Bioinformatics Advance Access originally published online on January 31, 2007
Bioinformatics 2007 23(7):875-881; doi:10.1093/bioinformatics/btm028
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Robustness analysis of the E.coli chemosensory system to perturbations in chemoattractant concentrations
Institute of Microbial Technology, Sector 39-A, Chandigarh-160 036, India
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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Motivation: Cells of Escherichia coli sense and move toward chemical attractants. This is done through an intricate sensory system that eventually directs the movements of flagellae which regulate the ‘runs’ and ‘tumbles’ of the cells. Under realistic conditions, chemical stimuli often fluctuate due to noise from the environment. The effect of noise on the chemosensory system has been investigated here through the sensitivity coefficients of the concentrations of four key proteins—the phosphorylated forms of CheA, CheB and CheY, and the FliM-CheY
P complex—that govern chemotactic motility. The letter P denotes phosphorylation.
Results: All sensitivities increased with time and then stabilized. However, the four sets of sensitivities differed in their magnitudes and the durations of their transient phases before stabilization. CheAP was the least sensitive and CheYP the most sensitive. Moreover, while the sensitivities of CheAP, CheBP and CheYP increased with chemoattractant concentration, that of the FliM complex decreased. These differences have been interpreted in terms of the mechanism of the chemosensory system and they have important implications for practical applications of chemotaxis.
Contact: pratap{at}imtech.res.in
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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When presented with a chemical stimulus, many bacteria orientate their normal paths of movement either toward (for a chemoattractant) or away from (for a chemorepellant) the stimulus. The mechanisms underlying such directed movement or chemotaxis have been analyzed and modeled in recent studies with Escherichia coli and Bacillus subtilis. There are both differences and similarities between these two organisms (Kirby et al., 1999; Rao et al., 2004; Sourjik and Berg, 2002a). The aim of this communication is not, however, to discuss these aspects but to analyze the robustness of the responses of one organism, E.coli, to perturbations in the concentration of a chemoattractant at different values. E.coli was chosen both because of the simplicity of its chemotactic network (Baker et al., 2006; Rao et al., 2004) and because of the many applications of E.coli in both fundamental research and industrial processes.
Two aspects of chemotactic motility constitute the essential background for an understanding of the present work. One aspect is the nature of chemotaxis. The cells of both E.coli and B.subtilis have long helical flagellae attached to rotary motors; these flagellae guide the movements of the cells, similar to the fins of fishes. E.coli typically has six or seven motors distributed over the cell surface. When the motors spin counterclockwise, they cause the cells to ‘run’ i.e. move along straight paths. Clockwise spins generate ‘tumbling’ of the cells, i.e. changes in the directions of the runs. Movements of cells thus comprise alternate phases of runs and tumbles (Fig. 1). In the absence of any stimulus, the movements are random, thereby creating a uniform distribution of cells in an environment. When a chemical stimulus is applied, the sensory system of a cell causes changes in the switching frequency between the two directional rotations of the flagellar motors, thereby changing the tumbling frequency. A chemoattractant generates changes that propel a swarm of cells toward the stimulus. A chemorepellant has the opposite effect.
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An intricate sensory system receives and transmits external stimuli to motors that control runs and tumbles. The response of the system to changes in a stimulus is thus a critical regulator of chemotactic motions. This is the second aspect of chemotaxis, and it is the subject of this work. The functioning of a cell's sensory system is briefly described in the next section. Using this description and its pictorial representation, we explore the robustness of the system to perturbations in the concentration of a chemoattractant. The perturbations are observed typically in the form of ‘noise’ in the temporal variations of key concentrations. Although the effect of noise on cellular behavior is widely acknowledged (Kaern et al., 2005; Raser and O'Shea, 2005), its role in bacterial chemotaxis has received limited attention (Korbokova et al., 2004; Samuel and Berg 1995). Since noise is commonly encountered in real situations, a quantitative analysis of chemotactic responses to perturbations is an important aspect of the understanding and control of cellular processes. The chemoattractant (or repellant) is the main driving force for cellular motions and, since it comes from the environment, it is also a dominant source of noise. Therefore the robustness of the E.coli chemosensory system to small disturbances in the concentration of a chemoattractant has been analyzed here.
