabolite repression only, model J includes inducer exclusion only, and model K includes both mechanisms.
B.1. Network Topology.
Models J and K encode glucose-control variants with inhibitory glucose-to-mRNA paths and positive feedback involving mRNA. Their wiring diagrams share topological features with model H.
- The Boolean functions for models J and K define their state updates under inducer exclusion alone or combined glucose control.
- Both models contain inhibitory paths from extracellular glucose to mRNA and positive feedback loops involving mRNA, with no negative feedback loops.
- Models H, J, and K share common topological features.
B.2. Dynamics.
The dynamics analysis evaluates four extracellular lactose/glucose cases for models J and K and finds OFF states when glucose is available, but ON states when lactose is available without glucose. Reduced models retain these dynamics.
- B.2. Dynamics.: For model J, the three cases with either absent lactose or available glucose have a single steady state corresponding to the operon being OFF.
- B.2. Dynamics.: For model H, J, and K, extracellular glucose availability or simultaneous absence of extracellular glucose and lactose predicts an OFF operon.
- B.2. Dynamics.: When extracellular lactose is available and extracellular glucose is absent, all models predict an ON operon, yielding two qualitative steady-state outcomes.
- B.2. Dynamics.: The reduced models j and k are obtained from J and K, and their wiring diagrams are presented as equal in the reduced representation.
Appendix C. Bistability
The paper defines stability for Boolean networks by requiring nearby trajectories to converge to a steady state. Nearby states are measured using Hamming distance one.
- The definition formalizes stability as convergence of all nearby trajectories to the same steady state.This is the discrete analogue of the nearby-trajectory criterion introduced before the formal definition.
- A Boolean-network steady state x satisfies S(x) = x.Stability is evaluated relative to states differing from x in at most one component.
- A steady state is stable when every state y at Hamming distance at most one reaches x after some number of updates.The Hamming distance counts the components in which two states differ.
C.1.2. The two steady states of the lac operon operon are stable.
For any parameter set, the Boolean model has one steady state in which the lac operon is either ON or OFF. Both states satisfy the model’s stability definition.
- For any parameter set, the lac operon has a single steady state that is either ON or OFF.The passage identifies these as the two possible operating states of the operon.
- The ON and OFF states are both stable under the model’s definition.Thus each of the two operon states qualifies as a stable steady state.
- The model therefore represents bistability through stable induced and uninduced states.This follows directly from the stated stability of the ON and OFF states.
C.2. Stochastic Model.
The stochastic model introduces variation in inducer uptake through a normally distributed variable that is thresholded into a Boolean state. Its update rules specify the network’s component transitions as functions of regulatory variables and parameters.
- Inducer uptake is modeled by Le ∼ N(µ, σ), then thresholded to 0 below 1 and 1 at or above 1.The thresholded function is called the stochastic model, S.
- The stochastic Boolean network defines updates for mRNA, proteins, regulatory states, lactose, and inducer-related variables.The listed rules combine logical operators with glucose, inducer, lactose, and regulatory inputs.
- Glucose and inducer parameters enter the network through rules for glucose-dependent regulation and stochastic inducer states.The model includes Ge as a binary parameter and Le as a thresholded random variable.
- Changing the inducer level means changing µ while retaining the stochastic model structure.The paper treats increases or decreases in inducer as corresponding changes in the normal distribution’s mean.
- With Ge = 0, µ = 1.1, and σ = .1, the model generates successive Boolean states through the specified update rules.The example evaluates transitions from s0 to s1, s2, and s3, including sampled inducer values.
C.3. Heat Maps.
The heat-map experiments examine bistability across inducer-noise settings and across models. The reported results indicate bistability in models J and K and stable behavior under both inducer-switching directions.
- C.3. Heat Maps: Starting from Le ∼ N(1.25, .1) or Le ∼ N(0.75, .1), populations are evaluated after 10 time units while the inducer is shifted by .5.The upper panel decreases inducer from 1.25 to 0.75, whereas the lower panel increases it from 0.75 to 1.25.
- C.3. Heat Maps: The experiment reports both stable steady states at the same time under opposite inducer-switching directions.The supplied passage states this conclusion for the population experiments.
- C.3. Heat Maps: Models J and K both exhibit bistability in the reported heat maps.Their parameters are stated to match those used for model H.
- C.3. Heat Maps: Figure 12 compares bistability heat maps for σ = .15, .2 and σ = .05, .03.The larger-noise settings appear in the top maps and the smaller-noise settings in the bottom maps.