Comparative analysis of prototype two-component systems with either bifunctional or monofunctional sensors: differences in molecular structure and physiological function

Authors

  • Rui Alves,

    1. Department of Microbiology and Immunology, University of Michigan Medical School, 5641 Medical Science Building II Ann Arbor, MI 48109–0620, USA.
    2. Grupo de Bioquimica e Biologia Teoricas, Instituto Rocha Cabral, Calçada Bento da Rocha Cabral 14, 1250 Lisboa, Portugal.
    3. Programa Gulbenkian de Doutoramentos em Biologia e Medicina, Departamento de Ensino, Instituto Gulbenkian de Ciencia, Rua da Quinta Grande 6, 1800 Oeiras, Portugal.
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  • Michael A. Savageau

    Corresponding author
    1. Department of Microbiology and Immunology, University of Michigan Medical School, 5641 Medical Science Building II Ann Arbor, MI 48109–0620, USA.
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    • Present address: Department of Biomedical Engineering, One Shields Avenue, University of California, Davis, CA 95616, USA.


Summary

Signal transduction by a traditional two-component system involves a sensor protein that recognizes a physiological signal, autophosphorylates and transfers its phosphate, and a response regulator protein that receives the phosphate, alters its affinity toward specific target proteins or DNA sequences and causes change in metabolic activity or gene expression. In some cases the sensor protein, when unphosphorylated, has a positive effect upon the rate of dephosphorylation of the regulator protein (bifunctional sensor), whereas in other cases it has no such effect (monofunctional sensor). In this work we identify structural and functional differences between these two designs. In the first part of the paper we use sequence data for two-component systems from several organisms and homology modelling techniques to determine structural features for response regulators and for sensors. Our results indicate that each type of reference sensor (bifunctional and monofunctional) has a distinctive structural feature, which we use to make predictions regarding the functionality of other sensors. In the second part of the paper we use mathematical models to analyse and compare the physiological function of systems that differ in the type of sensor and are otherwise equivalent. Our results show that a bifunctional sensor is better than a monofunctional sensor both at amplifying changes in the phosphorylation level of the regulator caused by signals from the sensor and at attenuating changes caused by signals from small phosphodonors. Cross-talk to or from other two-component systems is better suppressed if the transmitting sensor is monofunctional, which is the more appropriate design when such cross-talk represents pathological noise. Cross-talk to or from other two-component systems is better amplified if the transmitting sensor is bifunctional, which is the more appropriate design when such cross-talk represents a physiological signal. These results provide a functional rationale for the selection of each design that is consistent with available experimental evidence for several two-component systems.

Introduction

Two-component systems (TCS) are signal transduction modules that exist mainly in bacteria but have also been found in fungi and plants (recently reviewed in Hellingwerf et al., 1998; see also Parkinson, 1993; Hoch and Silhavy, 1995; Perego and Hoch, 1996; Schaller, 1997; Posas et al., 1998). Two-component systems differ from eukaryotic phosphorylation cascades like the MAP kinase cascade. In the former, ATP is only consumed in the first step of the cascade to provide for autophosphorylation of a histidine residue in the sensor, whereas in the latter, ATP is consumed at each step in the cascade to provide for phosphorylation of the protein at that step. More than 180 different TCS have been identified in bacteria (Kadner, 1996; see also http:www.genome.ad.jpkeggregulation.html) and over one thousand putative sensors and response regulators in over one hundred organisms have had their gene sequence determined (e.g. http:www.expasy.chsrs5http:www-fp.mcs.anl.govgaasterlandgenomes.html). Two-component systems are often involved in complex circuits that exhibit enormous variation in design. The physiological importance of TCS provides strong motivation to search for underlying design principles that would allow one to rationalize the variations in design. The search is in its infancy and there are undoubtedly many design principles that remain to be discovered.

Here we focus on a specific example of a simple design principle involving TCS composed of a sensor protein and a response regulator protein. These traditional TCS can be considered the prototype for a motif that has been elaborated on and modified in more complex cases that will not be considered here. [For example, we will not address cases in which there are specific aspartate phosphatases (e.g. Perego and Hoch, 1996; Blat et al., 1998) that are independent of the sensor protein.] In some prototype cases the sensor protein, when unphosphorylated, has a positive effect upon the rate of dephosphorylation of the regulator protein (bifunctional sensor) whereas in other cases it has no such effect (monofunctional sensor). For example, the TCS involved in the regulation of chemotaxis via the CheA/CheY cascade in Escherichia coli has a monofunctional sensor (reviewed in Eisenbach, 1996), whereas the TCS involved in the regulation of osmotic pressure via the EnvZ/OmpR cascade has a bifunctional sensor (reviewed in Pratt et al., 1996). These bifunctional and monofunctional sensor proteins exhibit specific differences in molecular structure that, by homology modelling, are found to reoccur in many other TCS.

In the first part of this paper we use sequence data for other TCS from several organisms, together with homology modelling techniques, to predict structural features and in turn functionality of their sensor proteins. These predictions are confirmed in a few cases for which there is independent biochemical or genetic evidence for functionality. Our results suggest an association between a sensor's mode of action (monofunctional or bifunctional) and the structure of its ATP-binding domain. In the second part of this paper we analyse and compare the functional effectiveness of TCS with either bifunctional or monofunctional sensors by using a technique known as Mathematically Controlled Comparison. This technique, which is analogous to a well-controlled experiment, determines the differences in the physiology between alternative designs that are otherwise equivalent. This approach reveals qualitative differences (independent of specific values for the parameters of the systems) as well as quantitative differences (statistical tendencies when a large sample of numerical values for the parameters is examined). We also obtain results that discriminate between different types of cross-talk to and from TCS. Our results suggest physiological situations that favour each of the two sensor designs. Hence, these results provide a functional rationale for the selection of each design. Experimental evidence for several TCS is discussed in the light of this rationale.

Results

Abstractions of the alternative designs for a prototype TCS are represented in Fig. 1. The unphosphorylated sensor protein (S,X3) becomes phosphorylated (S*,X1) in response to changes in a physiological signal (Q1,X5). Autophosphorylation of the sensor is achieved with consumption of ATP. Once phosphorylated, the sensor loses its phosphate group, either by auto-dephosphorylation (although this occurs on a time scale that is not of interest here) or by transfer of the phosphate to an aspartate residue in the regulator protein (R*,X2) (e.g. Weiss and Magasanik, 1988; Sanders et al., 1989; Hsing et al., 1998; Jung and Altendorf, 1998; Jiang and Ninfa, 1999). This covalent modification causes changes in the response regulator, thereby altering its affinity toward specific target proteins or DNA sequences (T), and this in turn leads to changes in metabolic activity or gene expression. The dephosphorylation of response regulators (e.g. Perego and Hoch, 1996) is, in some cases, enhanced by the unphosphorylated form of the sensor protein (bifunctional sensor, Fig. 1A). In other cases, the sensor protein lacks this ability (monofunctional sensor, Fig. 1B). In either case, the regulator protein also can be phosphorylated by other mechanisms (Q2,X6), e.g. by small phosphodonors like acetyl-phosphate, but not by ATP (e.g. Lukat et al., 1992; McCleary, 1996; Mayover et al., 1999). Cross-talk from the sensor proteins of other two-component systems has been demonstrated with purified proteins in vitro (Ninfa et al., 1988) and with overexpressed proteins in vivo (Grob et al., 1994), but the physiological significance of these results has been questioned (Wanner, 1992).

Figure 1.

Schematic representation of two-component system modules with either a bifunctional or a monofunctional sensor.

A. Module with a bifunctional sensor in which S* phosphorylates R and S increases the rate of dephosphorylation of R*.

B. Module with a monofunctional sensor in which S* phosphorylates R and S and has no direct effect on the rate of dephosphorylation of R*.

C. Example of a TCS module with a bifunctional sensor. EnvZ is the sensor protein of the module. It is a membrane protein that responds to changes in the osmotic pressure by changing its phosphorylation state. In its phosphorylated state (EnvZ∼P) it transfers phosphate to the response regulator OmpR. Phosphorylated OmpR represses expression of proteins that form large pores and induces expression of proteins that form small pores. Unphosphorylated EnvZ increases the rate of OmpR∼P dephosphorylation, repressing expression of small membrane pores and derepressing expression of larger membrane pores. OmpR can also be phosphorylated by other phospho-donors, for example acetyl-P.

D. Example of a TCS module with a monofunctional sensor. CheA is a membrane protein that is attached to a complex sensorial system. The system senses gradients of nutrients in the medium. When a nutritional gradient is sensed, CheA autophosphorylates and transfers its phosphate to the response regulator CheY. CheY can also be phosphorylated by other phospho-donors, for example acetyl-P. Phosphorylated CheY changes the tumbling movement of the cells into a directed motion towards the higher concentrations of nutrients by affecting flagellar proteins. Unphosphorylated CheA has no effect on CheY.

Symbols: S (X3) – unphosphorylated sensor protein; S* (X1) – phosphorylated sensor protein; R (X4) – unphosphorylated response regulator protein; R* (X2) – phosphorylated regulator protein; Q1 (X5) – signal that modulates the fraction of phosphorylated sensor; Q2 (X6) – signal that modulates the fraction of phosphorylated regulator independent of signalling by the cognate sensor; T– target for the response regulator.

Structure of regulators

Structures of some response regulators have been determined and can be used as templates to predict the 3D structure of other regulators by homology modelling. The known structures of response regulators have receiver domains composed of alternating α-helices and β-sheets. The 3D shape resembles a barrel composed of five α / β sequential segments, with parallel β-sheets (blue) in the middle, surrounded by the α-helices (red) shown in Fig. 2. In the protein classification database SCOP (Murzin et al., 1995) response regulators are classified within the flavodoxin-like folds. The aspartate residue that is phosphorylated is at the tip of one of the internal β-sheets, almost in the loop that connects it to the next α-helix (Fig. 2A). These structural features appear to be at least partially conserved in the receiver domains of all the response regulators we have been able to model (Fig. 2B, Table 1). Comparison of the experimentally determined structures of response regulators did not reveal structural features that might indicate any distinction between response regulators of TCS with either bifunctional sensors or monofunctional sensors. Similar results also were obtained when we examined the modelled structures for response regulators. Although the phosphorylation domain of all response regulators whose structure has been determined or modelled so far have similar structures this does not allow us to conclude that the hundreds to thousands more that are as of yet unresolved all share the same structure. There are known cases where protein domains with very similar sequence have very different folds and cases where protein domains with very different sequence have very similar folds. Table 1 may be used as a guide to indicate response regulators that are likely to have similar structures. In the effort to explore the fold space of response regulators, it will probably be more informative to concentrate on resolving structures of response regulators that are not present in Table 1.

Figure 2.

Three-dimensional structure for the receiver domain of response regulators.

A. Structure determined by X-ray crystallography for the Spo0F regulator from Bacillus subtilis (PDB reference identifier 1NAT).