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2 CHEMOSENSORY SYSTEM OF E.COLI |
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TOPABSTRACT1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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Figure 2 presents a protein network map of the E.coli chemotaxis system. The response of the system is encoded by six essential Che genes—CheA, CheB, CheR, CheW, CheY and CheZ—and five chemoreceptor genes (Baker et al., 2006; Wadhams and Armitage, 2004). The CheA protein catalyzes the transfer of phosphoryl groups from ATP to one of its histidine imidazole side-chains, from where it is subsequently transferred to an aspartyl side-chain in the CheY protein. The phosphorylated CheY (CheY
P) then dissociates from CheA, diffuses and binds to the flagellar motor switch. The bound CheYP functions as an allosteric regulator in shifting the bidirectional spin equilibrium of the motors toward clockwise rotation (Alon et al., 1998; Elowitz et al., 1999), thereby biasing the tumbling of the cells toward the chemoattractant. Fluorescence resonance measurements suggest that non-phosphorylated CheY does not bind to the flagellar motors (Sourjik and Berg, 2002b). Thus, CheYP is the main protein that regulates the chemotactic movements of the cells.
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The concentrations of CheY
P are modulated through the five chemoreceptors—Tsr, Tar, Tap, Trg, and Aer—which are present as large multimeric complexes with the CheA and CheW proteins (Francis et al., 2004). An E.coli cell typically has more than 10 000 receptor subunits linked with a comparable number of CheA and CheW subunits (Baker et al., 2006). As seen in Figure 2, the receptors are outside the cytoplasmic membrane whereas the rest of the chemosensory network is in the cytosol. The receptor complexes thus function as antennae that receive chemical signals and transmit them to the protein network, which then regulates the flagellar motors. A chemoattractant ligand binds to a receptor complex and causes reversible methylation of the receptor. Methylation contributes to the robust adaptation of bacterial chemotaxis. On this basis, Barkai and Leibler (1997) proposed in their two-state model that the methylated receptor (as a multimer with CheA and CheW) may exist in either of two functional states—active and inactive. By phosphorylating the response regulators, CheY, the active form sends a tumbling signal to the motors. Thus, the average number of receptors in the active state determines the tumbling frequency and hence the quantitative movement of an ensemble of cells toward a chemoattractant. Barkai and Leibler's model has been the origin of more elaborate models, one of which (Rao et al., 2004) is the basis of this work.
The functions of the six main Che proteins are summarized in Table 1. It should be clarified that this table and the description presented earlier contain a condensed account highlighting the essential features that are relevant to the present analysis. More detailed accounts are available elsewhere (Baker et al., 2006; Rao et al., 2004; Yi et al., 2000).
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3 CHEMOTAXIS MODEL AND ROBUSTNESS ANALYSIS |
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TOPABSTRACT1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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Rao et al. (2004) and Kollmann et al. (2005) have pointed out some limitations in Barkai and Leibler's (1997) model of E.coli chemotaxis. The former extended this model by incorporating Sourjik and Berg's (2002b) model for the phosphorylation cascade. Recall that in the two-state model the receptor complexes exist in either an active (TA) or an inactive (TI) state. The concentration of active receptors at any time t is:
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| (1) |
i(L) is the probability that the complex Ti is active when the concentration of the chemoattractant is L.
As a corollary of Equation (1), the concentration of inactive receptors is:
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| (2) |
According to the mechanism shown in Figure 2 and described previously, four key proteins are involved in the chemoresponse system: the phosphorylated forms of CheA (CheA), CheB (CheB) and CheY (CheY), and the complex of CheYP and the motor switching protein FliM. Their rates of change are described by the following equations (Rao et al., 2004).
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| (3) |
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| (4) |
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| (5) |
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| (6) |
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Methylation reactions are considered to follow Michaelis–Menten kinetics. In formulating these equations Rao et al. (2004) used Morton-Firth et al.'s (1999) observation that CheR binds only to inactive receptors and CheBP binds only to active receptors. Then the rate of methylation of the receptor Ti is
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| (7) |
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| (8) |
With Equations (7) and (8), simple mass balances yield the following differential equations for the receptor complexes.
