Bacterial strains, plasmids, and growth conditions. Bacterial strains and

Transcription

1 I Text I Materials and Methods acterial strains, plasmids, and growth conditions. acterial strains and plasmids used in this study are listed in I Table. almonella enterica serovar Typhimurium strains are derived from the wild-type strain 8s. Phage P- mediated transductions were performed as described (). acteria were typically grown at 7 C in L broth () or in N-minimal medium at ph 7.7 () supplemented with. % Casamino cids, 8 mm glycerol, µm or mm MgCl, and µm FeCl. Lac colonies were selected on N-minimal medium at ph 7.7 supplemented with 8 mm lactose, µm MgCl. Counterselections for tetracycline-sensitive (Tet ) colonies were performed as described (). mpicillin and anamycin were used at 5 µg/ml, tetracycline at.5 µg/ml and chloramphenicol at µg/ml. Construction of chromosomal mutants. train EG75, which has a tetr insertion in the pbgp promoter region, was constructed by the one-step gene inactivation method (5). The tetr genes were amplified using primers 65 and 66 and plasmid peg5 harboring the transposon Tn as template and recombined into the pbgp promoter region of the wild-type 8s strain. train EG7, which has a synthetic PhoP box instead of the original Pmr box at the pbgp promoter, was constructed by a combination of the one-step gene inactivation method and counter selection method for Tet s colonies. primer

2 partial homoduplex containing the synthetic PhoP box was generated using primers 67 and 68 and recombined into the chromosome, replacing the tetr gene at the pbgp promoter region in strain EG75. train EG76, which has a tetr insertion in the ugd promoter region, was constructed by the one-step gene inactivation method. The tetr genes were amplified using primers 6 and 6 and plasmid peg5 harboring the transposon Tn as template and recombined into the ugd promoter region in the wild-type 8s strain. train EG78, which has a Cm R cassette downstream of the tet gene in the ugd promoter region, was constructed by the one-step gene inactivation method. The Cm R cassette was amplified using primers 65 and 655 and pd as template, and recombined into the pbgp promoter region of the EG76 strain. train EG7, which has a synthetic PhoP box instead of the original Pmr box at the ugd promoter, was constructed by a combination of the one-step gene inactivation method and counter selection method for Tet s colonies. primer partial homoduplex containing the synthetic PhoP box was generated using primers 6 and 6 and recombined into the chromosome, replacing the tetr gene at the ugd promoter region in the strain EG78. ll strains constructed by using PCRs were analyzed by DN sequencing and confirmed that the DN regions generated by PCR had the predicted sequences and no unexpected substitutions. Primers used in the construction of chromosomal mutants are listed in I Table. Degradation rate analysis by quantitative Western blotting. acteria were grown as described above, in 5 ml media of low Mg culture for h; then,

3 chloramphenicol ( µg/ml) was added to inhibit protein synthesis. -ml aliquot of bacterial cell culture was collected at the indicated time points, mixed immediately with /5 volume of 5% phenol ph. (igma)/95% ethanol to inactivate cellular proteases, and ept on ice for at least min. fter spinning down, cells were resuspended in an appropriate volume of -PER solution (Pierce). whole-cell lysate ( µg of protein) was run on a is-tris % gel (Invitrogen) with ME D Running buffer, transferred to a PVDF membrane, and analyzed by Western blots using anti-pmrd polyclonal antibodies. Western blots were developed by using anti-rabbit IgG horseradish peroxidase-lined antibodies (mersham iosciences) and upersignal west femto (Pierce). Protein levels were digitized and quantified by using a film developed for Western blot, a fluorescence plate, and an FL-5 imaging system (Fuji Film). Description and Derivation of the Mathematical Models Our mathematical models for the ect and connector-mediated pathway dynamics describe temporal changes in the concentration of the pbgp mrn. ecause it is nown that the rrn concentration does not change when bacteria are growing at a constant rate (see, e. g., ref. 6), our models also describe the dynamics of the pbgp mrn concentration relative to that of 6 rrn, and can be fitted to the experimental data presented in Fig.. The ODEs (ordinary differential equations) constituting the model of the connector-mediated transcription control circuit are given below (under each equation, in gray boxes, we list the corresponding chemical reactions):

