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

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

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

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

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

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

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

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

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

Yang Yang, March APS Meeting, Denver, CO Parameter Surfaces Surface: 2D 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

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

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

Conditions for Perfect Adaptation

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

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

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

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

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

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

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

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