Materials and Methods

Transcription

1 55 Appendix A Materials and Methods All strains were constructed using standard B. subtilis protocols and molecular biology methods (Table A.1). The background of all strains used was B. subtilis PY79. For image segmentation, a constitutive promoter expressing fluorescent protein reporter RFP was chromosomally integrated into PY79 (LS1). Promoter fusions to fluorescent proteins were chromosomally integrated using Bacillus integration vectors pdl3, ECE174 (both from lab stocks), and per82 (kindly provided by Jonathan Dworkin). We also used the antibioticswitching plasmid ECE73 (cmr neor). The integration vector 174hs (from lab stock) containing the IPTG-inducible LacI system added to the ECE174 backbone was used to induce SpoF to different levels. For copy number perturbation, the plasmid php13 (from lab stock) was used. Based on these, other plasmids for promoter fusions and for circuit perturbations were constructed with Escherichia coli DH5α or DH5αZ1 by using standard methods of PCR, restriction enzyme digests, and ligations (Table A.2, A.3), i. The plasmid pss938 was constructed by ligating the EcoRI-BamHI fusion PCR fragment P A -YFP and pdl3 cut with EcoRI-BamHI. The fusion PCR fragment P A - YFP was made by fusing P A (primers oss8967, oss8949 and template PY79) and YFP (primers oss8944, oss8147 from a template plasmid containing YFP from lab stock) using primers oss8967, oss8147. ii. The plasmid pss925 was constructed by ligating the BamHI-EcoRI fusion PCR fragment P F -CFP and ECE174 cut with EcoRI-BamHI. The fusion PCR fragment P F - CFP was made by fusing P F (primers oss8973, oss8972 and template PY79) and CFP (primers oss8975, oss8146 from a template plasmid containing CFP from lab stock) using primers oss1436, oss8146.

2 56 iii. The plasmid pss711 was constructed by ligating the EcoRI-BamHI PCR fragment spof (primers oss897, oss8181 and template PY79) and php13 cut with EcoRI- BamHI. iv. The plasmid pss127 was constructed by ligating the EcoRI-BamHI fusion PCR fragment P hyp -CFP and ECE174 cut with EcoRI-BamHI. The fusion PCR fragment P hyp - CFP was made by fusing P hyp (primers oss181, oss185 and template MF2158) and CFP (primers oss184, oss182 from a template plasmid containing CFP from lab stock) using primers oss181, oss182. v. The plasmid pss916 was constructed by ligating the EcoRI-BamHI fusion PCR fragment P F -YFP and pdl3 cut with EcoRI-BamHI. The fusion PCR fragment P F - YFP was made by fusing P F (primers oss8973, oss1437 and template PY79) and YFP (primers oss1438, oss8147 from a template plasmid containing YFP from lab stock) using primers oss1436, oss8147. vi. The plasmid pss125 was constructed by ligating the EcoRI-BamHI fusion PCR fragment P F -YFP and per82 cut with EcoRI-BamHI. The fusion PCR fragment P F - YFP was made as above. vii. The plasmid pss131 was constructed by ligating the HindIII-NheI fusion PCR fragment SpoF-CFP and 174hs cut with HindIII-NheI. The fusion PCR fragment SpoF- CFP was amplified (primers oss198, oss195) from a vector already containing a SpoF-CFP fragment, which was made by fusing the SpoF gene (primers oss89, oss8913 and template PY79) and CFP (primers oss8912, oss182 from a template plasmid containing CFP from lab stock) using primers oss89, oss182. viii. The spof deletion PCR was made by fusing the gene conferring specr with flanking regions bearing homology upstream and downstream of the SpoF coding region. The primer pairs (oss8928, oss8924), (oss8922,oss8923) and (oss8925,oss8927) were used to PCR specr (pdl3 template), upstream homology region (PY79 template) and downstream homology region (PY79 template), respectively. These pieces were then fused using primers oss8922 and oss8927.

