Optimal Strategy in E. coli Chemotaxis: An Information Theoretic Approach
Lin Wang and Sima Setayeshgar
Department of Physics, Indiana University, Bloomington, Indiana 47405
Focus
Motivation
Biochemical signaling is the most fundamental level of information processing in
biological systems, where an external stimulus is measured and converted into a
response.
Photon counting in vision[1,2]
Photon
Δ[Ca2+]
Molecule counting in chemotaxis[3]
Attractant
E. coli varies its response to input signals with different statistics. Our goal is to
understand how signal transduction pathways, such as the chemotaxis network, may
adapt to the statistics of the fluctuating input so as to optimize the cell’s response. We
construct a measure of the information transmission rate and investigate the role of
varying response.
Effect of Correlation Time τ
Model Validation
Experiment
Simulation
Adaptation[9]
My first step is to investigate the effect of correlation time τ to the I/O mutual
information rate of the chemotaxis network.
Effect of τ on I/O relation
Response r(s) to signals: μ=1 μM, σ2 = μ, τ =
0.1, 0.3, 0.8, 1 sec, respectively. At τ > 0.8 sec,
the response does not change any more. (This
holds true for signals with different mean
values)
Δ[CheY-P]
Δ[Na+]
et al.
Response of drosophila photoreceptor
to single photon absorption.
Response of E. coli to external attractant.
We use the well-characterized chemotaxis network in E. coli as a prototype
for exploring general principles governing information processing in
biological signaling networks.
Numerical Implementation
Chemotaxis Network Equations and Parameters
E. coli Chemotaxis
Chemotaxis, a cell’s motion toward desirable chemicals (usually nutrients) and away
from harmful ones, is achieved through alternating ‘runs’ and ‘tumbles’. The mean
run-time is modulated in response to the cells’s measurement of the chemoattractant
concentration, resulting in a biased random walk up (down) chemoattractant (repellant)
concentration gradients.
Stimulus
Signal
Transduction
Pathway
[CheY-P]
Motor
Response
From R. M. Berry,
Encyclopedia of Life Sciences
Flagellar
Bundling
The chemotaxis signal transduction pathway in E. coli
– a network of ~50 interacting proteins – converts an
external stimulus (change in concentration of chemoattractant / repellent) into an internal stimulus (change
in concentration of intracellular response regulator,
CheY-P) which in turn interacts with the flagella motor
to bias the cell’s motion.
Table II: Activation
Probabilities
n
P1(n)
P2(n)
0
0.02
0.00291
1
0.17
0.02
2
0.5
0.17
3
0.874
0.5
4
0.997
0.98
Transition time to step change of external attractant.
Effect of varying response
Use r (s1) under input signal s1 (μ1=1 μM, σ12 = μ1, τ1 = 1 sec) to find P(r) for
different input signals, and calculate the mutual information between r (s1) and sk.
Molecule
Number
Concentration (μM)
Y
15684
18
Yp
0
0
R
250
0.29
E
6276
-
B
1928
2.27
Bp
0
0
Simulating Reactions
Reactions are simulated using Stochsim[5] package, a general platform for simulating
reactions stochastically.
Uni-molecular reaction
Symbols:
k
A 
B
p
kn(n  n0 )t
n0
Bi-molecular reaction
A  B 
C
p
kn( n  n0 ) t
2 N AV
n: Number of molecules in reaction system
n0: Number of pseudo-molecules NA:
Avogadro constant
p: Probability for a reaction to happen
Δt: Simulation time step
V: Simulation volume
Motor response
A simple threshold model[6] is used to model
motor response. The motor switches state
whenever CheY-P trace (blue trace) crosses
the threshold (red line).
Physical constants for motion:
Cell speed: 20-30 μm/sec
Mean run time: 1 sec
Mean tumble time: 0.1 sec
[5] C. J. Morton-Firth et al. 1998 J. Theor. Biol.. 192 117-128
[6]T. Emonet et al. 2005 Bioinformatics 21 2714-2721
P(r )   P( sk )
Adaptation
Mutual Information
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. In bacterial chemotaxis,
adaptation occurs when the response (e.g., running bias) returns precisely to the prestimulus level while the stimulus persists. It allows the system to compensate for the
presence of continued stimulation and to be ready to respond to further stimuli.
