1. Yang Yang & Sima Setayeshgar
Near-Perfect Adaptation in E. coli Chemotaxis Signal Transduction Network
Yang Yang & Sima Setayeshgar
Jan, 2007

2. E. coli lives primary in our intestines
1-3 microns long and 1 micron in diameter
Each cell has 4-6 flagella with approximately microns in length
Small genome (4288 genes), most of which encode proteins
ease of experimentation, through microscopy and genetic analysis
It is 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.
static.howstuffworks.com/gif/cell-ecoli.gif

3. Bacterial Chemotaxis
Increasing attractants or Decreasing repellents
Run Tumble
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” punctuated by “ tumbles”.

4. Chemotaxis signal transduction network in E. coli
W=CheW
Coupling CheA to MCPs
A=CheA
Histidine kinase
B=CheB
CheBp demethylate MCPs
R=CheR
Methylate MCPs
Y=CheY
In charge of the probality that flagella changing rotation direction
Z=CheZ
Dephosphoryte CheYp
With approximately 50 interacting proteins , the network converts an external stimulus into an internal stimulus 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.

5. Chemical reactions ligand binding Methylation Phosphorylation
T4p
T2p
T3p
LT3
LT4
LT4p
LT2
LT3p
LT2p
Full realistic model
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.

6. Perfect adaptation
It is an important and generic property of signaling systems, where the response returns precisely to the pre-stimulus level while the stimulus persists.
Steven M., et al. Journal of bacteriology. 1983
This property allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli.

7. Robustness
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.
U. Alon et al. Nature,1999
Recent works have highlighted the fact that this important feature of the network must be robust to changes in network parameters.
In engineered systems, this property is achieved through integral feedback control.
Tau-Mu Yi* et al. Biophysics,2000

8. shed light on values of unknown or partially known parameters
Motivation
QUESITON: The biochemical basis of robustness of perfect adaptation is not as yet fully understood.
we develop a novel method for elucidating regions in parameter space of which the E. coli chemotaxis network adapts perfectly:
shedding light on biochemical steps and feedback mechanisms underlying robustness
shed light on values of unknown or partially known parameters

9. Algorithm START with a fine-tuned model of chemotaxis network that:
reproduces key features of experiments
is NOT robust
AUGMENT the model explicitly with the requirements that:
steady state value of CheYp
values of reaction rate constants,
are independent of the external stimulus, s, thereby achieving robustness of perfect adaptation.
: state variables
: reaction kinetics
: reaction constants
: external stimulus
(adaptation times to small and large ramps, perfect adaptation of the steady state value of CheYp)

10. Augmented system
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:
: allows for near-perfect adaptation
= CheYp
Decretizing s
into H points

11. Implementation
Newton-Raphson, to solve for the steady state of augmented system:
Simplest multidimensional root finding method
Efficient way of converging to a root with a sufficiently good initial guess.
For multidimensional spaces, it is very easy to fail to find the solution, the version of Newton-Raphson code we use is very powerful that it can retrack the slope to get the solutions. x
f(x) 1 2
Works well unfortunate case fortunate case 3

12. Implementation
Dsode (stiff ODE solver), to verify Newton-Raphson result for different ranges of external stimulus by solving:

13. Working progress
Exploring the parameter spaces of E. coli chemotaxis signaling transduction network
Exploring the unknown parameter ranges of chemotaxis signaling transduction network which has two regulation proteins—CheY1 , CheY2

14. Exploring the parameter spaces of E
Exploring the parameter spaces of E. coli chemotaxis signaling transduction network
Exploring pairwise trajectory by setting two parameters as unknowns in the augmenting system
Exploring three dimensional parameter space by setting three parameters as unknowns in the augmenting system

15. Parameter spaces • less than 5% • less than 3% • less than 1%
E .coli
Pairwise result: d surface result:
Relative change of CheYp:
• less than 5%
• less than 3%
• less than 1%
• pairwise trajectory

16. Parameter spaces • less than 5% • less than 3% • less than 1%
E. coli
Pairwise result: d surface result:
Relative change of CheYp:
• less than 5%
• less than 3%
• less than 1%
• pairwise trajectory