3 57 Number Strain Reference/Source/Construction MF2158 PY79 spoa::specr::cmr P laci -LacI P hyp - Lab stock A sad67 LS1 PY79 ppsb::ermr P trpe -RFP Lab stock LS4 PY79 Lab stock SS813 PY79 ppsb::ermr P trpe -RFP MF2158 LS1 spoa::specr::cmr P laci -LacI P hyp -A sad67 SS139 PY79 saca::cmr P hyp -CFP pss127 LS4 SS16 PY79 saca::cmr::neor P hyp -CFP ECE73 SS139 SS175 PY79 ppsb::ermr P trpe -RFP SS16 SS813 spoa::specr::cmr P laci -LacI P hyp -A sad67 saca::cmr::neor P hyp -CFP SS1125 PY79 ppsb::ermr P trpe -RFP pss916 SS175 spoa::specr::cmr P laci -LacI P hyp -A sad67 saca::cmr::neor P hyp -CFP amye::specr P A -YFP SS117 PY79 ppsb::ermr P trpe -RFP amye::specr pss938 LS1 P F -YFP SS119 PY79 ppsb::ermr P trpe -RFP amye::specr MF2158 SS117 P F -YFP spoa::specr::cmr P laci -LacI P hyp - A sad67 SS116 PY79 ppsb::ermr P trpe -RFP amye::specr SS16 SS119 P F -YFP spoa::specr::cmr P laci -LacI P hyp - A sad67 saca::cmr::neor P hyp -CFP SS946 PY79 ppsb::ermr P trpe -RFP saca::cmr P F - pss925 LS1 CFP SS17 PY79 ppsb::ermr P trpe -RFP saca::cmr P F - pss916 SS946 CFP amye::specr P A -YFP SS121 PY79 ppsb::ermr P trpe -RFP saca::cmr::neor ECE73 SS17 P F -CFP amye::specr P A -YFP SS123 PY79 ppsb::ermr P trpe -RFP saca::cmr::neor pss711 SS121 P F -CFP amye::specr P A -YFP php13-spof SS745 PY79 spof ::specr spof deletion PCR LS4 SS843 PY79 ppsb::ermr P trpe -RFP spof ::specr SS745 LS1 SS137 PY79 ppsb::ermr P trpe -RFP spof ::specr pss125 SS843 amye::neor P F -YFP SS164 PY79 ppsb::ermr P trpe -RFP spof ::specr pss131 SS137 amye::neor P F -YFP saca::cmr P laci -LacI P hyp -SpoF-CFP Table A.1: List of strains.

4 58 Number pss916 pss925 pss711 pss127 pss938 pss131 pss125 Plasmid pdl3 P A -YFP ECE174 P F -CFP php13-spof ECE174 P hyp -CFP pdl3 P F -YFP 174hs SpoF-CFP per82 P F -YFP Table A.2: List of plasmids.

5 59 Number Primer Sequence (5-3 ) oss8146 ATA GAATTC AAAAGGCTGAACCCTAAGGT oss8147 GAT GGATCC GCAATGATGAACCAGTAAGAGTAGC oss8967 ACT GAATTC CAGAAGCAGGAATCGATATTTATGG oss8949 CAACGCCGGTGAACAGTTCTTCACCTTTGCTCAT GTTTCTTCCTCCCCAAATGTAGTTAA oss8944 GTGAATCCTGTTAACTACATTTGGGGAGGAAGAAAC ATGAGCAAAGGTGAAGAACTGTTC oss8973 GGCCTGCTGGTAATCGCAGGCCTTTTTATT AATCCTCCTTTATAACGTACAATATCAGTA oss1436 CAC GGATCC GGCCTGCTGGTAATCGCAGGCCTTTTTATTAATCCTCCTTTATAACGTACA oss8972 AACTCCAGTGAAAAGTTCTTCTCCTTTACGCAT ATTCATCATTTTACACCCCAATATTAT oss8975 CGAAAATCATAATATTGGGGTGTAAAATGATGAAT ATGCGTAAAGGAGAAGAACTTTTCA oss897 AGC GGATCC AAGTGAATCCTCCTTTATAACGTACAATA oss8181 ATA GAATTC GTCAGTTAGACTTCAGGGGCAGAT oss181 TAT GAATTC GACTCTCTAGCTTGAGGCATCAAATAA oss185 AACTCCAGTGAAAAGTTCTTCTCCTTTACGCAT AGTAGTTCCTCCTTATGTGTCGACTAA oss184 AATTAAGCTTAGTCGACACATAAGGAGGAACTACT ATGCGTAAAGGAGAAGAACTTTTCA oss182 ATA GGATCC AAAAGGCTGAACCCTAAGGT oss1436 CAC GAATTC GGCCTGCTGGTAATCGCAGGCCTTTTTATT AATCCTCCTTTATAACGTACA oss1437 AACGCCGGTGAACAGTTCTTCACCTTTGCTCAT ATTCATCATTTTACACCCCAATATTAT oss1438 ACGAAAATCATAATATTGGGGTGTAAAATGATGAAT ATGAGCAAAGGTGAAGAACTGTTC oss198 ATA AAGCTT ACATAAGGAGGAACTACT ATGATGAATGAAAAAATTTTAATCGTTG oss195 ATA GCTAGC AAAAGGCTGAACCCTAAGGT oss89 ATA GAATTC AGTGAATCCTCCTTTATAACGTACAATAT oss8913 TGAAAAGTTCTTCTCCTTTACGCATGTTAGACTTCAGGGGCAGATATTTT oss8912 AAAATATCTGCCCCTGAAGTCTAAC ATGCGTAAAGGAGAAGAACTTTTCA oss8922 TATCAGGATGAAGTGGTGTACGAGC oss8927 TAACTGTTTCTTTGATCGCTTCACG oss8924 CGAAAATCATAATATTGGGGTGTAAACATATGCAAGGGTTTATTGTTTTCTAA oss8923 TTAGAAAACAATAAACCCTTGCATATGTTTACACCCCAATATTATGATTTTCG oss8925 ATGTATTCACGAACGAAAATCGATCAAAAAGAAGAAACAAATGAATCATG oss8928 CATGATTCATTTGTTTCTTCTTTTTGATCGATTTTCGTTCGTGAATACAT Table A.3: List of primers.

