Optimal transcriptional regulation of dynamic bacterial responses to sudden drug exposures

Abstract

Cellular responses to the presence of toxic compounds in their environment require prompt expression of the correct levels of the appropriate enzymes, which are typically regulated by transcription factors that control gene expression for the duration of the response. The characteristics of each response dictate the choice of regulatory parameters such as the affinity of the transcription factor to its binding sites and the strength of the promoters it regulates. Although much is known about the dynamics of cellular responses, we still lack a framework to understand how different regulatory strategies evolved in natural systems relate to the selective pressures acting in each particular case. Here, we analyze a dynamical model of a typical antibiotic response in bacteria, where a transcriptionally repressed enzyme is induced by a sudden exposure to the drug that it processes. We identify strategies of gene regulation that optimize this response for different types of selective pressures, which we define as a set of costs associated with the drug, enzyme, and repressor concentrations during the response. We find that regulation happens in a limited region of the regulatory parameter space. While responses to more costly (toxic) drugs favor the usage of strongly self-regulated repressors, responses where expression of enzyme is more costly favor the usage of constitutively expressed repressors. Only a very narrow range of selective pressures favor weakly self-regulated repressors. We use this framework to determine which costs and benefits are most critical for the evolution of a variety of natural cellular responses that satisfy the approximations in our model and to analyze how regulation is optimized in new environments with different demands.

Significance

Cellular responses to changes in their environment require prompt expression of the appropriate genes. How can a cell optimize the regulation of its responses when the environment changes quickly? Here, we analyze a mathematical model of the dynamics of a typical drug response in bacteria: an enzyme regulated by a transcription repressor that senses the presence of the drug processed by the enzyme. We find that the choice of regulatory strategy depends on the selective pressures acting on each system. Our results establish a framework to understand the design of gene regulation in the context of dynamical responses, highlighting the importance of transcription factors in not only regulating steady-state expression levels but also in controlling gene expression outside of equilibrium.

Introduction

Cellular responses to changes in their environment, in fact cellular processes in general, require the coordinated expression of genes. When exposed to a chemical compound that demands urgent processing, the cell needs to sense its presence and synthesize precise amounts of correct enzymes in a timely fashion. Therefore, the optimization of cellular responses does not involve solely the evolution of enzymes but also the evolution of appropriate regulation that can manage the costs and benefits associated with the expression of these enzymes over the course of the response (1). Both of these aspects evolve in concert in the cell’s DNA sequence: while the coding regions (genes) contain information about the biochemical properties of enzymes and regulators, the noncoding regions contain information about DNA binding sites that together dictate the patterns of enzyme expression. Much attention has been given to the evolution of genes (2), while other studies recognize that the evolution of cellular responses typically involves changes in regulatory pathways (3,4). However, we still lack a framework to describe the evolution of cellular responses as dynamical processes (5, 6, 7, 8, 9) that can be directly inferred from genotype.

The balance of costs and benefits shaping the evolution of genes and their regulation depends on the particular context of each cellular response (10, 11, 12). The enzymes at play invariably interfere with other cellular processes, so their expression often poses trade-offs between their intended function and their unintended consequences. For instance, the presence of a toxin is harmful for the cell, but overproduction of the enzyme that rids the cell of this toxin can disrupt other cell functions, posing heavy metabolic burdens and impairing cell growth (13). In this case, the choice of regulation of such enzyme will depend strongly on the relation between the cost of the presence of the toxin versus the cost of enzyme side effects. Ultimately, the selective pressures guiding the evolution of a cellular response can be translated into a set of costs associated with the time-dependent concentration of each of the molecular components participating in the response. Evolution then proceeds to minimize the costs by optimizing the rates of the biochemical interactions involving these components.

Here, we analyze how different selective pressures shape the evolution of a simple response pathway: a transcription factor regulating an enzyme for a particular toxic substrate (drug), which is a typical mechanism of bacterial resistance to sudden shifts in the concentration of an antibiotic drug (14). Specifically, we base our model on the tetracycline resistance tet operon, a classical system regulating antibiotic resistance via drug efflux. The response to tetracycline is regulated by repressor TetR, part of the large TetR family of transcriptional repressors, which is known to regulate rapid cellular responses in unstable environments (15). We analyze the behavior of this system upon a sudden and sustained exposure to the drug, a situation where proper induction of the response is of utmost importance for the cell. In the absence of the drug, the expression of the enzyme is repressed by the transcription factor (Fig. 1 A). Upon exposure to the drug, as its concentration starts accumulating in the intracellular space, the transcription factor recognizes and binds the drug, losing affinity to the DNA and de-repressing the enzyme. The enzyme then proceeds to process the drug, whether through efflux or deactivation, until its intracellular levels begin to decrease. Finally, when the intracellular concentration of the drug returns to low levels, the transcription factor is released and resumes repression of the enzyme.

Figure 1.

General model of a cellular response to a drug exposure. (A) In the absence of drug, a transcription factor represses the expression of an enzyme and of itself. When drug is imported into the cell, it binds and inactivates the repressor, releasing expression of both genes, and the enzyme proceeds to process the drug. Cell components are degraded or diluted due to cell growth. (B) Measurements of cell growth in a population of 40 tetracycline-resistant single E. coli cells exposed to 70 μg/mL of tetracycline (around the half-maximal inhibitory concentration) at time zero. About half of the population survives exposure (22 cells). The average growth rate in the surviving subpopulation is plotted in bold. (C) Regulation of repressor and enzyme are modeled as on/off according to the concentration of free repressor. Repressor and enzyme are fully expressed at rates h_r and h_z when free repressor $R_{free}$ levels are below thresholds R_r and R_z, respectively, and are not expressed if free repressor concentration exceeds these thresholds. (D) A typical response to a sudden increase in extracellular drug concentration D_out at time zero, showing subsequent peaks in intracellular drug (D_peak) and enzyme (Z_peak) concentrations, before stabilizing to final values D_final, Z_final, and R_final. (E) Expression of efflux pump TetA and repressor TetR in the subpopulation of surviving E. coli cells during the tetracycline response. Intracellular free drug concentration was estimated from the initial decrease in cell growth upon drug exposure, assuming that tetracycline binds and deactivates ribosomes following Michaelis-Menten kinetics and that cell growth is proportional to the unbound fraction of ribosomes: $\lambda ={\text{λ}}_{0}K/(K+D_{free})$ (16). The calculated $D_{free}$ concentrations are normalized and therefore independent of $K$.

This model is especially relevant in situations where the cell needs to react quickly to rapidly changing environments (10,17,18), where a fast response brings the system out of equilibrium. Regulation of antibiotic resistance typically happens at the transcriptional level, and mechanisms where a single transcription factor both senses the drug and regulates gene expression provide particularly fast responses (14). However, indirect pathways for activation of resistance and alternative mechanisms of regulation, such as translational attenuation, are also common (14). In either case, if the drug (or another chemical signal) is sensed via ligand binding and is rapidly transduced into de-repression of the appropriate enzyme, the system will follow similar dynamics as described by our model. We describe the regulatory mechanism based on transcription repression rather than activation. Such mechanisms provide faster responses (19,20) and are typical in the regulation of antibiotic resistance (14,21). Antibiotic resistance enzymes act mainly by drug efflux through the cell membrane or drug deactivation, in which case they interact with the drug directly and follow Michaelis-Menten enzyme kinetics, as described in our model. However, some resistance enzymes act via drug target protection (14) and follow different kinetics. Therefore, we describe the response dynamics of a widespread mechanism of regulated antibiotic responses. Since the dynamics described in our model are not particular to antibiotic resistance mechanisms, many insights from this formulation can also be generalized to other transcriptionally repressed cellular responses that both sense and negatively act upon a chemical signal.

