Table 1.
Truth tables of transcriptional logic gates.
Figure 1.
Illustration of the model of transcription regulation.
The model describes the transcriptional regulation of a gene tf3 by two transcription factors, TF1 and TF2. In addition, auto-regulation is included: the gene product TF3 of tf3 can regulate the transcription of tf3. TFs act by binding to tf3's cis-regulatory region, represented as a string of located directly upstream of the start of transcription. The TF binding domains count
amino acids, and bind to binding sites of length
bp. TFs can bind anywhere on the cis-regulatory region but with varying affinity determined by the DNA sequence. When two TFs bind within a distance less than
, they interact with energy
, as is indicated schematically by a yellow connection between the TFs. This way cooperative binding is included. The core promoter, consisting of the
and
hexamers, is marked; when RNA polymerase (RNAP) binds to it, it blocks both hexamers and the spacer between them. When a TF binds close to the RNAP we assume an interaction energy
; thus the mechanism of regulated recruitment is included. The TF that binds to a site overlapping with the core promoter is red to indicate that it represses transcription by steric hindrance; the green TF is an activator, as it recruits RNAP from its binding site. The gray TFs bind too far upstream to influence the transcription rate.
Figure 2.
In some simulation results, auto-activation occurs only in the presence of another transcription factor (TF). We call this conditional auto-activation. The figure presents two examples. Fig. A: The promoter of an AND gate using conditional auto-activation and, for comparison, one using hetero-cooperative activation. Both designs emerged in the simulations. In the first case, the regulated gene tf3 codes for a transcription factor TF3 that binds to its own cis-regulatory region. However, from its binding site, TF3 can activate transcription only indirectly, by facilitating the binding of TF1 and TF2 to their binding sites. As a result, the auto-activation depends on the presence of TF1 and TF2 ( and
). Fig. B shows plots of the expression level of tf3 (fold-change
vs. the concentrations
and
of TF1 and TF2) resulting from the cis-regulatory regions in Fig. A. The red dots show the values of
that were used to evaluate the fitness of the gate (see Methods). Fig. C and D show the same mechanism in a simplified model inspired by the simulation results. Plot D compares the response functions corresponding to three activation systems depicted in C. In all cases, a single TF activates the expression of a gene tf3 coding for another transcription factor, TF3. The first two scenarios constitute conventional activation systems with one or two binding sites. In the third scenario, one binding site is replaced by an operator for TF3, which introduces a positive feedback loop depending on the presence TF1. The binding affinities of all sites are optimized using the fitness function described in the main text. The response of the conditional auto-activation system is clearly sharper/more sensitive than the one using a single activation site. Cooperative auto-activation by two sites, however, leads to a slightly sharper response. The results suggest that conditional auto-activation is an alternative design principle that can be used to sharpen responses.
Figure 3.
Sharp and complete repression using auto-activation.
In the simulations, auto-activation evolved in every gate that requires strong repression. This figure shows two examples in which auto-activation indeed aids sharp and thorough repression. Fig. A and B depict NAND gates resulting from the simulations. When auto-regulation is not allowed by the method, the input TFs have both activating and repressing binding sites, as reported earlier [15] (in Fig. A and C, red boxes represent repressor sites and green boxes activator sites). When auto-regulation is included (i.e., the regulated gene tf3 codes for a transcription factor TF3 that can bind to its own cis-regulatory region) auto-activation emerges. Gene tf3 is still repressed by a hetero-cooperative module consisting of binding sites for TF1 and TF2. At low concentrations of TF1 and TF2 the auto-activation counteracts the repression module; as a consequence, the response to the concentrations of TF1 and TF2 is very sharp and the fold-change high, as can be seen in Fig. B. In Fig. C and D we study the mechanism in a simpler model system. Fig. C shows the cis-regulatory regions of two slightly different repression systems. In both cases, a transcription factor TF1 represses a gene tf3, coding for a second transcription factor TF3, by binding cooperatively to a pair of repressor sites. In the second scenario, an auto-activation site for TF3 is present as well. Fig. D presents the steady-state expression level of tf3 as a function of the repressor concentration. In both alternatives we optimized the binding sites using the fitness function described in the main text. Clearly, the second scenario leads to a more sensitive repression curve than the first. The presence of auto-activation allows for stronger repressor sites; consequently, as the concentration of TF1 increases the displacement of RNAP from the promoter by the repressor is more effective (i.e., the remaining expression level at
is much lower than in the cooperative repression case).
Figure 4.
Linear repression using auto-regulation.
If in the simulations we selected for a linearly decreasing response function (a LIN gate), auto-activation emerged. The resulting cis-regulatory region is schematically depicted in Fig. A. Red boxes and green boxes represent repressor and activator sites, respectively. The corresponding response function is plotted in Fig. B, alongside the results of a simulation in which auto-regulation is excluded. The auto-activation indeed manages to straighten the repression curve. The seven red dots in Fig. B show the goal function that is used in the fitness function: gates are considered better if their response function fits these points better (see Methods). We again studied a simplified model in more detail; the cis-regulatory region and response function of this minimal model are also shown. Indeed, the simple model system with appropriate binding site affinities fits the goal points better than the design without auto-regulation.
Figure 5.
NAND gates at increasing selection pressure on response speed.
On the one hand, the sensitivity of the response function of a NAND gate is improved by auto-activation. On the other hand, the response speed of the gate is enhanced by auto-repression. Consequently, if selection acts both on the response function and the response time, the simulation results are a compromise and depend critically on the relative magnitudes of the two selection pressures. The figure shows representative response functions and promoter designs of NAND gates resulting from four values of the parameter , which controls the weight of the response speed in the total fitness function
. (The irrelevant constant
merely serves to ensure that
.) The average values of the measures
(measuring the response time in arbitrary units
) and
(the deviation of the response function from the goal function in units
) for each condition are also plotted. By definition, low values of
and
correspond to good performance. For the lowest value
the response function is optimized and shows an excellent NAND gate. Due to strong, cooperative auto-regulation, the response is very sharp and almost bistable in the transition region, but the response speed is low. At
the result is a compromise: the quality of the response function is clearly reduced but the response speed is higher. Still auto-activation evolves, but it is weaker and non-cooperative and combined with weak auto-repressing binding sites. At
auto-repression fully takes over; the response function is crippled but the response speed is high. If the selection pressure on response time is increased even further (
) the response speed is fully optimized by disabling the response altogether.