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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
Molecule counting in chemotaxis[3]
Δ[Ca2+],
Δ[Na+],
etc.
Attractant
Response of drosophila photoreceptor
to photon absorption
Model Validation
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 in maximizing this rate.
Response of E. coli to change in external
attractant concentration
Chemotaxis Network Equations and Parameters
Table I: Signal Transduction Network
E. coli is the best-studied organism in molecular biology, providing
an experimentally accessible basis for our understanding of
fundamental cellular processes.
Effect of τ on I/O relation
Adaptation[9]
Response r(s) to signals with μ=1 μM, σ2 = 1
μM2, τ = 0.1, 0.3, 0.8, 1 sec, respectively.
For τ > 1 sec, the response does not vary
significantly with . (This also holds true for
signals with different mean values).
1 μm in length, 0.4 μm in radius
10 μm long, 45 nm in diameter
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
Variable network adaptation time in response to different step changes of
concentration of external attractant, demonstrating increase in adaptation time with
increasing stimulus step size
Motor CCW and CW intervals[11]
Table III: Initial Protein
Levels
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 is achieved through a series of modulated
‘runs’ and ‘tumbles’, leading to a biased random walk in the desired direction.
Molecule
Stimulus
Signal
Transduction
Pathway
[CheY-P]
Motor
Response
Flagellar
Bundling
Motion
From R. M. Berry,
Encyclopedia of Life Sciences
The chemotaxis signal transduction pathway in E. coli is a network of
interacting proteins that converts an external stimulus (change in concentration
of chemo-attractant / 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 mean runtime. It is a model
system for studying the properties of the two-component superfamily of
receptor-regulated phosphorylation pathways in general.
We use the well-characterized chemotaxis network in E. coli as a
prototype for exploring general principles governing information
processing in biological signaling networks.
Adaptation
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.
Adaptation Time
Number
15684
18
Yp
0
0
R
250
0.29
E
6276
-
B
1928
2.27
0
0
Simulating Reactions
Reactions are simulated using Stochsim[5] package, a general platform for
simulating reactions stochastically.
Symbols:
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
Uni-molecular reaction
kn(n  n0 ) t
k
p
A 
B
n0
Bi-molecular reaction
kn( n  n0 ) t
p
2 N AV
k
A  B 
C
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).
[5] C. J. Morton-Firth et al. 1998 J. Theor. Biol.. 192 117-128
[6]T. Emonet et al. 2005 Bioinformatics 21 2714-2721
Adaptation to various step changes of
aspartate. Blue: 1 μM; Red: 100 μM
(simulation)
In bacterial chemotaxis, adaptation occurs when the steady state response
(running bias) returns precisely to the pre-stimulus 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.
Mutual Information
Furthermore, the dynamical properties of the network, such as the adaptation
time, vary for different inputs.
P(r ) 

P( s)
r(s)
s 
r
I  E[ P(r )]   P(r ) E[ P(n | r )]
r
E[ P(r )]   P log 2 PdP
Here,
s: input signal; P(s): probability distribution of signal
r: response; P(r): probability distribution of response
r(s): I-O relation, mapping s to r.
n: noise
P(n|r): probability distribution of noise distribution
conditioned on response
input, s : chemoattractant concentration
output, r : CheY-P concentration
[7] Spikes, Fred Rieke et al. 1997, p122-123
[8] N. Brenner et al. (2000) Neuron. 26 695-702
Effect of varying response
The response, r (s1), to input signal s1 (with μ1=1 μM, σ12 = μ1, 1 = 1 sec) is
used to map different input signals sk to output r’k (instead of using correct
response r(sk) to each sk). The mutual information between r’k and sk is
calculated as:
'

P(rk ) 
P( sk )
r(s
1) r
s 
I  E[ P (r )]   P(r ) E[ P(n | r )]
'
k
'
k
'
k
'
k
r
Distribution of CW (grey) and CCW (black) intervals in wild-type adapted cells
The I/O mutual information rate is
maximized when the response and the input
signal are matched.
Discussion: Our simulation results are in good agreement with experiments,
providing an experimentally faithful computational framework for bacterial
chemotaxis.
[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
Discussion: Intuitively, more information can be transmitted when input signal
changes slowly. We show that as the time scale of changes in the input signal
becomes comparable to the E. coli impulse response time (>0.8sec), the
information transmission rate approaches a constant asymptotic value.
Input-Output Relation
Utilizing this realistic numerical implementation, we explore the chemotaxis
network in E. coli from the standpoint of information-processing:
Signal
E. coli
chemotaxis
network
Input signal
Gaussian distributed time series for
chemoattractant concentration with
correlation time, :
2
As the statistics of the input stimulus varies, E. coli’s response adapts so as to
maximize the mutual information between the input signal and the output.
Conclusions
Output
Output
Number of CheY-P molecules
(s   )
p( s) 
exp(
)
2
2
2 2
<s(0)s(t)> ~ exp(-t / )
1
The average information that observation of Y provides about the signal X, is I,
the mutual information of X and Y[7]. I is minimum (zero) when Y is
independent of X, while it is maximum when Y is completely determined by X.
The Input-Output (I/O) mutual information rate, I, is given by:
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 μ=1, 3, 5, and
10 μM, respectively.
Concentration (μM)
Y
Bp
Physical constants of motion:
Cell speed: 20-30 μm/sec
Mean run time: 1 sec
Mean tumble time: 0.1 sec
Effect of τ on I/O mutual information
Table II: Activation
Probabilities
Courtesy of H. C. Berg lab)
[4] Sourjik et al. (2002) PNAS. 99 123-127
Simulation
Cell response (probability of CCW rotation of flagella, leading to running motion)
when exposed to a step change of aspartate from 0 to 0.1 mM (left), 10 μM (right)
beginning at 5 sec
Adaptation time[10]
Numerical Implementation
E. coli Chemotaxis
Perfect Adaptation[4]
Experiment
Δ[CheY-P]
[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
Body size:
Flagellum:
Varying Statistics of Input
The chemotaxis network in E. coli functions under varying environmental
conditions. We have shown that as the statistics of the input stimulus change,
the input-output relation varies. This adaptive behavior allows E. coli to extract
“as much information as possible” from the input signal (by maximizing the
mutual information between the input and output).
Future Work
Finding the input-output relation, r(s)
Work in progress includes investigation of:
Upper: Input = Gaussian
distributed signal as a
function of time {μ=3 μM, σ2
= μ, τ = 1 sec}.
Lower : Output = system
response to the input signal,
as a function of time.
Response function, r(s), as
a function of input signal.
Because of the stochastic
nature, response to input
signal varies. Each point
represents the average
value of response.
Response, r(si), to input
signals with
{μi, σi2, τ = 1 sec}
1)
Motor bias as the output of the chemotaxis network, by constructing a
more physically realistic description of the motor response based on the
statistical mechanics of switching between CW/CCW states.
2)
Role of the variable network adaptation time, from the standpoint of
optimizing information transmission.