B. Structure predicted by homology modelling for the ArcA regulator from Haemophilus influenza.

Table 1. . Putative response regulator proteins that yielded a predicted structure by homology modelling techniques.
OrganismSWISSPROT entry no.Name
Aeromas jandaei P97015P97015
P97018P97018
Agrobacterium
 tumefaciens
Q07783CHVI
P07545VIRG
Q44430NTRR
Alcaligenes eutrophus Q44006CZCR
P29267HOXA
P94153P94153
Anabaena sp. P39048PatA
O87395OrrA
Aquifex aeolicus O66551NtrC1
AQ1792NtrC2
O66596NtrC3
O66657PhoB
Arabidopsis thaliana Q9ZWS8Q9ZWS8
Q9ZWS7Q9ZWS7
Q9ZWK0Q9ZWK0
Q9ZWJ9Q9ZWJ9
Q9ZWS9Q9ZWS9
O80365O80365
O80366O80366
Q9ZQJ8Q9ZQJ8
O82798O82798
Archeoglobus fulgidus O29221CheB
O28035O28035
O28381O28381
O28799O28799
O28887O28887
O29012O29012
O29199O29199
O29220O29220
O29800O29800
O30252O30252
Azorhizobium
 caulonidans
P26487FixJ
Q04849NtrX
Azospirillum brasiliense P45671NtrC
Azobacter vinelandii O87478O87478
Bacillus brevis Q44929GTCR
P54662DEGU
P52929SP0A
Bacillus megaterium P39486YGD1
Bacillus subtillis P45709CCDB
 Q05522CHEB
 P37583CHEY
 P14204CMA1
 O34534CITT
 P13800DEGU
 Q02115LYTR
 O34804PHOP
 P35163RESD
 P06534SP0A
 P06535SP0B
 P06628SP0F
 Q9R9J8YBDG
 O31432YBDJ
 P40759YCBB
 P42244YCBL
 P70955YCCH
 P94413YCLJ
 O31517YESN
 P94439YFIK
 O07528YHCZ
 O34903YKOG
 O34723YOCG
 P54443YRKP
 O34951YTSA
 O05251YUFM
 O06978YVCP
 Q9L4F3YVFU
 O32192YVQA
 O32197YVQC
 P94504YVRH
 P42421YXDJ
 P55184YXJL
 P37478YYCF
Bacillus thurigiensis P52935Spo0A
P52942Spo0F
Bacteroides
 thetaiotamicron
Q45788RTEA
Q00936RTEB
Bordetella pertussis P16574BvgA
 O88117RisA
Borrelia burgdorferi O51380O51380
O51501O51501
O51517O51517
O51615O51615
O51704O51704
Bradyrhizobium
 japonicum
P72489RegR
P10576NTRc
Q45272NwsB
O52845PhoB
P31908HoxA
P23221FixJ
Brucella abortus Q9ZHS1CtrA
O67996BvrR
Brucella melitensis Q9XDD4Q9XDD4
Brucella suis O31409FEUP
Campylobacter jejuni P71129CheY
O68795O68795
Campobacterium divergens Q9ZEG3Q9ZEG3
Caulobacter crescentus Q45976DviK
Q46020PleD
Q45994SokA
Chlamydia trachomatis O84474O84474
Clostridium difficile P52938Spo0A
Clostridium innocuum P52939Spo0A
Clostridium pasteurianum P52940Spo0A
Clostridium perfringens Q9XDT7VirI
Cyanidium caldarium P48259P48259
P48359P48359
P28257YC27
Cyanophora paradoxa P48259Yc27
Desulfovibrio vulgaris P33394RRF1
Enterococcus faecalis Q06239VanRA
Q47744VanRB
Erwinia amylovora Q9X3S9HrpY
Erwinia carotovora O86197EXPM
O32556O32556a
Q9ZIN9Q9ZIN9
O08235RPFA
Escherichia coli P03026ARCA
 Q06065ATOC
 B2381B2381
 P30846BAER
 P30843BASR
 P78070CHEB
 P06143CHEY
 P76777CPXR
 P08368CREB
 P76794DCUR
 Q54149DPIA
 P30854EVGA
 P21502FIMZ
 P37055HNR
 P14375HYDG
 P77512YFHA
 P76822KDPE
 P10957NARL
 P31802NARP
 P06713NTRC
 P03025OMPR
 P08402PHOB
 P23836PHOP
 P14374RCSB
 P35163RESD
 P52108RSTA
 P56644SGAR
 P77344TORR
 P10940UHPA
 P07027UVRY
 P97172YEDW
 P33356YEHT
 P52076YGIX
 P77380YLCA
Eubacterium
 acidaminophilum
O86183NtrC
Fermiella diplosiphon O32610O32610
Q01473RCAC
Guillardia theta O78425YC27
Haemophilus influenza P44918ARCA
P76777CPXR
P44845NARP
P45189PHOB
P45337YGIX
Helicobacter pylori O25337CHEV
P71403CHEY
HP0703HP0703
HP1365HP1365
O25408O25408
O25684O25684
O25918O25918
O24973OMPR
Hyphomicrobium
 methylovorum
P56644SgaR
Klebsiela pneumoniae P03029NTRC
P45605PHOB
Lactobacillus sake Q9ZI91Q9ZI91
Q9ZI93Q9ZI93
Q9ZI95Q9ZI95
Q9ZI97Q9ZI97
Q9ZIA0Q9ZIA0
Lactococcus lactis Q9ZI77Q9ZI77
Listeria monocytogenes Q48767Q48767
Methylobacterium
 extorquens
O30796MXAB
Q49121Q49121
Mycobacterium leprae Q50136PHOP
Q4Z5G8Q4Z5G8
Q50136YV17
Mycobacterium
 tuberculosis
Q10531COPR
Q50447MTRA
 MTV43MTV43
 MTV44MTV44
 MTV25MTV25
 O50447O50447
 O50806O50806
 O50447O50447
 O50806O50806
 O53856O53856
 O53894O53894
 O69730O69730
 P71814PHOP
 Q50825Q50825
 Q11156RGX3
 O07776TCRA
 Q50806TRCR
Neisseria meningitides NMB0114NMB0114
 NMB0595NMB0595
Nostoc punctiforme Q51309Q51309
Plectonema boryuanum P51586YSO1
Porphyra purpurea P51358YC27
P51343YC29
Porphyrium aerugineum P28835YC27
Proteus vulgaris P28787NTRC
Providencia stuartii O85058O85058
O85059O85059
Pseudoalteromonas sp. O68498O68498
Pseudomonas aeruginosa P29369AGMR
 P23747ALGB
 P26275ALGR
 Q51455CHEY
 Q51454FLER
 Q51373GACA
 P72150GLTR
 O33493O33493
 O54039O54039
 Q04803PFER
 P23620PHOB
 P46384PILG
 P43501PILH
 Q00934PILR
Pseudomonas aureofaciens Q9XD07GACA
Pseudomonas pseudomallei O31395IRLR
Pseudomonas putida Q52201PPRA
Pseudomonas solanacearum O07846O07846
O07847O07847
Q52582VIRR
Q45417VSRD
Pseudomonas syringae Q02540COPR
Q52406CORP
Q52376GACA
Pseudomonas tolaasii O34175PHENa
Rhizobium leguminosarum Q9X574CELR
P10046DCTD
Q52852POPP
Rizhobium meliloti P50350CHVI
P13632DCTD
O54063EXSF
P10577NTRC
Q52990PHOB
Q52913Q52913
P55701Y4XI
Rhodobacter sphaeroides O33553CHERa
O30741DMSO
O32479DMSR
P95652P95652
Q53228REGA
Rhodobacter sulfidophilus O82868RegA
Rhodococcus erythropolis Q9ZNJ4Q9ZNJ4
Rickettsia prowazekii O05971OmpR
Saccaromyces cereviseae P38889SNK7
Salmonella dublin O85302CopR
Salmonella typhimurium P36556BASR
P04042CHEB
P06657CHEY
P26319FIMZ
P25852HYDG
Q56065MVIA
P41789NTRC
P41405OMPR
P06184PGTA
P14146PHOP
Q56127RCSB
Q56128RCSCa
P96058SIRA
P22104TCTD
P27667UHPA
Shigella dysenteriae P45606PhoB
Shigella flexneri P45607PhoB
Staphylococcus aureus P96456LYTS
Q9XCM7YYCF
Streptococcus pneumoniae Q54954CIAR
O54138KDPE
Q9X4S8PNPR
Streptomyces coelicolor O50496O50496
Streptomyces hygroscopicus Q54292Q54292
Streptomyces lividans P72471P72471
Q9X6J0Q9X6J0
Streptomyces peucetius Q54821Q54821
Streptomyces violaceoruber Q9ZA47Q9ZA47
Synechococcus sp. O68523NBLR
P72561P72561
Q56180PHOB
P39663SPHR
Q56003SRRB
Synechocystis sp. P72781P72781
P72790P72790
P72936P72936
P72948P72948
P73006P73006
P73036P73036
P73078P73078
P73104P73104
P73150P73150
P73175P73175
P73176P73176
P73243P73243
P73404P73404
P73927P73927
P73686P73686
P73864P73864
P73928P73928
P74014P74014
P74138P74138
P74139P74139
P74288P74288
P74294P74294
P74298P74298
P74314P74314
P74541P74541
P74626P74626
Q55733Q55733
Q55808Q55808
Q55890Q55890
Q55919Q55919
Q55933Q55933
Q55942Q55942
slr0474RCP1
S11797S11797
Thermotoga maritima Q9WY30Q9WY30
Q9WYN0Q9WYN0
Q9WYT9Q9WYT9
Q9WZY6Q9WZY6
Q9X181Q9X181
Q56312CHEY
P74922P74922
Q9WXY0Q9WXY0
Q9WXZ0Q9WXZ0
Thiocystis violacea P45365P45365
Vibrio cholerae O30664O30664
O68318O68318
O85089O85089
Q9X2S6Q9X2S6
Q9ZH73Q9ZH73
Vibrio harveyi P54299LUXO
Wollinela succinogenes P74968P74968
Xanthomas campestris Q04527Q04527
Yersinia pseudotuberculosis P74991PHOP

Structure of sensors

Partial structures for both types of sensors also have been determined and can be used as templates to predict the 3D structure of other sensors by homology modelling. Structures of the soluble domains of the CheA sensor (monofunctional, Fig. 3A) as well as the EnvZ sensor (bifunctional, Fig. 4A) have been determined. The CheA transmitter domain is of the type HPt, reminiscent of the PTS phosphotransferase systems. The EnvZ sensor has an HK transmitter domain. In the protein classification database SCOP (Murzin et al., 1995) sensor histidine kinases are classified within the ROP-like folds (‘four helices; dimers of identical alpha-hairpin subunits; bundle, closed, left-handed twist’). Even though the domains of the EnvZ sensor responsible for its bifunctionality are not defined, an HK transmitter domain appears to be essential for the increase in the rate of dephosphorylation of the response regulator (Zhu et al., 2000). In the same work, Zhu et al. (2000) also showed that this increase is much less for the bifunctional sensors if their catalytic ATP-binding domain is truncated from their transmitter domain.

Figure 3.

Three-dimensional structure for the catalytic domain of monofunctional sensors.

A. Structure determined by X-ray crystallography for the CheA sensor from Thermotoga maritima (PDB reference identifier 1B3Q).

B. Structure predicted by homology modelling for the CheA sensor from Escherichia coli.

Figure 4.

Three-dimensional structure for the catalytic domain of bifunctional sensors.

A. Structure determined by X-ray crystallography for the EnvZ sensor from Escherichia coli (PDB reference identifier 1BXD).