Materials and methods

All code used to generate the figures in this article is available at https://github.com/schultz-lab/Biophys-J-2022.

Pareto front

Pareto-optimal phenotype is defined as a phenotype whose multiple metrics (performance measures) cannot be all simultaneously improved within the allowed space of parameter values. Thus, improving one or more metrics would necessarily be accompanied by degrading one or more others (22). Pareto front is a set of all Pareto-optimal phenotypes. Since the topography of performance landscapes for individual performance measures are complicated, we computed Pareto front numerically, following (23). Here, we briefly describe our algorithm for the case described above, where we kept $h_z$ constant and only explored the two-dimensional (2D) parameter space of $h_r$ and $R_r$. We discretized the 2D parameter domain $0<h_r<h_r^{max}$, $0<R_r<R_r^{max}$ and computed five 2D arrays ${\mathit{Z}}_{initial},{\mathit{Z}}_{final},{\mathit{Z}}_{peak},{\mathit{D}}_{final},{\mathit{D}}_{peak}$ of our performance metrics $(Z_{initial},Z_{final},Z_{peak},D_{final},D_{peak})$ defined in every point $(i,j)$ in that domain. Then, for each point $(i,j)$ in that domain, we define a 2D binary array ${\mathit{Q}}_{ij}$ with elements

$$Q_{ij}\left(i^{\prime },j^{\prime }\right)=u\left[Z_{intial}\left(i,j\right)-Z_{initial}\left(i^{\prime },j^{\prime }\right)\right]\ast u\left[Z_{final}\left(i,j\right)-Z_{final}\left(i^{\prime },j^{\prime }\right)\right]\ast u\left[Z_{peak}\left(i,j\right)-Z_{peak}\left(i^{\prime },j^{\prime }\right)\right]\ast u\left[D_{final}\left(i,j\right)-D_{final}\left(i^{\prime },j^{\prime }\right)\right]\ast u\left[D_{peak}\left(i,j\right)-D_{peak}\left(i^{\prime },j^{\prime }\right)\right],$$

where $u\left[x\right]$ again denotes the Heaviside function, $u\left(x\right)=0$ for $x<0$, and $u\left(x\right)=1$ for $x\ge 0$, and $\ast$ stands for component-wise multiplication. If there is at least one element $Q_{ij}(i^{\prime },j^{\prime })$ of ${\mathit{Q}}_{\mathit{ij}}$ other than $Q_{ij}(i,j)$ that is equal 1, it means that all our metrics of the phenotype $(i^{\prime },j^{\prime })$ are lower (better) than of $(i,j)$, and so phenotype $(i,j)$ is dominated, or Pareto-suboptimal. Therefore, we discard a point $(i,j)$ if $ma{x}_{i^{\prime },j^{\prime }}\left\{\mathit{Q}ij\right\}>0$. The remaining points according to our definition belong to the set of Pareto-optimal phenotypes. As we expected, they indeed span region 4 (Figs. 2 A, 4 B, and S11).

Figure 2.

Distinct progressions of cellular responses. (A) Regions in the R_r × h_r plane that correspond to qualitatively different response progressions. Region 4, highlighted by red boundaries, contains the parameters resulting in all possible regulated responses, where the enzyme is repressed in the absence of drug and is induced in its presence. (B) Time courses of the responses corresponding to the different regions in the R_r × h_r plane in (A). For parameter combinations corresponding to points to the right of the diagonal, the threshold for repressor self-repression is increased further than the maximum expression of repressor, and therefore expression of repressor is constitutive. The system behavior for these points is similar to that for the corresponding points on the diagonal, where the threshold is the same as maximum repressor expression, for the same $h_r$.

Figure 4.

Changes in relative weights of selective pressures cause optimal regulatory strategy to switch. (A) Archetypes minimize the metric corresponding to a single selective pressure while disregarding the others, leading to extreme values mostly outside of the region corresponding to regulated responses. Archetypes are shown in yellow for each metric of the response. (B) The parameter region corresponding to regulated responses is the Pareto front of optimal solutions. The edges of this region each minimize one of the selective pressures of peak and final enzyme and drug concentrations, as indicated by the colors. The gray dot in the middle of the region indicates the generalist strategy, which optimizes the response when all four indicated selective pressures have the same weight ([w^z_peakw^d_peakw^z_finalw^d_final] = [1 1 1 1], while w^z_initial$\gg$ 1 and w^r_initial = w^r_peak= w^r_final = 0). (C–F) Tracking of optimal solutions when the weight corresponding to each selective pressure changes from 0.1 to 10, with all other weights held constant at 1. 200 points logarithmically spaced are shown. In each case, the regulatory strategy switches between a constitutively expressed and a self-regulated repressor, passing through the generalist strategy. The decision between prioritizing the optimization of final enzyme or drug values results in large changes in repressor expression h_r (large and small, respectively). Conversely, the decision between prioritizing the optimization of peak enzyme or drug values depends mostly on the threshold for repressor self-repression R_r (large and small, respectively). (G–J) Optimal parameters h_r and R_r as the weight corresponding to each selective pressure changes from 0.1 to 10, showing sharp transitions between constitutive repressor expression and strong self-repression of the repressor.

Regulatory parameters in natural circuits

Here, we calculate regulatory parameters ${h^{\prime }}_r=h_r/\left(\lambda R_z\right)$ and ${R^{\prime }}_r=R_r/R_z$ for a variety of E. coli responses that are reasonably characterized in the literature. When data are available on the expression of repressor with and without the presence of the drug/substrate (inducer), we assume the uninduced repressor expression level to be at the threshold for self-repression $R_r$, as explained in the Model assumptions. The fully induced level, in the presence of drug/substrate, is considered to be at maximal expression $h_r/\lambda$. The thresholds for enzyme repression were calculated using data on the affinity of repressor to its promoters, the kinetics of the response, or inferred from the architecture of the operons. The following are examples studied in the literature.

tet operon

Repressor TetR regulates efflux pump TetA, which provides resistance against tetracycline. Expression data during a response to tetracycline show that the fully induced levels of repressor TetR are about 5 times higher than uninduced levels, or $h_r/\lambda \approx$ 5$R_r$ (7). TetR is expressed by two promoters. Promoter $P_{R_1}$ is regulated by two TetR binding sites, in the same fashion as the TetA promoter $P_A$. However, promoter $P_{R_1}$ is regulated by only one of the binding sites, tet O₁, which has half the affinity to TetR as tet O₂ (24). Therefore, the threshold for TetR self-repression is twice as large as the threshold for TetA repression, or $R_r\approx 2R_z$. These values result in ${h^{\prime }}_r\approx 10$ and ${R^{\prime }}_r\approx 2$.

marRAB operon

Repressor MarR regulates global activator MarA, which induces a broad response against many stresses, including antibiotics. Expression data during a response to salicylate show that the fully induced levels of repressor MarR are about 20 times higher than uninduced levels, or $h_r/\lambda \approx$ 20$R_r$ (25,26). Since repressor MarR is expressed from the same transcript as regulators MarA and MarB, the threshold for their activation is the same, or $R_r=R_z$. These values result in ${h^{\prime }}_r\approx 20$ and ${R^{\prime }}_r\approx 1$.

