Comparison of Deterministic and Stochastic Models of the lac Operon Genetic Network  Michail Stamatakis, Nikos V. Mantzaris  Biophysical Journal  Volume 96, Issue 3, Pages 887-906 (February 2009) DOI: 10.1016/j.bpj.2008.10.028 Copyright © 2009 Biophysical Society Terms and Conditions

Figure 1 (a) Schematic representation of the interactions taken into account in the lac operon network. (b) Interaction diagram for the key species of the lac operon genetic switch (for species notation, see Table 1; + and − denote positive and negative feedback, respectively). Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 2 Bifurcation diagrams with respect to the extracellular IPTG concentration, [Iex], for (a) the total LacY concentration, (b) the intracellular IPTG concentration, and (c) the total repressor concentration (the dot-dashed line corresponds to the limiting [R]T as calculated from Eq. S46). Solid lines represent stable and dashed lines unstable steady states. For a complete list of parameters, see Table 2. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 3 Two parameter bifurcation diagrams for the turning points showing the effect of several biomolecular parameters on the bistable regime (enclosed between the curves). Shown are the effects of repressor-inducer complex degradation rate (a), lacI transcription rate (b), repressor-inducer dissociation rate (c), and total operator concentration (d). Unless otherwise noted, parameters are as given in Table 2. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 4 Transient and asymptotic effects of the alternative derepression mechanisms. (a and b) The mechanism where the inducer binds to the repressor-operator complex (thus, for the other mechanism, kdr1 and k−dr1 are both zero): loss of bistability and faster induction in the case of slower repressor-operator association and dissociation rates. [kr, k−r] ÷ 50 denotes that both kr and k−r have values 50-fold lower than those given in Table 2. (c and d) The mechanism where the inducer binds the free repressor (thus, kdr2 and k−dr2 are both zero): Shift of the bistable regime and slower induction for the case of slower repressor-inducer association and dissociation rates. The time course for [kdr1, k−dr1] ÷ 500 finally approaches the lower steady state. (e and f) The mechanism where the inducer binds to the repressor-operator complex (thus, kdr1 and k−dr1 are both zero), leading to broadening of the bistable regime and slower induction in the case of slower repressor operator complex-inducer association and dissociation rates. Parameters are as in Table 2, except as noted. For all induction transients, the initial condition is the corresponding steady state for [Iex] = 0 μM, and at t = 0 min, an extracellular IPTG concentration of [Iex] = 100 μM is imposed. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 5 Comparison of the stochastic average with the deterministic steady states. The error bars denote two standard deviations (i.e., the minimum and maximum of the error bars correspond to points μ ±σ). Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 6 Typical sample path for the stochastically simulated lac operon system for [Iex] = 20 μM (a), and the PMF for this parameter set (b). (c and d) [Iex] = 100 μM. Parameters are as in Table 2. Sample histograms for all the species can be found in Figs. S3 and S4. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 7 The stationary potential for [Y]T compared to the deterministic bifurcation diagram for a range of IPTG concentrations shows the effect of stochasticity on the behavior of the system in different parameter regimes. (a) Nominal parameter set (Table 2). (b) 100-fold faster lacY transcription (ks1MY = 50 min−1, ks0MY = 1 min−1) and slow translation (ksY = 0.3 min−1). (c) 100-fold faster lacI transcription (ksMR = 23 min−1) and slow translation (ksR = 0.15 min−1). (d) Simultaneous 100-fold faster lacY and lacI transcription and slow translation for both proteins (ks1MY = 50 min−1, ks0MY = 1 min−1, ksMR = 23 min−1, ksY = 0.3 min−1, ksR = 0.15 min−1). (e) 10-fold slower repressor-operator association and dissociation with only the first derepression mechanism, considering fast transcription and slow translation for lacY (kr = 96 nM−1·min−1, k−r = 0.24 min−1, kdr2 = 0 nM−2·min−1, k−dr2 = 0 nM−1·min−1, ks1MY = 50 min−1, ks1MY = 1 min−1, ksY = 0.3 min−1). (f) 10-fold faster repressor-operator association and dissociation with only the first derepression mechanism, considering fast lacY transcription and slow translation (kr = 9600 nM−1·min−1, k−r = 24 min−1, kdr2 = 0 nM−2·min−1, k−dr2 = 0 nM−1·min−1). The stochastic bistable (bimodal) regime is indicated by ●——● and the deterministic regime by ★——★. Surface color corresponds to stationary potential values and the bimodal regime is shaded green. Parameters are as in Table 2 unless otherwise noted. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 8 (a) MFPTs for the transition of [Y]T from the lower to the upper attracting vicinity (circles) or vice versa (triangles) for a range of extracellular IPTG concentrations. Fast lacY transcription and slow translation were considered. The minimum and maximum of the error bars correspond to the MFPT ± 1 SD. Superimposed for comparison is the corresponding deterministic bifurcation diagram for [Y]T (secondary y axis). (b) The probability density function for the FPT Tlow→up for [Iex] = 20 μM. The MFPT is 1/λ ≈ 179 min. Parameters are as in Fig. 7b. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 9 (a) Autocovariance function for extracellular IPTG concentration [Iex] = 10 μM. The autocovariance function decays exponentially fast, with an autocorrelation time of τc ≈ 25 min in this case. (b) Dependence of the autocorrelation times on the induction levels as quantified by [Iex]. Superimposed for comparison is the corresponding deterministic bifurcation diagram for [Y]T (secondary y axis). Parameters are as in Fig. 7b. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Figure 10 An IPTG molecule approaching the membrane of the cell. Collision will occur in the next δt time interval if the molecule is moving toward the membrane and the center of the molecule is inside δVcoll=δA·(−u∼IPTG,n∼A)·δt. See text for details. Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions

Biophysical Journal 2009 96, 887-906DOI: (10.1016/j.bpj.2008.10.028) Copyright © 2009 Biophysical Society Terms and Conditions