6 6 Appendix B Movie Protocols Time-lapse microcopy. Cells to be imaged were inoculated in CH media and grown overnight in a 3 C shaking water incubator. They were then diluted 1:4 into fresh CH media and grown for 2 3 hours so that the OD6 was in the range.5.7. Following this, they were resuspended 1:1 in Sterlini-Mandelstam (SM) media after two washes in SM..5ul of this was spotted on an appropriate pad (.5mm x.5mm), allowed to dry, and flipped onto a glass-bottom dish (Wilco). The dish was sealed with parafilm to reduce pad evaporation during imaging. The pads were made from SM media mixed with 1.5% Low Melting Point Agarose (Omni). Required amount of inducer IPTG was added to the pads. Progression to sporulation was imaged at 37 C using a microscope automated for timelapse fluorescence data acquisition. Images were acquired every 1 minutes on a Nikon Eclipse-Ti inverted microscope fitted with a perfect focus system, ASI motorized stage, Photometrix Coolsnap HQ2 camera, Sutter Lambda LS Xenon Arc lamp, and controlled from a computer using MetaMorph. Data analysis. The fluorescence values and lengths in single cells were extracted with customized segmentation and tracking algorithms coded in MATLAB and C. Promoter activity per unit length (P) was calculated from the measured mean fluorescence value (M) and cell lengths (L) using the formula P (t) = dm + 1 dl L M(t). The derivative was computed from a smoothened version of the mean fluorescence and cell length. For this, mean fluorescence was smoothened with a moving average filter and length

7 61 was fitted to a second-order polynomial. To plot the transcriptional bandpass characteristics, time traces up to the initiation of sporulation, as seen from the appearance of a fluorescent dark spot at the tip of the cell, were used. For the post-translational bandpass, the peak P F pulse amplitude was measured from T = minutes to T = 6 minutes, and the steady state SpoF-CFP level was measured at T = 5 minutes. For measurements of phase shift between P A and P F, promoter activity pulses with peak amplitude greater than a threshold (.1a.u.) were used.

8 62 a 4 Promoter Activity P A b 6 A sad67 Calibration Reporter corresponding to P A Surrogate Cells Difference (a.u.) y = x Surrogate Cells Difference (a.u.) 4 2 y = x Sister Cells 2 25 Difference (a.u.) 5 1 Sister Cells Difference (a.u.) c Promoter Activity P F d 25 A sad67 Calibration Reporter corresponding to P F Surrogate Cells Difference (a.u.) 1 5 y = x Surrogate Cells Difference (a.u.) y = x 5 1 Sister Cells Difference (a.u.) Sister Cells Difference (a.u.) Figure B.1: Variability between sister cells is typically smaller than between randomly chosen sister cell pairs. Difference between a cell and its sister and between the same cell and a randomly chosen surrogate sister cell is calculated for traces of (a) P A, (b) SpoA sad67 calibration reporter corresponding to the P A bandpass measurement, (c) P F, and (d) SpoA sad67 calibration reporter corresponding to the P F bandpass measurement. Each dot represents one cell, and has co-ordinates (x, y), where x is the difference between sister cells and y is the difference between surrogate cells. Difference metric for two given traces u(t) and v(t) is, d(u, v) = T t= u(t) v(t), where T is the minimum of the durations of the two traces. In each case, most points lie above the straight line y = x, indicating that the difference between sister cells is smaller than between surrogate sister cells.

