Different Regulatory Networks Can Yield the Same Optimal Enzyme Expression Level as a Function of Inducer Concentration.

<p>This is illustrated for two regulatory networks of the <i>lac</i> system, which differ in the dissociation constants of lactose-repressor binding and repressor-operator binding. Panels (A) and (B) show the response functions at two different stages of the <i>lac</i> regulatory network, while panel (C) shows the resulting optimal enzyme expression level as a function of lactose concentration. (A) The fraction of repressor that is not bound by lactose, <i>X</i>_free/<i>X</i>, as a function of lactose concentration for two different lactose-repressor binding constants. (B) The corresponding response curves of the enzyme expression level as a function of the fraction of free repressor. The total expression level of repressor is chosen to correspond to the optimal growth rate (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000125#pcbi-1000125-g002" target="_blank">Figure 2</a>). (C) The resulting optimal enzyme expression level as a function of the lactose concentration, as predicted by Equation 20 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000125#pcbi.1000125-Dekel1" target="_blank">[12]</a>.</p

Relative Change in the Growth Rate as a Function of the Average Repressor Concentration.

<p>The growth rate is averaged over different lactose concentrations in the environment (see Equation 17), for two different lactose concentration distributions in the environment.</p

The Optimal Design of the <i>lac</i> Regulatory Network Is Determined by the <i>lac</i> Repressor Copy Number and the Repressor–Lactose Binding Constant.

<p>Contour plot of the growth rate as a function of the repressor copy number <i>X</i> and repressor-lactose binding constant <i>K</i>_D. The weighting of the lactose levels is nonuniform. Lower binding constants allow for higher optimal growth rates at lower optimal expression levels for the repressor.</p

A Sketch of the Instantaneous Growth Rate λ of a Single Cell as a Function of the Concentration <i>X</i> of Component X.

<p>If the average expression level <i>X</i>_s is close to the optimal expression level <i>X</i>_opt, biochemical noise will always decrease the growth rate. If, however, the average expression level deviates sufficiently from the optimal expression level (i.e. if <i>ax</i>><i>bx</i>² in Equation 11), then fluctuations can enhance the growth rate of the population, even when the growth rate λ of a single cell is linear in <i>X</i>, i.e. if <i>b</i> = 0. The reason is that fast growing cells dominate the population.</p

Examples of Complex E. coli Promoters

<p>(A–C) Taken directly from the EcoCyc database [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b008" target="_blank">8</a>]. (D) Described in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b025" target="_blank">25</a>]. Green blocks denote TF binding sites that have an activating effect; red blocks denote repressor sites. Brown sites can both activate and repress transcription. Note that repetitive and overlapping binding sites occur frequently. Understanding these kinds of promoters requires detailed quantitative information about binding affinities and interactions.</p

Table Summarizing Which Homo-Cooperative or Hetero-Cooperative Activation or Repression Modules Are Needed to Obtain a Particular Transcriptional Logic Gate

<p>The table consists of four quadrants, corresponding to different TF concentrations <i>c</i>₁ and <i>c</i>₂ (each being low or high). Each quadrant is divided into two parts (white and gray), corresponding to the alternative promoter states (on or off). As an example, the AND gate is on only if both TF1 and TF2 are present; this requires a hetero-cooperative activation module. In contrast, an OR gate should be on if either TF1 or TF2 is present. This requires homo-cooperative activation modules for each of the species, because the promoter is weak (the gate must be off when both species are absent); however, since the activation modules do not compete with one another, a hetero-activation module is not required: the homo-cooperative activation modules also turn the gate on when both TFs are present. In general, the design can be most easily understood by first considering the design constraints when both TFs are absent, then the requirements when one of the two are present, and lastly the design constraints when both TFs are present. The EQU and XOR gates discussed in the main text illustrate this perhaps most clearly. Note that the EQU gate is an example of a gate in which a hetero-activation module is required, despite the fact that the promoter is strong; the hetero-activation module is needed to counteract the two homo-cooperative repression modules when both TFs are present.</p

Cartoons of <i>cis</i>-Regulatory Constructs Emerging from Our In Silico Design of Transcriptional Logic Gates

