Overlaid positive and negative feedback loops shape dynamical properties of PhoPQ two-component system

Bacteria use two-component systems (TCSs) to sense environmental conditions and change gene expression in response to those conditions. To amplify cellular responses, many bacterial TCSs are under positive feedback control, i.e. increase their expression when activated. Escherichia coli Mg2+ -sensing TCS, PhoPQ, in addition to the positive feedback, includes a negative feedback loop via the upregulation of the MgrB protein that inhibits PhoQ. How the interplay of these feedback loops shapes steady-state and dynamical responses of PhoPQ TCS to change in Mg2+ remains poorly understood. In particular, how the presence of MgrB feedback affects the robustness of PhoPQ response to overexpression of TCS is unclear. It is also unclear why the steady-state response to decreasing Mg2+ is biphasic, i.e. plateaus over a range of Mg2+ concentrations, and then increases again at growth-limiting Mg2+. In this study, we use mathematical modeling to identify potential mechanisms behind these experimentally observed dynamical properties. The results make experimentally testable predictions for the regime with response robustness and propose a novel explanation of biphasic response constraining the mechanisms for modulation of PhoQ activity by Mg2+ and MgrB. Finally, we show how the interplay of positive and negative feedback loops affects the network’s steady-state sensitivity and response dynamics. In the absence of MgrB feedback, the model predicts oscillations thereby suggesting a general mechanism of oscillatory or pulsatile dynamics in autoregulated TCSs. These results improve the understanding of TCS signaling and other networks with overlaid positive and negative feedback.


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Bacteria use two component systems (TCSs) to sense and respond to environmental 14 stimuli [1,2]. TCSs are also widely used in synthetic biology applications to sense 15 specific stimuli and control gene expression [3][4][5]. A TCS consists of a sensor kinase 16 often located on the inner membrane and a cognate response regulator protein located 17 in the cytoplasm. The sensor kinase senses environmental stimulus and responds by Normalized reporter output from P mgrB saturates as phoPQ operon transcription is increased. (C) Steady state normalized reporter output (P mgrB ) plateaus as Mg 2+ decreases, but increases further at growth limiting conditions (hypothetical normalized reporter output at growth limiting Mg 2+ , red square). Plot recreated from [19] explicitly include negative feedback regulation. 58 Notably, robustness of PhoP activity to elimination of PhoPQ autoregualtion is no 59 longer observed in growth-limiting Mg 2+ levels (< 10 −3 mM) [19]. Furthermore, in these 60 conditions, the PhoP activity greatly exceeds the activity observed over a range of low 61 but not growth-limiting (between 1 and 0.01 mM) Mg 2+ concentrations. Notably: PhoP 62 activity is nearly the same over that range of Mg 2+ levels forming a plateau between 1 63 and 0.01 mM Mg 2+ following a gradual increase from 100 to 1 mM Mg 2+ . Such a plateau 64 has been observed for multiple promoters with varying affinities to PhoP-P, suggesting 65 this pattern is not a property of one particular promoter [9]. Interestingly, at 0.01 mM 66 Mg 2+ the levels of PhoP-P are such that the promoters remain far from saturation [9]. concentration of free MgrB depends on concentration of PhoQ in general, there is no 139 robustness. However, when MgrB is in large excess of PhoQ, the function simplifies to 140 Eq. 3.

