A single phosphatase can convert a robust step response into a graded , tunable or adaptive response

Received 17 January 2013 Accepted 17 May 2013 Oxford Centre for Integrative Systems Biology, Department of Biochemistry, South Parks Road, Oxford OX1 3QU, UK Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Graduate Institute of Biomedical Informatics, College of Medical Science and Technology, Taipei Medical University, 250 Wu-Hsing Street, Taipei 11031, Taiwan, ROC


INTRODUCTION
Synthetic biology aims at designing new or redesigning existing biological circuits for a particular purpose (Bashor et al., 2010;Lim, 2010).For many biological systems, the response to an input step change is independent of the copy number of the components in the system or other environmental factors (Batchelor & Goulian, 2003;Shinar et al., 2007;Shinar & Feinberg, 2010).Sometimes, however, this 'robustness', which makes the system insensitive to perturbations, also makes it challenging to redesign.In such cases, structural modifications are necessary to destroy the original robustness and allow engineers to modify the response as desired.Two-component signalling systems are widely used by bacteria and some eukaryotes to sense and respond to environmental signals.These systems typically consist of a sensory histidine protein kinase (HPK) and a response regulator (RR).The HPK can receive stimuli, autophosphorylate and transfer the phosphoryl group to the RR.
The phosphorylated RR can be dephosphorylated by either a bifunctional HPK or a separate phosphatase enzyme or through the RR's own intrinsic autophosphatase activity.The actual response of the system is regulated by the phosphorylation levels of the RR in a cell.The vast variety of modularized receptors, coupled with the speed and high level of specificity of signal transduction, makes twocomponent systems an attractive target to engineer.
One of the best-characterized two-component systems is the EnvZ/OmpR osmosensory system from Escherichia coli (Forst & Roberts, 1994;Casino et al., 2009).In this system, the bifunctional HPK, EnvZ, senses changes in osmolarity and alters its kinase/phosphatase activity, and consequently regulates the phosphorylation levels of the RR, OmpR.The phosphorylated OmpR (OmpR-P) binds to the promoters of genes encoding the outer membrane porins OmpF and OmpC, regulating their transcription in response to the input signal (Yoshida et al., 2006).In early attempts to redesign this system (Utsumi et al., 1989;Jin & Inouye, 1993, 1994;Yoshida et al., 2007), the periplasmic ligand binding domain of the chemoreceptor Tar was fused with the cytoplasmic kinase/phosphatase domain of EnvZ, thus allowing the system to sense aspartate.The resulting chimeric receptor, Taz, was able to activate OmpR in response to the extracellular levels of aspartate.Previous models of the EnvZ/OmpR system suggested that the steady-state IP: 54.70.40.11On: Sat, 20 Oct 2018 13:59:34 output of the system is robust over circa 10-fold changes of the total concentrations of EnvZ or OmpR in an E. coli cell (Batchelor & Goulian, 2003).This makes tuning or redesigning the input-output response of this signalling pathway challenging, as the major control 'knobs' that synthetic biologists can implement in vivo are to regulate the amount of protein made by changing the strength of the promoter or the efficiency of the ribosome-binding site.In this paper we develop a model for the Taz/OmpR system, and show both computationally and experimentally that the regulated expression of a truncated version of Taz, serving as a single extra phosphatase, can convert the original robust switch-like system into a graded, tunable redesigned system, and an adaptive system (see Fig. 1).

