1. Near-Perfect Adaptation in Bacterial Chemotaxis
Yang Yang & Sima Setayeshgar 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.
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
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
Demethylation rate/mmethylation rate is proportional to autophosphorylation rate:
T3 autophosphorylation rate
T3 demethylation rate/ T2 methylation rate[3]
T4 autophosphorylation rate
T4 demethylation rate/ T3 methylation rate[3]
LT3 autophosphorylation rate
LT4 autophosphorylation rate
LT4 demethylation rate/ LT3 methylation rate[3]
CheB phosphorylation rate (kb) / literature value[3]
CheY phosphorylation rate (ky) /
literature value[3]
(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
Conditions for Perfect Adaptation (cont’d)
CheB, CheY phosphorylation rate is proportional to autophosphorylation rate:
Ligand binding
We used 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, 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
T4 autophosphorylation rate (k10)
T4 autophosphorylation rate (k10)
3%<<5%
1%<<3
0%<<1%
Parameter Surfaces
LT2 methylation rate (k3c)
T4 demethylation rate (k-2)
Methylation
Phosphorylation
k10 (s-1)
These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state receptor model[6].
k3c (s-1)
[6] B. Mello et al. Biophysical Journal 84, 2943 (2003).
k-2 (M-1s-1)
k-2 (M-1s-1)
Extensions of Chemotaxis Systems
Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue.
Diversity of chemotaxis systems: 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 CheY1
Faster phosphorylation rate of CheB
CheY1p (µM)
Time(s)
Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:
Literature value[4]
[4] P. A. Spiro et al. Proc. Natl. Acad. Sci. USA 94, 7263 (1997)
Time (s)
Concentration (µM)
Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:
k3c= 5 s-1
k10 = 101 s-1
k-2 = 6.3e+4 M-1s-1
Validation
[4] P. A. Spiro et al. Proc. Natl. Acad. Sci. (USA ) 94, 7263 (1997)
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)
Exact adaptation in the chemotaxis network involves rapid response to step change in external signal - in form of 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)
Robust Perfect Adaptation
Numerical 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 CheY-P)
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 achieving robustness of perfect adaptation.
: state variables
: reaction kinetics
: reaction constants
: external stimulus
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
k3c (s-1)
k-2 (M-1s-1)
Figure from [1] V. Sourjik et al. Proc. Natl. Acad. Sci. 99, 123 (2002).
Literature value[4]
[4] P. A. Spiro et al. Proc. Natl. Acad. Sci. USA 94, 7263 (1997)
Violating and Restoring Perfect Adaptation 1%
k1c : 0.17 s-1  1 s-1
k8 : 15 s-1  12.7 s-1 9%
Step stimulus from 0 to 1e-6M at t=250s
(1,15)
(1,12.7)
T2 Methylation rate (k1c)
T2 autophosphorylation rate (k8)
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
Conclusions
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
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.
Discretizing s
into H points
Augmented System
Figure from [2] U. Alon et al. Nature 397, 168 (1999)
Attractant: 30 μM aspartate. Repellent: 100 μM NiCl2
T3 demethylation rate (k-1)
T3 autophosphorylation rate (k9)
T4 autophosphorylation rate (k10)
T4 demethylation rate (k-2)
LT3 autophosphorylation rate (k12)
LT3 demethylation rate (k-3)
LT4 autophosphorylation rate (k13)
LT4 demethylation rate (k-4)
Demethylation rate is proportional to autophosphorylation rate2:
Conditions for Perfect Adaptation
T2 autophosphorylation rate (k8)
T2 Methylation rate (k1c)
T3 autophosphorylation rate (k9)
T3 Methylation rate (k2c)
Methylation rate is proportional to autophosphorylation rate:
LT2 autophosphorylation rate (k12)
LT2 Methylation rate (k3c)
LT3 autophosphorylation rate (k13)
LT3 Methylation rate (k4c)
Physical Interpretation of Parameter ε
Measurement of c = [CheY-P] by flagellar motor is constrained by diffusive noise. Relative accuracy[5] is given by:
Signaling pathway required to adapt “nearly” perfectly, to within this lower bound
[5] H. C. Berg et al. Biophys. Journal. 20, 193 (1997) .
: 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)
Work in Progress
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.