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[PhoP-P] ≈ C p Here C p is a combination of parameters as noted previously in refs [19,20]  Given that the previous model of autophosphorylation suppression by MgrB can explain 152 robustness of PhoP-P to total-PhoPQ levels, we explore steady state dose-response 153 behavior of the model. Specifically, we investigate whether the model can recreate a 154 biphasic dose-response, i.e. show an intermediate plateau ( Fig 1C). To compare with 155 experimental data of ref [19], we construct a detailed dynamic model with two reporter 156 proteins YFP and CFP [9] (S1 Appendix). To calibrate the model, we fit simulated values (Methods). 164 Steady state values of YFP:CFP over a range of Mg 2+ concentrations have also been 165 measured for wild-type cells (published in [19]). We use these measurements to tune 166 steady state parameters. In addition, we introduced a qualitative condition for greater 167 reporter output at very high stimulus to recapitulate effects at growth limiting 168 Mg 2+ [19]. Normalized experimental data used, simulation protocols and parameter 169 fitting procedure are described in Methods. To explain biphasic dose-response, we constructed a model of PhoQ with an explicit 177 Mg 2+ sensing mechanism (S1 Appendix). While understanding of how Mg 2+ modulates 178 PhoQ activity is still incomplete in E. coli, research in Salmonella has suggested that a 179 conformation change resulting from Mg 2+ binding to PhoQ increases phosphatase 180 activity [8]. Based on this finding we hypothesize two conformations of PhoQ: 181 phosphatase (PhoQ) and kinase (PhoQ*) (Fig 3A). Extracellular Mg 2+ binds to PhoQ* 182 and drives a transition to PhoQ thus shutting off PhoP-P activity. We assume that 183 extracellular Mg 2+ concentration does not change over time and include it in the rate 184 constant of switching from PhoQ* to PhoQ (Fig 3A) Fig 3A). Simulated steady state reporter output as a function of signal (k −1 ) shows two 207 distinct ranges of signal where output increases, separated by a plateau (Fig 3B). is insensitive to signal (k −1 ) in our model (Fig 4 D), we simplify the model so that  of signal can be found (Fig 4D). Taken together, strong suppression of kinetic rates by 255 excess MgrB and growth dilution shape biphasic dose response of PhoPQ. is shaped by overlapping feedback loops, we simulated steady state dose-response with 263 only one feedback present at a time. 264 We find a narrow range of signal sensitivity with negative feedback absent, while a 265 July 1, 2020 13/25 much wider range without positive feedback (Fig 5A and B). Thus, in addition to kinetic 266 advantages, negative feedback keeps the system sensitive to changes in magnesium over 267 a much wider range of concentrations. However, without positive feedback the 268 maximum output is much lower than with both feedback loops present, validating In addition to increasing range of sensitivity to signal, we unexpectedly find that in the negative feedback. This is why oscillations are not observed when MgrB is 297 upregulated by PhoP-P or when constitutive production of MgrB is too high (Fig 6B). 298 Thus, we find that autoregulated PhoPQ TCS may use negative feedback through MgrB 299 in order to avoid sustained oscillations in response to stimulus. Notably the oscillations 300 are not a consequence of any particular model assumption, but rather seem to stem 301 from a general mechanism that can be applicable to many autoregulated TCSs. It 302 remains to be seen if this mechanism can lead to oscillatory or pulsatile response for 303 systems where it is physiologically beneficial.

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The output of some two-component systems with bifunctional kinases -phosphorylated 306 response regulator protein -displays robustness to overexpression of the two proteins. 307 Here We propose that Mg 2+ binds to PhoQ and promotes the phosphatase conformation.

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Limitation of Mg 2+ then drives a change in conformation of PhoQ to the kinase form. 316 We also hypothesize that MgrB suppresses the rate of this conformation change and the 317 autokinase activity of PhoQ. Next, we find approximate analytical solutions for PhoP-P 318 at different ranges of Mg 2+ concentration. In our models we find that strong PhoPQ expression becomes negligible. This condition for robustness is not implausible 330 within E. coli since P mgrB is one of the strongest PhoP-activated promoters [16] and is 331 likely weaker than P phoPQ . PhoQ, like many TCS sensor kinases, is expressed at low 332 concentrations, estimated 50 fold less than PhoP [19,23]. In fact, measurement of 333 protein synthesis rates puts the ratio of MgrB to PhoQ between 3 and 23 depending on 334 the culture medium [23]. It is possible that over the range of induction rates (upto 4x 335 wild type) MgrB remains in excess of PhoQ.

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Multiple PhoP-P dependent promoters plateau over a range of 1 to 0.01 mM 337 Mg 2+ while remaining far from saturation [9], only to be stimulated strongly when and Salmonella typhimurium [10,14,18]. Interestingly, PhoP-P regulons in these species 355 also encode molecules that limit PhoQ activity [18]. function of independent induction of phoPQ operon (Fig 1B) is conducted using a strain 392 expressing mgrB constitutively, we predict that robustness should not be observed often activated by a ligand that drives a conformational change in sensory kinase [28,29]. 416 Therefore this oscillatory or pulsatile response dynamics may be observed in the systems 417 where it is of physiological benefit and could be used in synthetic biology applications. 418 collaborators [19]. Gene transcription regulation is modeled by phenomenological 427 models of (Hill-function) dependence on [PhoP-P].