4 d dt ( ) ; ( ) ( ) ( ) pbgp promoter pbgp mrn ( ) (degradation dilution) d dt PhoP - P ; PhoP - P PhoP-P pmrd promoter PmrD ( PmrD ( ) Pmr - P ( PhoP-P / ) PmrD/Pmr - P ( ) ) (degradation dilution) d dt ; Pmr (const) Pmr - P ( PmrD ( ) Pmr - P ( ) (dephosphorylation degradation dilution) / ) PmrD/Pmr - P ( ) d dt ) (. PmrD ( ) PmrD/Pmr - P ( Pmr - P ( / ) PmrD/Pmr - P ( ) ) (degradation dilution) The correspondence between the concentration variables i and the chemical species is as follows: ~ pbgp mrn; ~ PmrD (protein); ~ Pmr-P (phosphorylated protein); ~ Pmr-P/PmrD (protein complex). In Eqs., it is assumed that Pmr-P and Pmr-P/PmrD interact with DN only as homodimers (Pmr-P) and (Pmr-P/PmrD). We also considered the situation when functional heterodimers can form (see Model fitting below);

5 this situation is described by a system consisting of I Eqs., and I Eq. modified as follows: d dt ( ) 5 ( ) The concentration of PhoP-P, which appears in I Eq., is the external variable representing the input of the transcription control system. The output of the system is the concentration of the pbgp mrn ( ). In the above equations, the constant represents the transcription rate for the pbgp inducible promoter (under full induction conditions), and the constant is the production rate for PmrD (for a fully induced pmrd gene). The constants and are the rates of exponential depletion (due to biochemical degradation and dilution resulting from cell growth) of pbgp mrn and the PmrD protein, respectively. The constants and are the production and depletion rates for Pmr-P; and are the forward and bacward rates of the reaction PmrDPmr-P PmrD/Pmr-P. is the rate of exponential depletion of Pmr-P due to protein degradation and dilution. The constants and are the affinities (association constants) for Pmr-P binding to the pbgp promoter and PhoP-P binding to the pmrd promoter, respectively. In the above model, we assume that only phosphorylated regulator proteins activate transcription (7-9), and that they interact with the corresponding binding sites in the form of dimers (9, ). The latter assumption is supported by the facts that the PhoP and Pmr proteins recognize distinct tandem repeats in promoters (9, ). The dynamic equation for the ect regulation circuit is 5

6 d dt PhoP - P. 6 PhoP - P Here, is the affinity of PhoP-P binding to the pbgp promoter; analogous to and. and are The expressions for the first- and second-order terms on the right-hand sides of the model equations I Eqs. 6 follow ectly from our description of the regulatory circuits. In the models, we focus only on the major components and neglect intermediate stages of reactions. It is assumed that the concentrations of all the chemical substances not explicitly presented in the model stay constant (e.g., Pmr const, and the phosphorylation of Pmr occurs at a constant rate). To simplify the connector-mediated pathway model, we do not include the variable describing changes in the concentration of pmrd mrn. Taing the approach described in the literature (-5), we assume that the rate of PmrD production is proportional to the activity of the pmrd promoter. We also neglect the pmrd transcription repression by Pmr-P, because this effect is nown to be relatively wea (6). Our description of transcription, translation, and mrn decay processes as simple zero- and first-order reactions is a simplification justified by our purpose to consider only the most essential parts of the expression control systems. The approach we used to model transcriptional control is as follows: we assume that two activators, and, can competitively bind to DN binding sites,, with affinities and, respectively; that the activators bind DN as dimers; and that each binding site is attached to a promoter. We denote by r the rate of transcription (micromoles/liter of transcripts initiated per minute) 6

7 7 assuming all binding sites are occupied; in reality, only a fraction,, p, of all activator sites are bound. We assume that transcription initiation is possible only for promoters bound to activator proteins. Therefore, the general expression for the transcription initiation rate is pr. Our objective is to find an expression for p in terms of the concentrations of the activator proteins. The activator protein binds to the activator site as a result of the reaction ; the binding reaction for is similar (while this reaction is a simplification of reality, we assume that the form (bound monomers) never accumulates significantly; as a consequence, the equilibrium equations for more detailed binding mechanisms are well approximated by the one for the reaction above (7)). It can be assumed that the binding reactions are in equilibrium at all times because protein DN binding is much faster than other reactions in the system (transcription, translation, phosphorylation). Therefore, the concentrations of the reacting species satisfy the equations, Using these equations, we obtain Total ound p. If there exists a positive such that, we can use the approximation ) ( ) ( Total ound p.

