Near-Perfect Adaptation in E. coli Chemotaxis Signal Transduction Network

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

2. E. coli static.howstuffworks.com/gif/cell-ecoli.gif • 1-3 microns long • 1 micron in diameter • 4-6 flagella • Small genome (4288 genes) www.hatetank.dk

3. Bacterial Chemotaxis Increasing attractants or Decreasingrepellents Run Tumble http://www.rowland.harvard.edu/labs/bacteria/index_movies.html

4. Chemotaxis signal transduction network in E. coli well-characterized model system CheW: Coupling CheA to MCPs CheB: CheBp demethylate MCPs CheR: Methylate MCPs CheY: flagella motor regulator protein CheZ: Dephosphoryte CheYp; CheA: Histidine kinase

5. T2p T3p T4p T2 T3 T4 LT2p LT3p LT4p LT2 LT4 LT3 • ligand binding • Methylation • Phosphorylation Chemical reactions phosphorylation methylation Ligand binding Full realistic model

6. Perfect adaptation This property allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli Steven M., et al. Journal of bacteriology. 1983

7. Robustness proteins input output Reaction rates Can be achieved by Tau-Mu Yi* et al. Biophysics,2000 U. Alon et al. Nature,1999

8. Motivation QUESITON: basis of robustness of perfect adaptation? • shedding light on biochemical steps and feedback mechanisms underlying robustness • shed light on values of unknown or partially known paramters SOLVE: we develop a novel method for elucidating regions in parameter space allowing perfect adaptation.

9. START with a fine-tuned model of chemotaxis network that: : state variables : reaction kinetics : reaction constants : external stimulus • 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. Algorithm

10. Decretizing s into H points 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 Augmented system

11. f(x) 1 f(x) 2 1 x x 2 f(x) 1 2 3 x Newton-Raphson, to solve for the steady state of augmented system: Implementation • multidimensional root finding method • Efficient way of converging to a root with a sufficiently good initial guess. Works well unfortunate case fortunate case

12. Implementation Dsode (stiff ODE solver), to verify Time dependent behavior of proteins 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 of species with multiple CheYs

14. parameter spaces of E. coli E .coli Pairwise result: 3D surface result: Relative change of CheYp: • less than 5% •less than 3% •less than 1% • pairwise trajectory

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

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

17. Violating and restoring perfect adaptation (1,15) (1,12.7) 1% 9% k1c : 0.17 s-1 1 s-1 k1c : 0.17 s-1 1 s-1 k8 : 15 s-1 12.7 s-1 At 250s, giving step stimulus from 0 to 1e-6M

18. Consistency with recent work done by Bernardo A. mello and Yuhai Tu • They list a series of conditions which allow near-perfect adaptations • They are a active-ependent model which the receptors are either in active or inactive state Our parameter space remarkable shows the same consistency with their predictions about the relationships of the parameter values although we are using a active-independent model.

19. List of conditions: The timescale for ligand binding is much shorter than the methylation and phosphorylation timescale. This condition allows us to neglect ligand-binding/unbinding kinetics. The association rates between the receptor and the methylation/demethylation enzymes, CheR and CheB-P, are linearly related to the activity of the receptor and are zero for n = 4 and n = 0, respectively: and : The dissociation rates of the enzyme receptor bound states are independent of λ. The receptor activities of the nonmethylated and the maximally methylated receptors are independent of l: P0v = P0o, P4v = P4o. The ratios between the CheR catalytic rate kRn and the CheB-P catalytic rate of the next methylation level kBn+1 are the same for all methylation states n: kBn+1 / kRn= const: The phosphate transfer rates from CheA to CheB or CheY are proportional to CheA autophosphorylation rate: The explicit dependence on [TFn] distribution can be removed from the expression this condition can only be strictly satisfied when

20. Condition 2

21. P0v = P0o, P4v = P4o Condition 3

22. kBn+1 / kRn= const Condition 4

23. Condition 5 kb /k8-13 and ky/k8-13 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) while the inset is not since the literature value is zero for k11.

24. Condition 6

25. kb /k8-13 and ky/k8-13 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) while the inset is not since the literature value is zero for k11.

26. Two CheY system • Rhodobacter sphaeroides, Caulobacter crescentus have multiple CheYs while lack of CheZ protein. • Similar chemotaxis behaviors.

27. Two CheY system Our work: Reproduce the key feature of chemotaxis behavior in two CheY system by replacing CheZ with CheY2.

28. Two CheY Parameter spaces of two CheY system • Introducing [CheY2] and CheY2 (de-)phosphorylation rates. • Exploring the parameter values which can give perfect adaptation. Relative change of CheYp: • less than 5% •less than 3% •less than 1% Other parameter value were set as the literature value except Kb= 1e+6 M-1s-1 instead of 8e+5 M-1s-1.

29. parameter spaces comparison of two and single CheY case The parameter space for single CheY case seems more restrict than the two CheY case

30. 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 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

31. Future work • Applying the method to other cellular signal transduction networks exhibiting robust homeostasis, such as phototransduction Signal flow in visual transduction, Leon Lagnado and Denis Baylor,Neuron,1992

32. Thanks and comments!

34. 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

35. *Computational models of chemotaxis signal transduction network

36. 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).