article: Several key transcription factors have unusually short half-lives compared to other cellular proteins. Here, we explore the utility of active degradation in shaping how the multiple antibiotic resistance activator MarA coordinates its downstream targets. MarA controls a variety of stress response genes in Escherichia coli. We modify its half-life either by knocking down the protease that targets it via CRISPRi or by engineering MarA to protect it from degradation. Our experimental and analytical results indicate that active degradation can impact both the rate of coordination and the maximum coordination that downstream genes can achieve. In the context of multi-gene regulation, trade-offs between these properties show that perfect information fidelity and instantaneous coordination cannot coexist.
  Active degradation is a rare feature in bacteria, affecting only 2–7% of total cellular proteins in Escherichia coli [1,2]. Active degradation is accomplished by ATP-dependent proteases such as ClpXP, HflB, and Lon [3]. These proteases play a role in protein quality control by degrading misfolded proteins [4–6], but they can also degrade functional proteins. Active degradation of functional proteins can serve as a type of post-translational regulation to modify protein concentration [7]. This can be important in time-sensitive processes when removal through dilution is insufficient. Metabolic processes that incorporate active degradation include DNA damage repair, gene expression during stationary-phase, cell division, filamentation, and accelerated removal of certain regulatory proteins [8,9]. Although active degradation can play a regulatory role, this comes with an energy cost due to the ATP necessary to activate the proteases, and a metabolic investment required to replace degraded proteins [10]. Active degradation occurs in many other microorganisms as well, such as with the ComK regulator of sporulation in Bacillus subtilis [11] and in many regulatory proteins in humans including the p53 tumor suppressor [12,13] and IκBα, which dictates NF-κB activity and cellular stress response [14,15]. The refractory period of NF-κB pulses is determined by this degradation rate, which dictates the limit of information transmission in the system [16]. In eukaryotes, the percentage of proteins that are primarily removed through active degradation ranges from ~15% in yeast [17] to ~50% in humans [18]. Therefore, despite active degradation’s role in regulation across domains of life, it is still comparatively rare in bacteria due to their rapid growth rate, which produces stronger dilution effects. The relative rarity coupled with the cost implicit to active degradation makes the presence of any short half-life protein in bacteria conspicuous. What potential utility could necessitate the use of this rare design feature? To answer this, we focused on the multi-antibiotic resistance activator (MarA) in E. coli as a case study. MarA has a short half-life even by the standards of an actively degraded protein, with an estimated half-life of 1–3 minutes due to Lon protease activity [19,20]. Moreover, MarA regulates over 60 downstream targets involved in a variety of antibiotic resistance roles, from genes encoding efflux pump components like acrAB and tolC to stress response proteins like inaA [21,22]. The majority of the downstream genes encode for stable proteins, therefore MarA has a much shorter half-life than most of its downstream targets [21]. MarA is part of a greater regulon, where three homologous proteins—MarA, Rob, and SoxS—regulate an overlapping suite of downstream genes [23]. However, MarA’s clear correlation with phenotypic outcomes make it an ideal target for studying why evolution would select for actively degraded regulatory molecules [24]. Recent studies have revealed that marA is expressed stochastically, creating phenotypic diversity within isogenic populations. This noisy expression is linked to transient antibiotic resistance [24]. Thus, stochastic expression of marA can impact whether a cell lives or dies under antibiotic exposure. At the population level, there is a distribution of MarA levels where some cells are antibiotic resistant and others are susceptible. This may serve as a bet hedging strategy, where cells with high marA expression act as insurance policy to protect against the sudden appearance of antibiotics or other stressors. In conditions with small numbers of cells, the rate at which diversity in MarA levels is generated is important. Starting with a single cell at a single time point, there is no diversity. As the cell grows and divides, diversity develops within the population. The rate at which this diversity develops depends on the time scale of the stochastic dynamics of the genes being expressed, including their rate of degradation [11,25]. We measure diversity as the distribution of distinct protein