B. Structure predicted by homology modelling for the putative O66656 sensor from Aquifex aeolicus.

We have made predictions regarding the monofunctional or bifunctional character of the sensor for many TCS (Table 2). In most cases, TCS have been identified by sequence similarity, and definitive biochemical evidence regarding the monofunctional or bifunctional character of their sensor is lacking. Our predicted molecular models for these sensors reveal a consistent difference in the folding of their catalytic domain: the fold resembles either that of the monofunctional template or that of the bifunctional template.

Table 2. . Putative sensor proteins that yielded a predicted structure by homology modelling techniques.
OrganismSWISSPROT entry no.NamePrediction
  • a

    . These are partially modelled structures that closely resemble the ATP-binding domain of sensor EnvZ. The resulting model has the ATP lid and a portion, but not all, of the remaining ATP binding domain of the sensor EnvZ.

  • b

    .These are partially modelled structures that only slightly resemble the ATP-binding domain of sensor EnvZ. The resulting model has an ATP lid similar to that of EnvZ, but lacks the remainder of the EnvZ-like ATP binding domain.

  • c . These are partially modelled structures obtained from a composite TCS. The resulting model has the ATP lid and a portion, but not all, of the remaining ATP binding domain of the sensor EnvZ, as in a.

  • d . These are partially modelled structures obtained from a composite TCS. The resulting model has an ATP lid similar to that of EnvZ, but lacks the remainder of the EnvZ-like ATP binding domain, as in b.

  • e

    . This is a case in which the sensor protein, NRI, is monofunctional unless it is bound with a second protein, PII, in which case it becomes bifunctional.

Agrobacterium tumefaciens Q07737ChvGBi
Aquifex aeolicus O66656O66656Bi
O66597O66597Bi
Arabidopsis thaliana P49333ETR1Lidd
O22267O22267Lidd
O48834O48834Lidd
O82798O82798Lidd
Archeoglobus fulgidus O28171O28171Lidb
O28653O28653Lidb
O28800O28800Lidb
O28789O28789Lidb
O29222O29222Mono
O30214O30214Lidb
Azorhizobium caulonidans P26489FixLLidb
Q04850NtrYBia
Azospirillum brasiliense P45670NtrBMonoe
Bacillus subtillis P29072CheAMono
P37599CheVLidd
Q03069DegMLidb
P16497KinALidb
Q08430KinBBia
P39764KinCLidb
P23545PhoRBi
P35164ResELidb
Q9Z2K7SpaFLidb
P33113SpaKBi
P94414YclKBi
BG11896YxjMBi
O34638YkoHBi
O32193IvqBBi
O34989YvrGLidb
P42422YxdKBi
Q45614YycGBi
Bordetella bronchiseptica P26762BvgSBic
Bordetella parapertusis P40330BvgSBic
Bordetella pertussis P16575BvgSBic
Borrelia burgdorferi Q44737CheAMono
O51381O51381Lidd
Q44875Q44875Mono
Bradyrhizobium japonicum P23222FixLLidb
P10578NtrBMonoe
Calothrix viguieri O52937O52937Bi
Candida albicans O74171O74171Lidd
O42696Canik1Lidd
O42695Casln1Lidd
O59892HK1Lidd
Caulobacter crescentus Q9X688CckALidb
Q03228DivJBi
P37894PleCLidb
Colletotrichum gloeosporioides P79086CHK1Lidb
Deinococcus DR1175DR1175Bic
radioduransDR1606DR1606Lidb
 DR0892DR0892Bia
 DR2416DR2416Lidb
 DR205DR205Lidd
 DR744DR744Bi
Dictyostelium discoideum O15763DhkBLidd
Q23901HkALidd
 O15783O15783Lidd
 O15784HkCLidd
Enterococcus faecalis Q47745VanSBBi
Escherichia coli P22763ARCBMono
Q06067ATOSBi
P30847BAESBia
P26607BARALidd
P30844BASSBia
P08336CPXABi
P08401CRECBia
P77644EVGSLidd
P14377HYDHBi
P21865KDPDLidb
P06712NTRBMonoe
P23837PHOQBia
P76457RCSCLidd
P35164RESELidb
P18392RSTBBia
P75887TORSBia
P77485YBCZBia
P76339YEDVBia
P76587YFHKBia
Fusarium solani O94094FikSLidd
Haemophilus P44578ARCBBia
influenzaP71380PHORMono
 P45336YGIYBi
Halobacterium salinarium Q48297CheA likeMono
Helicobacter pylori Q9ZKE9Q9ZKE9Mono
O25153O25153Mono
Herbaspirillum seropedicae O86056O86056Lidb
Klebsiela pneumoniae P06218NTRBMonoe
P45608PHORLidb
Lactobacillus sake Q9ZI92Q9ZI92Bi
Q9ZI94Q9ZI94Lidb
Lactococcus lactis Q48675NisKBi
O07382O07382Monoe
O07383O07383Bia
O07384O07384Bia
O07386O07386Bia
Listeria monocytogenes Q48768CheAMono
Mastigocladus laminosus O07114O07114Mono
Methanobacterium thermoautotrophicum MT902MT902Lidd
MT901MT901Lidd
O26540O26540Lidd
O26544O26544Lidb
O26545O26545Lidd
O26546O26546Lidd
O26547O26547Lidd
O26557O26557Lidd
O26648O26648Lidd
O26879O26879Lidd
O26913O26913Lidd
O26987O26987Lidd
O26988O26988Lidd
O27644O27644Lidd
O27792O27792Lidd
Methanococcus jannaschii MJECL24Spo0JLidb
Methylobacterium extorquens Q49120Q49120Lidb
Mycobacterium leprae P54883YV16Bi
Mycobacterium tuberculosis P96372KDPDBi
MT1302MT1302Lidb
MT3764MT3764Lidb
MT490MT490Lidb
MT902MT902Bi
MT982MT982Lidb
MT600MT600Bi
O53895O53895Lidb
P71815P71815Bi
mtu:Rv0758PhoRBi
Q10560Q10560Bi
Q11155Sex3Lidb
Y902Y902Bi
Neisseria meningitides Q9K1K2NtrXLidd
NMB114NMB114Bi
NMB1792NMB1792Bi
NMB594NMB594Bi
Neurospora crassa O93851NIK1Lidd
Q01318OS1PLidd
Nostoc punctiforme Q9Z693Q9Z693Lidb
Proteus mirabilis O85662RcsBLidd
Pseudomonas Q51453FLESLidb
aeruginosaO34206KINBBi
 Q04804PFESBia
 P33639PILSBia
 O68597PIRSBi
 Q51453PSEABia
 Q51420Q51420Lidb
 O34206KINBBia
 O68596O68596Lidd
Pseudomonas pseudomallei O31396IrlSBi
Pseudomonas putida O07831BziPLidb
Pseudomonas solanacearum O07845O07845Bia
Pseudomonas syringae O06437CVGSLidd
P48027GACSLidb
Q9WWH5Q9WWH5Lidd
Q9WWH9Q9WWH9Lidd
Pseudomonas tolaasii Q9ZNQ1RtpALidb
Pyrococcus horikoshii O58192CheAMono
Rathayibacter rathayi O34971KdpDBi
Rhizobium leguminosarum P10047DctBBia
P41503NtrBMonoe
Rizhobium meliloti Q52880CHEAMono
P13633DCTBBia
P10955FIXLLidb
Q52912Q52912Bi
Rhodobacter capsulatus P09431NTRBMonoe
P37739DCTSBia
Rhodobacter sphaeroides Q53135CheAMono
Q53163HupSLidd
O33554O33554Mono
Q53162Q53162Bia
Rhodospirillum centenum Q9W2W8Q9W2W8Lidd
Rickettsia prowazekii Q9ZDU5BarABi
Riftia pachyptila symbiont O33541RSSALidd
O33542RSSBLidd
Saccaromyces cereviseae P39928SLN1Lidd
Salmonella dublin O85303CopSBia
Salmonella typhimurium P36557BASSBi
P09384CHEAMono
P41406ENVZBi
P37461HYDHLidb
Q9ZHD4Q9ZHD4Bia
Scizosaccaromyces O14002MCS4Lidd
pombe O74539O74539Lidd
Shigella dysenteriae P45609PHORLidb
Shigella flexneri O31140EnvZBi
Staphylococcus aureus Q9XCM6YYCGBi
Streptococcus pneumoniae Q54955Q54955Bi
Streptomyces hygroscopicus Q54293Q54293Bi
Synechococcus sp. Q56181PHORBia
P72560P72560Lidb
P20169P20169Bi
P72728P72728Lidb
P72791P72791Lidb
P73025P73025Bia
P73035P73035Bia
P73184P73184Lidb
P73337P73337Lidb
P73687P73687Lidb
P73828P73828Lidd
P73865P73865Bia
P73926P73926Lidd
P73932P73932Lidd
P74004P74004Lidb
P74015P74015Lidb
Synechocystis sp. P74622P74622Bia
 Q55475Q55475Lidb
 Q55693Q55693Lidb
 Q55718Q55718Lidb
 Q55783Q55783Lidb
 Q55838Q55838Lidb
 Q55918Q55918Bia
 Q55932Q55932Bia
 Q55941Q55941Bia
 Q73276Q73276Lidb
 Q74111Q74111Lidb
 Q74137Q74137Bi
 Q9ZAL8Q9ZAL8Lidd
 slr 2099slr 2099Lidb
 P39664P39664Lidb
 Y437Y437Bia
Thermotoga maritima Q56310CheAMono
P74923P74923Lidb
Q9WYN1Q9WYN1Bi
Q9WZV7Q9WZV7Bi
Q9X180Q9X180Lidb
Vibrio alginolyticus P19906NtrBMonoe
Vibrio cholerae O30663FlrBLidb
O68317O68317Bi
O85090PhoRBi
Vibrio harveyi P54302LUXQLidd
P54301LUXNLidd
Xanthomas campestris P49246RpfCLidd

We also have tested our predictions with the TCS for which biochemical and genetic evidence regarding the monofunctional or bifunctional character of their sensor is available and homology modelling was possible (Table 3). The sensor protein has an ATP-binding domain that contains a small sequence of amino acids giving rise to a 3D structure known as the ATP lid. This structure is highly organized and its position shifts, enclosing the nucleotide or releasing it from its binding site. The secondary structure of the ATP lid is mostly α-helical. If the sensor is known to be bifunctional, we find that its ATP lid resembles that of EnvZ (Fig. 3B). Approximately in the middle of the amino acid sequence, the α-helix fold is interrupted by a small non-α-helical T-loop. If the sensor protein is known to be monofunctional, we find a different folding of the ATP lid in which the three-dimensional structure resembles the ATP-binding domain of CheA (Fig. 4B). In this case, the α-helical structure of the ATP lid is interrupted by two small non- α-helical loops, with the α-helix stretch in between the two loops laying almost perpendicular to the other two α-helical stretches.

Table 3. . Sensor proteins observed and predicted, on the basis of homology modelling techniques, to be either monofunctional or bifunctional.
Sensor nameFunctionalityReference
PredictedObserved
  • a

    . Inferred from gene expression results.

  • b

    . Shows an increase in the rate of dephosphorylation only for the soluble domain of PhoR in the presence of ADP or ATP.

