1. Marcus Tindall Centre for Mathematical Biology Mathematical Institute 24-29 St Giles’ Oxford. E-mail: tindallm@maths.ox.ac.uk. PESB, Manchester, 2007. Spatiotemporal Modelling of Intracellular Signalling in Bacterial Chemotaxis

2. Outline Bacterial chemotaxis. Intracellular signalling in E. coli. A mathematical model of intracellular signalling in E. coli. Intracellular signalling in R. sphaeroides. PESB, Manchester, 2007. A spatiotemporal model of signalling in E. coli. Determining reaction rates from in vitro data. Future work

3. PESB, Manchester, 2007. 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.

4. PESB, Manchester, 2007. 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?

5. PESB, Manchester, 2007. Intracellular Signalling in E. coli

6. PESB, Manchester, 2007. Intracellular Signalling in E. coli ProcessReactionDetails Autophosphorylation PhosphotransferCheA P to CheY. CheA P to CheB. DephosphorylationDephosphorylation by CheZ. Natural dephosphorylation.

7. PESB, Manchester, 2007. Intracellular Signalling in E. coli RateDescriptionValueReference k1k1 Autophosphorylation of CheA.34s -1 Francis et al. (2002) Shrout et al. (2003) k2k2 Phosphotransfer from CheA P to CheY.1 x 10 8 (Ms) -1 Stewart et al. (2000) k3k3 Phosphotransfer from CheA P to CheB.1.5 x 10 7 (Ms) -1 Stewart (1993) k4k4 CheY P dephosphorylation by CheZ.1.6 x 10 6 (Ms) -1 Li and Hazelbauer (2004) Sourjik and Berg (2002a) k5k5 CheB P natural dephosphorylation0.7s -1 Stewart (1993) k6k6 CheY P natural dephosphorylation.8.5 x 10 -2 s -1 Smith et al. (2003) Stewart and van Bruggen (2004). CheY, CheY P diffusion coefficients.10μm 2 s -1 Elowitz et al. (1999) Segall et al. (1985) CheB P diffusion coefficient. 7 μm 2 s -1 Falke et al. (1997) ATAT Total CheA concentration in an E. coli cell.7.9μmBray website data. (www.pdn.cam.ac.uk/groups/ comp-cell/Rates.html) YTYT Total CheY concentration in an E. coli cell.9.7μmBray website data. BTBT Total CheB concentration in an E. coli cell.0.28μmBray website data. ZTotal CheZ concentration in an E. coli cell.3.8μmBray website data

8. What is the importance of protein spatial localisation within a bacterial cell? PESB, Manchester, 2007.

9. A Spatiotemporal Model of Intracellular Signalling in E. coli Consider a 2-D model of a cell.

10. PESB, Manchester, 2007. A Spatiotemporal Model of Intracellular Signalling in E. coli In the regions Ω 2 and Ω 3 and in Ω 1

11. PESB, Manchester, 2007. A Spatiotemporal Model of Intracellular Signalling in E. coli Boundary conditions Initial conditions We assume no flux boundary conditions on The flux of CheY, CheY P CheB and CheB P is taken to be continuous between each of the three regions Ω 1, Ω 2 and Ω 3. In Ω 1 we have and in Ω 2 and Ω 3

12. PESB, Manchester, 2007. A Spatiotemporal Model of Intracellular Signalling in E. coli Solution method Numerical solutions using Femlab. Non-dimensionalise system of equations. Transient and steady-state analysis.

13. PESB, Manchester, 2007. A Spatiotemporal Model of Intracellular Signalling in E. coli Change in CheYp concentration

14. Intracellular Signalling in Rhodobacter sphaeroides PESB, Manchester, 2007.

15. Intracellular Signalling in Rhodobacter sphaeroides PESB, Manchester, 2007. CheA 3,CheA 4 CheA 2 Consider subnetwork of CheA 2, CheA 3, CheA 4, CheY 1 -CheY 6, CheB 1 and CheB 2. How does spatial localisation of the proteins and their reactions effect the concentration of CheY 6 (dynamically and in steady-state)?