concentrations present in a population. The half-life of the protein impacts the rate of diversity generation and the maximum diversity achieved (Fig 1A). For a regulatory protein such as MarA (X), phenotypic diversity can propagate to its downstream targets such as AcrAB and InaA (Y and Z) (Fig 1B). The rate at which diversity appears in downstream genes is impacted by how the upstream regulator changes with time. Active degradation affects the variability in the activator concentration—how quickly the concentration changes in one cell over time, as well as the diversity in concentrations between cells. This produces a spectrum of phenotypes where individual cells exhibit a range from low to high expression for all downstream targets (Fig 1C). This coordinated diversity in expression of genes that share a master regulator is different from the uncoordinated diversity in two uncoupled genes (Fig 1D). We used mutual information to quantify increasing coordination of downstream genes. Mutual information describes how well knowing the concentration of one of these proteins allows us to predict the concentration of the other [26]. It measures their co-expression properties even if the correlation between them is nonlinear, an important feature because transcriptional dose response curves typically saturate, following the shape of a Hill function [27]. S1 Movie shows an animation of how stochastic expression of an activator produces coordinated diversity within downstream targets. Coordination as a function of a shared master regulator is important for establishing synergistic effects between downstream targets. For instance, MarA activates expression of genes in the AcrAB-TolC efflux pump, the small RNA micF which decreases porin expression, and other stress response genes such as inaA. Conceptually, expressing multiple genes is like locking both the front door and the back door to a home in order to protect it; locking just one door makes the home only marginally safer and expends energy. Thus, coordinating expression of stress response genes is essential. Here, we investigate the role of activator half-life in controlling how a stochastic regulator generates coordinated diversity in multiple downstream genes. We do this by controlling the half-life of MarA, either by modulating the concentration of Lon protease or by protecting MarA itself from proteolytic degradation. We then evaluate the impact of MarA’s half-life by computing both the maximum amount of coordinated diversity that MarA can achieve as well as the rate at which it achieves it.
  We first developed an analytic model of how noise in MarA propagates to downstream genes as a function of its half-life to generate predictions that we could test experimentally. In the model, an upstream regulator (X) activates two downstream proteins (Y and Z). We modeled noise in X using an Ornstein-Uhlenbeck process [27–29]. The simulation and analytical results from this model produce results in terms of units of concentration of X. The model has two free parameters: τ, which is a scaled version of the half-life of the molecule (τ = λ/ln (2), where λ is the half-life) and σ, which specifies the noise in protein expression (Methods). Using this model, τ changes the dynamics of the process as well as the population distribution that such a process can produce (Fig 2A). The stochastic simulations demonstrate that having a longer half-life causes the protein concentration to change more slowly–negative values indicate a lower concentration than the mean, positive values indicate a higher concentration. Additionally, the longer half-life allows the trajectory of protein concentration to wander farther away from the mean, increasing the standard deviation (Fig 2A). In this case, σ = 1 and the standard deviation scales as τ/2 [28]. Analytical solutions to these functions show that as the half-life of a molecule increases, it takes longer to reach the maximum variability (Fig 2B). Here, variability is measured as variance because it is a zero mean process (S1 File). Variability in a regulatory molecule propagates to the downstream genes it controls. To quantify the correlation between the concentrations of two downstream proteins (Y and Z) regulated by X, we measured the mutual information between their concentrations as a function of time. First, we observed an apparent advantage to longer half-life activators, as increasing half-life increases the mutual information between two downstream targets (Fig 2C). Large variability in the activator concentration allows for coordinated diversity in the downstream genes, and a longer half-life molecule produces greater information between the two downstream genes. The decreased mutual information of short half-life activators is due to filtering of high-frequency changes. Downstream genes function as low-pass filters in which signals that change quickly are