8 This expression is used to model competitive binding of PmrD/Pmr-P and Pmr-P to the binding sites upstream of the pbgp promoter (6) in the connectormediated regulation model (I Eq. ). The use of the relation is justified by our assumption that PmrD/Pmr-P and Pmr-P have approximately the same DN binding properties. We also postulate that the concentration of free binding proteins is considerably larger than the concentration of binding sites, which is valid because the bacterial cell normally has only one copy of a gene, but hundreds or even thousands (8) of copies of regulator proteins. This postulation allows us to disregard changes in the concentrations of free proteins (and complexes thereof) resulting from the formation of protein DN complexes, and thus ignore the details of the binding mechanism. If in the regulation model described above we consider only one activator protein, then we will arrive at the expression for transcription control by PhoP-P which was used in I Eqs. and 6. If we consider the possibility of DN binding by the heterodimers, then we will obtain the expression used in I Eq. 5. teady-tate Equations and Induction Ratios The steady-state equations for the models are obtained from I Eqs. 6 by setting the time derivatives to. I Eq. 6 for the ect regulation model has a unique solution: PhoP - P. 7 PhoP - P 8

9 The solution to I Eqs. can be found by solving I Eqs. and then substituting the solution into the steady-state version of I Eq.. The resulting steady-state expression for the pbgp mrn concentration reads: ( (PhoP - P) (PhoP - P)). 8 ( (PhoP - P) (PhoP - P)) For (PhoP - P), the steady-state concentration of pbgp mrn, the induction ratio is defined as ( PhoP - P) / ( P ), where P is the concentration of PhoP-P under repressing conditions ( P << PhoP - P ). From this definition and I Eqs. 7 8 it follows that, for both ect and connector-mediated regulation models, the induction ratio does not depend on and. The rate constant reflects the activity of the pbgp promoter, therefore, the models predict insensitivity of the induction ratios to variations in pbgp promoter activity for both ect and PmrDmediated control circuits. characteristic feature of the induction ratio curves corresponding to the experimental data is that they approach a plateau for ratios greater than (Fig. ). However, the curves for the corresponding models do not show this property (Fig. ). uch a discrepancy can be explained as follows. The steadystate induction ratio I (Mg ) can be represented as a composition of functions: I (Mg ) f ( g(mg )), where (Mg g ) PhoP - P. Fig. shows the functions I (Mg ), whereas Fig. shows the functions f (PhoP - P ). pproaching a plateau is equivalent to having small absolute derivative I (Mg ) ; using the chain rule for differentiation, we obtain I (Mg ) f ( g(mg )) g (Mg ). 9

10 When Mg is low, the system approaches its maximum capacity to generate PhoP-P, therefore, g (Mg ) should reach a plateau for low Mg. This assertion is supported by modeling (9). This implies that for low Mg, g (Mg ) is quite close to, and so is I (Mg ), although f ( g(mg )) is not. Cascade-Lie Properties of the Connector-Mediated Pathway s a result of model fitting for the connector-mediated pathway, we obtained a parameter set such that >> (the Pmr-P/PmrD complex formation rate is significantly larger than the complex degradation/dilution rate, I Table 5). Therefore, we can simplify the steady-state equations for the PmrD-mediated control circuit by dividing the equations by and using the approximate equality. With the notation ci i / for the coefficients, the / simplified system of equations reads: c ( ) c ( ) ; PhoP - P c c c c ; PhoP - P c c c c ; c c. Using the property c / c /, we can find the system s unique solution: i j i j

11 ( ) ; 9 ( ) PhoP - P PhoP - P ; ;. For the interval of PhoP-P concentrations presented in Fig., PhoP - P < (I Table 5). pplying the asymptotic formula a/( a) a a o(a ) to I Eq. and neglecting the small terms O( PhoP - P ), from I Eqs. we obtain, PhoP - P. Plugging this expression into I Eq. 9, we arrive at the simplified equation {( / ) PhoP - P }. {( / ) PhoP - P } For the fitted model,. (I Table 5), so we can neglect the term / O ( / ) and simplify even further: PhoP - P PhoP - P. For any, I Eqs. and are valid for sufficiently small PhoP - P. I Eq. is a Hill equation with Hill coefficient, whereas I Eq. 7 for the ect regulation circuit is a Hill equation with Hill coefficient (for a discussion of Hill equations and their properties, see refs. -). The increase in the Hill coefficient (compared to ect regulation) can be attributed to the presence of two stages of