IclR and aceBAK operon

Repressor IclR regulates the aceBAK operon that expresses enzymes for acetate metabolism. Expression data during a response to acetate show that fully induced levels of repressor IclR are about eight times higher than uninduced levels, or $h_r/\lambda \approx$ 8$R_r$ (27). Both the IclR promoter and the aceBAK promoter become active when IclR concentration drops below 3 pmol, so $R_r\approx R_z$ (28). These values result in ${h^{\prime }}_r\approx 8$ and ${R^{\prime }}_r\approx 1$.

mtlADR operon

Repressor MtlR regulates enzymes MtlA and MtlD for mannitol utilization. Enzyme activity data show a 14.5-fold increase in activity of enzyme MtlD upon induction with mannitol (29,30). Since MtlR is expressed from the same transcript as MtlD, we assume it is subjected to similar regulation. Therefore, full expression of MtlR is $h_r/\lambda \approx$ 14.5$R_r$, and the threshold for repression of repressor and enzyme are the same $R_r=R_z.$ These values result in ${h^{\prime }}_r\approx 14.5$ and ${R^{\prime }}_r\approx 1$.

GalS and mgl operon

Repressor GalS primarily regulates the mglABC transport system, which has high affinity for galactose and is utilized during low galactose concentrations, although there is overlap with regulation from repressor GalR. GalS is negatively auto-regulated, and induction by galactose analog D-fucose results in 3-fold higher expression from the galS promoter (31), so $h_r/\lambda =$ 3$R_r$. GalS affinity to the mgl promoter shows a dissociation constant of 18 nM (32), so we assume $R_z\approx$ 18 nM. Full expression of GalS dimers was estimated to be $h_r/\lambda$= 242 nM (assuming GalS is mostly dimerized) (33). These values result in ${h^{\prime }}_r\approx 13.4$ and ${R^{\prime }}_r\approx 4.5$.

GalR and gal operon

Repressor GalR primarily regulates galP and the galETKM operon for galactose transport and metabolism, which has lower affinity for galactose and is utilized in higher galactose concentrations. There is also overlap with regulation from repressor GalS. GalR is expressed almost constitutively, and induction by galactose analog D-fucose resulted in a small 1.1-fold increase in expression from the galR promoter (31), so $h_r/\lambda$ = 1.1$R_r$. GalR affinity to the galETKM promoter shows a dissociation constant of 1.3 nM (32), so $R_z\approx$ 1.3 nM (affinity to the galP promoter is 0.3 nM). Full expression of GalR dimers was estimated to be $h_r/\lambda$= 62.5 nM (assuming GalR is mostly dimerized) (33). These values result in ${h^{\prime }}_r\approx 48$ and ${R^{\prime }}_r\approx 44$.

LacI and lac operon

Repressor LacI regulates the lacZYA operon that expresses enzymes for lactose metabolism. Repressor LacI is expressed almost constitutively at about 25 tetramers per E. coli cell, so $h_r/\lambda \approx$ 25 nM. These tetramers have high affinity to the lac promoter, with a dissociation constant in the order of 0.4 nM, so $R_z\approx$ 0.4 nM (34). LacI is thought to show a negligible self-repression of around 5% ($R_r\approx$ 0.95 $H_r/\lambda$) (35). These values result in ${h^{\prime }}_r\approx 62.5$ and ${R^{\prime }}_r\approx 59.4$.

Results

General model of drug response

We model the cellular response by a system of three differential equations describing the concentrations of the enzyme $Z$, the repressor $R$, and the intracellular drug $D$ (36):

$$\begin{array}{l}\dot{Z}=H_z\left(R_{free}\right)-\lambda Z\\ \dot{R}=H_r\left(R_{free}\right)-\lambda R\\ \dot{D}=K_i\left(D_{out}\left(t\right)-D_{free}\right)-\frac{k_{cat}Z{D}_{free}}{K_M+D_{free}}-\lambda \text{D}\\ with{R}_{free}=R\frac{k_d}{k_d+D_{free}}\end{array}$$

The drug is imported into the cell with a rate proportional to the difference between its extracellular ($D_{out}$) and intracellular ($D$) concentrations with an import rate constant $K_i$, as is the case when the drug diffuses across the cell membrane. The drug is processed by the enzyme $Z$ following Michaelis-Menten kinetics, with a catalytic rate constant $k_{cat}$ and a Michaelis constant $K_M$. As the intracellular drug binds and inactivates the repressor, the synthesis of the enzyme and the repressor inside the cell is regulated by the amount of free (not bound to the drug) repressor $R_{free}$ according to regulatory functions $H_z$ and $H_r$, respectively (discussed below). Since the biochemical binding and unbinding of the drug to the transcription factor typically happen at a much faster rate than the aforementioned processes, we consider their unbound (free) forms ($D_{free}$, $R_{free}$) to be in chemical equilibrium with the bound form $\left[RD\right]$ with a dissociation constant $k_d$, such that $R_{free}{D}_{free}=k_d\left[RD\right]$. Therefore, the inactivation of repressor by the drug is included in the differential equations implicitly, with $R_{free}$ calculated from this binding equilibrium.

All intracellular components are diluted in the cytoplasm as the cell grows with a rate $\lambda$, which we assume constant for the duration of the response. Although antibiotics can certainly affect the growth, we base this assumption on the observation that severe reduction of growth and gene expression in the presence of antibiotics typically leads to cell death. Previous studies have shown that exposure to high drug doses causes a heterogeneous response in bacterial populations, with the coexistence of surviving and arrested cells (7,37,38). Throughout the response, surviving cells show only a modest reduction in growth (Fig. 1 B). Upon exposure to antibiotics, the expression of resistance depends on global effects of the drug action on cell growth and gene expression. Therefore, failure to quickly deploy resistance genes results in higher concentrations of intracellular drug and further reduction in expression of resistance genes and drug accumulation. This positive feedback mechanism prevents the survival of slow-growing cells, resulting in the coexistence of fast-growing and arrested cells. Therefore, maintenance of fast growth is required for the expression of resistance and cell survival during exposure to large drug concentrations.

