1. Near-Perfect Adaptation in Bacterial Chemotaxis
Yang Yang and Sima Setayeshgar
Department of Physics
Indiana University, Bloomington, IN
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

2. Chemotaxis Signal Transduction Network in E. coli
Pathway
Motor
Response
[CheY-P]
Stimulus
Flagellar
Bundling
Motion
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.
CheA: Histidine kinase CheB-P: Methylesterase
CheW: Couples CheA to MCPs CheY-P: Response regulator
CheR: Methyltransferase CheZ: Dephosphorylates CheY-P
It is used as a well-characterized model system for the study of properties of (two-component) cellular signaling networks in general.
Histidine kinase Methylesterase
Couples CheA to MCPs Response regulator
Methyltransferase Dephosphorylates CheY-P
CheB
CheA
CheW
CheZ
CheR
CheY
Run Tumble
March 8, 2007
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).
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

4. Yang Yang, March APS Meeting, Denver, CO
This Work: Outline
New computational scheme for determining conditions and numerical ranges for 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
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

5. E. coli Chemotaxis Signaling Network
Ligand binding
Methylation
Phosphorylation
phosphorylation
methylation
Ligand binding T3 T4 T2
T2p
T4p
T3p
LT3
LT4
LT4p
LT2
LT3p
LT2p
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.
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

6. Yang Yang, March APS Meeting, Denver, CO
Approach …
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
(adaptation times to small and large ramps, perfect adaptation of the steady state value of CheYp)
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

7. Yang Yang, March APS Meeting, Denver, CO
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}
March 8, 2007
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)
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

9. Yang Yang, March APS Meeting, Denver, CO
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:
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.
multidimensional root finding method
Efficient way of converging to a root with a sufficiently good initial guess.
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

10. Yang Yang, March APS Meeting, Denver, CO
Parameter Surfaces
Surface: D projections:
T4 autophosphorylation rate (k10)
LT2 methylation rate (k3c)
LT4 autophosphorylation rate (k10)
3%<<5%
1%<<3%
0%<<1%
T4 demethylation rate (km2)
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

11. Yang Yang, March APS Meeting, Denver, CO
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)
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

12. Violating and Restoring Perfect Adaptation
(1,15)
(1,12.7)
T2 Methylation rate (k1c)
T2 autophosphorylation rate (k8) 9%
k1c : 0.17 s-1  1 s-1 1%
k8 : 15 s-1  12.7 s-1
Step stimulus from 0 to 1e-6M at t=250s
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

13. Conditions for Perfect Adaptation

14. Methylation Rate Autophosphorylation Rate
T2 autophosphorylation rate (k8)
T2 Methylation rate (k1c)
T3 autophosphorylation rate (k9)
T3 Methylation rate (k2c)
LT2 autophosphorylation rate (k12)
LT2 Methylation rate (k3c)
LT3 autophosphorylation rate (k13)
LT3 Methylation rate (k4c)
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

15. Demethylation Rate Autophosphorylation Rate2
T3 autophosphorylation rate (k9)
T3 demethylation rate (km1)
T4 autophosphorylation rate (k10)
T4 demethylation rate (km2)
LT3 autophosphorylation rate (k12)
LT3 demethylation rate (km3)
LT4 autophosphorylation rate (k13)
LT4 demethylation rate (km4)
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

16. Demethylation Rate/Methylation Rate Autophosphorylation Rate
T3 autophosphorylation rate
T3 demethylation rate/ T2 methylation rate
T4 autophosphorylation rate
T4 demethylation rate/ T3 methylation rate
LT3 autophosphorylation rate
LT4 autophosphorylation rate
LT4 demethylation rate/ LT3 methylation rate
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

17. CheB, CheY Phosphorylation Rate Autophosphorylation Rate
CheB phosphorylation rate (kb) / literature value
CheY phosphorylation rate (ky) / literature value
(L)Tn autophosphorylation rate / literature value
(L)Tn autophosphorylation rate / literature value T2 T3 T4
LT3
LT4
CheB phosphorylation rate
LT2 autophosphorylation rate
CheY phosphorylation rate
LT2 autophosphorylation rate
March 8, 2007
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”
CheY1
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.
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.
Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue.
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

19. Yang Yang, March APS Meeting, Denver, CO
Two CheY System
Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:
CheY1p (µM)
Time(s)
Determination of numerical values and parameter dependences allowing exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ
Time(s)
Requiring:
Faster phosphorylation/autodephosphorylation rates of CheY2
than CheY1
Faster phosphorylation rate of CheB
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

20. Yang Yang, March APS Meeting, Denver, CO
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).
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO

21. Yang Yang, March APS Meeting, Denver, CO
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
March 8, 2007
Yang Yang, March APS Meeting, Denver, CO