investigation
Stochastic models were used to numerically predict how the number of plasmids in the cellular compartment affects the variability in GFP expression levels among an isogenic population of transformants. While the gene circuit’s average behaviour computed with deterministic and stochastic models is expected to coincide in linear systems, i.e. those described by zeroth and first order biochemical reactions, transitioning from a macroscopic to a microscopic picture of the biological model usually requires a proper scaling of the kinetic rates governing the system’s evolution over time. Indeed, state variables are conventionally expressed as molecules concentrations when writing rate equations, while in the microscopic description provided by the CME they appear as particles numbers. Having expressed the deterministic model’s state variables as molecules counts, stochastic models of the investigated gene circuits used the same parameters values identified in the fitting procedure previously outlined.
Using the Gillespie algorithm, trajectories of the CME providing a probabilistic picture of the TC circuit, cloned in both the pSB1A2 and pSB4A5 plasmid, were simulated using the set of reactions 2.1- 2.4. The steady-state mean and variance in GFP expression, averaged over 1000 trajectories, was computed varying the transcription rate of the TRANS-GFP sequence according to equation 2.5. The simulated dose-response curves, shown in panel A of Figure 2.8, correctly reproduce the experimental data for both plasmid contexts. In addition, the perfect match between the numerical and analytical dose-response curves ensured the correctness of the implemented algorithm. When considering the stochasticity in GFP expression, quantified by the coefficient of variation, the model describes the expected decrease in biological noise upon induction with IPTG. Administering the inducer causes an increase in TRANS-GFP transcription, leading to higher expression of the fluorescent reporter: a condition which limits the relevance of finite number effect. In addition, the numerical results suggest a higher stochasticity in GFP expression for the pSB4A5 plasmid. While this observation is in line with theoretical considerations, the obtained curves are not statistically different. A plausible explanation is the limited fractional change in counts between the two plasmids, which was revealed by the deterministic model.
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With analogous procedure, stochastic models based on the set of reactions 2.8-2.14 were developed for the pTC gene circuit. It is worth noting that the bimolecular nature of reaction 2.11, which embodies the post-transcriptional control mechanism, could lead to discrepancies in the dose- response curves simulated with the deterministic and stochastic version of the pTC gene circuit’s model. The good agreement between simulated dose-response curves and the experimental decrease in normalized fluorescence with IPTG indicates that the mean obtained with stochastic simulations equals the steady-state solution of the rate equations. When considering GFP expression’s variability, the model, in contrast with the results for the TC gene circuit, provides a coefficient of variation which is almost constant upon induction with IPTG. The difference in noise amplitude computed for the pSB4A5 and pSB1A2 cloning vectors is higher than the one estimated numerically for the TC circuit. While this supports the hypothesis that the limited difference observed for the gene circuit implementing transcriptional control in the expression of the fluorescent reporter is due to the similar number of plasmids estimated in the fitting procedure based on the deterministic model, this difference is relatively low and might not be detectable in experimental measurements.
Finally, a qualitative comparison among the CV estimated for the TC and pTC gene circuits, cloned in the high copy number plasmid, at maximum (IPTG = 200 M) and in absence of induction Figure 2.8: Results of the stochastic model for the TC gene circuit. In panel A the agreement between the dose-response curves simulated by stochastic simulations and the experimental data acquired on the TC gene circuit is shown. Experimental data are reported as mean ± standard deviation, using green upper triangles for the circuit cloned in pSB1A2 and blue lower triangles for the low copy number cloning vector. Experimental values are normalized by the average fluorescence measured in TC gene circuit, cloned in pSB1A2, at maximum induction (IPTG = 200 M). Panel B shows the trend of the numerical coefficient of variation (CV) with increasing IPTG for the pSB1A2 (green line) and pSB4A5 (blue line) plasmid backbones. In line with our expectations, the curves reproduce a reduction in GFP expression stochasticity upon induction: under this condition the higher synthesis of fluorescent reporter molecules constrains the relevance of finite number effect. When comparing the noise strengths originating from cloning vectors with different copy number, the numerical predictions suggest a higher variability in GFP expression levels for the TC gene circuit cloned in pSB4A5 plasmid.
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respectively –condition under which the mean normalized fluorescence has similar values (Figure 2.4) - would suggest that the post-transcriptional regulation in GFP expression produces a lower stochasticity than the one provided by the transcriptional control.
Figure 2.9: Results of the stochastic model for the pTC gene circuit. In panel A the agreement between the dose-response curves simulated by stochastic simulations and the experimental data acquired on the pTC gene circuit is shown. Experimental data are reported as mean ± standard deviation, using green squares for the circuit cloned in pSB1A2 and blue squares for the pSB4A5 cloning vector. Experimental values are normalized by the average fluorescence measured in pTC gene circuit, cloned in pSB1A2, in absence of induction. Panel B shows the trend of the numerical
coefficient of variation (CV) with increasing IPTG for the pSB1A2 (green line) and pSB4A5 (blue line)
plasmid backbones. The limited fractional change in GFP expression upon IPTG induction is probably responsible for a constant value of the CV with increasing inducer concentrations. When comparing the noise strengths originating from cloning vectors with different copy number, the numerical predictions suggest a higher variability in GFP expression levels for the pTC gene circuit cloned in pSB4A5 plasmid.
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