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| (9) |
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| (10) |
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| (11) |
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| (12) |
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| (13) |
The robustness of the variables Ap, Bp, Mp, Yp, T0, T1, T2, T3 and T4 to perturbations in a parameter may be expressed by their sensitivities (Kitano, 2004; Sourjik and Berg, 2002a). To compute these sensitivities, let equations (3)–(6)
and (9)–(13) be written compactly as
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| (14) |
denotes the vector of variables and 
the parameters. The sensitivity of yi with respect to the parameter pj is defined as:
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| (15) |
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| (16) |
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| (17) |
This equation states implicitly that sensitivities may change with time. This means the disturbed trajectory of a variable may not stay a constant distance away from the original trajectory. The time-dependence of the sensitivities has important implications for bacterial motility, and sometimes for the survival of the cells themselves (Baker et al., 2006; Kaern et al., 2005).
Jik is the (i, k)-th element of the Jacobial matrix of 
, i.e. the matrix of first partial derivatives:
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| (18) |
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| (19) |
Equations (3)–(6), (9)–(13) and (17) were solved with the parameter values and initial conditions for 
shown in Table 2. The superscript T here signifies the transpose of a vector. The (mean) concentration of the chemoattractant was varied from 0.001 to 5 nM, and the time-dependent sensitivities were determined for perturbations at each concentration. The initial conditions for Ap, Yp, Mp and Bp are zero since these proteins have not been phosphorylated at t = 0. Similarly, the absence of methylation at t = 0 is expressed by the zero initial conditions for T1 to T4 and T0 = 5, signifying five chemoreceptors.
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4 RESULTS AND DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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As explained in Sections 2 and 3, and portrayed in Figure 2, phosphorylated CheY (Yp in the model) controls the rotations of the flagellar motors and consequently the direction of movement of the bacteria. However, the concentrations of CheY
P are modulated by the chemoreceptors, which exist in complexation with CheA and CheW (Baker et al., 2006; Francis et al., 2004). The chemoreceptors are at the front end of the chemosensory system and it is their binding to chemical ligands that triggers the chemotactic responses (Barkai and Leibler, 1997; Sourjik and Berg, 2002b). This binding generates changes in the methylation of glutamate residues in the receptor domains, which are mediated by CheB and CheR (Zhulin, 2001). Therefore the concentrations of phosphorylated CheA, CheB and CheY, and the FliM–CheYP complex are fundamental determinants of chemotatctic responses. Hence the sensitivities of these four variables were studied as functions of time and the concentration, L, of the chemoattractant with respect to small disturbances in that concentration. The effect of CheR was not analyzed because it does not affect the adaptation of E.coli to chemoattractants (Alon et al., 1999). Although CheR mediates covalent receptor modifications that govern adaptation, the input–output changes occur over several seconds or minutes whereas changes in CheY phosphorylation frequency are at least ten times faster (Baker et al., 2006; Wadhams and Armitage, 2004).
The computed sensitivity profiles are presented in Figures 3–6
. It needs to be clarified here that the plots labeled L = 0 really correspond to L = 0.001 nM since obviously a zero concentration would generate no chemotaxis. We observe first that the sensitivities themselves differ significantly across the figures, both in magnitudes and in their variations with time. The CheAP protein is the least sensitive and CheYP the most sensitive, the latter sensitivities being an order of magnitude larger.
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We recognize at this stage that chemotactic motility in E.coli is regulated by controlling the phosphorylation of the CheY response regulator through the CheA histidine kinase (Alon et al., 1998; Rao et al., 2004; Sourjik and Berg, 2002b). The addition of an attractant inhibits the CheA kinase activity and results in a rapid decrease in the concentration of CheY
P (Berkovich et al., 1989). The cells adapt to the changed situation by altering the methylation state of the receptors (Barkai and Leibler, 1997; Kirby et al., 1999). The wide differences in the sensitivities of CheAP and CheYP even for small perturbations in chemoattractant concentration are commensurate with the accelerated reductions in CheYP concentrations triggered by much smaller changes in CheA kinase, as referred to above.