Although the mechanism described in this formulation is general and typical of antibiotic responses, which are usually controlled by only a few genes, the regulation of other cellular responses can be more complex (39). Metabolic responses, for instance, regulate the import of substrates in addition to the enzymes that process it. Regulation of responses can also depend on additional molecular interactions, such as phosphorylation or other forms of posttranslational modifications (40). The path between the sensing of the chemical signal and the action upon it can also involve extra steps, such as in eukaryotes, where signal transduction typically involves nuclear localization (41). Other responses are regulated by transcriptional activation instead of repression, which results in different dynamics (42). Here, we focus our analysis on the simple and ubiquitous mechanism where the dominant form of regulation is performed by a transcription repressor that both senses the chemical signal and regulates gene expression, but our framework can also be expanded to analyze these other scenarios.

Model assumptions

For simplicity, we focus our analysis on the evolution of the regulation of the response, assuming that the enzyme is already optimized to process the drug. This assumption is based on the observation that the short-term evolution of cellular responses in new environments proceeds primarily through the rewiring of regulatory interactions (3,43,44). However, our framework can easily accommodate additional parameters describing the evolution of the enzyme, which is particularly important for enzymes that are not optimized for a particular drug (such as enzymes with a broad range of substrates) or where processing of the drug brings heavy costs for the cell. We consider a simple “on/off” regulation of the synthesis rates of enzyme and repressor ($H_z$, $H_r$) by the concentration of free repressor $R_{free}$ (Fig. 1 C). Synthesis is kept at a constant rate ($h_z$, $h_r$) at low levels of free repressor and is shut down when the concentration of free repressor crosses a threshold for inhibition ($R_z$, $R_r$). This description results in four parameters describing gene regulation during the response, corresponding to promoter strengths and thresholds of inhibition of each the enzyme and its repressor. Although the on/off description misses finer details of the regulation (45,46) that are important in setting the steady-state concentrations of the molecular components of the response, it still captures essential aspects of the dynamics of cellular processes out of equilibrium. The Hill coefficient of cellular responses in real systems is typically in the order of four or higher, which yields qualitatively similar dynamics (35) (Fig. S9 A–C). “Leaky” repressors that allow higher basal levels of the enzyme in the absence of drug can also expedite the response at the expense of a potentially costly constant expression of enzyme, which is not captured by our model. On/off gene expression implements “bang bang” control, ubiquitous among cellular processes, which is frequently the optimal solution for minimum-time problems (17). For instance, the quickest way to reach a specific concentration of enzyme is to synthesize it at maximum speed until a switching point where expression is shut off. On/off expression also captures the temporal organization of gene expression in pulses, typical of cellular processes such as stress responses, signaling, and development (5,47, 48, 49, 50).

We can now numerically evaluate the time course of all the relevant concentrations in a typical response where the extracellular concentration of drug suddenly rises from zero to a high level $D_{out}$ at time zero (Fig. 1 D). Knowing how the system responds to a sudden input is important because it gives information on the stability of such a system and on its ability to reach one stationary state when starting from another. Besides, large deviations from the long-term steady state may have extreme effects on the overall system. In the absence of a drug, the concentration of repressor will sit at the threshold for its own repression $R_r$ (provided $h_r/\lambda >R_r$). Decreases in repressor concentration will activate its synthesis, while increases in its concentration will shut it off. Provided that $R_z<R_r$, the enzyme concentration will be zero. Once the drug is present, it starts to flow into the cell and bind the repressor, decreasing the amount of free repressor until it crosses the threshold for activation of enzyme $R_z$ at time $t_1$. The enzyme is then expressed, processing the drug until its intracellular concentration starts to decrease, after reaching a peak concentration $D_{peak}$. The concentration of free repressor then starts to increase until it crosses back the threshold for enzyme repression $R_z$ at time $t_z$, after which the concentration of enzyme begins to decrease. After a series of decaying oscillations in drug and enzyme concentrations, caused by inherent delays in the regulatory negative feedback loop, during which the enzyme synthesis is intermittently switched on and off, the levels of response components equilibrate to their final steady-state values. We note that the sustained decaying oscillations we observe are due to the description of regulation as strictly on/off, and real systems with smoother regulatory functions would reach equilibrium more quickly. In this process, free repressor concentration equilibrates to $R_z$, while the repressor synthesis is unchecked, reaching maximum levels. This time course of the response dynamics can reproduce experimental measurements of expression of resistance genes and cell growth, obtained in a resistant population of E. coli cells suddenly exposed to a large dose of tetracycline in a microfluidic device (data obtained from (7); Fig. 1 E).

We further consider two approximations. Inequality $D_{free}\ll K_M$ assumes that the concentration of intracellular drug does not reach high enough levels to saturate the enzyme. While a sharp increase in intracellular substrate concentration can exceed an enzyme’s Michaelis constant (51), this is not typically the case for the passive diffusion of drugs through the cell membrane (52,53). This regime where intracellular substrate concentration is kept low is typical of fast responses, such as drug responses, which are readily induced by the drug and are pressured to process it rapidly. Inequality $R_{free}\ll k_d$ assumes that the concentration of intracellular drug is much higher than the concentration of repressor, such that $D_{free}\approx D$. This means that regulation operates at relatively low levels of repressor, as is usually the case, and repressor binding does not significantly reduce the concentration of free drug. We test the limits of our approximations on on/off control, constant growth, and no enzyme saturation in Fig. S9.

We now rescale the variables and parameters of the system, as described in the supporting material and summarized in Table 1, to arrive at nondimensional equations:

$$\begin{array}{l}\dot{Z^{\text{'}}}=h_{z}u\left(1-{R^{\text{'}}}_{free}\right)-Z^{\text{'}}\\ \dot{R^{\text{'}}}=h_{r}u\left({R^{\text{'}}}_r-{R^{\text{'}}}_{free}\right)-R^{\text{'}}\\ \dot{D^{\text{'}}}=K_i\left({D^{\text{'}}}_{out}\left(t\text{'}\right)-D^{\text{'}}\right)-Z^{\text{'}}{D}^{\text{'}}-D^{\text{'}}\end{array}$$

with $R{\prime }_{free}=R^{\prime }/(1+D^{\prime })$ and $u\left(x\right)$ the Heaviside function. In these rescaled variables, time is rescaled by the growth rate of the cell ($t^{\text{′}}=\lambda t$), along with all reaction rates. The concentrations of enzyme and repressor are rescaled using parameters related to their “function” in the system, the enzyme catalytic rate ($Z^{\text{′}}=k_{cat}Z/\lambda K_M$), and the threshold for enzyme repression ($R^{\text{′}}=R/R_z$), respectively. The drug concentration is rescaled by its affinity to the repressor ($D^{\text{′}}=D/k_d$). Therefore, $R_{free}=1$ is the concentration of free repressor necessary to repress the enzyme, D = 1 is the drug concentration that binds half of the repressor, and Z = 1 is the enzyme concentration that processes one “functional unit” of drug per unit time (set by the cell growth rate). The regulation of the system is now described by three nondimensional parameters: $h{\text{′}}_z=k_{cat}{h}_z/\left({\lambda }^2{K}_M\right)$, which measures the drug processing by enzyme production; ${h^{\prime }}_r=h_r/\left(\lambda R_z\right)$, which measures enzyme repression by repressor production; and ${R^{\prime }}_r=R_r/R_z$, which measures the relative strength of enzyme repression and self-repression by the repressor (primes are dropped elsewhere for convenience).

Table 1.

