Optimal Strategy in E. coli Chemotaxis: An Information Theoretic ApproachLin Wang and Sima Setayeshgar

Department of Physics, Indiana University, Bloomington, Indiana 47405

Physical constants of motion: Cell speed: 20-30 μm/secMean run time: 1 secMean tumble time: 0.1 sec

E. coli is the best-studied organism in molecular biology, providing an experimentally accessible basis for our understanding of fundamental cellular processes.

Body size: 1 μm in length, 0.4 μm in radiusFlagellum: 10 μm long, 45 nm in diameter

MotivationBiochemical signaling is the most fundamental level of information processing in biological systems, where an external stimulus is measured and converted into a response.

E. coli Chemotaxis

AdaptationAdaptation 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.

Numerical Implementation

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.

Signal Transduction

Pathway

Motor Response

[CheY-P]

Stimulus

Flagellar Bundling

Motion

Photon counting in vision[1,2]

We use the well-characterized chemotaxis network in E. coli as a prototype for exploring general principles governing information processing in biological signaling networks.

[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

Perfect Adaptation[4]

Attractant: 30 μM aspartate. Repellent: 100 μM NiCl2

Adaptation to various step changes of aspartate. Blue: 1 μM; Red: 100 μM (simulation)

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

Chemotaxis Network Equations and Parameters

Table I: Signal Transduction Network

Table III: Initial Protein Levels

Table II: Activation Probabilities

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).

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

kA B⏐ ⏐→ 0

0

( )kn n n tp

n

+ Δ=

0( )

2 A

kn n n tp

N V

+ Δ=kA B C+ ⏐ ⏐→

Bi-molecular reaction

Uni-molecular reaction

FocusE. 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.

[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 InformationThe 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:

( )

2

( ) ( )

[ ( )] ( ) [ ( | )]

[ ( )] log

r ss 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): probability distribution of noise distribution conditioned on response

[7] Spikes, Fred Rieke et al. 1997, p122-123[8] N. Brenner et al. (2000) Neuron. 26 695-702

Adaptation Time

Molecule counting in chemotaxis[3]

Photon Δ[Ca2+],Δ[Na+],

etc.

AttractantΔ[CheY-P]

Response of drosophila photoreceptor to photon absorption

Response of E. coli to change in external attractant concentration

From R. M. Berry, Encyclopedia of Life Sciences

Courtesy of H. C. Berg lab)

Model Validation

Utilizing this realistic numerical implementation, we explore the chemotaxis network in E. coli from the standpoint of information-processing:

Input-Output Relation

Adaptation[9]

Motor CCW and CW intervals[11]

Adaptation time[10]

Discussion: Our simulation results are in good agreement with experiments, providing an experimentally faithful computational framework for bacterial chemotaxis.

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

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

Distribution of CW (grey) and CCW (black) intervals in wild-type adapted cells

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, r(si), to input signals with

{μi, σi2, τ = 1 sec}

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.

[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

E. coli chemotaxis networkSignal Output

2

22

1 ( )( ) exp( )

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

sp s

μσπσ

τ

−= −

Input signal Gaussian distributed time series for chemoattractant concentration with correlation time, τ:

OutputNumber of CheY-P

molecules

Simulation

Experiment

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).

Conclusions

Varying Statistics of Input

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 τ. (This also holds true for signals with different mean values).

Effect of τ on 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 μ=1, 3, 5, and 10 μM, respectively.

Work in progress includes investigation of:

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.

Future Work

Effect of τ on I/O relation

Effect of varying responseThe response, r (s1), to input signal s1 (with μ1=1 μM, σ1

2 = μ1,τ1 = 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:

The I/O mutual information rate is maximized when the response and the input signal are matched.

[4] Sourjik et al. (2002) PNAS. 99 123-127

Finding the input-output relation, r(s)

Here, input, s : chemoattractant concentrationoutput, r : CheY-P concentration

( )1

'

' ' ' '

( ) ( )

[ ( )] ( ) [ ( | )]

r sk k

s r

k k k kr

P r P s

I E P r P r E P n r

⏐ ⏐ ⏐ →

=

= −

∑

∑

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.

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 (τ>0.8sec), the information transmission rate approaches a constant asymptotic value.

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.

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.

Furthermore, the dynamical properties of the network, such as the adaptation time, vary for different inputs.