averaged out [30]. Moreover, because increasing the half-life increases the standard deviation of protein levels, the variability in downstream genes increases as well, allowing for a greater range of coordinated activation. These results show that longer half-life proteins may be better at generating coordinated diversity in downstream targets. However, this conclusion is dependent on allowing the standard deviation to scale freely with the half-life of the regulatory molecule. To validate these analytic predictions, we designed a genetic system to modify the concentration of Lon protease. Changing the concentration of Lon should change the statistics of MarA and the downstream genes it regulates. We used CRISPRi to knock down Lon expression, allowing us to control the level of MarA. We first transformed E. coli with dCas9 and a sgRNA targeting lon [31]. Lon protease knockouts have been shown to increase the half-life of MarA to ~30 minutes, therefore increasing the mean concentration MarA in E. coli, but this change affects other genes in addition to marA [19,32]. In order to avoid off-target pleiotropic effects from eliminating Lon protease such as cell filamentation [33], we designed the sgRNA to increase MarA levels, but to produce no qualitative morphological changes to the cell (S1 Fig). The half-life of MarA under these conditions should be increased above the wild type rate of 1–3 minutes but below the lon knockout rate of ~30 minutes. As in other studies, we were unable to quantify the exact half-life because the unusually fast degradation rate of MarA prevents observation via Western blot in the absence of overexpression (S2 Fig). We note that the lon knockdown may introduce more subtle changes, therefore we use this approach in tandem with computational modeling and an alternative experimental method protecting MarA from proteolytic degradation. Decreasing Lon protease is expected to affect the correlation time τ of MarA, as well as the standard deviation of its time trace (Fig 2D). The analytical modeling requires that all chemical species are abundant enough such that the propensity functions do not change at low copy numbers [34]. Here, we found that all fluorophores were well above background levels, indicative of non-zero gene product concentrations. We co-transformed cells with the CRISPRi construct with a plasmid containing transcriptional reporters for MarA and its downstream genes. To report MarA levels, the plasmid contains a modified marRAB promoter controlling cyan fluorescent protein (cfp). This cfp is modified by the addition of an ssrA tag in order to decrease its half-life to align it with the dynamics of the actively degraded MarA [35]. Our analytical results predict that extending the half-life of an activator should produce increased variability in growing microcolonies. By measuring the CFP levels of individual cells in the CRISPRi knockdown system and a wild type strain we found that the half-life of MarA determines how noise expands in growing microcolonies (Fig 2E). The analytical results predict that the rate of diversity generation should not be altered in growing microcolonies, but that the maximum diversity will be (as quantified by coefficient of variation). The coefficient of variation is used to measure variability in expression as the cellular fluorescence has a non-zero mean. Moreover, dividing by the mean removes the possibility that arbitrary adjustments to the gain of the microscope would impact the results. Consistent with our analytical predictions, we found that the slope of the curves was similar, but the maximum value changes. The distribution of CFP within microcolonies at the final time step of the experiments highlights the differences in maximum diversity (Fig 2F). As in the theoretical predictions, the mean and standard deviation of MarA in the CRISPRi knockdown system, which has a longer half-life, increase compared to the wild type system. In addition to the marA reporter, we also measured expression of two downstream genes: inaA, which is a pH-inducible gene involved in stress response [36] and acrAB, which is a component of the acrAB-tolC antibiotic efflux pump [37,38]. We selected these two genes because of their disparate functionality, yet similar dose response curves [27]. We used PinaA to control expression of yellow fluorescent protein (yfp) and PacrAB to control red fluorescent protein (rfp). By simultaneously measuring the dynamics of all three fluorophores (PmarA-cfp, PinaA-yfp, PacrAB-rfp) via time-lapse microscopy, we were able to quantify variability in the marA input and two outputs in both the CRISPRi and wild type systems. We used these fluorescence data to calculate the coordinated diversity in the microcolony by measuring mutual information over time. This metric captures the value of a large spectrum of correlations between two downstream genes without over-valuing