Variables and parameters used in the model

Variables		Dimensionless variables		Interpretation
t	time	t’	λ t	time scaled by growth rate
Z	enzyme concentration	Z′	kcatZ/(λKM)	enzyme concentration relative to its catalytic rate
R	total repressor concentration	R′	R/Rz	repressor concentration relative to the threshold for enzyme repression
D	intracellular drug concentration	D′	D/kd	intracellular drug concentration relative to the repressor dissociation constant
Dout	extracellular drug concentration	D’out	Dout/kd	extracellular drug concentration scaled by the repressor dissociation constant
Rfree	free (active) repressor concentration	R’free	Rfree/Rz	free repressor concentration relative to the threshold for enzyme repression
Dfree≈ D	free intracellular drug concentration	–	–	–
Parameters		Dimensionless parameters		Interpretation
λ	cell growth rate	–	–	–
Ki	drug import rate enzyme	K’i	Ki/λ	drug import rate relative to growth rate
$k_{cat}$	enzyme catalytic rate	–	–	–
$K_M\gg D_{free}$	enzyme Michaelis constant	–	–	–
$k_d\gg R_{free}$	repressor/drug dissociation constant	–	–	–
$h_z$	enzyme synthesis rate (when on)	h’z	kcathz/(λ2KM)	drug processing by enzyme production
$h_r$	repressor synthesis rate (when on)	h’r	hr/(λRz)	enzyme repression by repressor production
Rz	threshold for enzyme repression	–	–	–
Rr	threshold for repressor self-repression	R’r	Rr/Rz	relative strength of enzyme repression and self-repression by the repressor

We include variables in the system of differential equations, constants relating to the biochemical processes, and the normalizations leading to a dimensionless system of equations. Primes are dropped in the main text for simplicity.

We can now theoretically analyze the time course of a response to a sudden increase in extracellular levels of drug. For this, we can divide the time course of the response in different intervals, separated by the events when the concentration of free repressor crosses the thresholds for repression of either the enzyme ($R_z$) or the repressor itself ($R_r$). In the intervals between these events, the synthesis rates of both the enzyme and the repressor are constant, and the model equations can be integrated explicitly. We detail this procedure in the supporting material. Next, we consider the different regimes of the cellular response in the parameter space defined by $h_z$, $h_r$, and $R_r$.

Optimization of response dynamics

In this section, we will discuss how selective pressures in the cell environment determine the choice of parameter values that optimize the response. Let us fix the full enzyme expression level $H_z$ and only consider the two-parameter plane R_r × h_r, which describes the regulation by the transcriptional repressor (Fig. 2). This plane can be divided into regions where the progressions of the response are qualitatively different, and the boundaries between these different regimens can be used to constrain the search for optimal parameters. Constitutive enzyme expression occurs when h_r < 1, full expression of repressor does not reach the threshold for enzyme inhibition, and when R_r < 1, the repressor represses itself more strongly than it represses the enzyme. In both of these cases, the repressor is unable to regulate the enzyme, which is always expressed at a constant level $h_z$ (region 0). Constitutive repressor expression occurs when $R_r>h_r$, the threshold for repressor self-repression is higher than its full expression level. Since self-repression is never reached, the repressor is always expressed at a constant level $H_r$ (regions 1, 3, and 5). Notably, constitutive expression of repressor in this region is independent of $R_r$, so any point where $R_r>h_r$ is equivalent to the point on the boundary where $R_r=h_r$, and the constitutive repressor expression is already reached. At very high levels of repressor expression $h_r>{\alpha }_2=K_i{D}_{out}/(K_i+1)$ (or very low drug level $D_{out}$), the concentration of free repressor never drops below the threshold to activate the enzyme, so the response is never induced (regions 5 and 6). At lower levels of repressor expression $1<h_r<{\alpha }_1=K_i{D}_{out}/(K_i+h_z+1)$, repressor levels are insufficient to curb enzyme expression once the response is induced, leading to unnecessary full expression of enzyme (regions 1 and 2). Therefore, regardless of the selective pressures applied, optimized parameters for the response can be expected to be found in the region defined by $1<R_r<h_r$ and ${\alpha }_1<h_r<{\alpha }_2$ (region 4), with the remaining parameter space either being equivalent to the boundaries of this region or presenting extreme and obviously suboptimal solutions. We note that since the boundaries of region 4 depend on the extracellular drug concentration, the regulation of the response will be optimized according to the typical drug concentration to which the cell is exposed in its environment.

The selective pressures acting on the system during its evolution dictate the choice of parameters that maximize its efficiency. An ideal response should achieve maximal processing of the drug, while expending the least amounts of enzyme and repressor. Therefore, the optimization of the response involves the minimization of drug, enzyme, and repressor levels during the course of the response. We note (54) that minimization of drug levels is desirable because there is a cost associated with its toxicity, while minimization of enzyme and repressor levels are desirable both due to the metabolic burden of their syntheses and due to their possible interference with other cellular processes. The specific context of the system operation poses additional constraints that might prioritize the minimization of one of the response components over the other. For instance, the choice between minimization of drug or enzyme levels depends on which of them imposes higher costs for the cell. Strictly speaking, the system performance is characterized by the whole time courses of drug, enzyme, and repressor levels before, during, and after the response, which makes the optimization problem infinite dimensional.

We simplify our cost calculation by breaking down the cost associated with each component into three parts: (1) cost in the absence of drug, calculated from initial levels of the component. These costs are higher in environments where contacts with the drug are rare. (2) Costs upon exposure to the drug, calculated from maximum levels of the components reached during the response. These costs reflect high levels of the components that can be reached temporarily while concentrations are out of equilibrium but are not sustained. (3) Costs in the sustained presence of the drug, calculated from the end levels of the components. These costs are higher in environments where the drug is present for long periods of time. We note that calculating the cost by integrating concentrations along the whole time course of the response would make the cost of temporary high levels reached upon exposure (2) dependent on the duration of the response, which is not desirable. Therefore, to keep the dimensionality of the problem finite, we approximate the total cost of a given response using a finite set of key concentrations obtained from the dynamics of the response $c_i$: initial, maximum, and final levels for each of the three components (drug $D_{initial},D_{peak},D_{final}$, enzyme $Z_{initial},Z_{peak},Z_{final}$, and repressor $R_{initial},R_{peak},R_{final}$). Since, in the presence of drug, the concentration of free repressor equilibrates to $R_z$, below the threshold for its own repression, expression of repressor reaches full levels at equilibrium (above). Therefore, we always have $R_{peak}=R_{final}$. Furthermore, we always assume that $D_{initial}=0$, so we are left with seven output metrics. We then define the selective pressure acting on the system by a set of weights $w_i$ assigned to each of these metrics and compute the total cost of the response as a weighted sum of individual metrics $C=\sum _i{w}_i{c}_i$. The relative weights of different metrics represent the selective pressures of the environment, such as frequency and duration of drug exposure, and determine which set of regulatory parameters $(h_z,h_r,R_r)$ minimizes the total cost $C$. This approach assumes for simplicity that the cost grows linearly with each key concentration, which might not hold true, particularly in cases where the cost saturates at specific levels.

