Transcript E. coli - Department of Physics

Optimal Strategy of E. coli Chemotaxis Network from Information Processing View
Lin Wang and Sima Setayeshgar
Department of Physics, Indiana University, Bloomington, Indiana 47405
Focus
Model Validation
Effect of Correlation Timeτ
Chemical signaling cascade is the most fundamental information
processing unit in biological systems. Generally, it converts external
stimulus to change in concentration of intracellular signaling molecules.
E. Coli Chemotaxis[3]
Photoreceptor[1,2]
The preliminary result suggests that E. coli varies its response function
under signals with different statistics. My 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. My direction is to construct a measurement of information
transmission rate and investigate the role of varying response.
Adaptation
My first step is to investigate the effect of correlation time τ to the I/O
mutual information rate of the chemotaxis network.
Flagellar
Bundling
Motion
Uni-molecular reaction
k
A 
B
From R. M. Berry, Encyclopedia
of Life Sciences
Physical constants for motion:
Cell speed: 20-30 μm/sec
Mean run time: 1 sec
Mean tumble time: 0.1 sec
Adaptation
Run Bias
Adaptation is an important and generic property of signaling systems,
where the response (e.g. running bias in chemotaxis) returns precisely to
the pre-stimulus level while the stimulus persists. Adaptation functions
from short time scale (impulse) to long time scale (evolution). It allows
the system to compensate for the presence of continued stimulation and
to be ready to respond to further stimuli.
Adaptation variation[3]
Adaptation[3]

p
kn(n  n0 )t
n0
A  B 
C
p
kn( n  n0 ) t
2 N AV
Motor response[6]
1550
[3] Sourjik et al. (2002) PNAS. 99 123-127
[4] H. C. Berg, (1975) PNAS. 72 71-713
Adaptation to various step change
of attractant serine (mM).
1
0.17
0.02
2
0.5
0.17
3
0.874
0.5
4
0.997
0.98
Molecule
Number
Concentration (μM)
Y
15684
18
Yp
0
0
R
250
0.29
E
6276
-
B
1928
2.27
Bp
0
0
1350
1250
0
15
30
Time [sec]
45
Experiment: Transition time to step
change of external attractant (aspartate,
AIbu) and repellent (L-leucine)[10].
The average information that observation of Y provide 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].
P(r )   P( s)
I  E[ P (r )]   P (r ) E[ P (n | r )]
r
E[ P (r )]   P log 2 PdP
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): noise distribution conditioned on response
0
1
10
10
10
Step Size [ M]
2
10
NCheYp
= 0.1 sec
= 0.3 sec
= 0.8 sec
= 1 sec
Simulation: Adaptation time to step
change of concentration of aspartate.
Experiment: Distribution of wild-type E.
coli motor CW (grey) and CCW (black)
intervals[11].
5
6
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.
 = 1 M
10
-2
2
10
-3
1.5
10
-4
15
4
3.5
10
10
3
[Asp]  M
4
-1
5
2
4.5
0
CW and CCW intervals [sec]
1
5
10
0
0
Effect of τ in I/O mutual information
3
Motor CCW and CW intervals
S
3
 = 3 M
S
 = 5 M
2.5
S
 = 10  M
S
0
0.5
1
1.5
2
2.5
Correlation time of signal [sec]
3
3.5
20
Simulation: Distribution of wild-type
E. coli motor CW (grey) and CCW
(black) intervals.
Effect of varying response
Use found r(s1) under input signal 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.
P(r ) 
P( s )

