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Page 1: In-silico Implementation of Bacterial Chemotaxis

In-silico Implementation of Bacterial Chemotaxis

Lin WangAdvisor: Sima Setayeshgar

Page 2: In-silico Implementation of Bacterial Chemotaxis

Chemotaxis in E. coli

Dimensions: Body size: 1 μm in length

0.4 μm in radius Flagellum: 10 μm long

Physical constants: Cell speed: 20-30 μm/sec

Mean run time: 1 secMean tumble time: 0.1 sec

From Berg Lab From R. M. Berry, Encyclopedia of Life Sciences

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From Single Cells to Populations …Chemotactic response of individual cells forms the basis of macroscopic pattern formation in populations of bacteria:

Colonies

Pattern formation in E. coli:From H.C. Berg and E. O. Budrene,

Nature (1995)

Biofilms

Agrobacterium biofilm:From Fuqua Lab

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Motivation

Chemotaxis as a well-characterized “model” signaling network, amenable to quantitative analysis and extension to other signaling networks from the standpoint of general information-processing concepts, such as signal to noise, adaptation and memory

Chemotaxis as an important biophysical mechanism, for example underlying initial stages of biofilm formation

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Modeling Chemotaxis in E. coli

Signal Transduction

Pathway

Motor Response

[CheY-P]

Stimulus

Flagellar Response

Motion

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Outline

Chemotaxis signal transduction network in E. coli

Stochastic implementation of reaction network using Stochsim

Flagellar and motor response

Preliminary numerical results

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Chemotaxis Signal Transduction Pathway in E. coli

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Ligand Binding

E: receptor complexa: ligand (eg., aspartate)

Rapid equilibrium:

Rates1:

E: KD = 1.71x10-6 M-1

E*: KD = 12x10-6 M-1

f

r

k

kE a Ea

[ ]

[ ] D

ap

a K

[1] Morton-Firth et al., J. Mol. Biol. (1999)

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Receptor Activation

En: methylated receptor complex; activation probability, P1(n)Ena: ligand-bound receptor complex; activation probability, P2(n)En

*: active form of En En

*a: active form of Ena

Table 1: Activation Probabilities

n P1(n) P2(n)

0 0.02 0.00291

1 0.125 0.02

2 0.5 0.125

3 0.875 0.5

4 0.997 0.98

* * [0,4]n n n nE E E a E a n

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Methylation

R: CheREn(a): En, EnaEn

(*)(a): En, En*, Ena, En

*a

Rate constants:k1f = 5x106 M-1sec-1

k1r = 1 sec-1

k2f = 0.819 sec-1

(1)

(2)

1

1

2(*) (*)1

( ) ( )

( ) ( )

f

r

f

k

n nk

k

n n

E a R E a R

E a R E a R

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Demethylation

Bp: CheB-PEn

*(a): En*, En

*a

Rate constants:k1f = 1x106 M-1sec-1

k1r = 1.25 sec-1

k2f = 0.15484 sec-1

(1)

(2)

1

1

2

* *

(*) (*)1

( ) ( )

( ) ( )

f

r

f

k

n nk

k

n n

E a Bp E a Bp

E a Bp E a Bp

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Autophosphorylation* *fkE E p

E*: En*, En

*a

Rate constant:kf = 15.5 sec-1

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CheY Reactions1 2

1

f f

r

k k

kY Yp Yp Y

Y: CheYYp: CheY-P

Rate constants:k1f = 1.24x10-3 sec-1

k1r = 4.5x10-2 sec-1

k2f = 14.15 sec-1

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CheY Phosphotransfer21

2

3

3

ff

r

f

r

kk

k

k

k

Ep Y EpY E Yp

EY E Y

Rate constants:

k1f = 5x106 M-1sec-1

k2f = 20 sec-1

k2r = 5x106 M-1sec-1

k3f = 7.5 sec-1

k3r = 5x106 M-1sec-1

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CheB ReactionsfkBp B

B: CheBBp: CheB-P

Rate constant:kf = 0.35 sec-1

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CheB Phosphotransfer21

2

3

3

ff

r

f

r

kk

k

k

k

Ep B EpB E Bp

EB E B

Rate constants:k1f = 5x106 M-1sec-1

k2f = 16 sec-1

k2r = 5x106 M-1sec-1

k3f = 16 sec-1

k3r = 5x106 M-1sec-1

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Simulating Reactions

Stochastic2: Reaction has probability P of occurringa) Generate x, a uniform random number in [0, 1].b) x <= P, reaction happens.c) x > P, reaction does not happen.

