Table 1.
Parameters of the lac operon reaction model.
Fig 1.
Reduction of deterministic model.
Comparison between the full deterministic model (solid line) described by Eqs (2) with the reduced model (dashed line), which is described by Eq (7) for two different external IPTG concentrations leading to (a) an uninduced ([Iex] = 10μM) and (b) an induced state ([Iex] = 40μM). Kinetic constants are obtained from Table 1.
Fig 2.
Langevin vs Monte Carlo single-cell lac operon simulations.
Comparison between the Langevin approximation (black solid line) described by Eq (15) with Monte Carlo simulations (grey line with circles), using the Gillespie algorithm for the Master-equation of reactions Eqs (1) for two different external IPTG concentrations leading to (a) an uninduced ([Iex] = 15μM) and (b) an induced state ([Iex] = 45μM). 100 simulation copies are used to compute the average evolution.
Fig 3.
A schematic of the coarse time-stepper for the model of an isogenic cell population simulated by the CNMC algorithm.
Fig 4.
(a) Relative error between five “original” coefficients and coefficients computed after lifting. The solid lines correspond to relative errors, while dashed lines correspond to stochastic noise quantified by ; σαj is the standard deviation of the noise of coefficients αj computed using 50 direct CNMC copies. (b) The original (solid line) and the lifted (dashed line) ICDF distributions using 4 orthogonal basis functions at dimensionless time τ = 10.5.
Fig 5.
Comparison of full long temporal simulations with coarse steady state computations.
Solid lines with open triangles correspond to number density functions, n([Y]), obtained from long temporal simulations, who have practically reached a steady state. Dashed lines with open rectangles correspond to number density functions obtained from Newton-Rapshon coarse steady state computations. Good agreement is observed for both (a) low level and (b) high level lacY expression levels. Parameter values: K = 500, y* = 50, π = 0.03, m = 2, f = 0.5, N = 10,000 cells. We use 50 copies of CNMC simulations for stochastic noise reduction purposes.
Fig 6.
Effect of intrinsic heterogeneity.
(a) Steady state average expression level, ⟨[Y]⟩, as a function of the external IPTG concentration, [Iex]. The black lines (solid and dashed) correspond to the homogeneous model, the lines with open circles to the DCPB model; the lines with open squares correspond to the CNMC model neglecting the intrinsic source of heterogeneity and the lines with open triangles to the CNMC model incorporating intrinsic noise effects. (b) Steady state solutions of the number density function, n([Y]), corresponding to the upper stable solution branches of CNMC simulations for [Iex] ≈ 27μM, when intrinsic noise is incorporated (K = 500, y* = 50) and when neglected (K, y* → ∞).
Fig 7.
Eigenvalues of the matrix corresponding to (a) the upper branch stable steady state, (b) the intermediate branch unstable steady state, and (c) the lower branch stable steady state solution, for [Iex] = 28.8μM (ρ = 0.09). The loss of stability in (b) is marked by the eigenvalue crossing the unit circle (dashed line) in the complex plane. Parameter set values: f = 0.5, m = 2, π = 0.03, κ = 0.05, K = 500, y = 50, and N = 10,000 cells.
Fig 8.
Effect of operator fluctuations, K, and reference number of molecules, y*.
Effect of different sources of intrinsic heterogeneity on the average expression, ⟨[Y]⟩, as a function of the external IPTG concentration, [Iex]: (a) Effect of parameter K: lines with open circles correspond to K = 1000, lines with open squares correspond to K = 500 and lines with open triangles correspond to K = 250. (b) Effect of parameter y*: lines with open circles correspond to y* = 500, lines with open squares correspond to y* = 50 and lines with open triangles correspond to y* = 25. In both figures, solid and dashed lines denote stable and unstable steady state solutions, respectively. Parameter set values: f = 0.5, m = 2, π = 0.03 and κ = 0.05.
Fig 9.
Effect of intrinsic noise intensity.
Steady state average expression level, ⟨[Y]⟩, as a function of the external IPTG concentration, [Iex], for different K and y* values. The lines with full circles correspond to K = 1000 and y* = 500, the lines with full squares to K = 500 and y* = 50 and the lines with full triangles to K = 250 and y* = 25; the black lines correspond to the DCPB model. Stochastic simulations are performed with N = 10,000 cells (average of 50 copies for noise reduction). Parameter set values: f = 0.5, m = 2, π = 0.03 and κ = 0.05.
Fig 10.
Effect of partitioning asymmetry parameter.
Effect of the partitioning asymmetry parameter, f, in the average expression of lacY gene steady state (⟨[Y]⟩), as a function of the inverse IPTG concentration ([Iex]). (a) Lines with full circles correspond to symmetric partitioning (f = 0.5); lines with full squares correspond to f = 0.4 and lines with full triangles to f = 0.3. (b) Comparison between the CNMC model neglecting intrinsic source of heterogeneity with f = 0.3 (lines with open circles), the DCPB model (black lines (solid and dashed)) and the CNMC incorporating intrinsic noise effects (lines with full triangles). Parameter set values: m = 2, π = 0.03 and κ = 0.05. CNMC simulations are performed with N = 10,000 cells.
Fig 11.
Effect of the sharpness division parameter.
Effect of the sharpness division parameter, m, on the average expression of lacY gene steady state. (a) CNMC simulations of 10,000 cells with K = 500 and y* = 50. Lines with full squares correspond to m = 1, lines with full circles to m = 2 and lines with full triangles to the largest division rate, m = 3. (b) Comparison between the CNMC model with m = 3 (lines with full triangles) with the DCPB model (black lines (solid and dashed)) and the CNMC model neglecting intrinsic noise effects (lines with open circles). Parameter set values: f = 0.5, π = 0.03 and κ = 0.05.
Fig 12.
Noise induced phenotypic switching.
A single stochastic simulation starting from a coarse steady state at [Iex] = 30.8μM with ⟨[Y]⟩ = 157.2nM (upper branch). After the elapse of a long time interval (t ≈ 220hrs), stochastic noise induces a phenotypic switching towards an average value of ⟨[Y]⟩ = 7.9nM (lower branch). The inner figure shows the corresponding bifurcation diagram of the steady state average expression level of lacY as a function of [Iex], with the open circles representing the two different co-existing steady state solutions at [Iex] = 30.8μM. Parameter set values: m = 3, f = 0.5 π = 0.03, κ = 0.05, K = 500 and y* = 50. CNMC simulations are performed with N = 10,000 cells.
Fig 13.
Effect of intrinsic noise for low single cell division rates.
Steady state of average expression level, ⟨[Y]⟩, as a function of the external IPTG concentration for the case of f = 0.3 and m = 1. The lines with full circles correspond to the CNMC model with K = 500 and y* = 50; the lines with open circles correspond to the CNMC model neglecting intrinsic noise effects, and the black lines to the DCPB model. Parameter set values: π = 0.03 and κ = 0.05. CNMC simulations are carried out with N = 10,000 cells.