high correlations between a few cells, such as would occur during the beginning growth stages of a microcolony [26]. Mirroring the analytical results, we see that there is an advantage to increasing the half-life of MarA, where coordinated phenotypic diversity is higher between the two downstream genes in the CRISPRi strain (Fig 2G). To emphasize the differences, we compared single-cell data from the final time point of the movies. As expected, the mean concentration of both YFP and RFP increases in the CRISPRi system relative to wild type due to decreased MarA degradation (Fig 2H). Importantly, sample snapshots of the two conditions show greater correlation between the fluorophores (higher proportion of magenta cells) in the longer half-life CRISPRi system. This illustrates that in addition to increasing mean expression of the downstream genes, the correlation between downstream gene promoter activities increases with the half-life of the activator. These results show clear advantages for longer half-life activators (equivalent to conditions without active degradation), but they do so without constraints–the variance of the activator is allowed to increase without a limit. This may not be the case in natural systems which have a target concentration range for the activator molecule. In order to investigate this, we modified our mathematical model to create a system where the standard deviation does not increase as a function of the activator half-life. We did this by requiring the noise to be inversely related to the correlation time σ=2/τ. This represents a system where the total variability is constrained, but where the rate at which it grows is a free parameter. Three example trajectories from stochastic simulations with varying degradation rates but normalized standard deviations show different time series behaviors that produce identical steady state distributions (Fig 3A). Under this constraint, we see that the rate at which the variance of a population develops is a function of activator half-life (Fig 3B). Moreover, when the activator variance is constrained, a short half-life activator increases mutual information faster, but plateaus sooner and at lower values than a longer half-life activator (Fig 3C). Thus, faster-degrading activators are capable of generating coordinated diversity faster, but the maximum correlation that they can produce between two downstream genes is limited. Here is the first evidence of a potential trade-off as we increase activator half-life by inhibiting active degradation: maximum coordination between two downstream targets increases but the rate at which coordination is generated decreases. In order to verify these theoretical results experimentally, we developed a system in which we could control the mean activator concentration independently of the half-life. To this end, we employed a modified version of MarA that is resistant to degradation by Lon and placed it under the control of an inducible promoter. In this design, marA is translationally fused to cfp thereby protecting the C terminus from recognition and degradation by Lon protease [24,39]. This modified MarA has an estimated half-life of ~30 minutes compared to the wild type half-life of 1–3 minutes [24] (S2 Fig). Expressing the MarA fusion from the PlacUV5 promoter allowed us to tune its expression level using IPTG. Thus, we were able to tune the steady state concentration of MarA and the noise that scales with it. We transformed the MarA fusion plasmid into E. coli ΔmarRAB along with a two-color version of the downstream gene reporter plasmid (PlacUV5-marA-cfp with PinaA-yfp, PacrAB-rfp). By monitoring the real-time MarA level directly using CFP, we were able to quantify how much MarA was present in each cell. We compared this strain with altered MarA degradation to a system with wild type MarA under the control of the PlacUV5 promoter as well as the three-color reporter described previously (PlacUV5-marA with PmarA-cfp, PinaA-yfp, PacrAB-rfp). The two systems produce protein concentrations with different statistics over time (Fig 3D). The MarA fusion has a longer correlation time due to protection from Lon protease, however unlike in the CRISPRi system, the expression level can be tuned via IPTG induction. In experiments, we observed that wild type MarA achieved its maximum diversity faster than the MarA fusion (Fig 3E). This is in contrast to the CRISPRi knockdown system in which the rates did not change substantively as a function of half-life. This is an essential difference because the rate of activator diversity generation determines the rate of coordination in downstream targets. S2 and S3 Movies show how the growing microcolonies exhibit differences in how diversity is generated. Wild type MarA produces diversity quickly while the microcolony expressing the MarA fusion takes longer. Therefore, we see that both maximum