<p>The boxes indicate the TF binding sites; green indicates that a TF acts as an activator, red that it acts as a repressor, and brown that the action of the TF depends upon the concentrations of the two TFs. Weak binding sites (<i>K</i>_D > 2 × 10³ nM) have a light color, strong ones are dark. Yellow connections between TFs signify cooperative interactions. The designs show that the logic gates are constructed as overlapping arrays of cooperative binding sites. Each layer acts as a module, either activating or repressing transcription. Signals are integrated via the interplay between intramodular cooperativity and intermodular competition.</p

Illustration of the Model

<p>The <i>cis</i>-regulatory region consists of <i>N</i> = 100 bp directly upstream of the transcription start site. In <i>E. coli,</i> most TFs bind to this region, although binding sites are also found downstream of the transcription start site; mechanisms requiring such downstream sites are excluded by our model. A TF binding domain counts <i>M</i> amino acids, which can bind <i>M</i> = 10 bp [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b054" target="_blank">54</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b055" target="_blank">55</a>]. When two TFs bind within a distance less than <i>k</i> = 3 bp, they interact with energy <i>E</i>_TF−TF; this is indicated by a yellow connection between the TFs, although it should be realized that these cooperative interactions could also be mediated via the DNA. When a TF binds close to the RNAP, we assume an interaction energy <i>E</i>_TF−P. The core promoter, consisting of the −10 and −35 hexamers, is indicated; when the RNAP binds to it, it blocks both hexamers and the spacer between them. The TF that binds overlapping with the RNAP is red, to indicate that it represses transcription by steric hindrance; the green TF is an activator, since it recruits RNAP. The gray TFs bind too far upstream from the core promoter to influence the transcription rate. In our simulations, we used <i>k</i> = 3 and <i>E</i>_TF−TF<i>= E</i>_TF−P = 3.40 <i>k_BT</i> or 2.0 kCal/mol (so that <i>e</i>^{β<i>E</i>_TF−TF} = 30.0) [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b001" target="_blank">= 30 [1</a>].</p

Two-Gradient Model in <i>d</i> = 2

<div><p>(A) The mean threshold position fluctuates about <i>L</i>/2 due to the symmetry of the system.</p><p>(B) Variation of the width <i>w</i> as a function of averaging time.</p><p>(C) Data collapse of the width as a function of averaging time, at long times, for a range of parameter values. The full line shows <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030078#pcbi-0030078-e019" target="_blank">Equation 19</a> with <i>k~</i>_2<i>d</i> = 0.63 and Α~ = 2.5. * indicates the standard parameter values. For the other datasets, parameter values were changed as follows: open circle, <i>D</i> = 0.5 μm²s⁻¹; open square, <i>J</i> = 9 μm⁻¹s⁻¹; ×, Δ<i>x</i> = 0.02 μm; closed circle, <i>μ</i> = 1 s⁻¹; +, <i>μ</i> = 0.25 s⁻¹; diamond, <i>L</i> = 7.5 μm; and inverted triangle, <i>L</i> = 15 μm and Δ<i>x</i> = 0.02 μm. </p><p>(D) Plot of width as a function of decay length for averaging times: ×, <i>τ</i> = 30 s; open circle, <i>τ</i> = 45 s; and +, <i>τ</i> = 60 s. The full line shows the prediction from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030078#pcbi-0030078-e019" target="_blank">Equation 19</a>.</p></div

Repetitive and Overlapping Binding Sites

<div><p>(A) Histogram of the number of binding sites responsible for each interaction between a TF and an operon, according to the EcoCyc database [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b008" target="_blank">8</a>]. Note that multiple sites are common; the <i>cis</i>-regulatory region of <i>focA,</i> e.g., has as many as 11 binding sites for NarL.</p><p>(B) Histogram of the number of binding sites overlapping with each binding site [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020164#pcbi-0020164-b008" target="_blank">8</a>]. For example, bin 1 with height 300 should be interpreted as: there are 300 binding sites that overlap with exactly one other binding site. Overlap is common; some ArcA sites in the <i>sodA</i> regulatory region overlap with as many as 11 sites.</p></div