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All models used (S1 Appendix) were simulated to follow experimental protocol as In addition, steady state YFP:CFP is simulated at very high signal (equivalent of Parameters were fit to minimize the above squared residual error (E s + E t ) using 458 particle swarm optimization in MATLAB. Each particle swarm optimization run 459 resulted in one parameter set. The best fitting parameter sets were used for further 460 analysis. All codes and parameter/data sets can be found in the following GitHub (normalized to the ratio at 50mM) extracted from [16] and [19]. This normalized data 474 set was used to fit temporal and steady state parameters for all models described in this 475 paper. Data was extracted using image analysis in MATLAB (except figure f, which 476 was read out manually from ref [19]). All data except constitutive mgrB (top right  Figure 1 are analyzed for properties such as nodes, strong linkage classes and rank. The theorem given in ref [1] was used to identify absolute concentration robustness.  In this section we identify conditions under which PhoP-P is robust to overexpression of PhoP and PhoQ (Figure 2, Main text). We find an approximate analytical solution to This model is the simplest modification of the PhoPQ model created by Miyashiro and Goulian in [3]. Reactions and ODEs are described in Supplemental File 4. Phosphotransfer and phosphatase complexes at steady state give: Substituting complex concentrations from Equation 1:

Steady state expressions for [Q-P] and [QB-P] can be found by first setting
Plugging back into Equation 8, Negative transcriptional feedback dictates the following saturating relation between B-total and PhoP-P The set of equations 10,11,12 together form the complete model of PhoPQ TCS. If we set MgrB-total to zero, this set of equations reduces to the model in ref [1] and the concentration of [PhoP-P] is the negative solution of a quadratic as discussed in refs [1,3,5]. In general however, we solve equations 10,11,12 for [PhoP-P] assuming the following parameters: C p = 10C t , K D = 0.1, λ = 0.01, K B = 5. f B was measured in ref [4] at ∼60.
As [P] T increases, the term C t /[P] (dependent on autodephosphorylation) becomes less significant as explained previously in ref [1], leading to robustness of [PhoP-P] to total-PhoP,PhoQ. In the special case that C t = 0 which happens at k −ap = 0, [PhoP-P] outright does not depend on total-PhoP as discussed in detail in ref [5]. Figure 1: A -PhoPQ TCS models predict that PhoP-P output will not be robust to overexpression of PhoP and PhoQ if mgrB is expressed constitutively. Blue triangles and blue line represents experimental data and simulation of PhoP-P output to overexpression of phoPQ operon as described in [1]. Red line shows PhoP-P output as a function of phoPQ induction if mgrB is expressed from a constitutive promoter at twice the rate of the wild-type (at basal activity level). Inset: Models with mgrB expressed from P mgrB will show an increase in PhoP-P output at 100 fold overexpression of phoPQ B. Following a switch in Mg 2+ concentration from 50 to 1 mM, our PhoPQ models with constitutively expressed mgrB (at levels comparable to wild-type basal) predict oscillations in PhoP-P output. Each line represents a parameter set that explains wild-type data (including biphasic dose-response) as shown in S4 Fig   In order to understand the role of negative feedback in shaping the dynamical properties of the TCS, we constructed in-silico mutants of our wildtype models with negative feedback through mgrB -upregulation removed. We constructed in-silico mutants expressing mgrB constitutively. We find that models of wild-type PhoPQ TCS that fit well with temporal data as well as display biphasic dose-response can show limit cycle oscillations in PhoP-P if mgrB is expressed at a constant rate comparable to the basal mgrB expression rates of corresponding wild-type models, i.e. if mgrB upregulation by PhoP-P is removed from the wild-type model. We observe limit cycle oscillations at signal levels equivalent to ∼1 mM Mg 2+ ( Figure 1B).