9 63 a P B YFP b Mean Levels (a.u.) B 5 1 Time (minutes) Figure B.2: Mean fluorescence of a P B reporter changes less than twofold during sporulation initiation. (a) Schematic of the strain with a YFP fluorescent reporter fused to P B. The strain uses the plasmid ECE174 P B -YFP (from lab stocks) integrated into an LS1 background. Imaging was done as described in the methods section, but on an Olympus IX-81 inverted microscope fitted with an ASI motorized stage, Hamamatsu Orca-ER camera, Sutter Lambda LS Xenon Arc lamp, and controlled using a combined Visual Basic - ImagePro software. (b) Mean fluorescence levels are non-zero at the start and change less than twofold during sporulation initiation.

10 64 Appendix C Analysis of Models Mathematical modeling was done in MATLAB. Nullclines in the two-dimensional model were computed from the zero contour of the surface functions P A (A T, F T ) γa T and P F (A T, F T ) γf T. Ordinary differential equations were solved in MATLAB using integrator ode23s. The dependence of SpoA P on SpoF levels in the model without feedback and pulsing was computed by simulating the equations for a time longer than the timescale of the system. In the simulation of the two-dimensional model, A p was computed at each timestep by simulating the equation of the phosphorylated proteins using the current values of A T, B T, F T, K T, and k s (ON/OF F ). a KinA F B A RapB SpoE b KinA F B A RapB SpoE c A P /A T Total SpoF levels (nm) Figure C.1: Post-translational bandpass in the model is due to drain of SpoA phosphates via reverse phosphotransfer and SpoF phosphatases. Diagrams of the core phosphorelay with no reverse phosphotransfer (a, red) and no SpoF phosphatase (b, blue), for which the post-translational SpoF bandpass computation is performed from the simple phosphorelay model (no pulsing or transcriptional feedbacks). Both curves are plotted in (c), along with the curve from Fig. 2.4 (green).

11 65 Post-translational SpoF bandpass in a more complex model of the core phosphorelay. This supplementary information describes a model of core phosphorelay with a more complicated reaction scheme that that described in the main text. The posttranslational SpoF bandpass discussed in the main text is based on a simple model of the core phosphorelay. In principle, a more complicated reaction scheme involving reaction intermediates is also possible. Here, we verify the post-translational SpoF bandpass results in a model of the core phosphorelay that contains these reaction intermediates. Such reaction schemes have been previously used for phosphorelay-like signaling systems [54]. This model consists of 17 variables, of which eight correspond to the phosphoforms of the four phosphorelay proteins K, K p, F, F p, B, B p, A, A p. In addition there are two additional variables corresponding to the free levels of the phosphatases of SpoA and SpoF, denoted E and R, respectively. Finally, there are seven additional variables corresponding to the reaction intermediates involved in the phosphorylation, phosphotransfer, and dephosphorylation reactions [K p F ], [F p B], [B p A], [KT ], [KT F p ], [F p R], [A p E]. Here, the variable T denotes ATP, which is assumed to be maintained at a constant value by the cell. Consequently, [KT ] is the KinA-ATP intermediate that mediates autophosphorylation and [KT F p ] is the [KT ]-SpoF intermediate that mediates the dephosphorylation of SpoF. These variables are not independent as they are constrained by the total concentrations of these proteins. There are six such constraints, A T = A + A p + [B p A] + [A p E], B T = B + B p + [B p A] + [F p B], F T = F + F p + [F p B] + [K p F ] + [KT F p ] + [F p R], K T = K + K p + [KT F p ] + [K p F ] + [KT ], R T = [F p R] + R, E T = [A p E] + E. Here, A T, B T, F T, K T, R T, and E T denote the total concentrations of SpoA, SpoB, SpoF, KinA, RapB, and SpoE, respectively. Because of these six constraints, there are 11 free variables that described the system, and are chosen to be K, K p, F, F p, B, B p, A, A p, [K p F ], [F p R], [A p E]. The remaining variables can be expressed as a combination of