The sensitivities of phosphorylated CheB (Fig. 5) and the FliM complex with CheYP (Fig. 6) are between those of CheAP and CheYP. The signal flow diagram of the chemosensory network (Fig. 2) and the description in Section 2 indicate that the functioning of CheA and CheY are interrelated with those of two other Che proteins: CheB and CheW. The latter forms multimeric complexes with CheA and the chemoreceptors (Francis et al., 2004). Although the stoichiometry of this process is still not clear, it is known that CheW regulates CheA activity in a biphasic manner (Gegner et al., 1992). One result of this regulation is modulation of CheA activity in response to changes in the binding of the ligands of the chemical stimulant to the receptor complex. This implies that CheAP concentrations are likely to be more robust (i.e. less sensitive) to fluctuations in the chemoattractant concentration than are CheYP concentrations. Figures 3 and 4 conform to this mechanistic interpretation, thus establishing the credibility of the sensitivity plots.
The differences between the sensitivity plots suggest a correlation with the concentrations of the proteins themselves. Although measurements of transient protein concentrations in a noise-affected environment are not available, Figures 3–6, the steady state sensitivities reported by others (Alon et al., 1999; Rao et al., 2004) and known protein expressions (Table 1 of Alon et al., 1999) together indicate that proteins generated in low concentrations are more sensitive to external perturbations than those in higher concentrations. This observation extends to other bacterial systems also (Patnaik, 1999, 2005a), where the focus was on protein production rather than bacterial motility. These studies indicate that variables in high concentrations function effectively as sinks for incoming disturbances and are thus more robust to noise in the feed streams than those in low concentrations.
This rationale regarding robustness and fluctuations in the extra-cellular concentration of a chemoattractant blends with similar explanations for the creation of differences within a cell or between two cells in a population as a result of intra-cellular noise (Kaern et al., 2005; Korbokova et al., 2004; Raser and O'Shea, 2005). Molecules such as DNA, RNA and many proteins, which are present in small (numerical or gravimetric) concentrations, are also strongly susceptible to intra-cellular noise. These observations emphasize the critical effect of sensitivity to noise on cellular performance.
In the chemosensory system of E.coli, CheB provides a negative feedback loop since the rate of demethylation, which is catalyzed by CheB, is proportional to the activity of CheA (Anand and Stock, 2002). In many biological and engineering systems negative feedback has a stabilizing effect, thus increasing the robustness of the system to external disturbances (Kitano, 2004; Stelling et al., 2004). In E.coli chemotaxis, integral feedback control is a necessary and sufficient condition to explain robust adaptation (Yi et al., 2000). This feedback control of the methylation–demethylation equilibrium exerts a stronger influence on CheA than on CheY, as explained earlier (Barkai and Leibler, 1997; Borkovich et al., 1989; Kirby et al., 1999).
An interesting difference between Figures 3–5, on one hand, and Figure 6 on the other is that the former sensitivities increase with L, the concentration of the chemoattractant, whereas that of the FliM-CheYP complex decreases. While an elaborate account of the flagellar motor switching mechanism is outside the scope of this discussion, briefly this difference possibly arises because the binding of CheYP to FliM is much less cooperative than motor switching. On introducing a chemical attractant, the amount of CheYP bound to FliM decays exponentially (Sourjik and Berg, 2000, 2002b). From this perspective, the increase of sensitivities in Figures 3–5 with the attractant concentration is consistent with the opposite trend for the FliM -CheYP complex (Figure 6).