How to generate P from reaction rates?

[2] Morton-Firth et al., J. Mol. Biol. (1998)

][]][[][

11 ESkSEkdt

ESd

k

1

1

k

kE S ES

Two methods:Deterministic: ODE description, using rate constants,

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Stochsim Package

Stochsim package is a general platform for simulating reactions using a stochastic method.

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Pseudo-molecule

Pseudo-molecules are used to simulate unimolecular reaction.

Number of pseudomolecule in simulating system:

k1max: fastest unimolecular reaction ratek2max: fastest bimolecular reaction rate

1max

2max

(2 )A

kN INT N V

k

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From Rate Constant to Probability

Unimolecular reaction

kA B 0

0

( )kn n n tp

n

0( )

2 A

kn n n tp

N V

kA B C

n: number of molecules from reaction system n0: number of pseudomolecules NA: Avogadro constant

Bimolecular reaction

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Simulation Parameters

Reaction Volume: 1.41 x 10-15 liter

Rate constants given above.

Table 2: Initial Numbers of Molecules

Molecule Number Concentration (μM) Y 21284 25.07

Yp 0 0

R 200 0.24

E 4246 -

B 1928 2.27

Bp 0 0

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Output of Signal Transduction Network

Fig 1. Number of CheY-P molecules as a function of time, the trace is smoothed by an averaging window of 0.3 sec. The m

otor switches state whenever threshold (red line) is crossed. It’s assumed that there is only 1 motor/cell.

0 100 200 300 400 500 600 700 800 900 10001250

1300

1350

1400

1450

1500

Time [sec]

# of

Che

YP

mol

ecul

es

Threshold

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Flagellar Response

Flagellar state directly reflects motor state, except that 20% of the motor changing from CCW to CW is dropped3.

Assume there is only 1 flagellum/cell.

[3] Alon et al., The EMBO Journal (1998)

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Motion

Motion of the cell is determined by the state of flagellum.

CCW runCW tumble

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Run and Tumble Process

Run4

Tumble5

t t+Δt

α

v = 20 μm/sDr = 0.06205 s-1

γ = 4μ = -4.6β = 18.32

( ) 2 (0,1)rp tD N

[4] Zou et al., Biophys. J. (2003) [5] Berg and Brown, Nature (1972)

1( ) exp( )( )

( )p

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Some Simulation Results

Distribution of run and tumble intervals.

Diffusion of a population of cells in an unbounded region in the absence of stimulus.

Diffusion of a population of cells in a bounded region (z>0), with and without stimulus.

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Motor CW and CCW Intervals

Fig 2. Fraction of motor CW/CCW intervals of wild-type cell in an environment without ligand. Left: Experiment (Korobkova et al., Nature 2004); Right: Simulation

0 5 10 15 2010-4

10-3

10-2

10-1

100

CW and CCW intervals [sec]F

ract

ion

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Diffusion in Unbounded Region: No Stimulus

Fig 3. Mean-squared distance from initial position as a function of time (averaged over 540 cells). Diffusion constant is found to be 4.4 * 10-4 mm2/s, consistent with experimental results6.

[6] Paul Lewus et al., BioTech. and BioEng. (2001)

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Diffusion in Bounded Region (z>0)

Fig 4. Number of cells (out of a total of 540) above z=1.2 mm as a function of time. Red: constant linear gradient of aspartate 10-8 zM/μM; Blue: no aspartate.

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Future Directions

Optimal biochemical signal processing (role of “adaptive” network adaptation time)

Role of chemotaxis in initial stages of biofilm formation

Realistic description of chemotaxis in E. coli to explore:

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Thanks