Deterministic model of oscillator

Background:
We derive the set of delay-differential equation model for intracellular concentrations of LuxI (I), AiiA (A), internal AHL (H_i), and external AHL (H_e).

In the deterministic model, the concentration of the constitutively produced LuxR protein R is assumed constant.
In the first two equations, the Hill function

describes the delayed production of corresponding proteins, it depends on the past concentration of the internal AHL, . The delayed reactions include the complex processes (transcription, translation, maturation, etc.), leading to formation of functional proteins.

1. The factor describes slowing down of protein synthesis at high cell density d due to lower nutrient supply and high waste concentration.
2. γ_x describes enzymatic degradation of proteins and AHL by proteases inside of the cell due to their degradation tags(modeling these processes by Michaelis-Menten kinetics).
3. D describes diffusion of AHL by external fluid flow. The cell density (d) determines the amount of external AHL and thus affects the AHL decay rate.
4. The last term in equation for H_e describes the diffusion of external AHL.

Parameters set:
We use the following experimentally relevant scaled parameters in most of simulations.


Result:

Figure 2. The simulation result show that a typical time series of concentrations of LuxI (cyan circles), AiiA (blue circles), internal AHL (green line) and external AHL (red line). LuxI and AiiA closely track each other, and are anti-phase with the concentrations of external and internal AHL.

"degrade-and-fire" model

Our simplified model is based on apreviously described model for the oscillator. In addition to the H₂O₂ produce reactions reflected in that model. From the biochemical reactions, we derived a set of delay differential equations to be used as our model. These delayed reactions mimic the complex cascade of processes (transcription, translation, maturation, etc.) leading to formation of functional proteins.

Why should we simplify the model of oscillation
Deterministic model of oscillator requires a high cost of numerical method. And it’s hard to simulate the model when the oscillating colonies across a microfludic array.
To model the oscillating colonies across a microfluidic array, we set up a simplified "degrade-and-fire" model.

Background: Delayed Feedback Models
The delayed negative feedback-only (NFB) oscillator is modeled by two reactions describing delayed birth and death processes for repressor protein. The rates K₊ and K_- for production and degradation of repressor σ at the time t are

Where the subscript τ here and below denotes the value of a variable taken at time τ before the current time t, . Notice that a molecular derivation of the function F(ψ) may provide a form such as instead of , but we use the latter for simplicity. Equations (1)-(3) can be simulated with a modified Gillespie algorithm, described previously.
We use a similar model for coupled positive-negative feedback (PNFB) which is inspired by our recent experimental study. The rates K^(r) and K^(a) for repressor and activator, as the form

The τ_a is the delay time for the production of activator, and f is a parameter representing the strength of positive feedback. The activated rate for repressor production is α, and the corresponding rate is α/f.
The effective treatment of dilution with a first order dilution rate β in the above models warrants a brief discussion. Dilution in the context of gene circuits is a reduction of the intracellular concentration of a species due to an increase in cell volume. For the systems, the β-terms for the degradation rates adequately model the effect of dilution.

The delay-differential equation:

describe oscillations of individual biopixel as a combined effect of production and delayed autorepression of the colony-averaged LuxI concentration X and its enzymatic degradation by CloXP (second term). The first (production) term describes both delayed auto-repression of LuxI and its delayed activation by H₂O₂ proportional to its concentration P. Subscripts τ₁ and τ₂ indicate the delayed concentrations, and . The dynamics of P is described by the equation

Where the three terms describe the basal and induced production and degradation of H₂O₂. We use the following dimensionless parameters for most of our simulations:

Result:

Figure 3. The simulation result of oscillaiton through "degrade-and-fire" model

At the same time, the model is able to capture the alternating large and small amplitude oscillations observed in the ON/OFF biosensor. This behavior is seen when C₀ is increased 2-fold, capturing the decreased level of LuxR where it is the limiting factor for the oscillations.
The simulation result shows that the alternating oscillations vanish when LuxR is restored to its normal level in the model.

Reference:

[1] Danino. T, Mondragon-Palomino, O, Tsimring, L. & Hasty, J. A synchronized quorum of genetic clocks. Nature 463, 326-330 (2010).
[2] Arthur Prindle, Phillip Samayoa, Ivan Razinkov, Tal Danino, Lev S. Tsimring & Jeff Hasty. A sensing array of radically coupled genetic 'biopixels'. Nature 481, 39-44 (2012).
[3] William Mather, Matthew R. Bennett, Jeff Hasty and Lev S. Tsimring. Delay-induced degrade-and-fire oscillations in small genetic circuits. Phys. Rev. Lett. 102, 068105 (2009).