diversity and the rate of diversity generation can be altered by modulating the half-life of the molecule. As with the CRISPRi system, the changes in activator dynamics go on to alter how downstream genes are regulated, where increasing MarA’s half-life increases the maximum coordinated diversity between downstream genes (Fig 3F). However, unlike in Fig 2G, the rate at which the wild type and MarA fusion generate coordination between downstream genes is different. We observed that the wild type strain approached its maximum degree of coordination between the two genes faster than the MarA fusion. This is a reflection of activator diversity accumulating faster when the system produces wild type MarA. Also, in contrast to the CRISPRi knockdown example, here both strains express the same steady state levels for each of the downstream genes. This shows that even if the mean expression is equivalent, the coordination between genes can be different (S4 Fig). So far, we have compared the maximum coordinated diversity that a system can produce and the rate at which this diversity grows. The information rate is an important metric to consider because it explicitly outlines the role that a stochastic activator plays in managing the progressive development of phenotypic diversity among multiple genes. In order to conceptualize the difference between the information rate and total mutual information, consider Fig 4A showing two increasingly resolved maple leaves. We see that in the two rows, we must consider both the rate at which the image gains clarity as well as the maximum clarity of the image. Genetic networks may optimize for increased rate at the expense of low mutual information in time-sensitive conditions. Alternatively, a decreased information rate may be acceptable if time constraints are not an issue and total information is more important. To quantify this trade-off, we used the methods described in [40] to calculate the information rate for our model and compared that to the maximum information between two downstream genes. First, at extremely short half-lives both the information rate and the maximum information are very low. This is because near the uncorrelated white noise limit, the activator signal X is filtered by the downstream genes Y and Z. However, as we increase the activator half-life we see a marked increase in both information rate and mutual information. This is a function of a rapidly changing activator passing information to downstream targets, which allows them to coordinate. As the activator half-life increases further, the maximum information continues to increase monotonically while the information rate decreases. A slowly changing signal in X is passed faithfully to downstream targets but the rate of information is slow. Note however that information transmission rate is not simply the derivative of mutual information over time, as it is computed from the complete power spectra of the two signals and includes information passed with time delays [40] (Methods). We have demonstrated how active degradation of a master regulator can change the way a growing microcolony generates coordinated phenotypic diversity. Importantly, we have highlighted a role for active degradation in increasing the rate of information transmission. However, this increased information rate comes at the cost of decreased maximum mutual information between downstream targets. Together, these opposite design demands represent a trade-off that genetic networks can optimize based on individualized requirements.
  In E. coli, several key regulators are subject to active degradation. We show that in the case of MarA, the half-life of a regulatory molecule affects not only its time-series behavior and population statistics, but also its ability to coordinate downstream genes. This downstream coordination develops over time, with both a rate and maximum value. We focus on how individual founders of new bacterial populations can quickly establish a spectrum of coordinated phenotypes. We show that generating variability in downstream targets is limited by variability in the activators that control them. Also, signals from activators that change too quickly can be filtered, narrowing the spectrum of coordination in downstream targets. Our results indicate that there is a trade-off between information rate and maximum information in coordinating downstream targets. While this will depend on a number of system specific parameters, this trade-off shows that perfect information fidelity and instantaneous coordination cannot coexist. This is analogous to instructing two people to do a task in tandem–the faster you speak, the faster they can be instructed to coordinate and execute the objective, but at some point information is lost and mistakes will be made. Although we focused on how individual genes and proteins generate phenotypic diversity, extrinsic factors