METHODS
Mathematical analysis.All mathematical models were established using ordinary differential equations (ODEs) following mass-action law.The set of ODEs of the mathematical model are implemented and simulated in MATLAB 7.0 and SIMULINK.Tables 1 and 2 contain the parameters for model simulations.Sections S1-S3 of the Supplementary Material, available in Microbiology online, contain more detailed descriptions of the equations, derivations, mathematical analysis and modelling presented in the Results and Discussion.
Strains and plasmids.A list of the strains and plasmids used in the experiments and a summary of their main features are given in Table 3.The plasmids all contain modular and standardized parts, and a detailed description of the construction of novel plasmids is contained within Supplementary Material Section S4.
Measuring steady-state responses.All experiments were carried out in the E. coli strain MDG135 (Batchelor & Goulian 2003) which lacks the gene encoding EnvZ and has a construct allowing the expression of cyan fluorescent protein (CFP) to be induced by OmpR-P from the promoter for ompC on its genome.Taz protein was expressed from a tetracycline inducible promoter by the addition of anhydrotetracycline (aTc) to strains containing the plasmid pSPA3tet-Taz and pZS4Int-tetR (EXPRESSYS), which constitutively expresses the tetracycline promoter repressor TetR.Overexpression of OmpR was carried out from an IPTG-inducible promoter on the plasmid pSPY2lac-OmpR in MDG135 which also contained pSPA3tet- With the introduction of an additional independent phosphatase (Tazc) that dephosphorylates OmpRp, as the 'tuning knob' of the redesigned system, the step response of the system becomes more tunable and graded in response to input aspartate.(c) By using Tazc to construct a negative feedback, the redesigned system becomes a closed-loop feedback system which can exhibit an adaptive response in GFP fluorescence upon stimulation with input aspartate.Taz to allow Taz expression on addition of aTc and pZS4Int-laci/tetR (EXPRESSYS) to constitutively express both the LacI and the TetR repressor proteins.Expression of Tazc was carried out from an IPTGinducible promoter on the plasmid pSPY2lac-Tazc in MDG135 which also contained pSPA3tet-Taz to allow Taz levels to be varied by the addition of aTc and pZS4Int-laci/tetR (EXPRESSYS) to constitutively express both the LacI and the TetR repressor proteins.Taz expression was induced by the presence of 1-100 ng aTc ml 21 , OmpR expression by 10 mM to 1 mM IPTG and Tazc by 10 mM to 1 mM IPTG.Input aspartate was varied between 1 mM and 10 mM.
Measuring feedback responses.All experiments were carried out in the E. coli strain MSZ31 (Sato et al., 2000), which lacks the gene encoding EnvZ.Taz protein was expressed from a tetracyclineinducible promoter by the addition of 10 ng aTc ml 21 to strains containing the plasmid pSPA3tet-Taz and pZS4Int-tetR (EXPRESSYS), which constitutively expresses the tetracycline promoter repressor TetR.Both Tazc and the reporter green fluorescent protein (GFP) with an SsrA(Ala-Ser-Val) degradation tag rapid degradation tag are expressed from the OmpR-P-inducible ompC promoter on plasmid pSPU9ompC-Tc-G3ASV.Input a-methyl-DL-aspartate (Asp) concentrations were varied from 50 mM to 10 mM.
Sample preparation and plate reader protocol.Bacterial strains were cultured in Luria-Bertani medium with appropriate antibiotics at 37 uC for 16-18 h overnight.The overnight culture was inoculated at a ratio of 1 : 100 into minimal A medium (Miller, 1992) supplemented with 0.4 % (w/v) glucose, 0.2 % (w/v) Casamino acids, 1 mM MgSO 4 and 5 mg thiamine ml 21 .The cultures were grown with appropriate antibiotics and various levels of inducers (aTc and/or IPTG) at 37 uC until an OD 600 of 0.1 was reached, and then aliquoted into a Corning 3603 96-well black microplate.Each well contained 190 ml of the culture and various concentrations of 10 ml input Asp (Sigma-Aldrich).The plate was run in a TECAN Infinite 200 PRO plate reader at 37 uC with orbital shaking of 2 mm diameter.Data for absorbance (600 nm) and fluorescence (excitation 430±9 nm and emission 480±20 nm for CFP; excitation 488±9 nm and emission 535±20 nm for GFP) were acquired automatically every 15 min.Gain was set within the range 100-120.
Data processing.Absorbance and fluorescence data from the plate reader were analysed with MATLAB.The normalized per-cell fluorescence of the test cells is calculated as where A m and F m are absorbance and fluorescence of the wells with medium only, A n and F n are absorbance and fluorescence of the wells with negative control cells (e.g. cells with empty plasmids only), and A and F are absorbance and fluorescence of the wells with test cells.

