Reciprocal Regulation as a Source of Ultrasensitivity in Two-Component Systems with a Bifunctional Sensor Kinase

<div><p>Two-component signal transduction systems, where the phosphorylation state of a regulator protein is modulated by a sensor kinase, are common in bacteria and other microbes. In many of these systems, the sensor kinase is bifunctional catalyzing both, the phosphorylation and the dephosphorylation of the regulator protein in response to input signals. Previous studies have shown that systems with a bifunctional enzyme can adjust the phosphorylation level of the regulator protein independently of the total protein concentrations – a property known as concentration robustness. Here, I argue that two-component systems with a bifunctional enzyme may also exhibit ultrasensitivity if the input signal reciprocally affects multiple activities of the sensor kinase. To this end, I consider the case where an allosteric effector inhibits autophosphorylation and, concomitantly, activates the enzyme's phosphatase activity, as observed experimentally in the PhoQ/PhoP and NRII/NRI systems. A theoretical analysis reveals two operating regimes under steady state conditions depending on the effector affinity: If the affinity is low the system produces a graded response with respect to input signals and exhibits stimulus-dependent concentration robustness – consistent with previous experiments. In contrast, a high-affinity effector may generate ultrasensitivity by a similar mechanism as phosphorylation-dephosphorylation cycles with distinct converter enzymes. The occurrence of ultrasensitivity requires saturation of the sensor kinase's phosphatase activity, but is restricted to low effector concentrations, which suggests that this mode of operation might be employed for the detection and amplification of low abundant input signals. Interestingly, the same mechanism also applies to covalent modification cycles with a bifunctional converter enzyme, which suggests that reciprocal regulation, as a mechanism to generate ultrasensitivity, is not restricted to two-component systems, but may apply more generally to bifunctional enzyme systems.</p></div

Ultrasensitivity does not require both enzyme activities to be saturated.

<p>(A) As the phosphotransferase (PT) activity of the sensor kinase changes from saturation (blue curve) to non-saturation (red curve) the steady state response of as a function of remains ultrasensitive, but the transition point (), as defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e293" target="_blank">Eq. (38)</a>, is shifted to lower effector concentrations. Blue curve: , red curve: . In both cases . (B) Transient dynamics for (dotted line in A) indicating that the time-scale for reaching the steady state increases if the PT activity becomes non-saturated. Initial conditions: , , , all other concentrations were set to zero. Solid lines were computed from the full model in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e180" target="_blank">Eqs. (22)</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e187" target="_blank">(29)</a> with the parameters , , , (red curve) and , (blue curve). Other parameters: , , , , , so that and . Dashed lines in A correspond to the approximate expression for the stimulus-response curve in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e326" target="_blank">Eq. (40)</a>.</p

Stimulus-dependent concentration robustness in two-component systems.

<p>Steady state response curves according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e199" target="_blank">Eq. (31)</a> for (A and B) and (C and D). (A and C) exhibits a graded response as a function of . (B and D) exhibits stimulus-dependent concentration robustness as a function of . The dotted lines indicate the threshold concentrations beyond which becomes approximately constant. Note that, if (corresponding to the blue dotted line in A), increasing beyond does not lead to a higher phosphorylation level of the response regulator (B), which might explain why autoregulation in TCSs does not necessarily lead to a higher phosphorylation level of the response regulator (cf. Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Miyashiro1" target="_blank">[22]</a>). However, decreasing the effector concentration to (corresponding to the red dotted line in A) allows to increase as increases. Solid lines were obtained from simulations of the full model (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e180" target="_blank">Eqs. 22</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e187" target="_blank">29</a>) using the parameters: , , , , , (, cf. Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Lesley1" target="_blank">[46]</a>). (A and B) , (, ) and (C and D) , (, ). Dashed lines correspond to the approximate solutions in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e205" target="_blank">Eq. (33)</a> (A and B) and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e206" target="_blank">Eq. (34)</a> (C and D).</p

Autophosphatase activity of NRI may compromise ultrasensitivity in the NRII/NRI/PII system.