Next, we analyze the importance of the cost of individual components in determining the optimal response regulation. We start by finding a “reference” set of weights that does not prioritize any particular metric of the response. The initial levels of repressor $R_{initial}$ and enzyme $Z_{initial}$ are sustained indefinitely in the absence of drug. Therefore, in environments where the presence of drug is relatively rare and there is no demand for the enzyme otherwise, as is typically the case in antibiotic resistance, the cost of the initial level of enzyme should be high to reflect a selective pressure to keep the system repressed by default (otherwise, constitutive expression of appropriate levels of the enzyme would be optimal). As for the other metrics, since concentrations in our model are similarly scaled by relevant constants concerning their function, their respective weights are given in “cost per functional unit.” Therefore, the weights of each key concentration can be directly compared. Indeed, we find that picking one for the remaining weights results in optimized parameters in the center of region 4, from where slight variations in weight cause the resulting optimized parameters to diverge (as seen later in the section on Pareto-optimal solutions). We take this as our reference set of weights, and we analyze the effects of different selective pressures by varying the corresponding weights while keeping the others at one.

For a given set of weights describing a selective pressure and a given extracellular drug concentration, the optimal regulation is found within the 3D space defined by $R_r$, $h_r$, and $h_z$. If we disregard the costs of expressing repressor and consider only the costs associated with drug and enzyme, optimal solutions are always found at the highest enzyme expression rate $h_z$ physiologically permitted, since there is no cost in expressing repressor fast enough to ensure that enzyme production is shut down after only a short burst of expression. However, if there is a cost for expression of repressor, a local optimum can exist at an intermediate level of enzyme expression rate (Fig. 3). From here on, we will consider the maximal rate of enzyme expression $h_z$ fixed at a high level and focus on the design of regulation that optimizes the response in the $R_r\times h_r$ plane.

Figure 3.

Selective pressures to minimize enzyme or drug concentration result in qualitatively different optimal regulatory strategies. (A) Cost of the response across the R_r × h_r plane for K_i = 1, h_z = 25, D_out = 50, and weights [w^r_initialw^z_initialw^z_peakw^d_peakw^r_finalw^z_finalw^d_final] = [0 10 2 1 0 1 1], where peak enzyme concentration is relatively more costly for the cell than peak drug concentration. The cost of expressing enzyme in the absence of drug is very high, but there is no cost of regulation. Optimal parameters are indicated by the orange dot, located at the edge corresponding to a constitutively expressed repressor. The region corresponding to regulated responses (region 4 in Fig. 2A) is highlighted; increasing $R_r$ past the edge of this region does not change the dynamics of the response or its cost. (B) Cost of the response in the three-dimensional space showing the R_r × h_z plane for the same parameters as in (A). It is still possible to reduce the cost of the response by increasing h_z. (C) Time course of the response for the optimal parameters calculated in (A), with low peak concentration of enzyme. (D–F) Cost and time course of the response for the same parameters as in (A), except for weights [w^r_initialw^z_initialw^z_peakw^d_peakw^r_finalw^z_finalw^d_final] = [0 10 1 2 0 1 1], where peak drug concentration is more costly for the cell than peak enzyme concentration. The optimum is now located at the edge corresponding to a strongly self-repressed repressor (purple dot), and the time course of the response shows a low peak drug concentration. Intermediate optima found when weights shift linearly from values in (A) (orange dot) to values in (B) are shown in gray. (G–I) Cost and time course of the response for the same parameters as (A), except for weights [w^r_initialw^z_initialw^z_peakw^d_peakw^r_finalw^z_finalw^d_final] = [1 10 1 2 1 1 1]. Since there is a cost of regulation, the optimum (purple dot) is now found for an intermediate value of h_z=28.8 and is a local minimum in the three-dimensional space defined by $R_r$, $h_r$, and $h_z$.

Depending on the relative costs of drug and enzyme, the system is optimally regulated by qualitatively different strategies. Responses that prioritize minimization of enzyme levels show optimal responses with constitutive repressor expression (Fig. 3 A–C), while responses that prioritize minimization of drug levels require strong self-repression of the repressor (Fig. 3 D–F). When curbing expression of the enzyme is prioritized, the cell must keep high concentrations of repressors throughout the response to avoid any unnecessary enzyme expression. Therefore, selective pressures with a high cost of peak enzyme concentration find optima along the $R_r=h_r$ boundary, with high constitutive expression of the repressor. However, this solution comes at the expense of a slow induction of the expression of the enzyme, since intracellular drug levels need to rise further to bind and deplete a large initial pool of repressor. To avoid the resulting accumulation of drug, the response must follow the inverse logic. For a rapid response that prioritizes minimizing intracellular drug accumulation, the enzyme must be induced quickly upon exposure to the drug, so the initial concentration of repressor must be as low as possible. Therefore, selective pressures with a high cost of the maximal drug concentration find optima near the $R_r=1$ boundary ($R_r=R_z$, in dimensional units), where the threshold for repressor self-repression is just above the threshold for enzyme repression. This solution requires the strongest possible repressor self-repression while still being able to shut off the enzyme in the absence of drug.

Set of Pareto-optimal solutions shows distinct strategies of regulation

In order to determine the set of all possible optimized responses, regardless of selective pressures, we numerically determined the parameter sets corresponding to Pareto-optimal regimes (55, 56, 57). Previous studies have successfully used a similar optimal control framework to study regulation in a dynamical model of metabolic networks (9,58). In the Pareto analysis, phenotypes that perform best at one single metric while disregarding all other metrics are called archetypes and often involve extreme solutions (Fig. 4 A). A phenotype involving multiple metrics is said to be Pareto optimal if, due to constraints inherent to the system, it is impossible to find any other phenotype where all metrics are simultaneously improved. For a Pareto-optimal phenotype, improving one metric necessarily decreases the performance of at least one other metric. Since improving all metrics together is always desirable, phenotypes behind the Pareto front do just that, regardless of the selective pressures applied. Therefore, evolution of responses under any selective pressure should bring the system to the Pareto front that contains all Pareto-optimal solutions. Subsequent evolution of any given Pareto-optimal phenotype may further improve the overall fitness by balancing the trade-offs between different metrics, reflecting the specific selective pressures applied to the system. To identify the Pareto front in our system, we compute all metrics for each point in the parameter space and then numerically eliminate points for which all metrics can be simultaneously improved (23) (so-called dominated solutions; materials and methods). In agreement with our arguments above, the Pareto front containing all optimal solutions for any possible selective pressure applied to the system coincides with region 4 of the parameter plane (Fig. 4 B).

The Pareto set of optimal solutions shows general design principles of gene regulation. We calculated how each metric (each of the key concentrations) varies across the set, showing the trade-offs in performance among the Pareto-optimal solutions. A “generalist” response, which balances minimizing both maximum and final concentrations of both drug and enzyme (equal weights for $D_{peak}$, $D_{final}$, $Z_{peak}$, and $Z_{final}$), is optimized by parameters in the middle of the Pareto set, with a moderately expressed and weakly self-regulated repressor. However, we find that parameter sets that prioritize single traits are located along the boundaries of the region. The strength of repressor expression $H_r$ can be adjusted to control final levels of enzyme and drug, with weak repressor expression resulting in minimal levels of drug and strong repressor expression resulting in minimal levels of enzyme. Conversely, the threshold of repressor self-repression $R_r$ can be adjusted to optimize maximum levels of enzyme and drug, with strong self-repression ($R_r\approx 1$) resulting in minimal peak levels of drug and weak self-repression (constitutive repressor expression, $R_r\approx h_r$) resulting in minimal peak levels of enzyme (Fig. 4 B).