CheA
(E. coli)
MonoMono Ninfa et al. (1991 )
EnvZ
(E. coli)
BiBi Kanamaru et al. (1989)
KdpD
(E. coli)
BiBi Jung et al. (1997)
VanSA
(E. faecalis)
BiBi a Wright et al. (1993)
PhoR
(B. subtilis)
BiBi b Shi et al. (1999)

Qualitative functional differences

The concentrations of the input signal molecules (X5 primary input, such as a chemical gradient of nutrients in the CheA case or osmotic pressure changes in the EnvZ case, and X6 secondary input, such as acetyl phosphate) and the total concentrations of sensor protein (X7 = X1 + X3) and regulator protein (X8 = X2 + X4) are determined by external influences that are independent of changes within the system. Thus, these are defined as independent variables. By contrast, the concentrations of the sensor proteins (X1 phosphorylated and X3 unphosphorylated) and of the regulator proteins (X2 phosphorylated and X4 unphosphorylated) are determined by the values of the independent variables and by the system's internal dynamical behaviour. These variables are defined as dependent state variables in our models.

Protein synthesis and degradation occur on a time-scale that is much slower than that of phosphorylation, phosphotransfer and dephosphorylation in the TCS modules. Thus, on the time scale of interest here, one can ignore gene regulation and consider the total amount of sensor protein (STotal, X7) and the total amount of regulator protein (RTotal, X8) to be conserved quantities (i.e. X1 + X3 = X7 = constant and X2 + X4 = X8 = constant).

Thus, when considering changes in the dependent variables, in this and the following section, we need only emphasize the phosphorylated forms of the sensor protein X1 and the regulator protein X2. The corresponding changes in the unphosphorylated forms, X3 and X4, have the same magnitude but opposite sign. If for any reason, one of the X species in one of the conserved pairs increases by some amount, then the other X species of the pair must decrease by exactly the same amount. For example, amplification of signals at the level of the unphosphorylated proteins is equal to the negative of that at the level of the phosphorylated proteins. This amplification can be measured by the logarithmic gain L(Xk,Xj) (see Experimental procedures section), which is defined as the percentage change in a dependent variable Xk in response to a one per cent change in an independent variable Xj. Thus, L(X3,Xj) = − L(X1,Xj) and L(X4,Xj) =−L(X2,Xj), where Xj is any independent variable of the model. Similarly, the parameter sensitivities in the levels of the unphosphorylated proteins in response to parameter fluctuations are equal to the negative of those in the levels of the phosphorylated proteins. These parameter sensitivities can be measured by the expression S(Xk,pj) (see Experimental procedures section), which is defined as the percentage change in a dependent variable Xk in response to a one per cent change in a parameter pj. Thus, S(X3,pj) = − S(X1,pj) and, S(X4,pj) = − S(X2,pj) where pj is any parameter of the model.

The kinetic behaviour of the models in Fig. 1 can be described by a system of differential equations [Equations (1) (2) and (2’)], as outlined in the Experimental procedures section. Solving these equations allows us to quantify and to compare the systemic properties of the alternative designs shown in Fig. 1 based on the following functional considerations.

A signal transduction cascade should have a set of large logarithmic gains to amplify physiological signals and another set of small logarithmic gains that attenuate pathological noise. The cascade should be robust, i.e. it should function reproducibly despite perturbations in the values of the parameters that define the structure of the system. This is, by definition, equivalent to saying that the parameter sensitivities should, in general, be as low as possible. The steady state of the system should be stable and have a sufficient margin of stability, such that it will not become unstable when subjected to random fluctuations in the parameters of the system. If the margin of stability is small, a small change in a parameter of the system (e.g. ionic strength of the medium) may destroy the possibility of a stable steady state and make the system dysfunctional. Finally, the system should respond quickly to changes in its environment because otherwise the system is unlikely to be competitive in rapidly changing environments.

These properties, for each of the alternative models, are quantified and then compared by taking the ratio of the value for a property in the reference system (bifunctional sensor) to the corresponding value in the alternative system (monofunctional sensor). As we have analytical expressions for the steady-state properties, we can determine in some cases whether the ratio is always equal to one, less than one, or greater than one, independent of parameter values. With this approach, we have obtained the following qualitative results for steady-state concentrations and the amplification factors.

The concentrations (and rates of change) of the corresponding state variables in the two models can always be the same (see Calculating the constraints for external equivalence). Changes in the secondary signal X6 are amplified less in each of the corresponding state variables [i.e. each L(Xi,X6) for i = 1, 2, 3 and 4] with the bifunctional design. Similarly, these changes are amplified less in the flux through each of the sensor pools [i.e. L(Vi,X6) for i = 1 and 3] with the bifunctional design. This is shown in Table 4 by the ratio E, which is always less than 1. On the other hand, amplification of the phosphorylated regulator X2 in response to a percentage change in the primary input signal X5[i.e. L(X2,X5)] is always greater with the bifunctional design. This is shown in Table 4 by the ratio B, which is always greater than 1.0. In all other cases, the differences between the amplification factors of the alternative designs are dependent upon the specific numerical values of the parameters and can not be determined analyticaly.

Table 4. . Qualitative ratios of corresponding logarithmic gains for the reference system relative to the alternative system.
Systemic
property
Dependent variable of the system
X1X2V1V2
L(•, X5)AB > 1CD
L(•, X6)E < 1E < 1E < 1F
L(•, X7)GHJK
L(•, X8)LMNP

Quantitative functional differences

Although some functional differences between TCS with the alternative sensor designs are analyticaly indeterminate, numerical comparisons can be used to determine quantitative differences in function, and thus to establish statistical tendencies in the differences. With this approach we have obtained the following quantitative results for signal amplification, robustness, margin of stability, response time, and phosphorylation/dephosphorylation ratio.

Numerical results for signal amplification of concentrations are shown in Fig. 5 and described in detail in the next paragraph. The data are represented as Density of Ratios plots for moving medians (as described in Experimental procedures). The moving median of the ratio for the bifunctional design (reference) to monofunctional design (alternative), shown on the vertical axis, is a function of the moving median of the amplification for the bifunctional design, shown on the horizontal axis. [In symbolic terms, <L(Xi,Xj)Bi/L(Xi,Xj)Mono> is plotted as a function of <L(Xi,Xj)Bi>, where Xj indicates an independent variable, Xi a dependent variable and the angular brackets indicate averages.

Figure 5.

Comparison of signal amplification factors for concentrations in two-component systems with either a bifunctional or a monofunctional sensor. The logarithmic gains are compared in a density of ratios plot using moving medians (see Numerical analysis). On the x-axis are average values of a given logarithmic gain for the bifunctional design. On the y-axis are average values for the ratio of that logarithmic gain in the bifunctional design (Fig. 1A) over the corresponding logarithmic gain in the monofunctional design (Fig. 1B).

A. Logarithmic gain in phosphorylated sensor protein X1 with respect to changes in the primary input signal X5.

B. Logarithmic gain in X1 with respect to changes in the secondary input signal X6.

C. Logarithmic gain in X1 with respect to changes in the total amount of sensor, X7.

D. Logarithmic gain in X1 with respect to changes in the total amount of regulator, X8.

E. Logarithmic gain in phosphorylated regulator protein X2 with respect to changes in the primary input signal X5.

F. Logarithmic gain in X2 with respect to changes in the secondary input signal X6.

G. Logarithmic gain in X2 with respect to changes in the total amount of sensor, X7.

H. Logarithmic gain in X2 with respect to changes in the total amount of regulator, X8.

The pattern of responses exhibited by the sensor protein is as follows. Overall, values for the amplification of the phosphorylated sensor signal X1 in response to a percentage change in the primary input signal X5[L(X1,X5)] are similar in both designs. This is shown by the curve in Fig. 5A, which remains about 1.0. Values for the amplification of the phosphorylated sensor signal X1 in response to a percentage change in the secondary input signal X6[L(X1,X6)] are also smaller in the bifunctional design (the curve remains below 1.0 in Fig. 5B). On the other hand, values for the amplification of the phosphorylated sensor signal X1 in response to a percentage change in either the total concentration of sensor protein X7[L(X1,X7)] or the total concentration of regulator protein X8[L(X1,X8)] are similar for the two designs (curves remain very close to the y-axis value of 1.0 in Fig. 5C and D), except when signal amplification values are close to zero.

The regulator protein exhibits a different pattern of responses. Values for the amplification of the phosphorylated regulator signal X2 in response to a percentage change in the primary input signal X5[L(X2,X5)] are greater for the bifunctional design as can be seen by the curve in Fig. 5E, which is always above 1. For low median values of the gain L(X2,X5) in the bifunctional design, the differences in gain can be as large as 100%. For high median values, the differences are around 30%, with the gain in the bifunctional design being higher. This means that, for example, the bifunctional design will provide for larger changes in gene expression than would a monofunctional design for the same amount of osmotic pressure change in the EnvZ/OmpR system. Values for the amplification of the phosphorylated regulator signal X2 in response to a percentage change in the secondary input signal X6[L(X2,X6)] are smaller in the bifunctional design (Fig. 5F). On the other hand, values for the amplification of the phosphorylated regulator signal X2 in response to a percentage change in the total concentration of sensor protein X7[L(X2,X7)] can be larger in either alternative depending on the parameter values (Fig. 5G). If the logarithmic gain L(X2,X7) in the bifunctional design is negative, then the amplification in the bifunctional design is smaller (in absolute value) as can be seen in Fig. 5G. If the logarithmic gain in the bifunctional design is positive, then the amplification in the monofunctional design is smaller. Values for the amplification of the phosphorylated regulator signal X2 in response to a percentage change in the total concentration of regulator protein X8[L(X2,X8)] also can be larger in either alternative depending on the parameter values (Fig. 5H). If the logarithmic gain L(X2,X8) in the bifunctional design is negative, then the amplification in the bifunctional design is larger (in absolute value); if the logarithmic gain in the bifunctional design is positive, then the amplification in the monofunctional design is larger. Even though it seems that this shift occurs at values of about −0.3 for L(X2,X8) in this case, the actual values are about zero. Because of the moving averaging technique there are residual ratios that keep the median above 1.0. When the values for L(Xi,Xi) change sign, the curves tend to exhibit a dip, which is a consequence of the moving average technique when positive and negative values are being averaged.

Numerical results for signal amplification in flux are shown in Fig. 6. The pattern of responses exhibited by flux through the pools of sensor protein is as follows. Values for the amplification of the flux V1 in response to a percentage change in the primary input signal X5[L(V1,X5)] can be larger in either alternative depending on the parameter values (Fig. 6A). If the logarithmic gain L(V1,X5) in the bifunctional design is negative, then the amplification in the bifunctional design is larger (in absolute value); if the logarithmic gain in the bifunctional design is positive, then the amplification in the monofunctional design is larger. Values for the amplification of the flux V1 in response to a percentage change in the secondary input signal X6[L(V1,X6)] are smaller in the bifunctional design (Fig. 6B). Values for the amplification of the flux V1 in response to a percentage change in the total concentration of sensor protein X7[L(V1,X7)] in the monofunctional design are always larger in absolute value (Fig. 6C). Values for the amplification of the flux V1 in response to a percentage change in the total concentration of regulator protein X8[L(V1,X8)] can be larger in either alternative depending on the parameter values (Fig. 6D). If the logarithmic gain L(V1,X8) in the bifunctional design is negative, then the amplification in the monofunctional design is larger (in absolute value); if the logarithmic gain in the bifunctional design is positive, then the amplification in the bifunctional design is larger.

