Marcus Tindall

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PESB, Manchester, 2007.
Spatiotemporal Modelling of Intracellular
Signalling in Bacterial Chemotaxis
Marcus Tindall
Centre for Mathematical Biology Mathematical
Institute 24-29 St Giles Oxford. E-mail
tindallm_at_maths.ox.ac.uk.
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PESB, Manchester, 2007.
Outline

• Bacterial chemotaxis.

• Intracellular signalling in E. coli.

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

• A spatiotemporal model of signalling in E. coli.

• Intracellular signalling in R. sphaeroides.

• Determining reaction rates from in vitro data.

• Future work

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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.

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PESB, Manchester, 2007.
Bacterial chemotaxis.

• 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.

• 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.

Why?
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PESB, Manchester, 2007.
Intracellular Signalling in E. coli
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PESB, Manchester, 2007.
Intracellular Signalling in E. coli
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PESB, Manchester, 2007.
Intracellular Signalling in E. coli
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PESB, Manchester, 2007.
What is the importance of protein spatial
localisation within a bacterial cell?
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PESB, Manchester, 2007.
A Spatiotemporal Model of Intracellular
Signalling in E. coli

• Consider a 2-D model of a cell.

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PESB, Manchester, 2007.
A Spatiotemporal Model of Intracellular
Signalling in E. coli
In the regions O2 and O3
and in O1
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PESB, Manchester, 2007.
A Spatiotemporal Model of Intracellular
Signalling in E. coli
Boundary conditions
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 O1, O2 and O3.
Initial conditions
In O1 we have
and in O2 and O3
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PESB, Manchester, 2007.
A Spatiotemporal Model of Intracellular
Signalling in E. coli
Solution method

• Non-dimensionalise system of equations.

• Numerical solutions using Femlab.

• Transient and steady-state analysis.

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

• 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)?

CheA2
CheA2
CheA3,CheA4
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PESB, Manchester, 2007.
In vitro Reaction Data
Porter, S. and Armitage, J.P. (2002).
Phosphotransfer in Rhodobacter sphaeroides
chemotaxis, J. Mol. Biol., 324, 35-45.
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PESB, Manchester, 2007.
Determining reaction rates from in vitro data

• Many of the in vitro reactions are of the form

Autophosphorylation
Phosphotransfer
Dephosphorylation
where when i1, j1, 2, 3 and 5 and when i2,
j1,..6.

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

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

• Governing ODE equations (assuming mass action
  kinetics) are

with
and

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

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

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

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

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

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

• 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.

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PESB, Manchester, 2007.
Determining reaction rates from in vitro data
Example CheA2P to CheY6
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PESB, Manchester, 2007.
Determining reaction rates from in vitro data
Example CheA2P to CheY6
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PESB, Manchester, 2007.
Determining reaction rates from in vitro data
Example CheA2P to CheY1
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PESB, Manchester, 2007.
Determining reaction rates from in vitro data
Example CheA2P to CheY1

• Best fit from using case (2)(ii), but
  asymptotically determine k21.50x10-2 from inner
  solution then use this to determine
  k-29.31x10-11 using BM.

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PESB, Manchester, 2007.
Determining reaction rates from in vitro data
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.
(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.
Review all results with the experimentalists!
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PESB, Manchester, 2007.
Determining reaction rates from in vitro data
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PESB, Manchester, 2007.
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.

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PESB, Manchester, 2007.
Publications

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

Acknowledgements

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

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

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