Numerical Approach

1. 1% The steady state concentration of proteins in the network must satisfy: The steady state concentration of CheY-P must satisfy: At the same time, the reaction rate constants must be independent of stimulus: : allows for near-perfect adaptation = CheY-P • Ligand binding • 3%<<5% • 1%<<3 • 0%<<1% Near-Perfect Adaptation in Bacterial Chemotaxis Yang Yang & Sima SetayeshgarDepartment of Physics, Indiana University, Bloomington, Indiana 47405 (1,15) • T2 • T3 • T4 • LT3 • LT4 • T2 • T3 • T4 • LT3 • LT4 Verify steady state NR solutions dynamically using DSODE for different stimulus ramps: Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation) T3 demethylation rate/ T2 methylation rate[3] T4 demethylation rate/ T3 methylation rate[3] (1,12.7) Parameter Surfaces T3 Methylation rate (k2c) T3 demethylation rate/ T2 methylation rate[3] LT4 demethylation rate/ LT3 methylation rate[3] T2 Methylation rate (k1c) T4 autophosphorylation rate (k10) CheB phosphorylation rate (kb) / literature value[3] k3c= 5 s-1 k10 = 101 s-1 k-2 = 6.3e+4 M-1s-1 T4 autophosphorylation rate (k10) k1c : 0.17 s-1 1 s-1 k8 : 15 s-1 12.7 s-1 CheY phosphorylation rate (ky) / literature value[3] T3 demethylation rate (k-1) Precision of adaptation insensitive to changes in network parameters (robustness) T3 autophosphorylation rate CheY phosphorylation rate 9% T4 autophosphorylation rate LT3 autophosphorylation rate LT4 autophosphorylation rate CheB phosphorylation rate T2 autophosphorylation rate (k8) Concentration (µM) T3 autophosphorylation rate (k9) 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. (L)Tn autophosphorylation rate / literature value (L)Tn autophosphorylation rate / literature value k1c : 0.17 s-1 1 s-1 T2 Methylation rate (k1c) CheY1p (µM) Time (s) CheR fold expression LT2 methylation rate (k3c) T4 demethylation rate (k-2) Time(s) FRET signal [CheY-P] Methylation Validation • STARTwith a fine-tuned model of chemotaxis network that: Phosphorylation Numerical Approach : state variables : reaction kinetics : reaction constants : external stimulus k10 (s-1) These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state receptor model[6]. Run Tumble • reproduces key features of experiments (adaptation times to small and large ramps, perfect adaptation of the steady state value of CheY-P) • is NOT robust k3c (s-1) [6] B. Mello et al. Biophysical Journal 84, 2943 (2003). k-2 (M-1s-1) k-2 (M-1s-1) Literature value[4] 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: [4] P. A. Spiro et al. Proc. Natl. Acad. Sci. USA 94, 7263 (1997) Robust Perfect Adaptation • steady state value of CheY-P • values of reaction rate constants, are independent of the external stimulus, s, thereby achieving robustness of perfect adaptation. Violating and Restoring Perfect Adaptation [4] P. A. Spiro et al. Proc. Natl. Acad. Sci. (USA ) 94, 7263 (1997) AugmentedSystem k10 (s-1) Adaptation is an important and generic property of biological systems. Adaptive responses occur over a wide range of time scales, from fractions of a second in neural systems, to millions of years in the evolution of species. Discretizing s into H points k3c (s-1) Conditions for Perfect Adaptation k-2 (M-1s-1) Figure from [1] V. Sourjik et al. Proc. Natl. Acad. Sci. 99, 123 (2002). Extensions to Other Chemotaxis Systems Literature value[4] Diversity of chemotaxis systems: In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present. [4] P. A. Spiro et al. Proc. Natl. Acad. Sci. USA 94, 7263 (1997) Step stimulus from 0 to 1e-6M at t=250s There are n system variables, m system parameters and 1 parameter, ε,to allow near perfect adaptation, giving a total of (n+m+1)H equations and (n+m+1)H variables. Methylation rate is proportional to autophosphorylation rate: Eg.,Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteriapossess multiple CheYs while lacking of CheZ homologue. LT3 Methylation rate (k4c) LT2 Methylation rate (k3c) LT3 demethylation rate (k-3) LT4 demethylation rate (k-4) T4 demethylation rate (k-2) Physical Interpretation of Parameter ε Phosphate “sink” Response regulator Exact adaptation in the chemotaxis network involves rapid response to step change in external signal - in the form of a change in concentration of the response regulator CheY-P and corresponding change in running versus tumbling bias - followed by return back to the prestimulus 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[3]. [3] N. Barkai et al. Nature 387, 855 (1997) LT2 autophosphorylation rate (k12) LT3 autophosphorylation rate (k13) 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) Figure from [2] U. Alon et al. Nature 397, 168 (1999) • Measurement of c = [CheY-P] by flagellar motor is constrained by diffusive noise. Relative accuracy[5] is given by: • Signaling pathway is required to adapt “nearly” perfectly, to within this lower bound. • [5] H. C. Berg et al. Biophys. Journal. 20, 193 (1997) . Conditions for Perfect Adaptation (cont’d) Attractant: 30 μM aspartate. Repellent: 100 μM NiCl2 Demethylation rate/mmethylation rate is proportional toautophosphorylation rate: E.Coli Chemotaxis Signaling Network : 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) • Requiring: • Faster phosphorylation/autodephosphorylation rates of CheY than CheY1 • Faster phosphorylation rate of CheB 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: CheB, CheYphosphorylation rate is proportional to autophosphorylation rate: Conclusions Work in Progress • We have demonstrated: • 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 • numerical ranges for unknown or partially known kinetic parameters • Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs • Extension to other signaling networks: • vertebrate phototransduction • mammalian circadian clock • allowing determination of • parameter dependences underlying robustness of response • plausible numerical values for unknown network parameters We use the well-characterized chemotaxis network in E. coli as a prototype for exploring physical principles underlying the design of biological signaling networks from the standpoint of adapation and robustness. 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 We use a detailed mechanistic model of the chemotaxis network[4], which is not based on the two-receptor model 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, in this model values of reaction rates and protein concentrations are fine-tuned to achieve perfect adaptation of the response.