12 transcriptional regulation in the connector-mediated pathway, and is a rather general property of multi-stage regulatory cascades whose stages have sigmoidal signal response curves (). Let I (PhoP - P) (PhoP - P), ( P ) I (PhoP - P) (PhoP - P) ( P ) be the induction ratios for the ect and connector-mediated pathways (as in teady-tate Equations and Induction Ratios above, P << PhoP - P is the concentration of PhoP-P under repressing conditions). It follows from I Eqs. 7 and 9 that, if (the PmrD/Pmr-P complex formation rate) is large enough (which is true for our fitted model), lim PhoP-P I I (PhoP - P) (PhoP - P) ( P ). 5 ( P ) From I Eq. we have ( PhoP - P) ( / ) O(PhoP - P ) O (PhoP - P) PhoP - P as PhoP - P. Consequently, if P is small enough and ( / ) P ) / 6 O is negligible compared to O ( ), then P ) / ( ) P ( P is large (>> ). In this case, I Eq. 5 gives I (PhoP - P) lim >, 7 PhoP-P I (PhoP - P) ~ meaning that there exists a concentration of PhoP-P, P P, ), such that I ~ ~ ( P) I ( P), I (PhoP - P) > I (PhoP - P) for all PhoP - P > P ~.

13 In principle, we can choose different high values of Mg (and, therefore, different small values of PhoP-P) to represent repressing conditions, ~ P P (Mg ). For some choices of P, we will have P P (such as in Fig. ); for other choices, we may have P ~ > P. It should be noted that, because in reality, for vanishingly small PhoP - P / >, the terms O( / ) /PhoP - P ) ( cannot be neglected and I Eq. 7 will not hold. However, the use of vanishingly small P is not biologically meaningful. Computational Procedures Dynamics and steady-state computations. The model trajectories in Fig. were generated using MTL s imiology function sbiosimulate. The steady-state solutions for the ect regulation model were computed using I Eq. 7. The steady states for the connector-mediated pathway were found as follows. First, the steady-state versions of I Eqs. were reduced to a quadratic equation in, which was solved using MTL s roots function. If the solution produced one positive root (or two positive roots very close to each other) such that the resulting and were also positive, then these values were used to find (from the steady-state version of I Eq. ). If it did not, the steady state was found by solving I Eqs. (using MTL s function ode5s) on increasing time intervals until convergence, similarly to the way described in ().

14 Model fitting. Model fitting with models defined by I Eqs. and I Eq. 6 was performed as described in Materials and Methods of the main article; the fitting results are shown in Fig., and the resulting parameter sets are given in I Tables 5 and 6. The data on relative PhoP-P levels for pathway activation were taen from ref. 5 and normalized so that the pea level has the value ; this normalization allowed us to avoid unrealistically high values for the PhoP-P interval in Fig. and I Fig. 6, and unrealistically low values for and. The fitting method ga was empirically found to give better fits for the connectormediated pathway model, while the method fmincon wored better for the ect regulation model. For the PmrD pathway model, we performed fitting experiments and selected the best final fit. For the ect regulation model, we also performed fitting experiments; the resulting fits gave parameter sets quite similar to each other, so we randomly selected one of them to be the final fit. t the randomized stage of our fitting procedure, the fitting routine was initialized with different parameter sets, in which some of the parameters were selected at random. t the initial value adjustment stage, we performed 5 nonrandomized fitting adjustment iterations. In the fitting procedure for the ect regulation model, we used the condition that the depletion rate. 5, where. 5 corresponds to cell growth rate with doubling time 5 min. In the fitting procedure for the connectormediated pathway, we assumed that >>, and >>. We thus chose,, 5,. ; these values were neither randomized nor modified in the process of fitting. Other fixed values were 5,.. To

15 estimate (PmrD translation rate under full activation), we used the data from ref. 6 according to which the average time between two consecutive translation initiation events on one mrn molecule is s, which gives the initiation rate / s min ; it is also nown that translation initiation rates vary by several orders of magnitude (7). The estimate for (mrn degradation rate) was based on the article (8), which states that 8% of E. coli mrn have half-lives between and 8 min (exponential decay rate. min corresponds to the halflife of ~.5 min); in general, mrn half-lives vary from s to min (9). fter fitting the connector-mediated pathway model (I Eqs. ) to the experimental data in Fig. and generating Fig., we used the obtained parameter values to generate analogous curves for the PmrD regulation model with formation of active heterodimer (Pmr-P and PmrD/Pmr-P) (I Eqs. 5). The difference between these curves and the curves in Fig. was negligible, which suggests that heterodimer formation does not play an important role in the connector-mediated pathway dynamics. 5