Supplementary Information for: Control and Regulation of Pathways via Negative Feedback from Control and regulation of pathways via negative feedback

The biochemical networks found in living organisms include a huge variety of control mechanisms at multiple levels of organization. While the mechanistic and molecular details of many of these control mechanisms are understood, their exact role in driving cellular behaviour is not. For example, yeast glycolysis has been studied for almost 80 years but it is only recently that we have come to understand the systemic role of the multitude of feed-back and feed-forward controls that exist in this pathway. In this article, control theory is discussed as an approach to dissect the control logic of complex pathways. One of the key issues is distinguishing between the terms control and regulation and how these concepts are applied to regulated enzymes such as phosphofructokinase. In doing so, one of the paradoxes in metabolic regulation can be resolved where enzymes such as phosphofructokinase have little control but, nevertheless, possess significant regulatory influence

Control vector analysis for noise reduction.

<p>The noise level of a concentration (variance divided by mean squared) is aimed to be reduced while its mean level does not change. When parameter perturbations are performed within the space perpendicular to the control vector for the mean level, the mean concentration does not change. A control vector for the concentration noise level is projected onto the perpendicular space. The projected vector is denoted by . When parameter perturbations are directed along (the opposite direction of ), the noise level will decrease while the mean concentration does not change.</p

Noise control in a metabolic network under end-product inhibition.

<p>(a) The metabolic network is under stochastic fluctuations of an enzyme level . Other enzyme level fluctuations are neglected for simplicity. (b) Control analysis was applied to decrease the noise level of the end product, , without changing its mean level. Iterative small perturbations reduced the noise level significantly with a minor change in the mean level, as shown by the change in the probability distribution functions of . The original parameter values can be found in the caption of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi-1002344-g007" target="_blank">Fig. 7</a>.</p

Efficiency and strength of orthogonal control in the HIV-1 expression system (<b>Fig. 2a</b>).

<p>All possible two-parameter controls were considered. The efficiency and strength were computed by using Eqs. (6) and (7). (a) Among noise reduction control schemes, the most efficient and strongest one was related to gene activation and translation . (b) Among mean-level reduction controls, the most efficient and strongest control schemes were related to translation (collapsed data points).</p

Linear metabolic pathways.

<p>For the network A and B, the enzyme level was fixed to . For the network C, the enzyme level was allowed to fluctuate due to its random synthesis and degradation. The parameter values: , , , , , , . Here, we consider the number of molecules is dimensionless.</p

Orthogonal control of mean and noise levels in the HIV-1 LTR-promoter expression.

<p>(a) The HIV-1 model vector with a green fluorescence protein (GFP) gene that is transfected to Jurkat cells <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-Singh1" target="_blank">[42]</a> is considered. (b) The promoter inactive and active states are explicitly represented by the two-state model <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-Raser1" target="_blank">[4]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-Murphy1" target="_blank">[41]</a>. Based on the values of the control coefficients (provided in (c)), <i>in silico</i> perturbation experiments were designed. (d) The noise level was reduced without changing its mean level. The translation rate was decreased 10 times and one of the reactions among transcript degradation, gene deactivation, and protein degradation was decreased 10 times, or one of the reactions among gene activation and transcription was increased 10 times. (e) The mean level was reduced without changing the noise level either by decreasing the translation 10 times (Translation), or by increasing the gene activation by twice and protein degradation 10 times (Gene-Activate + Protein-Deg). Two orthogonal control schemes were combined so that both the noise and mean levels were simultaneously controlled. The combined control was performed by decreasing the translation rate 100 times and increasing the gene activation 10 times (Combined Control).</p

Visualization of Evolutionary Stability Dynamics and Competitive Fitness of Escherichia coli Engineered with Randomized Multigene Circuits

