Approach…  START with a fine-tuned model of chemotaxis network that:  reproduces key features of experiments (adaptation times to small and large ramps, perfect adaptation of the steady state value of CheYp)  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 Time (s) Concentration (µM) Verify steady state NR solutions dynamically using DSODE for different stimulus ramps: k 3c = 5 s -1 k 10 = 101 s -1 k -2 = 6.3e+4 M -1 s -1 ValidationViolating and Restoring Perfect Adaptation 1% k 1c : 0.17 s -1  1 s -1 k 8 : 15 s -1  12.7 s -1 9% k 1c : 0.17 s -1  1 s -1 Step stimulus from 0 to 1e-6M at t=250s (1,15) (1,12.7) T 2 Methylation rate (k 1c ) T 2 autophosphorylation rate (k 8 ) Demethylation rate/mmethylation rate is proportional to autophosphorylation rate: T 3 autophosphorylation rate T 3 demethylation rate/ T 2 methylation rate [3] T 4 autophosphorylation rate T 4 demethylation rate/ T 3 methylation rate [3] LT 3 autophosphorylation rate T 3 demethylation rate/ T 2 methylation rate [3] LT 4 autophosphorylation rate LT 4 demethylation rate/ LT 3 methylation rate [3] CheB phosphorylation rate (k b ) / literature value [3] CheY phosphorylation rate (k y ) / literature value [3] (L)T n autophosphorylation rate / literature value ● T2 ● T3 ● T4 ● LT3 ● LT4 ● T2 ● T3 ● T4 ● LT3 ● LT4 CheB phosphorylation rate LT 2 autophosphorylation rate CheY phosphorylation rate LT 2 autophosphorylation rate Conditions for Perfect Adaptation(con’d) CheB, CheY phosphorylation rate is proportional to autophosphorylation rate: Diversity of Chemotaxis Systems Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue. In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present. Response regulator CheY1 CheY2 Phosphate “sink” Requiring: Faster phosphorylation/autodephosphorylation rates of CheY than CheY 1 Faster phosphorylation rate of CheB CheY 1p (µM) Time(s) Exact adaptation in modified chemotaxis network with CheY 1, CheY 2 and no CheZ: Near-Perfect Adaptation in Bacterial Chemotaxis Yang Yang & Sima Setyeshgar Department of Physics, Indiana University, Bloomington, Indiana Run Tumble A network of interacting proteins converts an external stimulus (attractant/repellent) into an internal signal - the phosphorylated form of the Y chemotaxis protein - which in turn interacts with the flagellar motor to bias the cell’s motion between runs and tumbles. The chemotaxis signal transduction network is a well-characterized model system for studying the properties of the two-component superfamily of receptor-regulated phosphorylation pathways in general. Chemotaxis signal transduction network in E. coli Chemotaxis in E. coli - motion toward desirable chemicals and away from harmful ones - is an important behavioral response also shared by many other prokaryotic and eukaryotic cells. It consists of a series of modulated runs and tumbles, leading to a biased random walk in the desired direction. Fast responseSlow adaptation [1] V. Sourjik et al., (2002), PNAS, 99, 123 [2] U. Alon et al., (1999), Nature, 397, 168 FRET signal [CheY-P] CheR fold expression Adaptation Precison Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation) Precision of adaptation insensitive to changes in network parameters (robustness) [2] It is an important property of the chemotaxis network: rapid response - in the form of change in concentration of intracellular response regulator and corresponding change in running versus tumbling bias - to a step change in external signal, followed by exact adaptation back to the pre-stimulus value. Recent work has highlighted the fact that the underlying design of the chemosensory pathway is such that exact adaptation is "robust" or insensitive to changes in network parameters such as total protein concentrations and reaction rates. Robust Perfect Adaptation T 4 autophosphorylation rate (k 10 ) ● 3%<  <5% ● 1%<  <3 ● 0%<  <1% Parameter Surfaces LT 2 methylation rate (k 3c )T 4 demethylation rate (k -2 ) 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 [5-7]] numerical ranges for unknown or partially known kinetic parameters Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs [5] N. Barkai et al., (1997), Nature, 387,855 [6] T. M. Yi et al., (2000), PNAS,97,4649. [7] B. A. Mello et al., (2003), Biophys. J., 84, 2943 Conclusion Work in progress Extension to other signaling networks: vertebrate phototransduction mammalian circadian clock allowing determination of parameter dependences underlying robustness plausible numerical values for unknown network parameters Ligand binding We begin with a detailed model of the chemotaxis pathway in E. coli, including ligand binding, methylation/demethylation and phosphorylation steps. This model is not assume the two-state active/inactive description of the receptor complex: instead receptor activity is allowed to be graded through the variable autophosphorylation rate of the histidine kinase, CheA. Although capturing the main features of the chemotactic response, this model is "broken" in that the values of reaction rates and protein concentrations are fine- tuned to achieve perfect adaptation of the response. E.Coli Chemotaxis Signaling Network Phosphorylation Methylation 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 This work: outline 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 There are n system variables, m system parameters and 1 small variable to allow near perfect adaptation, giving a total of (n+m+1)H equations and (n+m+1)H variables. Discretizing s into H points Augmented system Physical Interpretation of Parameter ε Measurement of c = [CheY-P] by flagellar motor constrained by diffusive noise Relative accuracy [3], Signaling pathway required to adapt “nearly” perfectly, to within this lower bound [3] H. C. Berg et al., (1977), Biophys. Journal. 20, 193. : 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) 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: Implementation T 3 demethylation rate (k -1 ) T 3 autophosphorylation rate (k 9 ) T 4 autophosphorylation rate (k 10 ) T 4 demethylation rate (k -2 ) LT 3 autophosphorylation rate (k 12 ) LT 3 demethylation rate (k -3 ) LT 4 autophosphorylation rate (k 13 ) LT 4 demethylation rate (k -4 ) Demethylation rate is proportional to autophosphorylation rate 2 : Conditions for Perfect Adaptation T 2 autophosphorylation rate (k 8 ) T 2 Methylation rate (k 1c ) T 3 autophosphorylation rate (k 9 ) T 3 Methylation rate (k 2c ) Methylation rate is proportional to autophosphorylation rate: LT 2 autophosphorylation rate (k 12 ) LT 2 Methylation rate (k 3c ) LT 3 autophosphorylation rate (k 13 ) LT 3 Methylation rate (k 4c ) [1] Literature value [3] k 10 (s -1 ) k -2 (M -1 s -1 ) k 3c (s -1 ) k -2 (M -1 s -1 ) k 10 (s -1 ) k 3c (s -1 ) k -2 (M -1 s -1 ) Literature value [3] [3] P. A. Spiro et al., (1997), Proc. Natl. Acad. Sci. USA, 94, 7263