16. In vitro Reaction Data PESB, Manchester, 2007. Porter, S. and Armitage, J.P. (2002). Phosphotransfer in Rhodobacter sphaeroides chemotaxis, J. Mol. Biol., 324, 35-45.

17. Determining reaction rates from in vitro data PESB, Manchester, 2007. Many of the in vitro reactions are of the form Autophosphorylation Phosphotransfer Dephosphorylation where when i=1, j=1, 2, 3 and 5 and when i=2, j=1,..6. Similar for CheB 1 and CheB 2. CheA 3 and CheA 4 are more complex reactions.

18. Determining reaction rates from in vitro data PESB, Manchester, 2007. Governing ODE equations (assuming mass action kinetics) are with Rates of autophosphorylation of CheAs (k 1 ) are known from experiment. and

19. Determining reaction rates from in vitro data PESB, Manchester, 2007. Rate of CheY dephosphorylation (k 3 ) can be determined by adding eqns (1) and (2) to obtain ProteinCheY 1 CheY 2 CheY 3 CheY 4 CheY 5 CheY 6 CheA 1 9.43x10 - 3 4.021.88x10 - 1 -2.69x10 - 1 - CheA 2 2.01x10 - 2 4.04x10 - 2 5.39x10 - 2 3.36x10 - 2 5.20x10 - 2 1.76x10 - 1 ProteinCheB 1 CheB 2 CheA 2 4.70x10 - 3 2.63x10 - 2

20. Determining reaction rates from in vitro data PESB, Manchester, 2007. 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 k 1 and use k 3 determined from CheA 1 transfer and use BM to determine k 2 and k -2. (2)(ii) Fix k 1 and use k 3 determined from CheA 2 transfer and use BM to determine k 2 and k -2. (3) Fix k 1 and allow BM to determine all remaining parameters. We have also used asymptotic estimates where appropriate.

21. Determining reaction rates from in vitro data PESB, Manchester, 2007. Example: CheA 2P to CheY 6

22. Determining reaction rates from in vitro data PESB, Manchester, 2007. Example: CheA 2P to CheY 6 MethodologyResiduek1k1 k2k2 k -2 k3k3 (1)0.0841.59x10 -3 8.59x10 -3 7.69x10 -3 9.59x10 -3 (2)(i)----- (2)(ii)0.1675.86x10 -3 2.03x10 -3 2.27x10 -3 1.76x10 -2 (3)0.1555.86x10 -3 2.05x10 -3 2.23x10 -8 2.15x10 -1

23. Determining reaction rates from in vitro data PESB, Manchester, 2007. Example: CheA 2P to CheY 1

24. Determining reaction rates from in vitro data PESB, Manchester, 2007. MethodologyResiduek1k1 k2k2 k -2 k3k3 (1)0.1033.60x10 -3 2.47x10 -2 7.30x10 -9 1.32x10 -3 (2)(i)0.8214.22x10 -3 3.33x10 -2 2.01x10 -8 9.43x10 -3 (2)(ii)0.1144.22x10 -3 3.38x10 -2 1.40x10 -9 2.01x10 -2 (3)0.1134.22x10 -3 3.49x10 -3 7.53x10 -10 2.04x10 -2 Example: CheA 2P to CheY 1 Best fit from using case (2)(ii), but asymptotically determine k 2 =1.50x10 -2 from inner solution then use this to determine k -2 =9.31x10 -11 using BM.

25. Determining reaction rates from in vitro data PESB, Manchester, 2007. Methodology for determining ‘best fit’ phosphotransfer rates. (1) Use fixed k 1 and k 3. If not good graphical fit then proceed to (2). (2) Determine if asymptotics useful to help in determining either k 2 or k -2. Review all results with the experimentalists! (3) If (2) not possible then determine next case best fit from k 3 as free parameter. (4) If still poor fit then determine validity of all parameter fit.

26. Determining reaction rates from in vitro data PESB, Manchester, 2007.

27. Future Work PESB, Manchester, 2007. 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.

28. PESB, Manchester, 2007. 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. Submitted to 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).