like growth rate have the potential to affect this as well. For instance, varying the cell division time can affect how variability in MarA forms (S5 Fig). In effect, the analytical solutions to the variance functions represent the upper limit on how diversity develops, and decreasing the growth rate scales this curve. Moreover, the greater complexity of the combined mar-rob-sox regulon may affect the rate at which their shared downstream genes coordinate [23]. Expanding this study to investigate the role of each component may reveal more complex emergent effects. While our experiments and models focused on downstream genes where the typical concentrations of MarA fall within their linear regime of activation, it should be noted that this is not true for all downstream targets of MarA. Previous research has highlighted that each downstream target possesses a unique nonlinear dose response curve as a function of MarA activation [27,32]. It may be possible for growing cells to exploit the inherent nonlinearity of each downstream gene’s activation curve to better achieve the stoichiometric balance required by downstream targets with wide ranging functions. That is to say, while multiple downstream targets may be necessary to mount an effective barrier to stressors such as antibiotic challenge, the optimal activation curves of each may be unique. This flexibility of downstream activation allows for this balance to occur despite signals coming from a single shared activator. Alternatively, it may be possible that the promoters compete for a shared pool of regulatory molecules [41]. This would produce an anti-correlation between downstream gene responses, unlike the correlated responses we observed. Still in other systems or possibly in other regimes of MarA expression, this phenomenon may be observable. Moreover, we have focused on the role of MarA as a stochastic activator, however the concentration of MarA is also dependent on the environment [42]. Interestingly, the natural degradation rate of MarA may be below our calculated optimum for balancing between rate and information. In this case, the utility of active degradation may play a role in signal transmission as well by decreasing the response time [43,44]. For instance, active degradation may allow for a transcription factor to more accurately track a time-varying environmental signal. By erasing the memory of a previous protein level, active degradation may allow for a more responsive and agile signaling network. Alternatively, the capacity of a degraded activator to generate phenotypic diversity rapidly may be useful for a few remaining cells to regenerate diversity following antibiotic challenge. Using a combination of single-cell time-lapse movies with multiple fluorescent reporters and analytical models we have demonstrated a role that protein half-life plays in generating single-cell variability and the ability of an activator to produce coordinated diversity in downstream genes. Our results show that both maximum coordinated diversity as well as the rate of coordination between multiple targets is dependent on activator half-life. This suggests a possible design advantage for active degradation and sheds light on how cells grow from individuals to diverse populations, while coordinating expression of related processes.
  We used two strains for the experiments: wild type E. coli MG1655 and E. coli MG1655 ΔmarRAB from [24]. In order to construct the three color reporter plasmid, we modified pNS2-σVL from [29] by placing PmarA, PinaA, and PacrAB upstream of cyan fluorescent protein (CFP), venus yellow fluorescent protein (YFP), and mCherry red fluorescent protein (RFP), respectively. All other plasmids were derived from the BioBrick library described in [45]. For the CRISPRi knockdown of Lon protease, we transformed the plasmid system from [31], using one plasmid containing the constitutively expressed lon targeting sgRNA and another plasmid containing dCas9 under the control of a tetracycline inducible promoter. For the MarA fusion experiments, we cotransformed the IPTG inducible MarA-CFP translational fusion plasmid from [24] with a two-color reporter plasmid bearing PinaA and PacrAB controlling YFP and RFP into E. coli ΔmarRAB. Additional details on plasmid construction are available in S1 File. Cultures were inoculated from single colonies and grown overnight at 37˚C with 200 rpm shaking in Luria-Broth (LB) medium. All strains were grown in 30 μg/ml kanamycin. In addition, strains containing the CRISPRi knockdown system were grown in the presence of 100 μg/ml carbenicillin and 30 μg/ml chloramphenicol. Strains containing the wild type or MarA fusion inducible plasmids were grown with 100 μg/ml carbenicillin in addition to 30 μg/ml kanamycin. Overnight cultures were diluted 1: 100 in selective LB medium. For the wild type or inducible MarA experiments, we added 0 μM IPTG for the MarA-CFP fusion and 10 μM IPTG for wild type MarA and grew cultures for four hours. The differences in IPTG induction allowed for similar mean concentrations with different degradation kinetics (S4 Fig). For the other experiments containing CRISPRi knockdown plasmids, cultures were refreshed for 4 hours without addition of inducer. For microscopy movies, we placed cells on 1. 