In-silico Implementation of Bacterial Chemotaxis

Lin WangAdvisor: Sima Setayeshgar

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

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

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

Modeling Chemotaxis in E. coli

Signal Transduction

Pathway

Motor Response

[CheY-P]

Stimulus

Flagellar Response

Motion

Outline

Chemotaxis signal transduction network in E. coli

Stochastic implementation of reaction network using Stochsim

Flagellar and motor response

Preliminary numerical results

Chemotaxis Signal Transduction Pathway in E. coli

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)

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

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

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

Autophosphorylation* *fkE E p

E*: En*, En

*a

Rate constant:kf = 15.5 sec-1

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

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

CheB ReactionsfkBp B

B: CheBBp: CheB-P

Rate constant:kf = 0.35 sec-1

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

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,

Stochsim Package

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

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

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

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

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

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)

Motion

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

CCW runCW tumble

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

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.

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

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)

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.

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:

Thanks