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

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.


S1.1 Derivation of Eq. (1)
The Batchelor-Goulian model, as depicted in Fig. 2B, is described by the system of ordinary differential equations Under steady state conditions the left-hand sides in Eqs. (S1) -(S4) can be set to zero. For Eqs. (S3) and (S4) this leads to where the Michaelis-Menten constants are defined by Addition of Eqs. (S2) and (S4) yields Similarly, addition of Eqs. (S1) and (S4) yields and, after substituting the expressions in Eqs. (S8) and (S9), Finally, replacing [HK] by the expression in Eq. (S10) the factor [ HK P ] cancels on both sides of the equation and one obtains .
By defining the rescaled Michaelis-Menten constants C p and C t through K t this equation can be written in the form which directly leads to Eq. (1) of the main text.

S1.2 Derivation of Eqs. (2) and (3)
Within the Batchelor-Goulian model (Fig. 2B) the steady state concentration of the phosphorylated form of the response regulator (RR P ) is determined by the quadratic equation (Eq. 1 of the main text) which is valid if the concentration of the response regulator is much higher than that of the sensor kinase (R T ≫ H T ). In the following, approximate solutions of this equation are derived, which are either valid in the limit In the first case (C t ≪ C p ), it is advantageous to introduce the following dimensionless quantities through which Eq. (S11) becomes The solution of this equation is sought in the form Inserting this expansion into Eq. (S13) and equating terms of equal order of magnitude leads, to lowest order (ε = 0), to the quadratic equation which has the two solutions By definition, x = [RR P ]/R T must remain within the range 0 ≤ x ≤ 1 which requires to choose the '−' sign in Eq. (S15). Hence, the physiologically reasonable solution is given by The O(ε)-equation has the solution (S17) Combining Eqs. (S16) and (S17) shows that the solution of Eq. (S13) can be approximated (up to terms of O(ε 2 )) by Switching back to original variables using Eq. (S12) yields Eq.
(2) of the main text.
In the limit C t ≫ C p , one may rewrite the exact solution of Eq. (S11) in terms of the dimensionless quantities which yields . (S19) Using that √ 1 − εx ≈ 1 − εx/2 for ε ≪ 1, one may then expand the square root in Eq. (S19) which leads to where the first approximation (with the ε-term in the denominator) has been used in Eq.
(3) of the main text (after switching back to original variables using Eq. S18).

S1.3 Asymptotic analysis of Eq. (15)
To find an approximate solution of Eq. (15) of the main text one may employ the same expansion as in Eq. (S14). Inserting this expansion into Eq. (15) leads, to lowest order (ε = 0), to the quadratic equation which has the two solutions Its solutions are given by The approximate expression for [E K ], as used in Eq. (16) of the main text, is The same replacement, together with ε = K d /E T , yields the following expression for

S1.4 Derivation of Eq. (40)
Eq. (35) of the main text can be written as a quadratic equation where α is given by To derive an approximate expression for the solution of Eq. (S27), which is valid under the condition max(K app t , K p ) ≪ R T (Eq. 37), it is advantageous to introduce the dimensionless quantities through which Eq. (S27) becomes Eq. (S30) has been analyzed previously for the special case K = 1 [1]. Following the same steps as in Ref. [1] it is straightforward to show that, in the limit ε ≪ 1, the solution of Eq. (S30) can be approximated (up to terms of O(ε 2 )) by Using the definition of α in Eq. (S28) the conditions α > 1 (α < 1) become L T < L * T (L T > L * T ) where L * T is defined by (cf. Eq. 38 of the main text)