Adaptation variation
Adaptation[4]
The average information that observation of Y provides about the signal X, is I, the
mutual information of X and Y[7]. I is at minimum, zero, when Y is independent of X,
while it is at maximum when Y is completely determined by X. The I/O mutual
information rate can be calculated by the following equation[8].
[4] Sourjik et al. (2002) PNAS. 99 123-127
Adaptation to various step change of
aspartate. Blue: 1 μM; Red: 100 μM.
(simulation)
s: input signal; P(s): probability distribution of signal
r: response; P(r): probability distribution of response
P(r )   P( s )
s r
r(s): I-O relation, mapping s to r.
I  E[ P (r )]   P (r ) E[ P (n | r )]
n: noise
r
E[ P (r )]   P log 2 PdP
P(n|r): probability distribution of noise distribution
conditioned on response
Input to our system (E. coli chemotaxis network) is the concentration of
attractant, and the output is the number of CheY-P molecules.
[7] Spikes, Fred Rieke et al. 1997, p122-123
[8] N. Brenner et al. (2000) Neuron. 26 695-702
s r
Distribution of wild-type E. coli motor CW (grey) and CCW (black) intervals.
The calculated I/O mutual information rate
of E. coli chemotaxis network maximizes
under the condition that the response and the
input signal matches.
Discussion: the simulation results are in good agreement with experiments, although
the adaptation times differ by a small factor.
[9] S. M. Block et al. 1982 Cell 31 215-226
[10] H. C. Berg et al. 1975 PNAS 72 3235-3239
[11] T. Emonet et al. 2005 Bioinformatics 21 2714-2721
Conclusions
Input-Output Relation
By utilizing this realistic and stochastic numerical implementation, we explore E. coli
chemotaxis network from the standpoint of general information-processing concepts.
Signal
Input signal
Artificially generated Gaussian
distributed time series with
correlation time τ.
1
Output
Output
Number of CheY-P molecules
exp(
The chemotaxis network is able to extract as much as information possible once the
input signal varies slower relative to the response time of the chemotaxis network.
Under an input signal with specific statistics, the chemotaxis network varies its
response to optimize the cell’s performance, maximizing the mutual information
between input signal and output response.
E. coli
chemotaxis
network
p( s) 
Attractant: 30 μM aspartate.
Repellent: 100 μM NiCl2
The I/O mutual information rate of E. coli
chemotaxis network is plotted as a function
of correlation time τ. The Gaussian
distributed signals used here have means of 1,
3, 5, and 10, respectively.
Table III: Initial Protein
Levels
k
Motion
Effect of τ in I/O mutual information
Motor CCW and CW intervals[11]
Fluorescently labeled E. coli (Berg lab)
Chemotaxis network
Adaptation time[10]
Table I: Signal Transduction Network
[1] R. C. Hardie et al. (2001) Nature 413, 186-193
[2] M. Postma et al. (1999) Biophysical Journal 77 1811-1823
[3] S. M. Block et al. 1982 Cell 31 215-226
The chemotaxis signal transduction pathway in E. coli is one of the
best-characterized chemotaxis network, all of the genes and proteins
involved in its chemotaxis network are known and most of them
have been crystallized.
Body size:
1 μm in length, 0.4 μm in radius
Flagellum:
10 μm long, 45 nm in diameter
Cell response when exposed to a step change of aspartate from 0 to 0.1 mM (left),
10 μM (right) beginning at 5 sec.
Future Work
Use a realistic description of motor to replace the simple threshold model of motor
response.
(s   )
)
2
2
2
2 2
<s(0)s(t)> ~ exp(-t / )
Take into account the clustering effect among trans-membrane aspartate receptors to
improve the performance of the numerical implementation.
Investigate role of adaptation time.
Upper: Gaussian distributed
signal (μ=3 μM, σ2 = μ, τ =
1 sec)
Lower panel: Response to
the input signal.
The response is the
average of responses in
each bin of signal.
I/O relation under signals
with different statistics.
(τ = 1 sec)
Acknowledgment
I thank Sima Seteyashgar for the help in preparing this poster, and thank Xianfeng for
useful suggestions.