RESULTS AND DISCUSSION
The basic Taz/OmpR system is robust to perturbations We have developed an ODE model of the Taz/OmpR system based around the reactions shown in Fig. 2(a).The analytical solution of the steady-state level of OmpRp in this system (for full derivation see Supplementary Material Section S1.1) is given as: where u denotes the input aspartate and Rp stands for the phosphorylated response regulator OmpR-P.This solution comprises only kinetic constants and one system state and so the general form of Rp can be represented as: where C 1 is a constant and u is the level of aspartate not bound to the receptor Taz.
Using the nomenclature from Fig. 2(a), then, T is the sensor Taz, R is the response regulator OmpR and T or R with a lower case p means the protein is phosphorylated.Analytically, if U total ..T total .uT+uTp+uTpRthen Where two values are cited in the literature we considered both possibilities, although the simulation results did not show any obvious differences between them.U total # u and Rp # C 1 ?U total , which is robust to the variations of R total and T total .However, if C 1 ?U total .R total then Rp will be limited by R total and thus Rp5R total , which is robust to U total and T total but sensitive to R total .

Reactions
From the mathematical analysis we find that, as with the EnvZ/OmpR system (Batchelor & Goulian, 2003;Shinar et al., 2007), the Taz/OmpR system is generally robust to changes in the total concentrations of Taz or OmpR and only sensitive to [Asp] T (here [ ] denotes concentration and subscript T denotes Total) when [Taz] T is low and [OmpR] T is high, despite their input mechanisms being different.
Simulations were carried out using the parameters given in Tables 1 and 2, in which the input aspartate was varied from 1 mM to 100 mM (Fig. 2b).Experiments were also designed (Fig. 1a) in which Taz protein was expressed from the tetracycline-inducible promoter on plasmid pSPA3tet- Fluorescence/absorbance (a.u.) Fluorescence/absorbance (a.u.) log([IPTG]), mM Taz by addition of 1-100 ng aTc ml 21 and CFP production from the OmpR-P-inducible ompC promoter on the genome of strain MDG135 (Batchelor & Goulian 2003) was measured in response to 1 mM to 10 mM input Asp.Our results show that the system is activated at 50-100 mM Asp, with a narrow transition region between 500 mM and 1 mM, and is saturated by 1-5 mM Asp (Fig. 2b).Varying the levels of Taz or OmpR, either in the model or when experimentally expressed from the inducible plasmids pSPA3tet-Taz or pSPY2lac-OmpR, respectively, in MDG135, , μM [OmpRp] , μM [OmpRp] , μM [OmpRp] , μM [OmpRp] , μM [OmpRp] , μM fluorescence/absorbance (a.u.) φ φ confirmed that around the regions of low [Taz] T and medium to high [OmpR] T the system is robust (Fig. 3).An exception is that experimentally the output rises significantly when OmpR is induced with 1 mM IPTG.This result is similar to that for the EnvZ/OmpR system observed by Batchelor and Goulian (2003), who suggested that extremely high levels of unphosphorylated OmpR can bind to and activate expression from the ompC promoter, a factor that is not included in the model.However, both the model and the experimental data suggest that it would not be possible to accurately tune the steady-state output levels of OmpR-P for a given input [Asp] T without expressing either very high levels of Taz, which may become toxic to the cell (Wagner et al., 2006(Wagner et al., , 2007)), or expressing very low levels of OmpR, which would limit the output from the system (Shinar et al., 2007).We therefore considered alternative ways to engineer the system to allow the output to be tuned.

