in silico
Phase 1: In Silico Validation
- Objective
- To determine whether the predicted Hopf bifurcation and limit-cycle oscillations occur in a stochastic, delayed differential equation model of the minimal negative feedback circuit, and to identify the parameter regimes (τ_total, n, γ) that maximize oscillation robustness.
- Estimated cost
- €500-1500
- Estimated duration
- 4-6 weeks
- Success criteria
- Existence of a Hopf bifurcation boundary · A clear boundary in (τ_total, n, γ) space where the steady state becomes unstable and limit-cycle oscillations emerge · (AUTO-07p continuation analysis: detection of a pair of complex conjugate eigenvalues crossing the imaginary axis)
- Oscillation period scaling · T = 2·τ_total ± 20% for at least 3 values of τ_total (e.g., 30, 60, 90 min) · (Peak-to-peak interval from stochastic simulations, averaged over 10 cycles)
- Amplitude dependence on n · For n=1, amplitude A < 10% of the steady-state concentration for all τ_total < 120 min; for n=3, A > 50% for τ_total > 45 min · (Peak-to-trough amplitude from stochastic simulations)
- Effect of degradation rate γ · Increasing γ by 3x increases τ_crit by √3 (e.g., from 45 min to 78 min) · (Bifurcation analysis: τ_crit as a function of γ)
- Go if
- All four success criteria are met (Hopf boundary exists, period scales, n>1 required, γ effect confirmed)
- No-go if
- No Hopf bifurcation detected for any parameter set within the physiological range (τ_total < 120 min, n ≤ 4, γ ≥ 0.01 min⁻¹)
- Pivot if
- Oscillations are observed but with period T significantly different from 2·τ_total (e.g., T > 3·τ_total) — then refine the model to include distributed delays or additional feedback loops
- Risks
- Stochastic noise destroys oscillations (coherence too low)Probability: mediumMitigation: Increase system size (volume) in simulations; use a Langevin approximation; if noise is too high, consider a cell-free system with higher molecular counts
- Cell division (dilution) interrupts oscillations before they stabilizeProbability: highMitigation: Simulate with periodic dilution; if oscillations are lost, consider a slower-growing organism (e.g., yeast) or a microfluidic chemostat with constant cell density
- Bifurcation analysis fails due to stiff DDEsProbability: lowMitigation: Use adaptive time-stepping (PyDDE’s dde23 solver) or switch to a delay differential equation solver in Julia (DelayDiffEq.jl)