Near-Perfect Adaptation in Bacterial Chemotaxis

1. Near-Perfect Adaptation in Bacterial Chemotaxis Yang Yang and Sima Setayeshgar Department of Physics Indiana University, Bloomington, IN Yang Yang, March APS Meeting, Denver, CO

2. Chemotaxis Signal Transduction Network in E. coli Stimulus Signal Transduction Pathway [CheY-P] Motor Response Flagellar Bundling Histidine kinase Methylesterase Couples CheA to MCPs Response regulator Methyltransferase Dephosphorylates CheY-P Motion CheW CheR CheB CheZ CheA CheY Run Tumble Yang Yang, March APS Meeting, Denver, CO

3. Robust Perfect Adaptation From Sourjik et al., PNAS (2002). Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation) Precision of adaptation insensitive to changes in network parameters (robustness) Adaptation Precison FRET signal [CheY-P] CheR fold expression Fast response Slow adaptation From Alon et al.,Nature (1999). Yang Yang, March APS Meeting, Denver, CO

4. This Work: Outline • New computational scheme for determining conditions and numerical rangesfor parameters allowing robust (near-)perfect adaptation in the E. coli chemotaxis network • Comparison of results with previous works • Extension to other modified chemotaxis networks, with additional protein components • Conclusions and future work Yang Yang, March APS Meeting, Denver, CO

5. T2p T3p T4p T2 LT2p LT3p LT4p LT2 LT4 LT3 • Ligand binding • Methylation • Phosphorylation E. coli ChemotaxisSignaling Network T3 T4 phosphorylation methylation Ligand binding Yang Yang, March APS Meeting, Denver, CO

6. 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 CheY-P • values of reaction rate constants, • are independent of the external stimulus, s, thereby explicitly incorporating perfect adaptation. : state variables : reaction kinetics : reaction rates : external stimulus Approach … Yang Yang, March APS Meeting, Denver, CO

7. Augmented System The steady state concentration of proteins in the network satisfy: The steady state concentration of = [CheY-P] must be independent of stimulus, s: where parameter allows for “near-perfect” adaptation. Reaction rates are constant and must also be independent of stimulus, s: Discretize s in range {slow, shigh} Yang Yang, March APS Meeting, Denver, CO

8. Physical Interpretation of Parameter, : Near-Perfect Adaptation • Measurement of c = [CheY-P] by flagellar motor constrained by diffusive noise • Relative accuracy*, • Signaling pathway required to adapt “nearly” perfectly, to within this lower bound • (*) Berg & Purcell, Biophys. J. (1977). : diffusion constant (~ 3 µM) : linear dimension of motor C-ring (~ 45 nm) : CheY-P concentration (at steady state ~ 3 µM) : measurement time (run duration ~ 1 second) Yang Yang, March APS Meeting, Denver, CO

9. Implementation • Use Newton-Raphson (root finding algorithm with back-tracking), to solve for the steady state of augmented system, • Use Dsode (stiff ODE solver), to verify time- dependent behavior for different ranges of external stimulus by solving: Yang Yang, March APS Meeting, Denver, CO

10. Parameter Surfaces Surface: 2D projections: T4autophosphorylation rate (k10) LT2methylation rate (k3c) LT4 autophosphorylation rate (k10) • 3%<<5% • 1%<<3% • 0%<<1% T4 demethylation rate (km2) Yang Yang, March APS Meeting, Denver, CO

11. Validation Verify steady state NR solutions dynamically using DSODE for different stimulus ramps: {k3c= 5 s-1, k10 = 101 s-1, km2 = 6.3e+4 M-1s-1} Concentration (µM) Time (s) Yang Yang, March APS Meeting, Denver, CO

12. Violating and Restoring Perfect Adaptation (1,15) (1,12.7) 1% 9% T2 autophosphorylation rate (k8) k1c : 0.17 s-1 1 s-1 k8 : 15 s-1 12.7 s-1 k1c : 0.17 s-1 1 s-1 T2Methylation rate (k1c) Step stimulus from 0 to 1e-6M at t=250s Yang Yang, March APS Meeting, Denver, CO

13. Conditions for Perfect Adaptation

14. Methylation Rate Autophosphorylation Rate LT3 Methylation rate (k4c) T3 Methylation rate (k2c) T2 Methylation rate (k1c) LT2 Methylation rate (k3c) LT3 autophosphorylation rate (k13) T3 autophosphorylation rate (k9) T2 autophosphorylation rate (k8) LT2 autophosphorylation rate (k12) Yang Yang, March APS Meeting, Denver, CO

15. Demethylation Rate Autophosphorylation Rate2 T4 demethylation rate (km2) LT4 demethylation rate (km4) LT3 demethylation rate (km3) T3 demethylation rate (km1) T4 autophosphorylation rate (k10) LT4 autophosphorylation rate (k13) T3 autophosphorylation rate (k9) LT3 autophosphorylation rate (k12) Yang Yang, March APS Meeting, Denver, CO

16. Demethylation Rate/Methylation Rate Autophosphorylation Rate T3 demethylation rate/ T2 methylation rate T4 demethylation rate/ T3 methylation rate T3 autophosphorylation rate T4 autophosphorylation rate T3 demethylation rate/ T2 methylation rate LT4 demethylation rate/ LT3 methylation rate Yang Yang, March APS Meeting, Denver, CO LT4 autophosphorylation rate LT3 autophosphorylation rate

17. CheB, CheY Phosphorylation Rate Autophosphorylation Rate • T2 • T3 • T4 • LT3 • LT4 • T2 • T3 • T4 • LT3 • LT4 CheYphosphorylation rate (ky) / literature value CheBphosphorylation rate (kb) / literature value CheB phosphorylation rate CheY phosphorylation rate LT2 autophosphorylation rate LT2 autophosphorylation rate (L)Tn autophosphorylation rate / literature value (L)Tn autophosphorylation rate / literature value Yang Yang, March APS Meeting, Denver, CO

18. Diversity of Chemotaxis Systems In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present. Response regulator Phosphate “sink” Eg.,Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteriapossess multiple CheYs while lacking of CheZ homologue. CheY2 CheY1 Yang Yang, March APS Meeting, Denver, CO

19. Two CheY System Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ: CheY1p (µM) CheY1p (µM) Time(s) • Requiring: • Faster phosphorylation/autodephosphorylation rates of CheY2 than CheY1 • Faster phosphorylation rate of CheB Time(s) Yang Yang, March APS Meeting, Denver, CO

20. Conclusions • Successful implementation of a novel method for elucidating regions in parameter space allowing precise adaptation • Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of • conditions required for perfect adaptation, consistent with and extending previous works [1-3] • numerical ranges for unknown or partially known kinetic parameters • Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs [1] Barkai & Leibler, Nature (1997). [2] Yi et al., PNAS (2000). [3] Tu & Mello, Biophys. J. (2003). Yang Yang, March APS Meeting, Denver, CO

21. Future Work • Extension to other signaling networks • vertebrate phototransduction • mammalian circadian clock • allowing determination of • parameter dependences underlying robustness • b) plausible numerical values for unknown network parameters Yang Yang, March APS Meeting, Denver, CO