Comment on agreement: the simulation results are in good
agreement with experiments, although, the adaptation differ by a
factor of unit in time scale.
s r
[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
Input-Output Relation
k
4
The calculated I/O mutual information rate
of E. coli chemotaxis network maximizes
under the condition that the response and the
input signal matches.
3
2
1
0
0
5
10
15
<Signal> [ M]
20
25
Utilizing this realistic and stochastic numerical implementation that is
consistent with experiments, we explore E. coli chemotaxis network from
the standpoint of general information-processing concepts.
Conclusions
E. coli
chemotaxis
network
Signal
Input signal
Artificially generated Gaussian
distributed time series with
correlation time τ.
1
(s   )2
p( s) 
exp(
)
2
2
2
2
<s(0)s(t)> ~ exp(-t / )
[5] C. J. Morton-Firth et al. 1998 J. Theor. Biol.. 192 117-128
[6]T. Emonet et al. 2005 Bioinformatics 21 2714-2721
Mutual Information
1300
50
Information rate (bit)
0.00291
Motor response
A simple threshold model is used to model motor
response. The motor switches state whenever
CheY-P trace (blue trace) crosses the threshold
(red line)
1450
s r
Attractant: 30 μM aspartate.
Repellent: 100 μM NiCl2
0.02
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
Bi-molecular reaction
k
Number of CheYp Molecules
The chemotaxis signal transduction
pathway in E. coli – a network of ~50
interacting proteins – 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 motion.
0
Simulating reactions
Reactions are simulated using Stochsim[5] package, a general platform for
simulating reactions stochastically. Reactions have a probability p to
occur.

1400
Information Rate [bit]
Stimulus
Numerical
P2(n)
Table III: Initial Protein Levels
Fluorescently labeled E. coli (Berg lab)
Body size:
1 μm in length, 0.4 μm in radius
Flagellum:
10 μm long, 45 nm in diameter
Signal
Transduction
Pathway
[CheY-P]
Motor
Response
P1(n)




100
0 -2
10
Fraction
Chemotaxis, motion toward desirable chemicals (usually nutrients) and
away from harmful ones, is achieved through continuous ‘runs’ and
‘tumbles’.
Chemotaxis network
Probability CCW
Table II: Activation Probabilities
n
Signals: μ=1 μM, σ2 = μ and τ = 0.1, 0.3, 0.8,
1 sec, respectively.
At τ > 0.8 sec, the response does not change
any more.
(This holds for signals with different mean
values)
1450
1350
Table I: Signal Transduction Network
E. coli Chemotaxis
55
Adaptation time
Parameter values of chemotaxis network
[1] R. C. Hardie et al. (2001) Nature 413, 186-193
[2] J. Oberwinkler et al. (2000) PNAS 97, 8578-8583
20
40
Time [sec]
4
Output
Output
Number of CheY-P molecules
is used as the system output.
1500
1550
1450
1500
2
0
0
15
2
4
6
Time [sec]
8
10
x100
13
1400
1350
14
0
2
4
6
Time [sec]
8
10
1300
2
4
6
[Asp]  M
8
10
1.Signal is binned.
Upper:
Gaussian distributed signal 2. response is the average of
(μ=3 μM, σ2 = μ, τ = 1 sec) responses to signals falls
within each bin.
Lower panel:
Response to the input signal.
[7] Spikes, Fred Rieke et al. 1997, p122-123
[8] N. Brenner et al. (2000) Neuron. 26 695-702
ltiply100
Under an input signal with specific statistics, the chemotaxis network
varies its response to optimize the cell’s response, maximizing the
mutual information between input signal and output response.
Use a realistic description of motor to Replace the simple threshold
model of motor response.
 =1
 =3
 =5
1450
 =10
1400
1350
0
The chemotaxis network is able to extract once the input signal varies
slower relative to the response time of the chemotaxis network.
Future Work
NCheYp
Use E. coli chemotaxis network as a prototype to explore the
general information processing principle in biological systems.
Numerical Implementation
0
Simulation: Cell response when
exposed to a step change in aspartate
from 0 to 10 μM, beginning at 5 sec.
Adaptation Time [sec]
Response of E. coli to external attractant.
Yellow: CheY-P relative level.
Experiment: Cell response when expose
to a step change of aspartate from 0 to 0.1
mM, beginning at 5 sec[9].
Effect of τ in I/O relation
0.5
0
NCheYp
Response of drosophila photoreceptor
to photon absorption.
Δ[CheY-P]
[As]  M
Δ[Ca2+]
Attractant
1
NCheYp
Photon
Ca2+ Fluorescence
Motivation
1300
0
5
10
15
20
[Asp]  M
25
30
I/O relation under signals
with different statistics.
(τ = 1 sec)
Taken into account the clustering effect among trans-membrane aspartate
receptors to improve the performance of the numerical implementation.
Role of adaptation time.