<p>(A) Comparison of experimental data (filled boxes, data taken from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi-1003614-g004" target="_blank">Fig. 4A</a> of Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Jiang3" target="_blank">[27]</a>) with the steady state response curve calculated from the extended Batchelor-Goulian model in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e180" target="_blank">Eqs. (22)</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e187" target="_blank">(29)</a> with an extra term ‘’ added to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e180" target="_blank">Eq. (22)</a>, which accounts for autodephosphorylation of NRI-P. The blue dashed line represents the best fit obtained for , , and . The other parameters were kept fixed: , , , so that and corresponding to a half-life of 5 minutes <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Jiang3" target="_blank">[27]</a>. (B) As the autodephosphorylation rate constant of NRI-P is lowered (bottom to top: , , , ) the response curve becomes more and more ultrasensitive (solid lines). Note that ultrasensitivity is restricted to the region as predicted by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e293" target="_blank">Eq. (38)</a>. The dashed (blue) lines in (A) and (B) are identical.</p

Ultrasensitivity in covalent modification cycles with a bifunctional converter enzyme.

<p>(A) Reaction scheme: A substrate molecule () undergoes reversible phosphorylation by a bifunctional converter enzyme which can exist in two activity states. Binding of the allosteric effector inhibits the kinase activity () and, concomitantly, activates the phosphatase activity () of the enzyme. (B) As the value of the dissociation constant is lowered from to (from right to left) the steady state curve becomes ultrasensitive near the transition point , as defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e175" target="_blank">Eq. (21)</a>. The solid lines were computed from the full model using <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e096" target="_blank">Eqs. (4)</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e105" target="_blank">(7)</a>. Dashed lines were computed from the reduced models using <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e142" target="_blank">Eq. (14)</a> (right curve) and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e159" target="_blank">Eq. (18)</a> (left curve). Parameters: , , so that , and (for ) or (for ).</p

Experimental observations of concentration robustness in TCSs.

<p>Comparison between predictions of the Batchelor-Goulian model and measurements in the PhoR/PhoB <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Gao1" target="_blank">[26]</a> and NRII/NRI systems <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Jiang3" target="_blank">[27]</a>. (A) Symbols denote measurements of PhoB-P as a function of total PhoB amounts in the wild-type system (open squares) and in a mutant strain (filled circles) (data were taken from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi-1003614-g004" target="_blank">Fig. 4C</a> in Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Gao1" target="_blank">[26]</a>). Solid lines were calculated from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e199" target="_blank">Eq. (31)</a> with pmol, pmol and pmol, pmol. Note that (dotted lines) determines both, the threshold amount of total PhoB beyond which PhoB-P becomes constant as well as the value of that constant, as expected from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e027" target="_blank">Eq. (2)</a>. (B) Symbols denote <i>in vitro</i> measurements of NRI-P as a function of total NRI (reproduced from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi-1003614-g004" target="_blank">Fig. 4A</a> in Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Jiang3" target="_blank">[27]</a>). Solid line represents the best fit of the data to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614.e028" target="_blank">Eq. (3)</a> with and , which indicates that the NRII/NRI system operates in the regime .</p

Reciprocal regulation in two-component systems.

<p>(A) Schematic representation of reciprocal regulation in the PhoQ/PhoP <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Chamnongpol1" target="_blank">[20]</a> and NRII/NRI systems <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Jiang2" target="_blank">[21]</a>. In both cases, an allosteric effector ( or PII) inhibits autophosphorylation of the sensor kinase and increases the enzyme's phosphatase activity. (B) Batchelor-Goulian model <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi.1003614-Batchelor1" target="_blank">[11]</a> based on the three activities of the sensor kinase (cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003614#pcbi-1003614-g001" target="_blank">Fig. 1</a>): (1) Autophosphorylation of the sensor kinase (HK), (2) phosphotransfer to the response regulator (RR) and (3) dephosphorylation of the RR. Cofactors such as ATP are assumed to be constant. (C) Extension of the Batchelor-Goulian model to include reciprocal regulation of the HK's activities as schematized in (A). Binding of the allosteric effector (4) inhibits autophosphorylation (1) and activates the phosphatase activity (3) of the sensor kinase. For simplicity, the free form of the enzyme () is assumed to have no phosphatase activity whereas the effector-bound form () is assumed to have no autokinase activity.</p