Figure 6.

Comparison of signal amplification factors for fluxes in two-component systems with either a bifunctional or a monofunctional sensor. The logarithmic gains are compared in a density of ratios plot using moving medians (see Numerical analysis). On the x-axis are average values of a given logarithmic gain in the bifunctional design. On the y-axis are average values for the ratio of that logarithmic gain in the bifunctional design (Fig. 1A) over the corresponding logarithmic gain in the monofunctional design (Fig. 1B).

A. Logarithmic gain in flux through the pool of phosphorylated sensor protein V1 with respect to changes in the primary input signal X5.

B. Logarithmic gain in V1 with respect to changes in the secondary input signal X6.

C. Logarithmic gain in V1 with respect to changes in the total amount of sensor, X7.

D. Logarithmic gain in V1 with respect to changes in the total amount of regulator, X8.

E. Logarithmic gain in flux through the pool of phosphorylated regulator protein V2 with respect to changes in the primary input signal X5.

F. Logarithmic gain in V2 with respect to changes in the secondary input signal X6.

G. Logarithmic gain in V2 with respect to changes in the total amount of sensor, X7.

H. Logarithmic gain in V2 with respect to changes in the total amount of regulator, X8.

The pattern of responses exhibited by the flux through the pools of regulator protein is as follows. Values for the amplification of the flux V2 in response to a percentage change in the primary input signal X5[L(V2,X5)] can be larger in either alternative depending on the parameter values (Fig. 6E). If the logarithmic gain L(V2,X5) in the bifunctional design is negative, then the amplification in the bifunctional design is larger (in absolute value); if the logarithmic gain in the bifunctional design is positive, then the amplification in the monofunctional design is larger. Even though it seems that this shift occurs at values of about −0.8 for L(V2,X5) in this case, the actual values are about zero. Because of the moving averaging technique there are residual ratios that keep the median above 1.0. Values for the amplification of the flux V2 in response to a percentage change in the secondary input signal X6[L(V2,X6)] can be larger in either alternative depending on the parameter values (Fig. 6F). If the logarithmic gain L(V2,X6) in the bifunctional design is negative, then the amplification in the monofunctional design is larger (in absolute value); if the logarithmic gain in the bifunctional design is positive, then the amplification in the bifunctional design is larger. Values for the amplification of the flux V2 in response to a percentage change in the total concentration of sensor protein X7[L(V2,X7)] is on average larger in the monofunctional design (Fig. 6G). Values for the amplification of the flux V2 in response to a percentage change in the total concentration of regulator protein X8[L(V2,X8)] can be larger in either alternative depending on the parameter values (Fig. 6H). If the logarithmic gain L(V2,X8) in the bifunctional design is negative, then the amplification in the monofunctional design is larger (in absolute value); if the logarithmic gain in the bifunctional design is positive, then the amplification in the bifunctional design is larger. Even though it seems that this shift occurs at values of about 0.5 for L(V2,X8) in this case, the actual values are about zero. Again, because of the moving averaging technique there are residual ratios that keep the median above 1.0. When the values for L(Vi,Xj) change sign, the curves tend to exhibit a dip, which is a consequence of the moving average technique when positive and negative values are being averaged. The dip for the logarithmic gains in flux tends to be more pronounced than for the logarithmic gains in concentration.

Numerical results for robustness of the alternative designs are shown in Fig. 7. The data for aggregate parameter sensitivities are represented as Density of Ratios plots for moving medians. The moving median of the ratio for the bifunctional design (reference) to monofunctional design (alternative) is on the vertical axis, and the moving median of aggregate parameter sensitivity for the bifunctional design is on the horizontal axis. [In symbolic terms, <S(Xi)Bi/S(Xi)Mono> as a function of <S(Xi)Bi>, or <S(Vi)Bi/S(Vi)Mono> as a function of <S(Vi)Bi>, where Xi indicates a dependent concentration, V i a dependent flux and the angular brackets indicate averages.] On average, the aggregate sensitivities of the sensor signals (Fig. 7A and C) are less in the bifunctional design than in the monofunctional design when the values for the parameter sensitivities are low, whereas the aggregate sensitivities of the regulator signals (Fig. 7B and D) are greater in the bifunctional design than in the monofunctional design under these conditions. The aggregate sensitivities of all the signals are lower in the monofunctional design when the values for the parameter sensitivities are high, which is the less physiologically relevant case. However, in all cases, the average differences in corresponding sensitivities between the monofunctional and the bifunctional system are very small. In the case of the sensor signals, this difference is negligible and the median ratio is for all practical purposes 1.0.

Figure 7.

Comparison of robustness for two-component systems with either a bifunctional or a monofunctional sensor. The aggregate parameter sensitivities (see Steady-state solution and key systemic properties) are compared in a density of ratios plot using moving medians. On the x-axis are average values of a given aggregate sensitivity in the bifunctional design. On the y-axis are average values for the ratio of that aggregate sensitivity in the bifunctional design (Fig. 1A) over the corresponding aggregate sensitivity in the monofunctional design (Fig. 1B).

A. Aggregate sensitivity of phosphorylated sensor protein X1.

B. Aggregate sensitivity of phosphorylated regulator protein X2.

C. Aggregate sensitivity of flux through the pool of phosphorylated sensor protein V1.

D. Aggregate sensitivity of flux through the pool of phosphorylated regulator protein V2.

Numerical results for the stability margins of the alternative designs are shown in Fig. 8. These magnitudes, which correspond to the two critical Routh Criteria for stability, provide a measurement for the amount of perturbation that the system will tolerate before the steady state becomes unstable (see the Experimental procedure section). The bifunctional design has a larger margin of stability with respect to the first of the critical Routh criteria when this margin is small, which is when this margin is most important. When this margin becomes large, and its value becomes less important, the two designs have essentially the same value (Fig. 8A). The bifunctional design also has a larger margin of stability with respect to the second of the critical Routh criteria (Fig. 8B).

Figure 8.

Comparison of stability margins for two-component systems with either a bifunctional or a monofunctional sensor. The stability margins determined by the two critical Routh criteria for local stability (see Steady-state solution and key systemic properties) are compared in a density of ratios plot using moving medians. On the x-axis are average values of a given stability margin in the bifunctional design. On the y-axis are average values for the ratio of that stability margin in the bifunctional design (Fig. 1A) over the corresponding stability margin in the monofunctional design (Fig. 1B).

A. First critical Routh criterion.

B. Second critical Routh criterion.

Numerical results for the temporal responsiveness of the alternative designs are shown in Fig. 9. The response time, τ, is defined as time required for the return to a steady state following a perturbation (see the Experimental procedures section). The data are represented as Density of Ratios plots for the raw data (Fig. 9A) and for moving medians (Fig. 9B). The ratio for the bifunctional design (reference) to monofunctional design (alternative) is on the vertical axis, and the response time for the bifunctional design is on the horizontal axis. [In symbolic terms, <τBi/τmono as a function of <τBi, where τ is the response time and the angular brackets indicate averages.] On average, the response time is slightly less for the bifunctional design when response times are small, but the differences between designs become insignificant as the response times become larger.

Figure 9.

Comparison of response times for two-component systems with either a bifunctional or a monofunctional sensor. The response times (see Steady-state solution and key systemic properties), τ, are compared in a density of ratios plot. On the x-axis are values of response times for the bifunctional design. On the y-axis are values for the ratio of response times for the bifunctional design (Fig. 1A) over the response times for the monofunctional design (Fig. 1B).

A. Raw data.

B. Average values in a moving median plot.

In this work we allowed the sensor and regulator each to have steady-state operating levels of phosphorylation between 0% and 100%. In practice, the random values for the ratios ƒ31 = − X10 / X30 and ƒ42 = − X20 / X40 in our ensemble range between −0.0000001 and −999. The amplification properties of the TCS with a bifunctional design are not influenced to any large extent by the steady-state operating value for phosphorylation of either the sensor or the regulator (data not shown), although more significant changes can be seen in some cases when the operating value for phosphorylation drops to near 0%.

Discussion

In this work we examined alternative designs for the sensor proteins of prototype TCS. A bifunctional sensor is characterized by two functions: (i) when phosphorylated, the sensor transfers its phosphate group to the response regulator; (ii) when unphosphorylated the sensor increases the dephosphorylation rate of the response regulator. A monofunctional sensor has only the first of these functions. Our results have identified both structural and functional attributes of these alternative designs.

Structural differences

The results of our homology modelling show a highly conserved structural feature (‘ATP lid’) that appears to be distinctive for each of the alternative designs (Figs 3 and 4). Whether or not the characteristic ATP lid is responsible for the bifunctionality of a sensor is unknown, but recent experiments suggest that this part of the sensor has a role in enhancing the rate of dephosphorylation of the response regulator for EnvZ/OmpR (Zhu et al., 2000) and for NRII-PII/NRI (Pioszak and Ninfa, 2003). An experiment that exchanges the ATP lid of the EnvZ and CheA sensor proteins and assays the resulting proteins for bifunctional behaviour would help to resolve this issue. Whether this structural feature is responsible for the bifunctionality of a sensor or not, our rigorous comparative analysis of physiological function demonstrates that there is a clear basis for selection of monofunctional and bifunctional sensor proteins.

Functional differences

Protein levels for the majority of TCS are regulated on a slow time-scale by mechanisms affecting transcription. Regulation by these mechanisms, which have been studied elsewhere (Hlavacek and Savageau, 1995, 1996, 1997), is beyond the scope of this article, which focuses on the more rapid time scale of regulation within the TCS. Nevertheless, the results for the logarithmic gains in concentrations (Fig. 5C, D, G and H) and fluxes (Fig. 6C, D, G and H) with respect to changes in the total concentration of sensor (X7) and regulator (X8) provide some insight regarding the influence of changing protein levels.

The mathematically controlled comparisons in our study show that monofunctional and bifunctional designs for signalling within TCS differ on the basis of several criteria for functional effectiveness. The model with a bifunctional sensor has higher signal amplification in response to changes in the primary signal (Fig. 5E), X5, as represented by L(X2,X5). It has lower signal amplification in response to changes in the secondary signal (Fig. 5F), X6, as represented by L(X2,X6). If X5 is considered to be the physiological signal for a TCS, then this implies that the model with the bifunctional sensor is more effective in responding to this signal. Robustness, except in the case of V2, tends to be similar when the values for the parameter sensitivities are low (Fig. 7). Although, robustness for all the variables is greater in the bifunctional design when the values for the parameter sensitivities are high, systems with high parameter sensitivities are less likely to be biologically significant. The margins of stability for the steady state, as measured by both Routh criteria, are larger for the bifunctional design (Fig. 8). Response times are similar for the alternative designs (Fig. 9). These functional differences have implications for cross-talk among TCS.

Cross-talk to and from a TCS

The specificity of sensors and regulators is not absolute. Regulator proteins can respond to signals other than those transmitted by their cognate sensor, and sensor proteins can transmit signals to destinations other than their cognate regulator. In some cases, this cross-talk might be undesirable noise that should be minimized. Sensors that are homologous to the cognate sensor but are involved in distinct physiological responses may represent such a case. In other cases, this cross-talk might represent the physiological co-ordination of several processes that needs to be enhanced. Chemotaxis represents such a case, where the state of cellular metabolism needs to be taken into account before the cell migrates towards nutrient sources. It is less important for a cell that is already well fed to spend energy in migrating towards nutrients than if the cell is starving. The results in the previous sections allow us to identify designs that are appropriate for dealing with cross-talk in each of these contexts.

