Marcus Tindall Centre for Mathematical Biology Mathematical Institute 24-29 St Giles’ Oxford....

Marcus Tindall

Centre for Mathematical Biology Mathematical Institute

24-29 St Giles’Oxford.

E-mail: [email protected].

PESB, Manchester, 2007.

Spatiotemporal Modelling of Intracellular

Signalling in Bacterial Chemotaxis

Outline

• Bacterial chemotaxis.

• Intracellular signalling in E. coli.

• A mathematical model of intracellular signalling in E. coli.

• Intracellular signalling in R. sphaeroides.


• A spatiotemporal model of signalling in E. coli.

• Determining reaction rates from in vitro data.

• Future work


Bacterial chemotaxis.

• Bacteria commonly 2-3μm in length, 1μm wide.

• Respond to gradients of attractant and repellent.

• In absence of stimulus default setting is short runs with random reorientating tumbles.

• Detection of attractant gradient leads to extension of runs (chemotaxis).

• E. coli is one of the most commonly studied systems.

• Bacterial chemotaxis is a paradigm for systems biology.

• Mathematical modelling (single and population scale) has aided in understanding experimental observations for the past 35 plus years.


Bacterial chemotaxis.

• There exist a number of different species of bacteria which respond to stimuli in a similar way, but which have very different intracellular signalling dynamics.

• Bacterial response is by detection of attractant gradient by receptor clusters at certain regions in the cell.

• Movement is initiated by rotation of flagella at opposing end of bacterium.

• Signalling between receptors and flagella motors is by a series of intracellular phosphotransfer reactions.

Why?


Intracellular Signalling in E. coli



P

k

P CheYCheACheYCheA2

P

k

P CheBCheACheBCheA3

CheZCheYCheZCheY4k

P

CheBCheB5k

P

CheYCheY6k

P

Process Reaction Details

Autophosphorylation

Phosphotransfer CheAP to CheY.

CheAP to CheB.

Dephosphorylation Dephosphorylation by CheZ.

Natural dephosphorylation.

Natural dephosphorylation.

P

k

CheACheA1



Rate Description Value Reference

k1 Autophosphorylation of CheA. 34s-1 Francis et al. (2002)

Shrout et al. (2003)

k2 Phosphotransfer from CheAP to CheY. 1 x 108(Ms)-1 Stewart et al. (2000)

k3 Phosphotransfer from CheAP to CheB. 1.5 x 107(Ms)-1 Stewart (1993)

k4 CheYP dephosphorylation by CheZ. 1.6 x 106(Ms)-1 Li and Hazelbauer (2004)

Sourjik and Berg (2002a)

k5 CheBP natural dephosphorylation 0.7s-1 Stewart (1993)

k6 CheYP natural dephosphorylation. 8.5 x 10-2 s-1 Smith et al. (2003)

Stewart and van Bruggen (2004).

CheY, CheYP diffusion coefficients. 10μm2 s-1 Elowitz et al. (1999)

Segall et al. (1985)

CheBP diffusion coefficient. 7μm2 s-1 Falke et al. (1997)

AT Total CheA concentration in an E. coli cell. 7.9μm Bray website data.

(www.pdn.cam.ac.uk/groups/

comp-cell/Rates.html)

YT Total CheY concentration in an E. coli cell. 9.7μm Bray website data.

BT Total CheB concentration in an E. coli cell. 0.28μm Bray website data.

Z Total CheZ concentration in an E. coli cell. 3.8μm Bray website data

PBD

PYY DD ,

What is the importance of protein spatial

localisation within a bacterial cell?



A Spatiotemporal Model of Intracellular Signalling in E. coli

• Consider a 2-D model of a cell.



)6(

)5(

)4(

)3(

)2(

)1(

532

532

6422

6422

321

321

PPPBP

PPB

PPPPYP

PPPY

PPP

PP

BkBAkBDt

B

BkBAkBDt

B

YkZYkYAkYDt

Y

YkZYkYAkYDt

Y

BAkYAkAkt

A

BAkYAkAkt

A

P

P

)10(

)9(

)8(

)7(

52

52

62

62

PPBP

PB

PPYP

PY

BkBDt

B

BkBDt

B

YkYDt

Y

YkYDt

Y

P

P

In the regions Ω2 and Ω3

and in Ω1



Boundary conditions

Initial conditions

1

.0),(ˆand0),(ˆ,0),(ˆ,0),(ˆ tBtBtYtY PP xnxnxnxn

We assume no flux boundary conditions on

The flux of CheY, CheYP CheB and CheBP is taken to be continuous between each of the three regions Ω1, Ω2 and Ω3.

,0)0,(and0)0,(,0)0,(,)0,( 0 xxxx PP BBYYY

In Ω1 we have

and in Ω2 and Ω3

.0)0,(and)0,(,0)0,(,)0,(,0)0,(,)0,( 000 xxxxxx PPP BBBYYYAAA



Solution method

• Numerical solutions using Femlab.

• Non-dimensionalise system of equations.

• Transient and steady-state analysis.