17. Parameter spaces • less than 5% • less than 3% • less than 1%
E .coli
Pairwise result: d surface result:
Relative change of CheYp:
• less than 5%
• less than 3%
• less than 1%
• pairwise trajectory

18. Consistency with recent work by Bernardo A. mello and Yuhai Tu
In order to hold the perfect adaptation, their simulation (perfect
and near-perfect adaptation in a model of bacterial chemotaxis,
biophysical journal, 2003) shows that:
the phosphorylation rates of CheB(kb) or CheY(ky) are proportional to CheA autophosphorylation rate(k8-13).
methylation rates of n+1 methylation level (k1c-4c) are proportional to demethylation rates of n methylation level (km1-m4)
Our simulation of parameter space also shows linearly or near-linearly relationship as indicate above although we are using a different model of the chemotaxis transduction network*.

19. CheB(kb) and CheY(ky) are proportional to CheA autophosphorylation rate(k8-13)
*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997) while the inset is not since the literature value is zero for k11.

20. k1c/km1, k2c/km2, k3c/km3, k4c/km4 are linearly related:
*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997).

21. Exploring unknown parameter values for signal transduction network for two CheY system
Rhodobacter sphaeroides, Caulobacter crescentus, and several nitrogen-fixing rhizobacteria have multiple cheY which one of the CheY functions as the primary motor-binding protein while the others work as a phosphatase sink in order to compensate the lack of CheZ protein.
The Two CheY system shows chemotaxis behaviors which is similar to E. coli.
Our work:
Reproduce the key feature of chemotaxis behavior in two CheY system by replacing CheZ with CheY2.
CheZ protein is responsible for CheY dephosphorylation , but in some cases, there is no CheZ protein in the system. In order to keep the concentration of CheY regulator in the right range, the system produces two CheYs: CheY1 and CheY2. CheY1 has the same function as single CheY case which interact with the flagella motor while CheY2 only works as a phosphatase sink.

22. Simulation result for two CheY system
Modify the augment system by introducing [CheY2] and CheY2 (de-)phosphorylation rates.
Exploring the parameter values of CheY2 (de-)phosphorylation rates which can give perfect adaptation.
Relative change of CheYp:
• less than 5%
• less than 3%
• less than 1%
• pairwise trajectory
Other parameter value were set as the literature value except
Kb= 1e+6 M-1s-1 instead of 8e+5 M-1s-1.

23. Conclusions Work in progress
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 of E. coli and two CheY system
Work in progress
Complete construction of manifolds in parameter space, allowing insight into parameter dependence giving rise to robustness

24. Future work
This method should have applicability to other cellular signal transduction networks and engineered systems that exhibit robust homeostasis, such as phototransduction.
The molecular mechanism underlying light adaptation may be discussed in the context of the reaction governing the cGMP in the photo receptor cytoplasm:
Signal flow in visual transduction, Leon Lagnado and Denis Baylor,Neuron,1992

25. Thanks and comments!

27. Physics limitation in signal sensing
25 years ago Berg and Purcell had showed that the physics limitation of the single celled organism.
The derivation is mainly assumed a perfect measurement device and they determined the relative measurement accuracy is :
But for multiple and noninterating receptors shaped as a ring, the formula is derived by Willam and Sima recently as:
With know parameter value, we can get the actual physics limit to measurements of CheYp concentration corresponds to
: diffusion constant
: device size
: average concentration
: sampling time
: receptor numbers
: single receptor size
: geometric factor of order unity

28. *Computational models of chemotaxis signal transduction network
Activity dependent model
Barkai & Leibler (1997)
The concept of robustness in biochemical networks introduced, showing how it may arise in bacterial chemotaxis through activity-dependent kinetics. The chemoreceptor is either in active state or inactive state. Simulations show that precision of adaptation is a robust property, while adaptation time is not, and that adaptation time is inversely proportional to receptor-complex activity.
It did not show how the parameter space will change which is very important for understanding the robustness mechanism.
Activity independent model
Spiro et al. (1997)
Simplified three-methylation-state model, fine-tuned by trial and error, simulates ramp, step and saturation responses to aspartate.
Although it can not achieve the robust perfect adaptation, but it is a more realistic model without assuming any of the activity dependent parameter values.
And our work is start from implementing this fine-tuned model.