Although generalist responses may be optimized by a weakly self-repressed repressor, most responses are optimized by either constitutive or strongly self-repressed repressors. Analyzing the locations of optima within the Pareto front for different combinations of weights, we note that they typically occur along the edges: either at the line $R_r=1$ (strong self-repression) or along the diagonal $R_r=h_r$ (constitutive repressor expression). Varying the weight of each of the metrics $D_{peak}$, $D_{final}$, $Z_{peak}$, and $Z_{final}$, we find that the optimum starts by moving along one edge, then quickly transitions through the generalist optimum toward the opposite edge (Fig. 4 C–J). Optimal parameters in the interior of the Pareto front, while possible, require careful balancing of the weights for all metrics and appear to be much less typical. Two different trade-offs in the response metrics emerge: one between the peak concentrations of drug and enzyme, and another between their final concentrations. The trade-off between final drug and enzyme levels mostly involves changes in the repressor expression $h_r$, while the self-repression threshold $R_r$ remains at relatively low values for most optimal solutions. Responses with either a high cost of final drug or a low cost of final enzyme are optimized by constitutive expression of low levels of repressor, while responses with either a low cost of final drug or a high cost of final enzyme are optimized by expression of high levels of repressor and strong self-repression. Conversely, the trade-off between peak drug and enzyme mostly involves changes in the self-repression threshold $R_r$, while the repressor expression $h_r$ stays at intermediate values. Responses with either a high cost of peak enzyme or a low cost of peak drug are optimized by constitutive expression of moderate to high levels of repressor, while responses with either a high cost of peak drug or a low cost of peak enzyme are optimized by expression of moderate levels of repressor and strong self-repression. Of particular note, responses minimizing peak drug levels are optimized by $h_r\approx 10$ and $R_r\approx 1$ for any weight $w_{peak}^d$ larger than 1 (when extracellular drug levels are ${\text{D}}_{\text{out}}=50$), suggesting that the same regulatory strategy can be widely used.

Stress and metabolic responses are regulated differently

We next characterize the regulatory strategies of several types of cellular responses by calculating the regulatory parameters from data on protein expression and placing them into our optimization framework (Fig. 5; details in the materials and methods and supporting material). Although there have been recent efforts to characterize all transcription factor in E. coli systematically (59, 60, 61), we still lack a database describing the affinities of repressors to the different promoters they regulate, with and without the presence of their ligands. Therefore, we only calculate the regulatory parameters of transcription repressors that are well described in the literature, case by case, as described in the materials and methods and supporting material. Attending only to the regulation of the response, described by the $R_r\times h_r$ plane, we can compare systems even with slightly different regulatory architectures.

Figure 5.

Natural examples of cellular response systems show different strategies of regulation. Regulatory parameters were calculated for a variety of cellular responses. Stress responses (purple dots), such as the tetracycline resistance tet operon repressed by TetR and the multidrug resistance marRAB operon repressed by MarR, typically show strong repressor self-repression and moderate repressor expression. Metabolic responses (yellow dots) show diverse strategies of regulation. The acetate utilization aceBAK operon and the mannitol utilization mtlADR operon are repressed by strongly self-repressed repressors IclR and MtlR, respectively, as in antibiotic responses. The lactose utilization lac operon is repressed by a constitutive repressor LacI, expressed much above the threshold for repression of the lac genes. Galactose utilization is mediated by two mechanisms. The mglABC transport system, active in low galactose concentrations, is regulated by strongly self-repressed repressor GalS. Meanwhile, the galETKM operon, active at high galactose concentrations, is regulated by constitutively expressed repressor GalR. The edges of the Pareto front that minimize each of the selective pressures of peak and final enzyme and drug are indicated by the colors described in the legend.

Metabolic responses challenge many of the assumptions made in our model. We distinguish between “stress responses,” where the presence of a toxic drug poses a direct threat to the cell, and “metabolic responses,” where there is no inherent cost associated with the presence of the substrate in the intracellular space but only an opportunity cost associated with an unprocessed nutrient. While it can be assumed that cells do not actively import antibiotics, cells frequently respond to the presence of beneficial compounds by also regulating its active import, which is typically the case in metabolic responses. This rapid influx of nutrient substrate often also leads to the saturation of the enzyme. Additionally, the “opportunity cost” of unprocessed nutrients does not increase monotonically with its concentration but saturates when no more nutrient is needed. Therefore, caution is required when applying our simplified model directly to metabolic responses. Although these differences can lead to different dynamics, the overall progression of the response is still similar (Fig. S9). Since metabolic responses are typically regulated by similar mechanisms of transcriptional repression, subject to evolutionary pressures of similar nature, we can still calculate regulatory parameters and qualitatively compare general strategies of optimization.

Metabolic responses show diverse strategies of regulation, which seem to depend on the level of urgency of the response. When it is important to process the drug quickly, such as when resources are scarce, metabolic responses can be regulated similarly to stress responses. For example, in the transition to stationary phase, E. coli cells induce a metabolic response that allows the consumption of the acetate secreted as a by-product during exponential growth. This response is regulated by repressor IclR, which represses the aceBAK operon necessary for growth on acetate. IclR strongly represses its own expression (27,28). In this case, both the toxicity of high acetate levels and the scarcity of alternative carbon sources typical of the stationary phase increase the costs associated with drug levels, so the acetate metabolic response has similar regulation to antibiotic or stress responses. The mtlADR operon for mannitol utilization also shows similar regulation (29,30).

Other metabolic responses are regulated by constitutively expressed repressors, which minimizes peak enzyme levels at the expense of a slower response and a higher peak of drug concentration (62). Such is the case of the lac operon, which produces enzymes LacZ, LacY, and LacA necessary for lactose metabolism. These enzymes are repressed by repressor LacI, which is expressed nearly constitutively (35). Although failure to quickly process lactose might result in a competitive disadvantage for the cell, high intracellular concentration of a nutrient such as lactose is not costly per se, as it is not toxic. Therefore, unlike the case of antibiotic resistance, induction of the response is less urgent. Consequently, in metabolic responses such as lactose utilization, the cell prioritizes controlling enzyme levels over substrate levels, adopting a regulatory strategy that minimizes both peak and steady-state levels of the enzyme (63).

There are also examples of mixed strategies in the utilization of carbon sources, such as the two mechanisms of galactose transport and utilization in E. coli. The mglABC system has high affinity for galactose and is utilized during low galactose concentration, while the galP and galETKM system has lower affinity for galactose and is utilized in higher galactose concentrations (32). The mglABC system, which is deployed when galactose is a scarce resource, is regulated primarily by self-repressed repressor GalS, as in stress responses. Meanwhile, the galP and galETKM system, deployed when galactose is abundant, is regulated by constitutively expressed repressor GalR, as in the lac operon. There is also a significant level of crosstalk between GalS and GalR regulation, which suggests that these two mechanisms coevolved to be complementary to each other and optimize galactose utilization under different circumstances.