Cross-talk to a TCS module is represented by any secondary input signal (Q2) coming from sources other than the regulator's cognate sensor that causes a change in phosphorylation level of the module's response regulator (R*). The schematic diagrams in Fig. 1 explicitly represent the case in which the secondary signal is a small phosphodonor like acetyl-phosphate, but not ATP. Cross-talk in this case is less amplified by the module with a bifunctional sensor (Fig. 1A) than by the module with the monofunctional sensor (Fig. 1B). Thus, the design with a bifunctional sensor is better at attenuating the cross-talk to the module, and this is appropriate when the cross-talk is physiologically undesirable. Conversely, the design with a monofunctional sensor is better at amplifying the cross-talk to the module, and this is appropriate when the cross-talk is a relevant physiological signal. A TCS with a design that could change from bifunctional, when only its primary signal conveyed physiologically relevant information, to monofunctional, when other signals should also be considered, would have an advantage in dealing with more complex situations in which there is a changing requirement for suppression or integration of secondary signals. (The NRI/NRII system of E. coli, which will be discussed below, may be one such example.)

Cross-talk to the module also can result from secondary input signals originating from other sensors. There are several formal possibilities shown schematically in Fig. 10. The analysis of these possibilities yields results similar to those already described (data not shown). A controlled comparison of the alternatives in Fig. 10A and B shows that a bifunctional design for the cognate sensor (S1) is better at enhancing amplification of the regulator (R) response to the primary input signal (Q1) while it is better at suppressing noise represented by the secondary input signal (Q2). A controlled comparison of the alternatives in Fig. 10C and D shows a similar result. Thus, regardless of the design for the non-cognate sensor, a bifunctional design for the cognate sensor results in better amplification of the primary input signal and better suppression of the secondary input signal. A controlled comparison of the alternatives in Fig. 10B and D shows that a bifunctional design for the non-cognate sensor (S2) results in better amplification of the regulator (R) response to the secondary input signal (Q2). Taken together, these results suggest that the design in Fig. 10C is preferred for enhancing amplification of the primary input signal while promoting attenuation of the secondary input signal. However, the design in Fig. 10A is preferred for enhanced cross-talk to achieve a more balanced integration of the two input signals.

Figure 10.

Cross-talk to a common response regulator from two distinct sensor proteins.

A. Both sensors are bifunctional with respect to the common response regulator.

B and C. One sensor is bifunctional and the other is monofunctional with respect to the common response regulator.

D. Both sensors are monofunctional with respect to the common response regulator. See text for discussion.

Cross-talk from the module occurs when the cognate sensor transmits its signal to a non-cognate regulator protein. Again, there are several formal possibilities, shown schematically in Fig. 11, that we have analysed (data not shown). A controlled comparison of the alternatives in Fig. 11A and B shows that the design with a sensor that is bifunctional with respect to its cognate regulator (R1) is better in two respects. It is better at amplifying the signal that is transmitted from the input Q to the primary output (T1) and better at suppressing the signal that is transmitted to the secondary output (T2). A controlled comparison of the alternatives in Fig. 11C and D shows a similar result. Thus, regardless of the design with respect to the non-cognate regulator, the design with a sensor that is bifunctional with respect to its cognate regulator is better at amplifying the signal transmitted to the primary output and at suppressing that to the secondary output. A controlled comparison of the alternatives in Fig. 11B and D shows that the design with a sensor that is bifunctional with respect to the non-cognate regulator (R2) is better at amplifying the signal transmitted from the input Q to the secondary output (T2). Taken together, these results suggest that the design in Fig. 11C is more effective in suppressing cross-talk from the module to the response regulators of other TCS. However, the design in Fig. 11A is more effective in enhanced cross-talk to achieve a more balanced set of response in both regulators.

Figure 11.

Cross-talk from a common sensor protein to two distinct response regulator proteins.

A. The sensor is bifunctional with respect to both response regulators.

B and C. The sensor is bifunctional with respect to one response regulator and monofunctional with respect to the other.

D. The sensor is monofunctional with respect to both response regulators. See text for discussion.

Examples

The results in the previous sections suggest a rationale for selection of the two alternative sensor designs based on the physiology of the system in which the TCS is embedded. Assume that the output of the module is the phosphorylation level of the response regulator (X2). Phosphorylation levels change as a response to changes in the input signals X5 and X6. Usually, X5 is thought of as the input signal and X6 is not considered. However, X6 is also an input signal because it changes the phosphorylation level of the response regulator. Thus, we can consider the TCS in Fig. 1 as integrators of two signals, X5 and X6. When compared to the TCS with a monofunctional sensor, those with a bifunctional sensor maximize amplification of the signal X5 and minimize amplification of the signal X6. On the other hand, the TCS with a monofunctional sensor maximize X6 amplification and minimize X5 amplification when compared to otherwise equivalent TCS with a bifunctional sensor. Thus, in systems for which X5 is the major signal and the influence of other signals (represented by X6) needs to be minimized, the design with a bifunctional sensor should be selected; in systems for which the other signals need to be taken into account and integrated, the design with a monofunctional sensor should be selected. We will clarify these notions with a few examples.

The regulation of pore size in bacteria by changes in the osmolarity of the medium is mediated by a TCS with a bifunctional sensor. The pores in their cell membrane are composed of two different proteins. Subunit OmpF forms large pores, whereas OmpC forms smaller pores. EnvZ (X1) is a membrane protein and the sensor for a TCS module. Changes in the osmolarity of the medium lead to changes in the EnvZ protein. In a high osmolarity medium, EnvZ increases its autophosphorylation rate. This in turn leads to a transfer of phosphate from EnvZ to the response regulator of the TCS, OmpR (X2). OmpR is a transcription factor that binds DNA either in its phosphorylated (high affinity) or unphosphorylated (low affinity) form. In its phosphorylated form, OmpR dimerizes to increase its interaction with DNA. This leads to increased expression of OmpC and to decreased expression of OmpF (Pratt et al., 1996 for a review). The sensor in this system, EnvZ, is bifunctional (Igo et al., 1989), which can be rationalized in terms of our results as follows. Pore size should be determined exclusively by differences in osmolarity between the intracellular and the extracellular medium. If pore size were to be affected by other signals, such as changes in the levels of small phosphodonors, then osmotic balance could not be maintained and cell viability would be diminished. Thus, the bifunctional sensor design used in this TCS module is the one that maximizes amplification of changes in the osmotic pressure (X5) and minimizes amplification of changes in other spurious signalling processes (X6).

The VanS/VanR module in Enterococus faecalis is another example of a TCS with a bifunctional sensor (Wright et al., 1993, Arthur et al., 1997). This TCS regulates synthesis of proteins responsible for the organism's resistance to antibiotics. Again, as the function of the proteins is to confer resistance to antibiotics, it should not be activated by other signals, because this would unnecessarily increase the protein burden of the cell. Thus, selection of the design with a bifunctional sensor is to be expected.

Chemotaxis in E. coli is mediated by a TCS module with a monofunctional sensor (see Eisenbach, 1996; for a review). In this case, the CheA sensor protein (X1) transfers its phosphate to either of two response regulators, CheY or CheB (X2). CheB is responsible for desensitizing the cell to chemical gradients, whereas CheY is responsible for changing the rate of cell tumbling so as to promote movement towards favourable concentrations. CheA is a monofunctional sensor, which means that CheY and CheB can be more effectively phosphorylated by other sources (X6), either by phospho-donors or by other sensors, than they could if CheA were a bifunctional sensor. Thus, the internal metabolism of the bacterium regulating the levels of these phospho-donors is more likely to be involved in determining whether the cell will search for nutrients than it would be if CheA were a bifunctional sensor. [It must be emphasized that the chemotaxis system in E. coli has a phosphatase protein, CheZ, that acts downstream of both response regulators. Another case where this situation also occurs is in the Spo0 phosphorelay in B. subtilis. These phosphatases are also the subject of regulation. However, in this work our goal has been to evaluate the effect of alternative sensor design on signal transmission within the prototype module and downstream aspects of design will not be considered here.]

FlbE/FlbD in Caulobacter crescentus is a presumptive TCS module involved in flagellum assembly and cell cycle regulation (e.g. Wingrove and Gober, 1996). We are not aware of studies showing whether FlbE is a bifunctional or monofunctional sensor, but based on its involvement in the cell cycle we would predict that it would have a monofunctional sensor as several signals must be integrated to co-ordinate the timing of cell cycle events. Similar arguments apply to the co-ordination of sporulation events in Bacillus subtilis. The Spo phosphorelay system appears to have a monofunctional sensor and to include different phosphatases that are specific for the different components of the relay (Perego and Hoch, 1996 for a review).

PhoR/PhoP is a TCS module with a monofunctional sensor involved in regulating expression of genes responsible for the transport of phosphate in B. subtilis. This module transduces signals generated by phosphate starvation (X5). There is also cross-regulation between this module and PTS sugar systems (X6) (Hulett, 1996; for a review). Thus, it is important for this TCS to sense other signals, beside the one coming from PhoR, and the design with a monofunctional sensor should be favoured. In fact, it has been shown that membrane-bound PhoR does not seem to influence the dephosphorylation rate of PhoB significantly (Shi et al., 1999) and thus it borders on the design of a monofunctional sensor. However, from Table 3, the PhoR sensor is predicted to be bifunctional which seem to contradict this result. A careful analysis of Shi et al. (1999) explains the apparent contradiction. A soluble version of PhoR (without the membrane spanning domains) enhances the dephosphorylation rate of PhoP only slightly in the absence of ATP or ADP but much more significantly in the presence of either of these molecules. Thus, the design of soluble PhoR is that of a bifunctional sensor as predicted from the homology modelling; however, this bifunctionality is effectively inhibited and transformed into monofunctionality by locating PhoR in the membrane.

If one accepts the rule we have suggested for the selection of monofunctional and bifunctional sensors, and the supportive evidence in the above cases where the physiological context can be interpreted in a fairly straightforward fashion, then one can go on to apply this rule in the interpretation of more complex systems. Two such cases in E. coli are considered below, the NRI/NRII system involved in nitrogen fixation and the NarX/NarL and NarQ/NarP systems involved in nitrate and nitrite dependent gene expression.

