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f(x). 1. x. 2. Increasing attractants or Decreasing repellents. The main idea is to: START with a fine-tuned model of chemotaxis network that:. : state variables : reaction kinetics : reaction constants : external stimulus. Augmented system. Discretizing s into H points.

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Increasing attractants or Decreasingrepellents

  • The main idea is to:
  • STARTwith a fine-tuned model of chemotaxis network
  • that:

: state variables

: reaction kinetics

: reaction constants

: external stimulus

Augmented system

Discretizing s

into H points

  • reproduces key features of experiments (adaptation times to small and large ramps, perfect adaptation of the steady state value of CheYp)
  • is NOT robust

Demethylation rate (km1,km2) and methylation rate (k1c,k2c) versus phosphorylation rate(k9)

There are n system variables, m system parameters and 1 small variable to allow near perfect adaptation, giving a total of (n+m+1)H equations and (n+m+1)H variables.

Demethylation rate (km1,km2) and methylation rate (k1c,k2c) versus dephosphorylation rate(kmy)

  • AUGMENT the model explicitly with the requirements that:
  • steady state value of CheYp
  • values of reaction rate constants,

The steady state concentration of proteins in the network must satisfy:

The steady state concentration of CheYp must satisfy:

At the same time, the reaction rate constants must be independent of stimulus:

are independent of the external stimulus, s, thereby achieving robustness of perfect adaptation.

  • Chemical reactions:
  • Ligand binding
  • Methylation
  • Phosphorylation








: allows for near-perfect adaptation

= CheYp



works well unfortunate case unfortunate case

Demethylation rate (km1,km2) and methylation rate (k1c,k2c) versus dephosphorylation rate(kmb)

  • Newton-Raphson, to solve for the steady state of augmented system


  • Dsode (stiff ODE solver), to verify Newton-Raphson result for different ranges of external stimulus
  • Simplest multidimensional root finding method
  • Efficient way of converging to a root with a sufficiently good initial guess.

Robust Perfect Adaptation in Bacterial Chemotaxis

Yang Yang & Sima Setayeshgar Department of Physics, Indiana University, Bloomington

My research: I am a physics graduate student working in theoretical biophysics.

What is E coli ? Why study it?

E. coli is a single-celled organism that lives primary in our intestines. It is approximately 1-2 microns long and 1 micron in diameter, and weights 1 picogram. Each cell has 4-6 flagella, approximately 10-20 microns long, driven by an intracellular rotary motor operated by the protonmotive force.


The biochemical basis of robustness of perfect adaptation is not as yet fully understood. In this work, we develop a novel method for elucidating regions in parameter space of which the E. coli chemotaxis network adapts perfectly:

  • The shapes of resulting manifolds determine relationships between reaction parameters (for example, methylation and phosphorylation rates) that must be satisfied in order for the network to exhibit perfect adaptation, thereby shedding light on biochemical steps and feedback mechanisms underlying robustness.
  • Given lack of complete data on values of in vivo reaction rates, the numerical ranges of the resulting manifolds will shed light on values of unknown or partially known parameters.
  • It is considered to be an ideal model organism for understanding the behavior of cells at the molecular level from the perspectives of several scientific disciplines-anatomy, genetics, chemistry and physics- because of :
  • Ease of experimentation, through microscopy and genetic analysis
  • Small genome (4288 genes), most of which encode proteins

Broader impact



This method should have applicability to other cellular signal transduction networks and engineered systems that exhibit robust homeostasis.

Ref: H. C. Berg, “Motile behavior of bacterial”, Physics Today, January 2000


Computational results

What is the chemotaxis signal transduction network in E. coli?

Some examples of the parameter space giving perfect adaptation

Obtained for sslow< s < shigh

E. coli exhibits an important behavioral response known as chemotaxis - motion toward desirable chemicals (usually nutrients) and away from harmful ones - which is also shared by various other prokaryotic and eukaryotic cells. The cell’s motion consists of series of “runs” puntuated punctuated by “ tumbles”.

Verification for different stimuli

s > shigh

The chemotaxis signal transduction pathway in E. coli – a network of ~50 interacting proteins – converts an external stimulus (change in concentration of chemoattractant / repellent) into an internal stimulus (change in concentration of intracellular response regulator, CheYp) which in turn interacts with the flagella motor to bias the cell’s motion.

It is used as a well-characterized model system for the study of properties of (two-component) cellular signaling networks in general.

Chemotaxis in E. coli involves temporal measurement of the change in concentration of an external stimulus. This is achieved through the existence of fast and slow reaction time scales, in the chemotaxis signal transduction network: fast measurement of the current external concentration is compared with the cell’s “memory” of the concentration some time ago to determine whether to extend a run in a given direction or to tumble, thereby randomly selecting a new direction.



Ref: P. A. Spiro, J. S. Parkinson, and H. G. Othmer, “A model of excitation and adaptation in bacterial chemotaxis”,

Proc. Natl. Acad. Sci. USA 94, 7263(1997)

What is prefect adaptation? Why is it important?

Perfect adaptation is an important and generic property of signaling systems, where the response (e.g. running bias in chemotaxis) returns precisely to the pre-stimulus level while the stimulus persists.

This property allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli.

Thus, E. coli is able to respond to changes in chemoattractant concentrations spanning 5 orders of magnitude! Similarly, the vertebrate visual system responds to changes in light intensity spanning 10 orders of magnitude during the night-day cycle.

Single result from fine-tuned model




Ref: P. A. Spiro, J. S. Parkinson, and H. G. Othmer, “A model of excitation and adaptation in bacterial chemotaxis”, Proc. Natl. Acad. Sci. USA 94, 7263(1997)

  • Conclusions
          • Successful implementation of the augmented model of the chemotaxis signal transduction network in E. coli that explicitly takes into account robust perfect adaption
          • Preliminary results on projections of robustness manifolds in parameter space

What is robustness? Why is it important?

The E. coli chemotaxis signal transduction network exhibits robust perfect adaptation, where the concentration of CheYp returns to its prestimulus value despite large changes in the values of many of the biochemical reaction rate constants. These rate constants depend on concentrations of enzymes, which are often present in small copy numbers, making fluctuations in their numbers significant.

  • Work in progress
          • Complete construction of manifolds in parameter space, allowing insight into parameter dependence giving rise to robustness.
          • Extend to other signaling systems, such as phototransduction.



Ref: N.Barkai & S. Leibler, “Robustness in simple biochemical network”, Nature 387, 913(1997)