Generating a synthetic, graded tunable system
It has been previously shown that the isolated cytoplasmic domain A of EnvZ exhibits constitutive phosphatase activity towards OmpR-P both in vitro and in vivo (Zhu et al., 2000).
As aspartate causes the Taz protein to switch from its phosphatase-predominant to kinase-predominant conformations, we truncated the sensory and the transmembrane domains from Taz to introduce an additional independent intracellular phosphatase, Tazc, as a 'tuning knob' into the Taz/OmpR system.This produced a 'futile cycle' found in many natural metabolic and signalling systems where two proteins have antagonistic effects on a central molecule.
The original Taz/OmpR model was extended to include this extra phosphatase (Fig. 4a), and the analytical solution (for full derivation see Supplementary Material Section S2) found to be: where u denotes the input aspartate, T is the sensor Taz, Tc is the truncated Taz and Rp is the phosphorylated response regulator OmpR-P.Therefore, Rp is related to u, T and Tc.
The general form of Rp can be represented as: where C 1 is a constant, and is the same as the C 1 found in the original Taz-OmpR system.
In this case if U total ..T total , then Rp5C 1 ?U total /(1+Tc/T), which is sensitive to U total , T and Tc.For a constant U total , the steady-state Rp is now determined by the ratio of Tc/T.Note that Rp is tunable only if Tc/T is not too small, which can be ensured by having a low level of T total .
From modelling and simulation results using the parameters in Tables 1 and 2, we found that increasing [Tazc] T allows the system to retain its ability to respond to the input [Asp] T but with reduced output levels of [OmpR-P] for a given [Asp] T (Fig. 4b).Surprisingly, the results also show that the redesigned system can respond in a more graded, linear manner than the response seen for the wildtype system (Figs 3 and 4b).Thus, the expression of a single phosphatase [Tazc] T allows the steady-state output [OmpR-P] to be easily tuned and converted from a robust step response to a graded sensor.
To validate the modelling results experimentally, Tazc was introduced into the previous Taz/OmpR system (pSPA3tet-Taz in MDG135) in vivo on the inducible plasmid pSPY2lac-Tazc (Fig. 1b), and [Tazc] T was varied by adding 10 mM to 10 mM IPTG.At steady state, the redesigned system responds to [Asp] with an activation threshold similar to that of the wild-type system of about 100 mM, but does not appear to saturate even at 10 mM aspartate (Fig. 4b).As was predicted by the model, the response becomes more graded with a much larger linear region, and hence the steady-state output of the redesigned system can be easily tuned by varying the level of [Tazc] T while still responding to changes in [Asp].This confirms that the Tazc 'knob' can be used to tune the response of the redesigned system.In the original system both the kinase and the phosphatase activities are carried out by the same protein (Taz) and are controlled by the same input [Asp] T .Decoupling the control of the phosphatase activity from the activation of the kinase activity by varying [Tazc] T results in a highly tunable redesigned system.