S3 Text: Model predictions and analysis of oscillations
To understand the mechanism that results in limit cycle oscillations following removal of negative feedback through MgrB upregulation, we find a minimal model of the TCS that still retains oscillations (Figure 2). The minimal model consists of two conformations of the kinase (Qkin and Qph). Each conformation catalyzes phosphorylation or dephosphorylation reactions with a saturating dependence on the substrate, P and P P respectively. We assume a constant amount of total-PhoP that is much greater than PhoQ as positive autoregulation of PhoP doesn't contribute to the oscillations significantly. We find that positive feedback through upregulation of PhoQ (synthesized in a kinase-biased conformation Qkin; Figure 2), together with a slow negative feedback through conversion of Qkin to the phosphatase conformation (Qph) is sufficient to yield oscillations. In the wild-type model, we find that upregulation of MgrB by PhoP-P reduces the delay by speeding up kinase to phosphatase switch, as well as increases the total amount of MgrB bound PhoQ (which contributes largely to phosphatase activity rather than kinase activity). (1 + f P 2 P /(K 2 0 + P 2 P ))). Qkin undergoes a reversible conversion to Qph. The rates used in this model were derived from the parameters used in the fully descriptive ODE model. B-The model yields limit cycle oscillations (blue). If the rate of kinase to phosphatase transition is modeled to increase as a function of P P (to recapitulate effects of MgrB upregulation), the oscillations are suppressed and result in a stable steady state (red). Rates (units: µM, s): k pd = 3.1 × 10 −4 , b = 1.2 × 10 −6 , f = 26, K 0 = 0.65, v k = 3, K k = 0.6, v P = 1.5, K P = 13.8, P T = 1.6, k f = 0.006, k r = 1.5 × 10 −6 S4 Text: A framework to examine steady state signal response for two-state model of PhoPQ-MgrB In this section, we look at how this particular model (Fig. 3 Main Text, Figure 2 in S1 Appendix, S4 Fig) that fits well to experimental data is able to simulate a biphasic dose-response. Therefore, the analysis here is not general but dependent on the particular parameter set resulting in simulations that match experimental data (See Fig. 3 Main Text, S4 Fig). We make many assumptions along the way to attempt to find why PhoP-P can show a biphasic response to decreasing k −1 . As we show in Fig. 4 (main text), the expressions we develop in this section for PhoP-P in ranges of k −1 corresponding to high (> 1 mM) and intermediate (1-0.01mM) Mg 2+ are good approximations of the steady state PhoP-P obtained from numerically solving the full model. We analyze the model with no autoregulation, since that strain shows no difference in output compared to wild-type over the plateau region, both in experiments and in this model [1]. The mechanism for plateauing however remains valid even with autoregulation of phoPQ. Consider 4 forms of PhoQ: kinase, phosphatase and MgrB bound kinase, phosphatase. So the ODE system for PhoQ concentrations reduces to: For simplicity, we do this analysis as λ →0 and f → 0. We assume all catalytic conversions within each form happen much faster than conversions between them. Thus the instantaneous concentrations of all sub-forms depend on the parent form in the following way: • Kinase: • Phosphatase: Now, we look at phosphorylation and dephosphorylation fluxes of PhoP to compute steady state PhoP-P.
• Dephosphorylation of PhoP-P  Fig. 4E). At steady state, To compute [PhoP-P] at various signal levels (i.e. k −1 values) using this flux balance equation (Equation 7, we find approximate expressions for relation between [Q*] and [QB ph ]. To this end, the set of equations in 1 can be solved at steady state to yield: Over this range,[QB ph ] can still be approximated as Q T and considered independent of signal (Main text Fig. 4E), In this expression, there is no dependence on k −1 , but on total MgrB. Total MgrB is a function of [PhoP-P], therefore steady state [PhoP-P] remains nearly independent of k −1 (Main text Fig. 4D). Promoter output plateaus over this range (Main text Fig. 3)

Transcription
Modeling of transcription and translation is the same for both models. PhoP-P binds promoter sites as a dimer, therefore we consider the following binding reactions. P mgrB + 2 P P We consider the binding reactions to be quasi-steady state. Considering 1 copy of promoter per cell, the occupancy fraction of the promoter can be calculated as: Here K 2 1 = k r1 k f 1 . P mgrB + P mgrB .P P2 = 1 Transcription from a promoter occurs at two rates: basal rate and active rate (when promoter is bound by the transcription factor). All transcripts degrade at a constant rate k md The same calculation gives a model for transcription for any mRNA (PhoP, PhoQ, MgrB or fluorescent reporter) from any PhoP-P dependent promoter (P mgrB or P phoP Q ). Model ODEs: PhoPQ model with single bifunctional kinase in-silico mutants In-silico mutants are modeled in the same way for both dynamical models.

Autoregulation deletion: Set
only so that total PhoP, PhoQ remains constant but reporters mRNA (mYFP1 ) and protein (YFP1) levels continue to report P phoP Q activity.
Model ODEs: PhoPQ model with two states of PhoQ Parameters: PhoPQ model with two states of PhoQ