The chemotaxis network is able to extract 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 response, maximizing the mutual information between input signal and output response.

Optimal Strategy of E. coli Chemotaxis Network from Information Processing View

Lin Wang and Sima SetayeshgarDepartment of Physics, Indiana University, Bloomington, Indiana 47405

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

Fluorescently labeled E. coli (Berg lab)Body size: 1 μm in length, 0.4 μm in radiusFlagellum: 10 μm long, 45 nm in diameter

MotivationChemical 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 ChemotaxisChemotaxis, motion toward desirable chemicals (usually nutrients) and away from harmful ones, is achieved through continuous ‘runs’ and ‘tumbles’.

AdaptationAdaptation 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.

Numerical Implementation

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.

Model Validation

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.

Input-Output Relation

Conclusions

Signal Transduction

Pathway

Motor Response

[CheY-P]

Stimulus

Flagellar Bundling

Motion

Photoreceptor[1,2]

Use E. coli chemotaxis network as a prototype to explore the general information processing principle in biological systems.

[1] R. C. Hardie et al. (2001) Nature 413, 186-193 [2] J. Oberwinkler et al. (2000) PNAS 97, 8578-8583

Chemotaxis network

Numerical

Adaptation[3]

[3] Sourjik et al. (2002) PNAS. 99 123-127[4] H. C. Berg, (1975) PNAS. 72 71-713

Attractant: 30 μM aspartate.

Repellent: 100 μM NiCl2

Adaptation to various step change of attractant serine (mM).

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

Molecule Number Concentration (μM)

Y 15684 18

Yp 0 0

R 250 0.29

E 6276 -

B 1928 2.27

Bp 0 0

Parameter values of chemotaxis networkTable I: Signal Transduction Network

Table III: Initial Protein Levels

Table II: Activation Probabilities

Motor response[6]

Motor responseA simple threshold model is used to model motor response. The motor switches state whenever CheY-P trace (blue trace) crosses the threshold (red line)

Simulating reactionsReactions are simulated using Stochsim[5] package, a general platform for simulating reactions stochastically. Reactions have a probability p to occur.

Symbols:n: Number of molecules in reaction systemn0: Number of pseudo-moleculesNA: Avogadro constantp: Probability for a reaction to happenΔt: Simulation time stepV: Simulation volume

kA B 0

0

( )kn n n tp

n

0( )

2 A

kn n n tp

N V

kA B C

Bi-molecular reaction

Uni-molecular reaction

FocusThe 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.

0 5 10 15 2010

-4

10-3

10-2

10-1

100

CW and CCW intervals [sec]

Fra

cti

on

10-2

100

101

102

103

0

50

100

Step Size [M]

Ad

ap

tati

on

Tim

e [

se

c]

Adaptation

Motor CCW and CW intervals

Adaptation time

[5] C. J. Morton-Firth et al. 1998 J. Theor. Biol.. 192 117-128[6]T. Emonet et al. 2005 Bioinformatics 21 2714-2721

Comment on agreement: the simulation results are in good agreement with experiments, although, the adaptation differ by a factor of unit in time scale.

0 20 40 550

0.5

1

Time [sec]

Pro

ba

bilit

y C

CW

Experiment: Cell response when expose to a step change of aspartate from 0 to 0.1 mM, beginning at 5 sec[9].

Experiment: Transition time to step change of external attractant (aspartate, AIbu) and repellent (L-leucine)[10].

Simulation: Cell response when exposed to a step change in aspartate from 0 to 10 μM, beginning at 5 sec.

Experiment: Distribution of wild-type E. coli motor CW (grey) and CCW (black) intervals[11].

0 15 30 45

1250

1350

1450

1550

Time [sec]

Nu

mb

er

of

Ch

eY

p M

ole

cu

les

0 2 4 6 8 100

2

4

Time [sec]

[As

] M

0 2 4 6 8 1013

14

15

Time [sec]

NC

he

Yp

ltiply100

x100

0 2 4 6 8 101300

1350

1400

1450

1500

[Asp] M

NC

he

Yp

0 5 10 15 20 25 301300

1350

1400

1450

1500

1550

[Asp] M

NC

he

Yp

=1=3=5=10

Upper:

Gaussian distributed signal

(μ=3 μM, σ2 = μ, τ = 1 sec)

Lower panel:

Response to the input signal.

I/O relation under signals

with different statistics. (τ = 1 sec)

1.Signal is binned.

2. response is the average of

responses to signals falls

within each bin.

Mutual InformationThe 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].

2

( ) ( )

[ ( )] ( ) [ ( | )]

[ ( )] log

s r

r

P r P s

I E P r P r E P n r

E P r P 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[7] Spikes, Fred Rieke et al. 1997, p122-123

[8] N. Brenner et al. (2000) Neuron. 26 695-702

Effect of Correlation TimeτMy first step is to investigate the effect of correlation time τ to the I/O mutual information rate of the chemotaxis network.

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)

Effect of τ in I/O mutual information

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.

Use a realistic description of motor to Replace the simple threshold model of motor response.

Taken into account the clustering effect among trans-membrane aspartate receptors to improve the performance of the numerical implementation.

Role of adaptation time.

Future Work

Effect of τ in I/O relation

Effect of varying responseUse found r(s1) under input signal μ1=1 μM, σ1

2 = μ1, τ1 = 1 sec to find P(r) for different input signals, and calculate the mutual information

between r(s1) and sk. ( ) ( )ks r

P r P s

0 5 10 15 20 250

1

2

3

4

<Signal> [M]

Info

rmat

ion

Rat

e [b

it]

The calculated I/O mutual information rate

of E. coli chemotaxis network maximizes

under the condition that the response and the

input signal matches.

[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

0 0.5 1 1.5 2 2.5 3 3.51.5

2

2.5

3

3.5

4

4.5

5

Correlation time of signal [sec]

Info

rma

tio

n r

ate

(b

it)

S = 1 M

S = 3 M

S = 5 M

S = 10 M

0 1 2 3 4 5 61300

1350

1400

1450

[Asp] M

NC

he

Yp

= 0.1 sec = 0.3 sec = 0.8 sec = 1 sec

E. coli chemotaxis networkSignal Output

2

22

1 ( )( ) exp( )

22<s(0)s(t)> ~ exp(-t / )

sp s

Input signalArtificially generated Gaussian distributed time series with correlation time τ.

OutputNumber of CheY-P molecules is used as the system output.

Simulation: Adaptation time to step change of concentration of aspartate.

Simulation: Distribution of wild-type E. coli motor CW (grey) and CCW (black) intervals.

Adaptation variation[3]

E. Coli Chemotaxis[3]

Photon Δ[Ca2+]

Ca2+ Fluorescence

AttractantΔ[CheY-P]

Response of drosophila photoreceptor to photon absorption.

Response of E. coli to external attractant. Yellow: CheY-P relative level.

Ru

n B

ias