Generating an adapting system
To further examine the uses of an independently controlled phosphatase in synthetic two-component systems, we utilized Tazc to convert the Taz/OmpR system into a closed-loop feedback system to examine whether it can exhibit adaptation (Yi et al., 2000;Wadhams & Armitage, 2004).To accomplish this we modelled a system in which the expression of Tazc is regulated by the ompC promoter (P ompC ), which is itself activated by the presence of OmpR-P, to form a negative feedback loop (Fig. 5a).The model was extended to include the expression of the phosphatase Tazc (Tc) being activated by OmpR-P (Rp) with a protein synthesis rate of k s .Each copy of the P ompC promoter is modelled as being activated by six molecules of Rp and Tc is degraded with a degradation rate k d (described in more detail in Supplementary Material Section S3).
The only two parameters that can be varied experimentally in this system are the stability of Tazc, which can be altered by the addition of specific targeted degradation tags, and the copy number of the plasmid carrying the coding gene P ompC -Tazc, which is determined by the origin of replication on the plasmid.For the simulations the parameters given in Tables 1 and 2 were used and the half-life of Tazc was set to 0 (undegradable, hypothetical), 24 h (hypothetical), 2 h (typical for a protein with an ssrA-ASV tag) and 40 min (for a protein with an ssrA-LVA tag).
The copy number of P ompC -Tazc was set to 1 (genomically encoded), 5 (pSC101 plasmid copy number), 25 (p15A/ pBBR1 copy number) and 200 (ColE1 plasmid copy number).The levels of input [Asp] T were varied from 1 mM to 10 mM.The simulation results shown in Fig. 5(b) and Fig. S3.2 demonstrate that the peak output response of the system is dependent on the input [Asp] T .They also show that higher copy numbers of P ompC -Tazc result in lower peak output values, but more rapid adaptation kinetics and better tolerance of less stable forms of Tazc.Taken together, these data imply that the adaptive response of this very simple feedback system is a trade-off between the peak output levels and the rate of adaptation, and is highly sensitive to the stability of the phosphatase protein Tazc.
As the actual stability of Tazc is not known, our experimental construction of the feedback loop used a high-copynumber ColE1-based plasmid containing P ompC -Tazc expressing no additional degradation tags to maximize the possibility of the system exhibiting adaptation.To measure the response dynamics we constructed a plasmid pSPU9ompC-Tc-G3ASV which contains the OmpR-Pinducible ompC promoter activating transcription of both the phosphatase tazc and a gfp reporter with an ASV rapid degradation tag (Fig. 1c).This plasmid (pSPU9ompC-Tc-G3ASV) was transformed into strain MSZ31 (Sato et al., 2000) which lacks envZ, along with the Taz producing plasmid pSPA3tet-Taz.The dynamic responses of the experimental system are shown in Fig. 5(c).As predicted, the system shows a response to the input [Asp] T followed by adaptation.At high levels of [Asp] T , the output adapted, but to a level higher than the starting values.In contrast, at very low levels of [Asp] T , adaptation occurred with an output level lower than the starting value.OmpR may be phosphorylated by sources other than Taz within the bacterial cell.Both the small molecule acetyl-phosphate (Alves & Savageau, 2003;Wolfe, 2005) and the HPK of the Cpx operon (Batchelor et al., 2005) have been shown to be alternative phosphodonors for OmpR.Prior to the Asp input, an initial output may already have been generated by OmpR phosphorylated by these alternative phosphodonors.After Asp stimulation, the accumulation of Tazc dephosphorylates any OmpR-P in the cell, including that caused by the alternative phosphodonors, and hence quenches the initial output level.The system therefore does not display perfect adaptation under experimental conditions.
Yi et al. (2000) showed that integral control is a necessary and sufficient condition for robust perfect adaptation so we determined whether the feedback loop of the Taz-OmpR system constructed in this section is an integral feedback or not (for full derivation see Supplementary Material Section S3.2).
In our system: dTc total /dt5k s ?D?H(Rp)2k d ?Tc total where Tc is Tazc, D is the DNA concentration of the P ompC promoter and H(Rp) is the Hill function.
Therefore, if k d is zero, which means that the truncated Taz is undegradable, then: Tc total 5#k s ?D?H(Rp) dt.
However, if k d is not exactly zero, then Tc total will eventually reach a fixed level at equilibrium.Therefore, for perfect adaptation, the closed-loop feedback system would require an infinitely stable phosphatase (i.e. with a zero degradation rate, which would correspond to integral feedback), which is biologically impossible.However, this simple system still shows good adaptation for input [Asp] T levels below 5 mM and is technically simple to implement.Depending on the degree of adaptation precision required by the specific application, it would be possible either to use the output from this system directly or to filter it with other components in a network.

CONCLUSIONS
In summary, our results show how the addition of a single phosphatase into a synthetic signalling pathway can convert a robust step-like response into a graded, tunable response or even an adaptive response.The tunable, graded system has been implemented and shown to function as the model predicted.The adaptive system shows basic adaptation as predicted by the models, although it requires biologically unachievable parameters to generate perfect adaptation.Overall, it is the sheer simplicity of implementation of these new design principles -the introduction of a single independently controllable phosphatase to generate these very different system properties, including synthetic feedback -which may herald the engineering of fast responding two-component signalling pathways as powerful tools for synthetic biology.

Fig. 1 .
Fig.1.Redesigning the Taz-OmpR system.(a) In the original Taz-OmpR system, the bi-functional sensory HPK, Taz, receives a step external stimulus of aspartate, phosphorylates OmpR to form OmpRp which then binds to the ompC promoter (PompC) and drives transcription of cfp which generates a robust, switch-like steady-state output level of CFP.(b) With the introduction of an additional independent phosphatase (Tazc) that dephosphorylates OmpRp, as the 'tuning knob' of the redesigned system, the step response of the system becomes more tunable and graded in response to input aspartate.(c) By using Tazc to construct a negative feedback, the redesigned system becomes a closed-loop feedback system which can exhibit an adaptive response in GFP fluorescence upon stimulation with input aspartate.

