Approach…

1. 1% 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: • Ligand binding : allows for near-perfect adaptation = CheYp • 3%<<5% • 1%<<3 • 0%<<1% Near-Perfect Adaptation in Bacterial Chemotaxis Yang Yang & Sima SetyeshgarDepartment of Physics, Indiana University, Bloomington, Indiana 47405 (1,15) Verify steady state NR solutions dynamically using DSODE for different stimulus ramps: • T2 • T3 • T4 • LT3 • LT4 • T2 • T3 • T4 • LT3 • LT4 T3 demethylation rate/ T2 methylation rate T4 demethylation rate/ T3 methylation rate (1,12.7) Parameter Surfaces T3 Methylation rate (k2c) T3 demethylation rate/ T2 methylation rate LT4 demethylation rate/ LT3 methylation rate Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation) T2 Methylation rate (k1c) T4 autophosphorylation rate (k10) T4 autophosphorylation rate (k10) k1c : 0.17 s-1 1 s-1 k8 : 15 s-1 12.7 s-1 T3 demethylation rate (km1) CheB phosphorylation rate (kb) / literature value T3 autophosphorylation rate CheY phosphorylation rate (ky) / literature value 9% T4 autophosphorylation rate LT3 autophosphorylation rate LT4 autophosphorylation rate Precision of adaptation insensitive to changes in network parameters (robustness) [2] CheY phosphorylation rate T2 autophosphorylation rate (k8) Concentration (µM) T3 autophosphorylation rate (k9) CheB phosphorylation rate T2 autophosphorylation rate (k8) CheY2 CheY1 Chemotaxis signal transduction network in E. coli T3 autophosphorylation rate (k9) Adaptation Precison LT2 autophosphorylation rate LT2 autophosphorylation rate 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. • {k3c= 5 s-1, k10 = 101 s-1, km2 = 6.3e+4 M-1s-1} k1c : 0.17 s-1 1 s-1 (L)Tn autophosphorylation rate / literature value (L)Tn autophosphorylation rate / literature value T2 Methylation rate (k1c) CheY1p (µM) FRET signal [CheY-P] Time (s) • Methylation LT2 methylation rate (k3c) T4 demethylation rate (km2) CheR fold expression Fast response Slow adaptation • Phosphorylation Time(s) Validation • STARTwith a fine-tuned model of chemotaxis network that: Approach… : state variables : reaction kinetics : reaction constants : external stimulus Run Tumble [1] V. Sourjik et al., (2002), PNAS, 99, 123 [2] U. Alon et al., (1999), Nature, 397, 168 • reproduces key features of experiments (adaptation times to small and large ramps, perfect adaptation of the steady state value of CheYp) • is NOT robust 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. • 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. Violating and Restoring Perfect Adaptation Augmentedsystem [1] Discretizing s into H points Conditions for Perfect Adaptation Diversity of Chemotaxis Systems In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present. Robust Perfect Adaptation 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. [1] Step stimulus from 0 to 1e-6M at t=250s Methylation Rate is proportional to Autophosphorylation Rate Eg.,Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteriapossess multiple CheYs while lacking of CheZ homologue. Physical Interpretation of Parameter ε LT3 Methylation rate (k4c) LT2 Methylation rate (k3c) LT3 demethylation rate (km3) LT4 demethylation rate (km4) T4 demethylation rate (km2) 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. Phosphate “sink” Response regulator LT2 autophosphorylation rate (k12) LT3 autophosphorylation rate (k13) E.Coli Chemotaxis Signaling Network Demethylation Rate is proportional to Autophosphorylation Rate2 Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ: LT4 autophosphorylation rate (k13) T4 autophosphorylation rate (k10) LT3 autophosphorylation rate (k12) • 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 . Conditions for Perfect Adaptation Demethylation Rate/Methylation Rate is proportional toAutophosphorylation Rate : 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) Implementation • Requiring: • Faster phosphorylation/autodephosphorylation rates of CheY than CheY1 • Faster phosphorylation rate of CheB • 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: This work: outline CheB, CheYPhosphorylation Rate is proportional to Autophosphorylation Rate Conclusion • 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 Work in progress • 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 [4-6]] • numerical ranges for unknown or partially known kinetic parameters • Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs • [4] N. Barkai et al., (1997), Nature , 387,855 • [5] T. M. Yi et al., (2000), PNAS ,97,4649. • [6] B. A. Mello et al., (2003), Biophys. J., 84, 2943 • Extension to other signaling networks: • vertebrate phototransduction • mammalian circadian clock • allowing determination of • parameter dependences underlying robustness • plausible numerical values for unknown network parameters 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.