Strain engineering for synthetic biology and metabolic engineering applications often requires the expression of foreign proteins that can reduce cellular fitness. In order to quantify and visualize the evolutionary stability dynamics in engineered populations of Escherichia coli, we constructed randomized CMY (cyan-magenta-yellow) genetic circuits with independently randomized promoters, ribosome binding sites, and transcriptional terminators that express cyan fluorescent protein (CFP), red fluorescent protein (RFP), and yellow fluorescent protein (YFP). Using a CMY color system allows for a spectrum of different colors to be produced under UV light since the relative ratio of fluorescent proteins vary between circuits, and this system can be used for the visualization of evolutionary stability dynamics. Evolutionary stability results from 192 evolved populations (24 CMY circuits with 8 replicates each) indicate that both the number of repeated sequences and overall expression levels contribute to circuit stability. The most evolutionarily robust circuit has no repeated parts, lower expression levels, and is about 3-fold more stable relative to a rationally designed circuit. Visualization results show that evolutionary dynamics are highly stochastic between replicate evolved populations and color changes over evolutionary time are consistent with quantitative data. We also measured the competitive fitness of different mutants in an evolved population and find that fitness is highest in mutants that express a lower number of genes (0 and 1 > 2 > 3). In addition, we find that individual circuits with expression levels below 10% of the maximum have significantly higher evolutionary stability, suggesting there may be a hypothetical “fitness threshold” that can be used for robust circuit design

Orthogonal control of mean and noise levels in the Gal10 promoter expression.

<p>(a) The yeast Gal10 promoter, expressing yeast-enhanced green fluorescent protein (yEGFP) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-Murphy1" target="_blank">[41]</a>, is considered and (b) mathematically described with the two-state model. (c) Control coefficients were computed for the wild-type and TATA-box mutated promoters. Based on the values of the control coefficients, <i>in silico</i> perturbation experiments were designed. (d) The noise level was reduced without changing its mean level. Gene-Activate: was increased 10 times. yEGFP-synthesis-deg: and were decreased 10 times. (e) The mean level was reduced without changing the noise level either by decreasing 10 times (), or by increasing 15 times and 225 times (TATA box). : The actual sum is zero, but the sum of the round-up control coefficient values (shown in (c)) is 0.02 due to a round-up error.</p

Control of p53 oscillations caused by DNA damage.

<p>(a) ATM protein kinases are activated in response to a DNA damage and phosphorylate p53, which activates the WIP1 gene that inhibits the ATM <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-Shreeram1" target="_blank">[68]</a>. The phosphorylated p53 activates <i>mdm2</i> at the transcription level and Mdm2 binds to p53 with the Mdm2-p53 complex undergoing enhanced degradation. These negative feedback loops among ATM, p53, and Mdm2 cause sustained noisy oscillation at the p53 level <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-GevaZatorsky1" target="_blank">[32]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-Batchelor1" target="_blank">[51]</a>. (b) Based on the model proposed by Geva-zatorsky et al. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344-GevaZatorsky1" target="_blank">[32]</a>, control coefficients were computed. (c) Accordingly perturbation experiments were designed. Autocorrelation functions of p53 showed damped oscillations and their amplitudes were increased by decreasing the effective degradation rate of p53 by 50%. (d) The oscillation period was increased by decreasing the inhibitory regulation of p53 on ATM 10 times. Refer to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344.s001" target="_blank">Text S1</a> for the details of the model. We note that the CCs for correlations also satisfy summation theorems (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002344#pcbi.1002344.s001" target="_blank">Text S1</a>), indicating nontrivial correlation among the sensitivities.</p

Visualization of Evolutionary Stability Dynamics and Competitive Fitness of Escherichia coli Engineered with Randomized Multigene Circuits

Strain engineering for synthetic biology and metabolic engineering applications often requires the expression of foreign proteins that can reduce cellular fitness. In order to quantify and visualize the evolutionary stability dynamics in engineered populations of Escherichia coli, we constructed randomized CMY (cyan-magenta-yellow) genetic circuits with independently randomized promoters, ribosome binding sites, and transcriptional terminators that express cyan fluorescent protein (CFP), red fluorescent protein (RFP), and yellow fluorescent protein (YFP). Using a CMY color system allows for a spectrum of different colors to be produced under UV light since the relative ratio of fluorescent proteins vary between circuits, and this system can be used for the visualization of evolutionary stability dynamics. Evolutionary stability results from 192 evolved populations (24 CMY circuits with 8 replicates each) indicate that both the number of repeated sequences and overall expression levels contribute to circuit stability. The most evolutionarily robust circuit has no repeated parts, lower expression levels, and is about 3-fold more stable relative to a rationally designed circuit. Visualization results show that evolutionary dynamics are highly stochastic between replicate evolved populations and color changes over evolutionary time are consistent with quantitative data. We also measured the competitive fitness of different mutants in an evolved population and find that fitness is highest in mutants that express a lower number of genes (0 and 1 > 2 > 3). In addition, we find that individual circuits with expression levels below 10% of the maximum have significantly higher evolutionary stability, suggesting there may be a hypothetical “fitness threshold” that can be used for robust circuit design