In-silico Implementation of Bacterial ChemotaxisLin 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/secMean 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. coliSignal
TransductionPathway
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 BindingE: receptor complexa: ligand (eg., aspartate)Rapid equilibrium:Rates1:E: KD = 1.71x10-6 M-1E*: KD = 12x10-6 M-1
f
r
k
kE a Ea
[ ][ ] D
apa K
[1] Morton-Firth et al., J. Mol. Biol. (1999)
Receptor ActivationEn: 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 0.002911 0.125 0.022 0.5 0.1253 0.875 0.54 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*aRate constants:
k1f = 5x106 M-1sec-1k1r = 1 sec-1k2f = 0.819 sec-1
(1)
(2)
1
1
2(*) (*)1
( ) ( )
( ) ( )
f
r
f
k
n nk
kn n
E a R E a R
E a R E a R
Demethylation
Bp: CheB-PEn*(a): En*, En*aRate constants:
k1f = 1x106 M-1sec-1k1r = 1.25 sec-1k2f = 0.15484 sec-1
(1)
(2)
1
1
2
* *
(*) (*)1
( ) ( )
( ) ( )
f
r
f
k
n nk
kn n
E a Bp E a Bp
E a Bp E a Bp
Autophosphorylation* *fkE E p
E*: En*, En*aRate constant:
kf = 15.5 sec-1
CheY Reactions1 2
1
f f
r
k k
kY Yp Yp Y
Y: CheYYp: CheY-PRate constants:
k1f = 1.24x10-3 sec-1k1r = 4.5x10-2 sec-1k2f = 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-PRate 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 ESkSEk
dtESd
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-moleculePseudo-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 )AkN INT N Vk
From Rate Constant to Probability Unimolecular reaction
kA B 0
0
( )kn n n tpn
0( )2 A
kn n n tpN 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 0R 200 0.24E 4246 -B 1928 2.27Bp 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 motor 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
esThreshold
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]Fr
actio
n
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