The NRI/NRII system in E. coli regulates nitrogen fixation and glutamine production. NRII is the sensor protein that phosphorylates the response regulator NRI. Under conditions of low nitrogen availability NRI upregulates the expression of glutamine synthase. This enzyme condenses nitrogen and glutamate to form glutamine. Glutamine increases the affinity of a third protein, PII, towards the sensor protein NRII, inhibiting NRII phosphorylation and creating a complex that binds phosphorylated NRI and increases NRI′s rate of dephosphorylation. Alone, NRII has little or no effect upon the rate of NRI dephosphorylation (Keener and Kustu, 1988). Thus, under normal nitrogen conditions (i.e. with normal levels of glutamine), this TCS is bifunctional and nitrogen fixation is more sensitive to regulation by glutamine levels than it would be if the module were monofunctional. Under nitrogen depletion (causing a decrease in the concentration of glutamine) PII does not bind NRII, which then becomes monofunctional. This causes the module to integrate the signals coming from glutamine/glutamate levels with those coming from other parts of metabolism, via the changes in the concentration of acetyl-phosphate, more efficiently than it would if NRII were bifunctional. Intuitively, one might think that, under nitrogen depletion, NRII should be bifunctional, in order to more efficiently respond to the nitrogen fixation needs of the cell and buffer against regulation of this fixation by other parts of metabolism. However, probably as a result of the central role of glutamate in amino acid biosynthesis, this is not so. Glutamate is a ubiquitous amino acid that is needed for the biosynthesis of all other amino acids and not just glutamine. Under nitrogen depleting conditions, it is important that the use of glutamate be co-ordinately regulated by the concentration of all amino acids, in order not to deplete the cell of some of them, by its overuse to produce glutamine. It has been reported that cell growth on glucose minimal medium containing arginine, a poor nitrogen source, is greatly decreased in mutants lacking either NRII or phosphate acetyl-transferase (E.C. 2.3.1.8) (Feng et al., 1992). This implies that phosphorylation by both NRII and acetyl-phosphate is important under these conditions, which agrees with a strong regulatory role for secondary signals in nitrogen fixation that get integrated through the NRI/NRII module. It is clear that the NR system is complex and that a more detailed model would provide additional insight.

Two TCS modules are involved in nitrate and nitrite dependent gene expression in E. coli, NarX/NarL and NarQ/NarP. NarX and NarQ are sensor proteins (X1) that independently recognize nitrate in the medium. Each of these sensors can transfer phosphate to both regulators NarL and NarP (X2) which have different specificities (Schroder et al., 1994; Chiang et al., 1997). Whereas NarL regulates expression of genes encoding nitrate and fumarate reductases as well as nitrite exporter proteins, both NarL and NarP regulate expression of genes encoding nitrite reductase and anaerobically expressed proteins. NarQ and NarX are both sensors for NarL. NarQ has a higher affinity than NarX for phosphorylating NarL. However, NarX is more effective in enhancing the rate of NarL dephosphorylation (Schroder et al., 1994). In light of our results this can be interpreted as a design that maximizes the signals transduced via NarX/NarL with respect to those transduced via NarQ/NarL, because NarX is bifunctional with respect to NarL whereas NarQ is not. It would be interesting to determine whether either of the sensors is bifunctional with respect to NarP. If not, this would indicate that NarP, in contrast to NarL, is designed to integrate signals coming from both NarX and NarQ.

It is clear from the examples above, and others, that two-component systems exhibit a diversity of design issues that are not well understood. Our analysis has contributed to the elucidation of just one of the variations in design, monofunctional versus bifunctional sensors. Based on our results, we have proposed the following rule: Relative to the transduction of cognate signals, bifunctional sensors enhance suppression of signals from non-cognate sources whereas monofunctional sensors enhance their integration. The experimental evidence we have discussed is consistent with this rule. This leads us to suggest that this rule may be useful in understanding whether other TCS are acting simply as transducers of the sensor signal or as integrators of signals coming from different sources.

Experimental procedures

Structures for TCS

Crystal or NMR structures have been determined for a limited number of TCS sensor and regulator proteins. In the PDB database one can find structures for a prototype bifunctional sensor (EnvZ: PDB reference identifiers 1BXD and 1JOY) and for a prototype monofunctional sensor (CheA: PDB reference identifiers 1A0O, 1B3Q and 1FWP). Partial structures have been determined for other sensor proteins (ArcB: PDB reference identifiers 1A0B and 2A0B; FixL: PDB reference identifiers 1D06, 1DRM, 1DRQ and 1EW0) and other response regulators (NarL: PDB reference identifiers 1A04 and 1RNL; CheY: PDB reference identifiers 1A0O, 1BDJ, 1C4W, 1D4Z, 1DJM, 1EAY, 1EHC, 1VLZ, 2CHE, 2CHF, 2CHY, 5CHY and 6CHY; Spo0A: PDB reference identifiers 1DZ3, 1FC3 and 1QMP; Spo0F: PDB reference identifiers 1FSP, 1SRR and 2FSP; Spo0B: PDB reference identifiers 1IXM and NAT; OmpR: PDB reference identifiers 1ODD and 1OPC (DNA binding domain; no structure for the phosphorylation domain); CheB: PDB reference identifiers 1A2O and 1CHD; DrrD: PDB reference identifier 1KGS; NtrC: PDB reference identifiers 1NTRC, 1NTC, 1DC7 and 1DC8; Etr1: PDB reference identifier 1DCF; FixJ: PDB reference identifiers 1D5W, 1DBW, 1DCK and 1DCM; PhoB: PDB reference identifiers 1B00, 1GXP, 1GXQ and 1QQI; Rcp1: PDB reference identifiers 1I3C and 1JLK). This is a very small portion of the total number of TCS with identified proteins. However, the known structures can be used in combination with sequence data and homology modelling techniques to predict the structures for these other proteins of TCS.

Sequence data for proteins of TCS

We searched GeneBank and Swissprot for Two Component Systems proteins. We also looked for proteins of TCS in the databases for sequenced microbial genomes (listed in MAGPIE at http:www-fp.mcs.anl.govgaasterlandgenomes.html). Whenever necessary, we translated the cDNA sequence into its corresponding amino acid sequence. The protein sequences that we extracted from the databases were then catalogued by organism. We did homology modelling of each protein by submitting the sequence to swissmodel (Guex and Peitsch, 1997). Those that were successfully modelled (at least for some domain of the protein) by this program are presented in Tables 1 and 2.

Homology modelling

The protein sequences were submitted via internet to the SwissModel server (http:swissmodel.expasy.org; Guex and Peitsch, 1997). As templates for the homology modelling we have used files from the PDB database (http:www.rcsb.orgpdbBerman et al., 2000). The server attempted to generate a three-dimensional model for each sequence and returned the results via E-mail. In many cases there was not enough homology to any known structure to create a working model. In the remaining cases we have modelled (at least partially) the catalytic domain of the sensor as well as the receiver and DNA-binding domains of the regulator. In some cases we were also able to model the transmitter or the signal-sensing domain of the sensors. We then visualized and manipulated the models using swissprotviewer (Guex and Peitsch, 1997).

Mathematical representations

We can describe the dynamical behaviour of a system exhibiting variations about any of its steady states by using a well-established power-law formalism for modelling biochemical systems (Savageau, 1969; Shiraishi and Savageau, 1992). For the models in Fig. 1, the independent variables of our models are the concentrations of the input signal molecules (X5 primary and X6 secondary) and the concentration totals for sensor protein (X7 = X1 + X3) and regulator protein (X8 = X2 + X4). The dependent state variables, which are dependent upon the values of the independent variables and the internal dynamics of the system, are the concentrations of the sensor proteins (X1 phosphorylated and X3 unphosphorylated) and of the regulator proteins (X2 phosphorylated and X4 unphosphorylated).

There is one equation for each of the four dependent variables. For each dependent variable Xi its change in concentration with time (dXi / dt) is given by the difference between the corresponding aggregate rate of production (Vi) and the aggregate rate of consumption (V–i). Each of these aggregate rates is represented by a mathematical rate law whose exact form is unknown, but whose arguments consist of all the variables that have a direct influence on the aggregate rate in question. For example, the aggregate rate law for consumption of the phosphorylated sensor protein in Fig. 1 is a function of its concentration X1 and of the concentration of the unphosphorylated regulator protein X4 acting as a co-substrate.

Each aggregate rate law can be represented by a product of power-law functions, one for each argument in the rate law. This representation is guaranteed to be an accurate representation for some region of operation about a nominal steady state (e.g. see Shiraishi and Savageau, 1992). Thus, the equations for the first two dependent variables of the TCS with the bifunctional sensor (Fig. 1A) are the following (the dependent variables X3 andX4 will be treated below).

image(1)
image(2)

The corresponding equations for the TCS with the monofunctional sensor (Fig. 1B) are the same, except for the last term in Equation (2):

image((2’) )

The primed parameters [in Equation (2’)], whose significance will become evident below, have values that in general are different from the corresponding unprimed parameters in Equation (2).

The multiplicative parameters (rate constants), the α's for production and the β's for consumption, influence the time scales of the reactions and are always non-negative. The subscript of a rate constant refers to the molecular species that is being produced or consumed. For example, α1 is the rate constant for the aggregate rate of production of X1 (phosphorylated sensor kinase). The exponential parameters (kinetic orders), g’s for production and h’s for consumption, represent the direct influence of each variable on each aggregate rate law. These parameters need not have integer values, but can assume real values (positive, negative or zero). The first subscript of a kinetic order refers to the molecular species that is being produced or consumed; the second refers to the variable that has a direct influence on the aggregate rate of production or consumption. For example, the influences on V−1, the aggregate rate of consumption of X1 (phosphorylated sensor protein), are represented by the kinetic orders h11, which is the kinetic order representing the influence of X1 (phosphorylated sensor protein) acting as substrate in the aggregate rate of its own consumption, and h14, which is the kinetic order representing the influence of X4 (unphosphorylated response regulator) acting as a co-substrate in the aggregate rate of consumption of X1.

It may seem more natural to represent the aggregate rate of consumption of X1 by two separate terms such as inline image. However, it has been demonstrated that this is less accurate than the simple product of power-law functions described above for small variations about the steady state; this also tends to be the case for large variations, although this is not necessarily true in general (Voit and Savageau, 1987).

Protein synthesis and degradation occur on a time-scale that is in much slower than that of the catalytic activities within the TCS modules. Thus, on the time scale of interest here, the total amount of sensor protein (STotal, X7) and the total amount of regulator protein (RTotal, X8) are considered to be conserved quantities. This permits one dependent variable to be expressed in terms of the conserved total minus the other dependent variable. Accordingly, one can write:

X3 = X7 − X1(3)
X4 = X8 − X2(4)

Because of these conservation relationships [Equations (3) and (4)], the rate of change in the concentration X3 is equal in amount and opposite in sign to that in the concentration X1 and the rate of change in the concentration X4 is equal in amount and opposite in sign to that in the concentration X2.

At any given steady state, one can represent the differences in Equations (3) and (4) by the following products of power-laws (see Sorribas and Savageau, 1989; for a more detailed explanation of the procedure)

image(5)
image(6)

The new parameters in Equations (5) and (6) are defined as follows:

ƒ37 = (X7 / X30)(dX30 / dX7) = X7 / X30, ƒ31 = (X10 / X30)(dX30 / dX10) = − X10 / X30, γ3 = X30 /inline image, ƒ48 = (X8 / X40)(dX40 / dX8) = X8 / X40, ƒ42 = (X20 / X40)(dX40 / dX20) = − X20 / X40, and γ4 = X40 /inline image. The additional subscript 0 indicates the operating point at which the representation is made (in this case, the steady state). It must be emphasized that this representation is exact at the operating point. The multiplicative and exponential parameters in Equations (5) and (6) are analogous to the rate-constant and kinetic-order parameters in Equations (1), (2) and (2’). Thus, the γ’s are non-negative and the ƒ’s are real (positive, negative or zero). The subscript of a γ parameter, and the first subscript of an ƒ parameter, refers to the dependent variable that is being represented in terms of its paired variable and their sum. The second subscript of an ƒ parameter refers to either the paired variable or the variable representing their sum.