Temporal hierarchy of substrate degradation.

<p><b>A, B, C</b>: Transient response upon substrate addition. At <i>t</i> = 0 two substrates, S1 and S2 (each 300nM), are added to a steady state mixture containing Cul1, Cand1 and SR1-SR3. The resulting decline of the total amount of substrates is displayed together with the <i>t</i>_1/2 (dotted lines). Substrates with a higher SR affinity (A), substrates for SRs with a higher affinity for Cul1 (B) and substrates for more abundant SRs (C) are preferentially degraded.<b>D, E, F</b>: Assembly and disassembly of SCF ligases upon substrate addition. Depicted are changes in the fraction of SRs that are bound in a SCF complex. The blue and violet curves correspond to ([Cul1.SR1] + [Cul1.SR1.S1])/SR1_T and ([Cul1.SR2] + [Cul1.SR2.S2])/SR2_T, respectively, whereas the light red curve denotes [Cul1.SR3]/SR3_T. In each case Cul1 is redistributed from Cul1.SR3 into Cul1.SR1(.S1) and Cul1.SR2(.S2). In (A-F) if not indicated otherwise reference parameters are: <i>K</i>_<i>S</i>1 = <i>K</i>_<i>S</i>2 = 10<i>nM</i> (<i>k</i>_<i>off</i> = 1<i>s</i>⁻¹), <i>K</i>_<i>sr</i>,1 = <i>K</i>_<i>sr</i>,2 = <i>K</i>_<i>sr</i>,3 = 0.225<i>pM</i>, SR1_T = SR2_T = 60<i>nM</i>. To preserve detailed balance has been increased by a factor of 5 in (B) and (E). SR3_T = 660<i>nM</i> − (SR1_T + SR2_T), Cand1_T = 400<i>nM</i>, <i>k</i>_<i>deg</i> = 0.004<i>s</i>⁻¹. Parameters other than those mentioned are listed in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005869#pcbi.1005869.t001" target="_blank">Table 1</a>.</p

Default parameter values.

<p>Default parameter values.</p

SCF-mediated substrate degradation and Cand1 cycle.

<p><b>A</b>: Scheme of SCF-mediated substrate degradation: (1) Substrate (S) binding to substrate receptors (Skp1/SR) and UBC12-mediated neddylation (N8) of Cul1, (2) E2 recruitment, ubiquitin (Ub) transfer by E2 to the substrate and Ub chain elongation, (3) substrate degradation by the 26S proteasome and (4) deneddylation of Cul1 by the COP9 signalosome. Relative sizes of protein subunits are not to scale. <b>B</b>: Model of the Cand1-mediated exchange cycle for two substrate receptors (Skp1/SR1 and Skp1/SR2). <i>K</i>_<i>sr</i>, <i>K</i>_<i>ca</i>, and denote dissociation constants whereas <i>k</i>_<i>sr</i>, <i>k</i>_<i>ca</i>, and are dissociation rate constants (cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005869#pcbi.1005869.t001" target="_blank">Table 1</a>). The parameter <i>η</i>, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005869#pcbi.1005869.e010" target="_blank">Eq (2)</a>, measures the preference of Cand1 and SR for binding to Cul1. Similarly, <i>α</i> and <i>β</i> account for relative differences in the dissociation rate constants for the binary complexes (<i>α</i>) and the ternary complex (<i>β</i>).</p