12 these variables and the total concentrations, 66 [B p A] = A T A A p [A p E], [F p B] = B T B B p [B p A], [KT F p ] = F T F F p [F p B] [K p F ] [F p R], [KT ] = K T K K p [KT F p ] [K p F ], R = R T [F p R], E = E T [A p E]. The equations corresponding to these reactions are dk dk p df df p db db p da da p d[k p F ] d[f p R] d[a p E] = T k 1 K + k 1p [KT ] + ν K [K p F ], = k s [KT ] k 2 K p F + k 2p [K p F ], = k 2 K p F + k 2p [K p F ] + ν P [KT F p ] + k 5 p[f p B] k 5 F B p + ν R [F p R], = ν K [K p F ] k 3 [KT ]F p + k 3p [KT F p ] k 4 F p B + k 4p [F p B] k 8 F p R + k 8 p[f p R], = k 4 F p B + k 4p [F p B] + k 7p [B p A] k 7 BA p, = k 5p [F p B] k 5 F B p k 6 B p A + k 6p [B p A], = k 6 B p A + k 6p [B p A] + ν E [A p E], = k 7p [B p A] k 7 BA p k 9 A p E + k 9p [A p E], = k 2 K p F (k 2p + ν K )[K p F ], = k 8 F p R (k 8p + ν R )[F p R], = k 9 A p E (k 9p + ν E )[A p E]. Here, the rate constants assigned to the different reactions are as follows, i. Autophosphorylation of K: k 1,k 1p are the association-dissociation rates for the reaction intermediate [KT ], and k s is the rate of formation of K p from this reaction intermediate. ii. Phosphotransfer from K p to F : k 2,k 2p are the association-dissociation rates for the reaction intermediate [K p F ], and ν K is the rate of formation of F p from this reaction

13 67 intermediate. iii. Dephosphorylation of F p by [KT ]: k 3,k 3p are the association-dissociation rates for the reaction intermediate [KT F p ], and ν P is the rate of formation of F from this reaction intermediate. iv. Phosphotransfer from F p to B: k 4,k 4p are the association-dissociation rates for the reaction intermediate [F p B] from the reactants F p and B. v. Reverse phosphotransfer from B p to F : k 5,k 5p are the association-dissociation rates for the reaction intermediate [F p B] from the reactants B p and F. vi. Phosphotransfer from B p to A: k 6,k 6p are the association-dissociation rates for the reaction intermediate [B p A] from the reactants B p and A. vii. Reverse phosphotransfer from A p to B : k 7,k 7p are the association-dissociation rates for the reaction intermediate [B p A] from the reactants A p and B. viii. Dephosphorylation of F p by R: k 8,k 8p are the association-dissociation rates for the reaction intermediate [F p R], and ν R is the rate of formation of F from this reaction intermediate. ix. Dephosphorylation of A p by E: k 9,k 9p are the association-dissociation rates for the reaction intermediate [A p E], and ν E is the rate of formation of A from this reaction intermediate. Parameter values are based on [54], k 1 = k 2 = k 3 = 1/(nM hr) = k 4 = k 5 = k 6 = k 7 = k 8 = k 9,k 1p = k 2p = k 3p = ν K = ν P = 1 3 /hr = k 4p = k 5p = k 6p = k 7p = k 8p = k 9p = ν R = ν E,k s = 1 2 /hr,a T = B T = F T = K T = 1nM,E T = R T = 1nM The steady state for different total concentrations of SpoF is computed by numerically integrating these equations using ode23s (Supp. Fig. C.2e). For the computation with no reverse phosphotransfer, the corresponding rate constants are set to zero, k 5 = k 7 = k 4p = k 6p =. Similarly, for the computation with no SpoF phosphatase, the corresponding parameters are set to zero, R T = k 8 = k 8p = ν R = k 3 = k 3 p = ν P =. In addition, there is an additional constraint as the reaction intermediate [KT F p ] does not exist. Enforcing this constraint ([KT F p ] = ) reduces the number of differential equations by one as the value of another reaction intermediate [K p F ] is already determined, [K p F ] = F T F F p [F p B] [F p R].