Fig. 2 .
Fig. 2. The original Taz-OmpR system.(a) The model of the system and its reactions.u denotes input aspartate, T is the sensor Taz and R is the response regulator OmpR.T or R with a lower case p means the protein is phosphorylated.The output is Rp.(b) Simulation (left) and experimental (right) results of the basic input-output response of the Taz-OmpR system.Taz expression is induced by 1-100 ng aTc ml "1 from pSPA3tet-Taz in MDG135 and CFP fluorescence from the P ompC -CFP construct on the genome measured as the output for input Asp from 1 mM to 10 mM.Error bars represent SD from three replicates.

Fig. 3 .
Fig. 3. Simulation (top and middle rows) and experimental (bottom row) results of the parametric responses varying [Taz] T and [OmpR] T .See Tables1 and 2for parameter settings.The dark red 'plateau' area indicates the 'robust region' in which the output level of OmpR-P is insensitive to the variations of [Taz] T and [OmpR] T .Taz expression is induced by 1-100 ng aTc ml "1 from pSPA3tet-Taz in MDG135 whereas OmpR expression is induced by 10 mM to 1 mM IPTG from pSPY2lac-OmpR in the presence of constant Taz expressed from pSPA3tet-Taz by 10 ng aTc ml "1 in MDG135.CFP fluorescence from the P ompC -CFP construct on the genome is measured as the output for input Asp from 10 mM to 10 mM.The same experimental data are plotted in different formats in the two graphs.Error bars represent SD from three replicates.

Fig. 4 .
Fig. 4. The redesigned graded tunable system including Tazc.(a) The extended model of the system.Reactions are as for Fig. 2(a) plus the light blue reaction showing dephosphorylation of OmpR-P by Tazc.u denotes input aspartate, T is the sensor Taz and R is the response regulator OmpR.T or R with a lower case p means the protein is phosphorylated.Tc denotes the truncated Taz, which is not able to receive the input signal u, but can still carry out binding/dissociation of OmpR and the phosphatase reaction as shown.(b) Simulation (top row) and experimental (bottom row) results of the parametric responses varying [Asp] T and [Tazc] T .Tazc expression is induced by 10 mM to 1 mM IPTG from pSPY2lac-Tazc in the presence of constant Taz expressed from pSPA3tet-Taz by 10 ng aTc ml "1 in MDG135.CFP fluorescence from the P ompC -CFP construct on the genome is measured as the output for input Asp from 10 mM to 10 mM.The same experimental data are plotted in different formats in the two graphs.Error bars represent SD from three replicates.

Fig. 5 .
Fig.5.The adaptive dynamic response of the redesigned closed-loop Taz-OmpR system.(a) The model of the system.Reactions are as for Fig.4(a) plus the orange and green reactions showing regulated synthesis of Tazc and its degradation, respectively.u denotes input aspartate, T is the sensor Taz and R is the response regulator OmpR.T or R with a lower case p means the protein is phosphorylated.Tc denotes the truncated Taz.(b) Simulation results using high-copy (ColE1, top rows) and medium-copy (p15A, bottom rows) plasmids with fast (ssrA-LVA tagged, left columns), medium (ssrA-ASV tagged, central columns) and slow (hypothetical, right columns) degradation rates for Tazc.The parameter settings are given in Tables1 and 2and more simulation results are shown in Fig.S3.2.(c) The experimental results using a high-copy-number ColE1 plasmid and untagged Tazc show imperfect adaptation under various levels of [Asp] T .Taz expression is induced from pSPA3tet-Taz by 10 ng aTc ml "1 and both Tazc and the GFP reporter are expressed from the OmpR-P-inducible PompC on pSPU9ompC-Tc_G3ASV in MSZ31.GFP fluorescence is measured over time from the addition of 50 mM to 10 mM input Asp to the cultures.Error bars represent SD from three replicates.

Table 1 .
Kinetic constants of binding/dissociation reactions (for more details see Supplementary Material Section S1.2)