Antibiotic responses, typical examples of stress responses, are very often regulated by strongly self-repressed repressors, which minimizes the peak antibiotic concentration reached during the response. In the tetracycline resistance tet operon, repressor TetR is expressed from a promoter that is marginally less repressed than the tightly controlled promoter that expresses efflux pump TetA (24). Therefore, when TetR levels drop slightly, the repressor pool is replenished before the costly induction of the efflux pump, protecting the system against unnecessary activation. Parameter $h_r$, which measures the expression of repressor in relation to its ability to repress enzyme, is typically not very high in antibiotic responses, since that would increase steady-state levels of drug inside the cell. Other stress responses, such as the broad antibiotic resistance marRAB operon, also follow the same pattern of strongly self-repressed repressors with a relatively low $h_r$.

Overall, the scatter of regulatory strategies over the parameter space reflects different optimization strategies according to trade-offs between costs of peak levels of drug and enzyme. Regulatory strategies exhibit either constitutive expression of moderate to high levels of repressor or expression of moderate levels of repressor with strong self-repression. This pattern of naturally occurring regulatory strategies suggests that peak levels of drug or enzyme obtained during the transient of the response are more critical than final steady-state levels in the optimization of response parameters. We found that self-repressed repressors are typically expressed in the h_r ≈ 10 to 20 range. According to our analysis, when peak drug level is the dominant burden, regulatory parameters depend only on the extracellular drug ${\text{D}}_{\text{out}}$ and not on the weight assigned to peak drug w^d_peak (Fig. 4 H). In this case, the expression levels found in natural systems correspond to regulatory architectures optimized for exposure to a range of extracellular drug D_out ≈ 20 to 200 (Fig. S10). Theoretically, responses could be optimized for higher levels of drug by increasing repressor expression $h_r$, although this would ultimately result in significant costs associated with repressor levels. Together, these results indicate a strong role of the transient dynamics of the response in shaping the evolution of regulation.

Discussion

Although it is not always possible to determine exactly which selective pressures result in each set of optimized regulation parameters, as different sets of selective pressures can result in similar regulation, the choice of regulatory strategy still reflects which costs are most critical in the evolution of a response. In some cases where only one key concentration is responsible for most the cost of the response, evolution of the enzyme’s regulation might not need to satisfy any trade-offs, evolving outside the Pareto front, as the archtype that optimizes that key concentration. For example, the gene ampC, which produces an enzyme that confers resistance against $\beta$-lactam antibiotics, is notably expressed constitutively in low levels in E. coli, suggesting that at typical drug concentrations for this organism, the level of enzyme necessary to inactivate the drug does not pose significant costs for the cell. Many pathogenic strains of E. coli, which are exposed to higher antibiotic concentrations, develop mutations that increase expression of AmpC, which incurs fitness costs for the cell (64,65). In typical responses where multiple costs are relevant, however, enzymes are regulated. Therefore, in many pathogen species, ampC is expressed much more strongly but is regulated by a self-regulated repressor ampR, which indicates that the cost of expressing the enzyme is relevant for these organisms (66,67). Conversely, repressor LexA of the SOS response is often lost in pathogenic spirochetes when these species are continuously exposed to DNA damage, which results in a constitutively expressed response (68,69).

Although a generalist regulation strategy, with relatively weak negative auto-regulation of the repressor, could in theory optimize the system in situations where all costs are equally important, it is rarely observed in nature. If the generalist strategy was to be adopted, moderate changes in the relative weights of the different costs would either prompt the cell to adopt stronger self-repression of the repressor or to get rid of self-regulation altogether. Moreover, self-repression or constitutive expression of the repressor does not simply depend on minor adjustments in parameters of the regulation but on a rearrangement of regulatory elements such as promoters and repressor binding sites. Therefore, strong differences in selective pressures would be necessary to switch from one strategy to the other.

Our framework provides a means to integrate several different environmental selective pressures into the choice of regulatory strategy of an enzyme and the optimization of its parameters. Previous work has noted that the choice of regulatory strategy depends on requirements of the response dynamics (70) or on the demand for the expression of the enzyme in the environment (71,72). Modeling the dynamics of cellular responses has allowed the evaluation of regulatory strategies in terms of effectiveness criteria such as decisiveness, efficiency, selectivity, robustness, stability, and responsiveness (73,74). Negative auto-regulation of a repressor has been shown to be a particularly effective regulatory strategy in a variety of conditions, significantly increasing the speed of transcriptional responses (which is essential to keep peak drug levels low) (19,70) and also decreasing noise and increasing the input dynamic range of responses (which makes the system more adaptable to different drug concentrations) (75, 76, 77). More recently, the combination of experimental evolution with Pareto optimality analysis has shed light into the constraints guiding the evolution of regulated cellular processes in different static environments (78), explaining the tuning of regulatory interactions and changes in network topology (16,78,79).

Here, we extend the notion of Pareto optimality on a constrained space to the evolution of dynamical regulated responses exposed to different environments. We find that the evolution of regulation depends not only on steady-state concentrations of intracellular components, whether in the presence of drug or not, but also on cellular states attained during transient dynamics following fast environmental shifts before the equilibrium is reached. In fact, peak levels of drug/substrate or enzyme reached during the response transient seem to be more important in guiding this evolution. These results suggest a strong role of the dynamics of the response in determining the choice of regulatory strategies. Furthermore, our dimensionless analysis can predict regulatory strategies and repressor expression rates that optimize the regulation of drug responses for specific conditions. We found the parameters calculated from well-known natural antibiotic responses to be consistent with predicted optimal regulatory parameters corresponding to a wide range of drug concentrations, although optimal parameters for high drug levels might be further constrained by costs associated to repressor expression. Our framework can therefore guide efforts in synthetic biology, aiding the design of circuits for dynamical applications.

While we focused our analysis on the case of an enzyme directly regulated by a transcription repressor, this framework can be extended to analyze dynamical models describing other types of regulatory architectures. Alternative regulatory designs can show different dynamics, such as responses involving additional feedback loops or regulation by transcription activators or by multiple regulators. This is also the case for responses where the transcription factor does not control enzymes directly, so drug levels only decrease later after the activation of enzymes further downstream. Additionally, there are plausible scenarios where the approximations used in our model would not be valid, such as when the intracellular drug concentration is low enough to be comparable to the repressor concentration or high enough to saturate the enzyme. While an analytical solution to some of these models might be challenging, most of these cases could still be described by a manageable number of parameters to permit optimization based on numerical simulations of the system. Therefore, this general framework can be used to study and categorize the vast collection of regulatory architectures found in natural systems.

Author contributions

D.S. and L.S.T. designed the study, and D.S., M.S., and L.S.T. analyzed data and wrote the manuscript.

Editor: Ramon Grima.

Supporting citations

References (80, 81, 82) appear in the supporting material.