14 68 a b c KinA KinA KinA F B RapB F B RapB F B RapB A SpoE A SpoE A SpoE d 2 Peak P F (nm/hr) e A P /A T Total SpoF levels (nm) Total SpoF levels (nm) Figure C.2: Post-translational SpoF bandpass in the full model with transcriptional feedback and pulsing or in the more complicated model of the core phosphorelay is due to the drain of SpoA phosphates via reverse phosphotransfer and SpoF phosphatase. Diagrams of the phosphorelay circuit (a) and versions with no reverse phosphotransfer (b, red) and no SpoF phosphatase (c, blue), for which the post-translational SpoF bandpass computation is performed from the full model described in the main text (including pulsing and transcriptional feedbacks, Fig. 2.7). Peak P F pulse amplitudes are plotted for different SpoF induction levels (at steady state) for three cases: base parameters (d, green), no reverse phosphotransfer (d, red), and no SpoF phosphatase (d, blue). The absence of SpoF phosphatases shifts the repression threshold to significantly higher SpoF values, representing drain of SpoF phosphates by dilution due to cell growth. (e) Green curve is the post-translational SpoF bandpass computed from a more complicated version of the core phosphorelay, described in the supplementary text. The computation is repeated in the absence of reverse phosphotransfer (red) and SpoF phosphatases (blue).

15 69 Nullclines in ON stage Total SpoA levels (nm) Total SpoF levels (nm) Figure C.3: Qualitative dynamic picture under the P kina = P A assumption is similar to the alternate P kina = P F assumption. Nullclines in the ON stage in the case when P kina (A p ) = P F (A p ) are similar to the case discussed in 2.3 with the assumption that P kina (A p ) = P A (A p ), suggesting that this assumption is justified. These nullclines are plotted for same parameters as used in the main text with k s (ON) = 11/hr.

16 7 a Both P A and P F bandpass, and reverse phosphotransfer (wild-type) Nullclines in ON stage b Promoter Activity (nm/hr) Promoter Activity (nm/hr) P A P F SpoA~P (nm) P A P F SpoA~P (nm) Total SpoA levels (nm) Total SpoA levels (nm) Total SpoF levels (nm) Both P A and P F bandpass, and NO reverse phosphotransfer Nullclines in ON stage Total SpoF levels (nm) c P A bandpass and P F lowpass, and reverse phosphotransfer Nullclines in ON stage Promoter Activity (nm/hr) P A P F SpoA~P (nm) Total SpoA levels (nm) Total SpoF levels (nm)

17 71 d P A bandpass and P F highpass, and reverse phosphotransfer Nullclines in ON stage Promoter Activity (nm/hr) P A P F SpoA~P (nm) Total SpoA levels (nm) Total SpoF levels (nm) e Promoter Activity (nm/hr) P F bandpass and P A lowpass, and reverse phosphotransfer P A P F 2 4 SpoA~P (nm) Total SpoA levels (nm) Nullclines in ON stage Total SpoF levels (nm) f Promoter Activity (nm/hr) P F bandpass and P A highpass, and reverse phosphotransfer P A P F 2 4 SpoA~P (nm) Total SpoA levels (nm) Nullclines in ON stage Total SpoF levels (nm)

18 72 Figure C.4: Effect of bandpass characteristics on the alternate state. The effect of bandpass characteristics on the alternate state are investigated by comparing ON phase planes in the following cases, (a) Both P A and P F bandpass, and reverse phosphotransfer (wild-type case from Fig. 2.8, considered in 2.3) (b) Both P A and P F bandpass, and NO reverse phosphotransfer: Transcriptional bandpasses remain the same as case (a), but there is no post-translational bandpass. Nullclines orient differently from case (a), so that the alternate state doesn t exist and its effect on dynamics is minimal. (c) P A bandpass, P F lowpass (only repression), and reverse phosphotransfer: Nullclines are slightly perturbed from case (a), with the appearance of a stable steady state, an unstable steady state (white circle), and the associated separatrix. The separatrix is computed by integrating equations backward in time starting from an initial condition near the unstable steady state. The additional stable steady state is similar in location to the alternate state. (d) P A bandpass, P F highpass (only activation), and reverse phosphotransfer: Nullclines are slightly perturbed from case (a), with the stable steady state situated like the alternate state. (e) P F bandpass, P A lowpass (only repression), and reverse phosphotransfer: Nullclines are slightly perturbed from case (a), but without significant change in the steady state location. (f) P F bandpass, P A highpass (only activation) and reverse phosphotransfer: Nullclines are slightly perturbed from case (a), but without significant change in the steady state location. In this case, the stable steady state is situated above the upper Y-axes limit.