The final form of the equations that will be used here is obtained by substituting the expressions in Equations (5) and (6) into Equations (1), (2) and (2’), and then redefining terms. Thus, one can write

image(7)
image(8)

and

image((8′) )

where the new parameters in Equations (7), (8), and (8′) are related to the original parameters in Equations (1), (2), (2’), (5) and (6) as follows: inline image , inline image, inline image, inline image, inline image, inline image, inline image, inline image, inline image, inline image, inline image, and inline image.

All of the parameters in Equations (7), (8) and (8′) are positive except for g11, h12, g22 and h21, which are negative. A negative exponent implies that an increase in the argument results in a decrease in the value of the power-law function. For example, an increase in X2 results in a decrease in the rate of consumption of X1. This apparent inhibition of V−1 by X2 is a result of the conservation relation among the different forms of the response regulator. Thus, an increase in X2 corresponds to an equivalent decrease in X4, which actually causes a de-activation of V−1. Similar explanations account for the behaviour associated with the other negative kinetic orders.

As a practical matter, the results for the monofunctional design can be obtained directly from those for the bifunctional design simply by making the following exchanges: h21 → 0, h27 → 0, and β20 →β20, and h22 → h′22. Hence, in the following sections we shall focus on the bifunctional design, make these exchanges to obtain the corresponding results for the monofunctional design, and then make the appropriate comparisons.

Steady-state solution and key systemic properties

The equations describing the dynamic behaviour of the model in Fig. 1A.[Equations (7) and (8)] can be solved analytically for the steady state (Savageau, 1969; 1971a), where the rates of aggregate production and aggregate consumption for each metabolite are the same. By equating these aggregate rates, taking logarithms of both sides of the resulting equations and rearranging terms, one can write the steady-state equations as follows:

a 11 Y 1 − h12Y2 = b1 − g15Y5 − g17Y7 + h18Y8(9)
a 21 Y 1 + a22Y2 = b2 − g26Y6 − h27Y7 − g28Y8(10)

where Yi = log Xi, bi = log(βi0 / αi0), and aij = gij – hij. Equations (9) and (10) are single linear algebraic equations that can be solved for the dependent variables Y1 and Y2 in terms of the parameters of the system, the input variables Y5 and Y6, and the total concentrations of sensor and regulator proteins Y7 and Y8. Thus,

image(11)
image(12)

Two types of coefficients, logarithmic gains and parameter sensitivities, can be used to characterize further the steady state of such models. Because steady-state solutions exist in explicit form [Equations (11) and (12)], we can calculate each of the two types of coefficients simply by taking the appropriate derivatives. Although the mathematical operations involved are the same in each case, it is important to keep in mind that the biological significance of the two types of coefficients is very different.

Logarithmic gains provide important information concerning the amplification or attenuation of signals as they are propagated through the system (Savageau, 1971a; Shiraishi and Savageau, 1992). For example,

image(13)

measures the percent change in the value of the dependent concentration variable Xi (or in Vi the flux through the pool of Xi) caused by a percentage change in the concentration of the input signal X5. A positive sign indicates that the changes are in the same direction (both increase or both decrease); a negative sign indicates that the changes are in the opposite direction (one increases while the other decreases).

Parameter sensitivities provide important information about system robustness, i.e. how sensitive the system is to perturbations in the structural determinants of the system (Savageau, 1971b; Shiraishi and Savageau, 1992). For example,

image(14)

measures the per cent change in the value of the dependent concentration variable Xi (or in Vi the flux through the pool of Xi) caused by a percentage change in the value of the parameter pj. Again, a positive sign indicates that the changes are in the same direction (both increase or both decrease); a negative sign indicates that the changes are in the opposite direction (one increases while the other decreases). The aggregate sensitivity of a given variable is defined as the Euclidean norm of the vector whose components are the individual parametric sensitivities for that variable. That is,

image(15)

These systems should be stable in the face of perturbations in their dependent state variables. That is, following a perturbation, the systems should return to their predisturbance state. The local stability of the steady state can be determined by applying the Routh criteria (Dorf, 1992). The magnitude of the two critical Routh conditions can be used to quantify the margin of stability (Savageau, 1976). The two critical Routh conditions are given by

F 1 a 11 + F2a22 < 0(16)
F 1 F 2(a11a22 + h12a21) > 0(17)

where F1 = V10 / X10 and F2 = V20 / X20 are the reciprocal of the turnover times for the X1 and X2 pools respectively.

Systems should respond quickly to changes in their environment (Savageau, 1975). Thus, another key property of the systems is their temporal response, which was determined by computer solution of the dynamic equations [Equations (7) (8) and (8’)]. At time zero, each intermediate concentration was set to a value 20% less than its steady-state value. The concentrations were then followed as a function of time from this initial condition, and the time for all the concentrations to settle within 1% of their final steady-state value was calculated and denoted by the symbol τ.

Mathematically controlled comparisons

To determine the differences in systemic behaviour between the reference model (bifunctional sensor, Fig. 1A) and the alternative model (monofunctional sensor, Fig. 1B) we use a technique known as mathematically controlled comparison (Savageau, 1972, 1976; Irvine and Savageau, 1985; Alves and Savageau, 2000a). This technique introduces mathematical controls to ensure that the differences observed in the systemic behaviour of alternative models are a result of the specific differences in the design and not some accidental difference. The parameters of the alternative model are fixed relative to those of the reference model by introducing constraints to ensure that the two models are as nearly equivalent as possible from both an internal and an external perspective.

Only the step that accounts for the dephosphorylation of the regulator is allowed to differ between the reference model and the alternative model. Therefore, to establish internal equivalence (Savageau, 1972, 1976; Irvine, 1991) between the two designs, we require the values for the corresponding parameters of all other steps in the two models to be the same. The step that accounts for the dephosphorylation of the regulator is then the only step that differs between the reference model and the alternative model. If we reason that loss or gain of an activation site on the regulator protein comes about by mutation, and that this mutation can cause changes in all the parameters of the process, then a mutation that converts a bifunctional sensor to a monofunctional sensor would change the parameters h22, h23, and β2 in Equation (2) to h’22, h’23 = 0, and β’2 in Equation (2). Alternatively, the parameters, h21, h22, h27, and β20 in Equation (8) would change to h’21 = 0, h’22, h’27 = 0, and β’20 in Equation (8).

Because we wish to determine the differences that are due solely to changes in the structure of the system, we need to specify values for h’22 and β’2 that minimize all other differences. This is accomplished by deriving the mathematical expression for a given steady-state property in each of the two models, equating these expressions to produce a constraint, and then solving the constraint equation for the value of a primed parameter (see Calculating the constraints for external equivalence). The process we have just described determines the maximal degree of external equivalence (Savageau, 1972, 1976; Irvine, 1991) between the two models. Once the two primed parameters have been determined in this manner, there are no more ‘free’ parameters that can be adjusted to reduce the differences, and all remaining differences can be attributed to the change in system structure from one with a bifunctional sensor to one with a monofunctional sensor. Having established the conditions on the parameters for maximal equivalence, we can proceed to analyse the two models and determine their remaining differences.

Calculating the constraints for external equivalence

Each constraint is established by requiring the value of a relevant systemic property to be the same in both designs. Because we have two primed parameters that need to be constrained, we require two different systemic properties to be invariant between the designs.

First, the value of h’22 is fixed by requiring the total gain of the system, L(X2,X5) + L(X2,X6) , to be the same in both designs. This total gain is determined from the explicit solution for X2[Y2 = log X2] in Equation (12)] by calculating the logarithmic gains as defined by the derivatives in Equation (13). Equating the results for each of the alternative designs allows the value of h’22 to be calculated as a function of the kinetic-order parameters in the reference model, which is taken to be the bifunctional case.

image(18)

By using this constraint we ensure that the output signal of the response regulator X2 is the same for both designs in responses to the aggregate of signals X5 and X6 that change its phosphorylation state. We also have constrained the alternative designs to have equal responses to the primary input signal X5[L(X2,X5)],or to the secondary input signal X6[L(X2,X6)]. The results of the comparative analysis in these other cases are qualitatively the same as those of the comparative analysis reported here (data not shown).

Second, the value of β’20 is fixed by requiring the steady-state concentrations of the corresponding variables to be the same in both designs. By equating the explicit solution for X2[Y2 = log X2 in Equation (12)] in each of the designs, and by utilizing the result in Equation (18), one is able to express the value of β’20 in terms of the independent variables and parameters of the reference model.

image(19)

In addition to making the steady-state value of X2 equal in the alternative designs, this value of β’20 also makes the steady-state value of X1[and X3 (= X7 − X1) and X4 (= X8 −X2)], and the steady-state values of the corresponding fluxes V2 (= V4) and V1 (= V3) equal in the alternative designs.

Numerical analysis

The analytical results give qualitative information that characterizes the effect of bifunctional versus monofunctional sensors in the models of Fig. 1. To obtain quantitative information about the comparisons, one must introduce specific values for the parameters and compare models (Alves and Savageau, 2000a). For this purpose we have randomly generated a large ensemble of parameter sets and selected 5 000 of these sets that define models consistent with various physical and biochemical constraints. These constraints include conservation of mass considerations, a requirement for positive signal amplification, and stability margins large enough to ensure local stability of the systems. A detailed description of these methods can be found in Alves and Savageau (2000b).

When applied to the current comparisons, the procedure is as follows. We select random values for all the unprimed parameters and for all the independent concentration variables (X5 through X8) in Equations (7) (8) and (8’). The values of the primed parameters in Equation (8’) are then fixed by the relationships in Equations (18) and (19). The analytical solution in Equations (11) and (12) determines the steady-state values for dependent state variables (X1 and X2), complementary variables (X3 and X4), fluxes, logarithmic gains and parameter sensitivities. This information in turn determines the values for all the parameters following Equations (6) and (8). Taken together, this information determines the values for all the parameters in the original equations [Equations (1) (2) and (2’)]. Mathematica™ (Wolfram, 1997) was used for all the numerical procedures.

To interpret the ratios that result from our comparative analysis we use Density of Ratios plots as defined in Alves and Savageau (2000c). The primary density plots from the raw data have the magnitude for some property of the reference model on the x-axis and the corresponding ratio of magnitudes (reference model to alternative model) on the y-axis. Secondary density plots are constructed from the primary plots by the use of moving quantile techniques with a window size of 500. The slope in the secondary plot measures the degree of correlation between the quantities plotted on the x- and y-axes.

Acknowledgements

This work was supported in part by a joint PhD fellowship PRAXIS XXI/BD/9803/96 granted by PRAXIS XXI through Programa Gulbenkian de Doutoramentos em Biologia e Medicina (R.A), U.S. Public Health Service Grant RO1-GM30054 from the National Institutes of Health (M.A.S.), and U.S. Department of Defense Grant N00014-97-1-0364 from the Office of Naval Research (M.A.S.). We thank Drs Susana Nery, Armindo Salvador, Vic DiRita and Alex Ninfa for critically reading early versions of this manuscript and making useful comments. We also thank an anonymous referee for a very careful and constructive review of the original manuscript. This work was completed while MAS was a guest at the Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France. He thanks Drs A. Carbonme. M. Gromov and F. Képès for providing resources, intellectual stimulation and generous hospitality.

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