Change in CheYp concentration

Intracellular Signalling in Rhodobacter sphaeroides


Intracellular Signalling in Rhodobacter sphaeroides


CheA3,CheA4

CheA2

CheA2

• Consider subnetwork of CheA2, CheA3, CheA4, CheY1-CheY6, CheB1 and CheB2.

• How does spatial localisation of the proteins and their reactions effect the concentration of CheY6 (dynamically and in steady-state)?

In vitro Reaction Data


Porter, S. and Armitage, J.P. (2002). Phosphotransfer in Rhodobacter sphaeroides chemotaxis, J. Mol. Biol., 324, 35-45.

Determining reaction rates from in vitro data


• Many of the in vitro reactions are of the form

ADPCheAATPCheAATPCheA iP

k

ii

1

k

k

jPijiP CheYCheACheYCheA2

2

k

k

PhosphateCheY0HCheY j2jP

3

k

Autophosphorylation

Phosphotransfer

Dephosphorylation

where when i=1, j=1, 2, 3 and 5 and when i=2, j=1,..6.

• Similar for CheB1 and CheB2. CheA3 and CheA4 are more complex reactions.



• Governing ODE equations (assuming mass action kinetics) are

)2()()(

)1()()()(

322

22'1

PPPTPTPp

PPTPTPPTp

ykyxxkyyxkdt

dy

yxxkyyxkxxkdt

dx

,0)0(and)0( 0 PPP yxxwith

• Rates of autophosphorylation of CheAs (k1) are known from experiment.

and

.[ATP]K

[ATP] and,

D1

'1

kkyyyxxx PTPT



• Rate of CheY dephosphorylation (k3) can be determined by adding eqns (1) and (2) to obtain

)3().('1

3 PSTPS

xxy

kk

Protein CheY1 CheY2 CheY3 CheY4 CheY5 CheY6

CheA1 9.43x10-3 4.02 1.88x10-1 - 2.69x10-1 -

CheA2 2.01x10-2 4.04x10-2 5.39x10-2 3.36x10-2 5.20x10-2 1.76x10-1

Protein

CheB1 CheB2

CheA2 4.70x10-3 2.63x10-2



• We determine the phosphotransfer rates using a data fitting program Berkeley Madonna (BM).

• We have utilised four strategies to determine the best data fit.

(1) Allow BM to determine all rates (assume none are known).

(2)(i) Fix k1 and use k3 determined from CheA1 transfer and use BM to determine k2 and k-2.

(2)(ii) Fix k1 and use k3 determined from CheA2 transfer and use BM to determine k2 and k-2.

(3) Fix k1 and allow BM to determine all remaining parameters.

• We have also used asymptotic estimates where appropriate.

Determining reaction rates from in vitro dataPESB, Manchester, 2007.

Example: CheA2P to CheY6



Methodology

Residue k1 k2 k-2 k3

(1) 0.084 1.59x10-3 8.59x10-3 7.69x10-3 9.59x10-3

(2)(i) - - - - -

(2)(ii) 0.167 5.86x10-3 2.03x10-3 2.27x10-3 1.76x10-2

(3) 0.155 5.86x10-3 2.05x10-3 2.23x10-8 2.15x10-1



Methodology Residue k1 k2 k-2 k3

(1) 0.103 3.60x10-3 2.47x10-2 7.30x10-9 1.32x10-3

(2)(i) 0.821 4.22x10-3 3.33x10-2 2.01x10-8 9.43x10-3

(2)(ii) 0.114 4.22x10-3 3.38x10-2 1.40x10-9 2.01x10-2

(3) 0.113 4.22x10-3 3.49x10-3 7.53x10-10 2.04x10-2


• Best fit from using case (2)(ii), but asymptotically determine k2=1.50x10-2 from inner solution then use this to determine k-2=9.31x10-11 using BM.



Methodology for determining ‘best fit’ phosphotransfer rates.

(1) Use fixed k1 and k3. If not good graphical fit then proceed to (2).

(2) Determine if asymptotics useful to help in determining either k2 or k-2.

Review all results with the experimentalists!

(3) If (2) not possible then determine next case best fit from k3 as free parameter.

(4) If still poor fit then determine validity of all parameter fit.

Future Work


• Finish determining reaction rates for R. sphaeroides.

• Use these in our reaction-diffusion model of intracellular signalling in R. sphaeroides.

• Consider experimentally re-determining reaction rates where necessary.


Acknowledgements

• Prof. Philip Maini, Mathematical Institute, University of Oxford.

• Prof. Judy Armitage, Dept. of Biochemistry, University of Oxford.

• Dr Steven Porter, Dept. of Biochemistry, University of Oxford.

Publications

Tindall, M., Maini, P., Porter, S., and Armitage, J., Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations. Submitted to the Bulletin of Mathematical Biology.

Tindall, M., Porter, S., Maini, P., Gaglia, G., and Armitage, J., Overview of mathematical approaches used to model bacterial chemotaxis I: The single cell. Submittedto the Bulletin of Mathematical Biology.

Tindall, M., Maini, P., Armitage, J., Singleton, C. and Mason, A., Intracellular signalling during bacterial chemotaxis in Practical Systems Biology (2007).

Marcus Tindall Centre for Mathematical Biology Mathematical Institute 24-29 St Giles’ Oxford....

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