1. 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.
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
T4 autophosphorylation rate (k10)
T4 autophosphorylation rate (k10)
3%<<5%
1%<<3
0%<<1%
Parameter Surfaces
Condition for Robust Perfect Adaptation
By varying 3 parameters(Ttot, k11, k3c) in the code to find a region where Ttot can vary a lot while the others remain constant.
L=0(solid)
L=1µM(dashed)
L=1mM(dashed dot)
Spiro model.
Barkai–Leibler model
Ref: P. A. Spiro, et al., 1997, Proc. Natl. Acad. Sci. USA 94, 7263
Ref: M.Yi, et al. 2000, PNAS, 97, 4649
Methylation
Phosphorylation
LT2 methylation rate (k3c)
T4 demethylation rate (km2)
Time (s)
Concentration (µM)
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}
Validation
Verifying the conditions for perfect adaptation of two-state model
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
Inactive active
Fast ligand (un-)binding reaction
Only acitive receptor can bind to CheB
Only inactive receptor can bind to CheR
Autophosphorylation rates of receptors are proportional to the activity
Phosphorylation transferrate form CheA to CheY and CheB are porportional to the activity
The ratios between the CheR catalytic rate and CheB-p catalytic rate of the next methylation level are the same for all methylation states.
Fast response
Slow adaptation
Ref: V. Sourjik et al.,2002, PNAS, 99(1), 123
Ref: 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)
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
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)
T2 demethylation catalytic rate
T1 methylation catalytic rate
T1 demethylation catalytic rate
T0 methylation catalytic rate
T3 demethylation catalytic rate
T2 methylation catalytic rate
T4 demethylation catalytic rate
T3 methylation catalytic rate
Verifying condition 6: reference value kb/kr=0.155/0.819=0.19
Slope=0.18
Slope=0.15
Slope=0.15
Slope=0.18
Slope=0.15
Slope=0.19
(1,12.7)
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
Ref: B. Mello et al, 2003, Biophysical Journal, 84, 2943
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)
Diversity of Chemotaxis Systems
Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:
In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present.
CheY1p (µM)
Time(s)
Response regulator
CheY1
CheY2
Phosphate “sink”
T3 demethylation rate (km1)
T3 autophosphorylation rate (k9)
T4 autophosphorylation rate (k10)
T4 demethylation rate (km2)
LT3 autophosphorylation rate (k12)
LT3 demethylation rate (km3)
LT4 autophosphorylation rate (k13)
LT4 demethylation rate (km4)
Demethylation Rate is proportional to Autophosphorylation Rate2
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
Ref: 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)
Motivation and Broader impact
The biochemical basis of robustness of perfect adaptation is not as yet fully understood. In this work, we develop a novel method for elucidating regions in parameter space of which the E. coli chemotaxis network adapts perfectly:
This method should have applicability to other cellular signal transduction networks and engineered systems that exhibit robust homeostasis.
The shapes of resulting manifolds determine relationships between reaction parameters (for example, methylation and phosphorylation rates) that must be satisfied in order for the network to exhibit perfect adaptation, thereby shedding light on biochemical steps and feedback mechanisms underlying robustness.
Given lack of complete data on values of in vivo reaction rates, the numerical ranges of the resulting manifolds will shed light on values of unknown or partially known parameters.
Requiring:
Faster phosphorylation/autodephosphorylation rates of CheY than CheY1
Faster phosphorylation rate of CheB
Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue.
Demethylation Rate/Methylation Rate is proportional to 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
Conclusion
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
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
CheB, CheY Phosphorylation Rate is proportional to 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
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
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