The prominent variations of the sensitivity profiles may appear to contradict numerous reports (Alon et al., 1999; Barkai and Leibler, 1997; Kollmann et al., 2005; Yi et al., 2000) that the chemotaxis of E.coli exhibits robust adaptation. Robustness indicates that certain characteristic functional features of a (biological) system are maintained in the presence of external and internal perturbations (Kitano, 2004; Stelling et al., 2004). However, it may be noted that while adaptation is robust, other network properties are not. For instance, the steady state concentrations of phosphorylated CheY and the adaptation time are not robust (Alon et al., 1999; Kollmann et al., 2005; Rao et al., 2004). Therefore the sensitivity plots are plausible biologically and do not contradict the robustness concept. Moreover, network design considerations other than adaptation are also important factors controlling chemotactic behavior (Rao et al., 2004). This compatibility is further underlined by the fact that the sensitivities in Figures 3–6 stabilize within time spans at least an order or magnitude smaller than the adaptation times for variations in the concentrations of the Che proteins and the chemoreceptors (Alon et al., 1999; Korbokova et al., 2004). This observation also contains a cautionary message: noise from the environment can upset the chemotaxis system much before the cells have been able to adapt to the evolving situation. If this happens, chemotactic motility may degenerate into psueudo-random behavior. This ability of environmental disturbances to displace a population of cells from one stationary state to a different kind of state or to stochastic chaos has been reported for fermentations with recombinant E.coli producing ß-galactosidase (Patnaik, 2003) and Saccharomyces cerevisiae cultivation for ethanol (Patnaik, 2005b). Such stochastic transitions caused by variable sensitivities may have strong practical consequences. For instance, stochastic resonance between external noise and genetic noise promotes the evolution of certain phenotypes that are resistant to external intervention (Kaern et al., 2005; Korobokova et al., 2004; Raser and O'Shea, 2005). This phenomenon has been invoked to explain the persistence of certain diseases even after treatment by antibiotics (Balaban et al, 2004; Seldman and Seldman, 2002).
The present results suggest that, besides affecting gene expression (Kaern et al., 2005) and product formation, externally induced fluctuations may also change cellular motility in a chemically nonuniform environment. Since product synthesis depends on the availability and dispersion of chemical nutrients, altered chemotactic behavior may change both substrate utilization and product formation patterns. Robustness results of the type presented here therefore have important implications in diverse areas such as pollution control, disease management and bioreactor optimization, where chemically induced cellular movements and metabolic changes are critical factors affecting the performance.
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NOMENCLATURE |
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TOPABSTRACT1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Alfonso Valencia
Received on September 13, 2006; revised on January 2, 2007; accepted on January 23, 2007
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REFERENCES |
|---|
|
TOPABSTRACT1 INTRODUCTION2 CHEMOSENSORY SYSTEM OF...3 CHEMOTAXIS MODEL AND...4 RESULTS AND DISCUSSIONNOMENCLATUREREFERENCES |
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Alon U, et al. Response regulator output in bacterial chemotaxis. EMBO J, ( (1998) ) 17, : 4238–4248.[CrossRef][ISI][Medline].
Alon U, et al. Robustness in bacterial chemotaxis. Nature, ( (1999) ) 397, : 168–171.[CrossRef][Medline].
Anand GS, Stock AM. Kinetic basis for the stimulatory effect of phosphorylation on the methyesterase activity of CheB. Biochemistry, ( (2002) ) 41, : 6752–6760.[CrossRef][Medline].
Baker MD, et al. Systems biology of bacterial chemotaxis. Curr. Opin. Microbiol, ( (2006) ) 9, : 187–192.[CrossRef][ISI][Medline].
Balaban NQ, et al. Bacterial persistence as a phenotypic switch. Science, ( (2004) ) 305, : 1622–1625.
Barkai N, Leibler S. Robustness in simple biochemical networks. Nature, ( (1997) ) 87, : 913–917..
Borkovich KA, et al. Transmembrane signal transduction in bacterial chemotaxis involves ligand–dependent activation of phosphate group transfer. Proc. Natl. Acad. Sci. USA, ( (1989) ) 86, : 1208–1212.
Elowitz MB, et al. Protein mobility in the cytoplasm of Escherichia coli. J. Bioteriol, ( (1999) ) 181, : 197–203..
Francis NR, et al. Three–dimensional structure and organization of a receptor/signaling complex. Proc. Natl. Acad. Sci. USA, ( (2004) ) 101, : 17480–17485.
Geevan CP, et al. A mathematical model for insulin kinetics III: sensitivity analysis of the model. J. Theor. Biol, ( (1990) ) 147, : 255–263.[ISI][Medline].