5% MGC low melting point agarose pads [29]. We used a Nikon Instruments Ti-E microscope to image cells at 100× magnification. Time-lapse movies were taken at a temporal resolution of every three minutes for 10 hours. We used Super Segger cell tracking software to extract fluorescence data from individual cells [46]. When computing the coefficient of variation and the mutual information of growing microcolonies of a given size, data from five replicates were combined and binned according to the number of cells in the microcolony (S3 Fig). We modeled the activator X and the downstream products Y and Z by: x˙=−xτx+σηx (1) y˙=1τy (gyx−y) +σηy (2) z˙=1τz (gzx−z) +σηz (3) where X represents the activator concentration and Y and Z are the concentrations of two downstream proteins. The system is assumed to start at equilibrium and models the relative changes in concentration away from that steady-state point. Moreover, the model uses continuous units of concentration, allowing the functions to be resolvable at arbitrarily small time steps [34]. X, Y, and Z mirror PmarRAB, PinaA, and PacrAB in our experimental system. τ is the correlation time of each of the downstream proteins and is proportional to the half-life (λ) of the molecule: τ = λ/ln (2). g is the gain of each downstream promoter. σ is the noise scaling term, which is set equal to one in the case where the variance is not normalized by the correlation time or is allowed to inversely scale with the correlation time: σ=2τ. η is a zero mean Gaussian distributed white noise random variable. To compute the analytical solutions to the variance over time we applied the methods from [28] to Eq 1: Var{X (t) }=τx2 (1−e−2tτx) (4) To compute the variance for systems that converge to the same final value, we modified Eq 1 to include a normalization term: x˙=−xτx+2τxηx (5) Computing the variance over time from this function yields: Var{X (t) }=1−e−2tτx (6) Derivations for variance calculations of downstream genes can be found in S1 File. Mutual information is calculated from the analytical solution to the correlation function between Y and Z: Corr (y, z) =Cov (y, z) Var (y) *Var (z) (7) I (y, z) =−12ln (1−Corr (y, z) 2) (8) The maximum mutual information between Y and Z is computed analytically by calculating the correlation using Eqs 2 and 3. The analytical calculations for mutual information over time in growing microcolonies can be found in S1 File. Mutual information was calculated from experimental fluorescence data using the YFP and RFP fluorescence data and k-nearest neighbor distances (k = 3) [47]. The information rate between downstream genes is computed using Eqs 2 and 3 and following the methods outlined in [40]. Syz is the cross power spectral density between Eqs 2 and 3, Syy is the power spectral density for Eq 2, and Szz is the power spectral density for Eq 3: R (y, z) =−14π∫−∞∞dωln[1−|syz (ω) |2syy (ω) szz (ω) ] (10) Computing the above power spectra and performing the integration yields the equation below, where R (y, z) is the information rate: R (y, z) =1τx (1+gy2τxτy+1+gz2τxτz+1+gy2τxτy+gz2τxτz−1) (11)
summary: Very few proteins are actively degraded in bacteria, and those that are tend to be regulatory molecules. One example is MarA, which is a master regulator of stress response in E. coli. To study the role of active degradation, we decreased the degradation of MarA two ways: either by knocking down the protease that degrades it, or by modifying MarA to make it resistant to degradation. We measured how changes to the half-life of MarA change the way MarA’s downstream genes are coordinated over time. Combining these results with a theoretical model shows that active degradation in MarA represents a tradeoff favoring an increased rate of coordination over high maximum coordination. By combining experimental results with a theoretical framework, this research provides new insights into the advantages and disadvantage of actively degrading MarA.
keywords: antimicrobials
  ecology and environmental sciences
  medicine and health sciences
  enzymes
  gene regulation
  drugs
  enzymology
  microbiology
  antibiotic resistance
  regulator genes
  luminescent proteins
  antibiotics
  yellow fluorescent protein
  gene types
  pharmacology
  ecological metrics
  antimicrobial resistance
  proteins
  gene expression
  species diversity
  biochemistry
  ecology
  genetics
  microbial control
  biology and life sciences
  proteases
title: Active degradation of MarA controls coordination of its downstream targets