Figure 1.
The Leader-Transcript of the trp Operon in E. coli
Attenuated transcription results in a 141-nucleotide (nt) transcript. Aborted transcription of the paused RNAP results in a 91-nt transcript. The transcript includes an open reading frame of 15 codons, encoding a very short-lived 14-residue leader peptide. The RNAP is released from the pause site when the seventh codon is read [17]. Two of the three codons in the control region (10 and 11) are trp codons. Also, ribosome stalling on the arg-codon (12) prevents I:II-hairpin formation and attenuation (see Figure 6). After reaching the stop codon, the ribosome dissociates in about 1 s. The II:III and III:IV conformations have similar stabilities, and the terminator is formed with 50% probability when the ribosome reaches the stop codon and dissociates from the transcript before region IV is available. This determines the basal read-through level of 10–15%.
Figure 2.
The Probability Q of Read-Through of the Attenuation Leader Is Plotted as a Function of k′ = 1/τ Where τ Is the Average Time for Translation of All the m Codons
The translation rates of the individual codons are k = k′ · m. The average time to transcribe the n nucleotides is kept constant at 1/q′ = 1s. The transcription rate for an individual nucleotide is q = q′ · n. For m = 1, n = 1 the response function is given by Equation 6; for m = 1, n = 50 the response function is given by Equation 8. These curves should be compared to the much more sensitive response that is reached with two competing multi-step processes, as in attenuation (e.g., m = 10, n = 50). Insert: The sensitivity amplification aQk is plotted as a function of n, while k is kept fixed, and q is changed to get the read-through probabilities (Q) corresponding to the different curves. The two curve families correspond to the ordinary multi-step mechanism with m = 1 (dashed) and the competing multi-step mechanisms with m = n (solid), respectively. The dashed curves approach ln(Q) as n →∞, whereas the solid curves all go to −∞ for all values of Q.
Figure 3.
The Probability Q of Read-Through of the Leader Is Plotted as a Function of the Signal s
For the curve with m = 10, k is given by Equation 9, and for the curves with m = 1, k is one-tenth of this value. q′ = 1s−1 for all curves. The usage frequency of the amino acid in proteins is f = 1/20, and the maximal rate of codon translation at full supply of amino acid is kmax= 20s−1. Insert: The contributions of the two factors to the sensitivity aQs= aQkaks is illustrated for the case with m = 10 and n = 50. aks approaches 1/f as s → 1. The asymptotic behavior of aQk is described in Figure 2.
Figure 4.
The Choice among Synonymous Codon
(A) The predicted concentration of aminoacylated tRNA isoacceptor A (solid) and B (dashed) is plotted as a function of the total concentration of isoacceptor B for a fixed but limiting supply of the cognate amino acid (s = 0.95). The codon usage of a and b are both equal to 2.5%, and the total concentration of tRNA isoacceptor A is kept constant at 3.33 μM. Insert: The sensitivity in rate of reading a codons by the A isoacceptor in response to a change in amino acid supply is plotted over the same interval as the main figure.
(B) The probability of read-through is plotted as a function of normalized amino acid supply rate. The attenuation control codon a is translated by tRNA isoacceptor A. The same amino acid is also encoded by another codon b, which is translated by tRNA isoacceptor B. The codon usage of a and b are both equal to 2.5%. The different curves correspond to different ratios between the total concentrations of isoacceptor A and B, as indicated in the figure. As m = 10, n = 50, and kmax = 20s−1, the situation where the concentrations are equal corresponds to the solid curve in Figure 3. Insert: The sensitivity aQs for the three cases. More details about the calculations are given in the Supporting Information and reference [16].
Figure 5.
The Sensitivity Amplification aQk Is Plotted as a Function of the Rate of Translation of Regulatory Codons
The different curves correspond to different number of regulatory codons in a control region that has in total ten codons. #i/#j indicates i regulatory codons out of j codons in the control region. The rate of translation for non-regulatory codons is 15s−1. The RNAP transcribes 50 nt s−1 (q = 50s−1) and the total number of RNAP steps is 80 (n = 80). Insert: The sensitivity amplification aQk is plotted as in the main figure, but now the rate of transcription (q) is scaled such that Q = 0.01 for each point.
Figure 6.
Attenuation Control of the his and trp Operons
(A) The trp and his attenuation leader regions. For the trp mechanism, codon 3 and 4 after the ribosome has released the RNAP at “rel” are trp codons [2,17]. Ribosome stalling on codon 3, 4, or 5 leads to anti-terminator conformation [42]. If the ribosome releases from the stop codon (codon 8) after segment III is available for secondary structure formation (RNAP has transcribed nt 28 + 8) but before more than half of segment IV is available (RNAP has transcribed nt 39 + 8), the probability for anti-terminator formation is 50% [18]. The “+ 8” are the bases that have been transcribed but are unavailable for secondary structure formation [43]. For the his mechanism, codon 2 to 8 after the ribosome has released the RNAP are his codons [2,44]. Ribosome stalling on any of these is assumed to lead to anti-terminator conformation. If the ribosome releases from the stop codon (codon 11) after segment III is available for secondary structure formation (RNAP has transcribed nt 7 + 8) but before more than half of segment IV is available (RNAP has transcribed nt 18 + 8), we assume a 50% chance to get anti-terminator formation. The transcription rate is 50 nt s−1 [45], the translation rate of codons other than trp or his is 15s−1 [46], and the rate of ribosome release from the stop codon is 1s−1.
(B,C) The probability of transcription (y-axis) past the leader for trp and his operons as a function of the rate of translation of respective trp and his codons (x-axis). The read-through probability is a sum of the probabilities for two mutually excusive events: ribosome stalling on the control codons while the RNAP escapes attenuation (dotted) or ribosome release from the stop codon before the termination hairpin is completed (dashed). In the trp case, the second event causes a high basal expression level.
(D) The probability of transcription as a function of normalized supply of his and trp, respectively. The codon usage frequency is 1% for trp and 2% for his [28]. kmax = 15s−1. Insert: The sensitivity aQs over the interval s = 0.8 to 1.0. The positive sign of the sensitivity in gene expression for an increase in histidine supply in the narrow range of s = 0.995–1.0 is due to reduced probability of ribosome release at the stop codon.