Gegner JA, et al. Assembly of an MCP receptor, CheW, and kinase CheA complex in the bacterial chemotaxis signal transduction pathway. Cell, ( (1992) ) 70, : 975–982.[CrossRef][ISI][Medline].
Kaern M, et al. Stochasticity in gene expression: from theories to phenotypes. Nature Rev. Genet, ( (2005) ) 6, : 451–464.[CrossRef][ISI][Medline].
Kitano H. Biological robustness. Nature Rev. Genet, ( (2004) ) 5, : 826–837.[CrossRef][ISI][Medline].
Kollmann M, et al. Design principles of a bacterial signalling network. Nature, ( (2005) ) 438, : 504–507.[CrossRef][Medline].
Korobokova E, et al. From molecular noise to behavioral variability in a single bacterium. Nature, ( (2004) ) 428, : 574–578.[CrossRef][Medline].
Kirby JR, et al. CheY–dependent methylation of the asparginase receptor, McpB, during chemotaxis in Bacillussubtilis. J. Biol. Chem, ( (1999) ) 274, : 11092–11100.
Morton-Firth CJ, et al. A free–energy–based stochastic simulation of the Tar receptor complex. J. Mol. Biol, ( (1999) ) 286, : 1059–1074.[CrossRef][ISI][Medline].
Patnaik PR. Spectral analysis of a nonideal bioreactor for a recombinant fermentation with run-away plasmids. Bioproc. Eng, ( (1999) ) 20, : 31–36.[CrossRef].
Patnaik PR. An integrated hybrid neural system for noise filtering, simulation and control of a fed-batch recombinant fermentation. Biochem. Eng. J, ( (2003) ) 15, : 165–175.[CrossRef].
Patnaik PR. Process analysis in a disturbed environment during oscillatorymetabolism of Saccharomyces cerevisiae. Ind. J. Biotechnol, ( (2005a) ) 4, : 201–208..
Patnaik PR. The extended Kalman filter as a noise modulator for continuous yeast cultures under monotonic, oscillating and chaotic conditions. Chem. Eng. J, ( (2005b) ) 108, : 91–99.[CrossRef].
Rao CV, et al. Design and diversity in bacterial chemotaxis: a comparative study in Escherichia coli and Bacillus subtilis. PloS Biol, ( (2004) ) 2, : 239–252.[CrossRef][ISI].
Raser JM, O'Shea EK. Noise in gene expression: origins, consequences, and control. Science, ( (2005) ) 309, : 2010–2013.
Samuel AD, Berg HC. Fluctuation analysis of rotational speeds of thebacterial flagellar motor. Proc. Natl. Acad. Sci. USA, ( (1995) ) 92, : 3502–3506.
Seldman JG, Seldman C. Transcription factor haploinsufficiency: when half a loaf in not enough. J. Clin. Invest, ( (2002) ) 109, : 451–455.[CrossRef][ISI][Medline].
Sourjik V, Berg HC. Localization of components of the chemotaxis machinery of Escherichia coli using fluorescent protein fusions. Mol. Biol, ( (2000) ) 37, : 740–745..
Sourjik V, Berg HC. Receptor sensitivity in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA, ( (2002a) ) 99, : 123–127.
Sourjik V, Berg HC. Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer. Proc. Natl. Acad. Aci. USA, ( (2002b) ) 99, : 12669–12674.
Stelling J, et al. Robustness of cellular functions. Cell, ( (2004) ) 118, : 675–685.[CrossRef][ISI][Medline].
Wadhams GH, Armitage JP. Making sense of it all: bacterial chemotaxis. Nature Rev. Mol. Cell. Biol, ( (2004) ) 5, : 1024–1037.[CrossRef][ISI][Medline].
Yi T-M, et al. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc. Natl. Acad. Sci. USA, ( (2000) ) 97, : 4649–4653.
Zhulin IB. The superfamily of chemotaxis transducers: from physiology to genomics and back. Adv. Microb. Physiol, ( (2001) ) 45, : 157–198.[CrossRef][ISI][Medline].
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