From coronal mass ejections to mouse keratinocytes

Transcription

1 From coronal mass ejections to mouse keratinocytes By Robin Thompson Supervised by Dr Ruth Baker and Christian Yates, Centre for Mathematical Biology (University of Oxford)

2 Where it all began... Find a technique to detect and track coronal mass ejections (CMEs) from the sun to Earth.

3 What is a CME? An eruption of plasma and magnetic field from the sun, travelling roughly radially outwards. Typical mass 1012 kg, typical speed km/s. Leading Edge Cavity Filament

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5 WHY BOTHER? Upon impact with Earth, interplanetary coronal mass ejections (ICMEs) can be responsible for severe space weather effects, e.g. Aurora enhancement Disruption of telecommunications facilities, power grids and spacecraft The development of more sensitive electronics means we now need greater understanding of CMEs, including predicting both their arrival and the consequences of their impact with Earth.

6 October 2003 Image CME in yellow circle. The darkened areas at centre and to the left are where SMEI does not take data because the Sun and Moon are in this area. The grid overlay indicates 60-degree increments in the fi eld of view. When the CME extends to the third ring degrees - it has reached the plane of the Earth. The far left and right points correspond with the anti-sun direction into deep space.

7 If we look at lots of these images over time, we can see CMEs move. We can subtract any constant/slow-moving known sources of light (e.g. stars/planets) from our images. LET'S PLAY... Spot the CME

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9 Raw SMEI image, May 2003 CME

10 Automatic Detection

11 The Tappin-Howard model

12 Our result We now have a completely automatic program that will warn us whenever a C M E is likely to hit the E arth, up to 12 hours in advance.

13 Cell adhesion Cells binding together with differing adhesiveness, due to different amounts of protein being present. During the development of the body's structure and organs, cells migrate to their target sites. Cellto-cell adhesion then enables them to aggregate and form cohesive tissues.

14 Further examples... Mouse keratinocytes grown under conditions that limit cell-to-cell adhesion. Keratinocyte skin cells adhere to form a protective layer against UV rays. Cancerous tumour invasion. Wound repair. Tissue engineering.

15 WHY BOTHER MODELLING? Puts cell migration and adhesion into framework where current hypotheses can be tested and refined, and experimentally verifiable predictions can be made. The differential adhesion hypothesis in action...

16 Types of model Cell-based (stochastic). Includes randomness often prevalent in biological systems. Population-based (deterministic). Less computational work, use tools from PDE analysis.

17 Within these two categories there are various sub-types. Cell-based models include: Lattice based. Velocity jump (run and tumble).

18 Our stochastic model Cell-based. Cells move on a fixed, one-dimensional lattice [0,1]. Cells jump to midpoint of neighbouring compartments only, with a cell in compartment i having probabilities Ti+ and Tiof moving up to i+1 and down to i-1 respectively. No flux boundary conditions. Adhesion (alpha is adhesion coefficient) and volume filling (carrying capacity S).

19 Stochastic simulation

20 The stochastic mean equations Derive a set of mean equations (k ODEs) that approximate the average behaviour of the stochastic system.

21 The continuum limit Taking the continuum limit in space (letting compartment size h tend to 0) leads to the following partial differential equation for the average behaviour of the stochastic system (can then use PDE analysis/only 1 equation):

22 Alpha < 0.75 We approximate the solution of the PDE system numerically. For small alpha, the deterministic system mimics the stochastic system well.

23 Alpha > 0.75 (diffusivity D(rho) can become negative) Certain density values leads to nonsensical (unphysical) solutions. PDE ill posed.

24 Solution of mean equations Instead use finite difference methods to numerically approximate solutions of the mean equations themselves. Solutions mimic the stochastics even for large alpha.

25 Varying the number of compartments and ICs Different types of patterning (metastable steady states). Highly dependent on ICs. All mimicked by mean equations.

26 Differences between stochastic and deterministic systems Can find ICs such that, for high alpha, the mean equations and stochastic system produce different solutions (although the average stochastic behaviour is still that of the mean equations). Can see none/one/two peaks. For these ICs, PDE behaviour sensitive to perturbations (random perturbations given by stochastic system).

27 The story so far... Can simulate our cell adhesion model. Have deterministic systems that mimic the average behaviour of the stochastics both in the small alpha and large alpha regimes. For certain ICs in high alpha regime, individual stochastic simulations produce different behaviour to deterministic systems. What next...?

28 Implementation on a growing domain The traditional compartment splitting method allow a randomly chosen 'compartment split' event instead of a cell movement. Old parent compartment replaced by two daughter compartments of the same size, with cells redistributed binomially. Many problems including that which compartment splits hugely affects the behaviour of the system, so little point looking at deterministic systems (average behaviour of the stochastic system).

29 New method for implementing domain growth Compartments grow by a small amount at a time (1 cell size). When double the original size, the compartment splits into two compartments (to stop cell jumps becoming unphysical), and the cells are redistributed. No sudden large change in cell density at any point on the domain.

30 3 Scenarios (exponentially growing domain) Small growth rate still see patterning. Large growth rate patterning destroyed. Medium growth rate patterning still evident but eventually destroyed.

31 Implementation of my method Can derive stochastic mean equations and PDE with domain growth incorporated. PDE still ill posed in the high alpha regime. Difficulty in how to implement mean equations over long timescales.

32 Future work Extend model to 2 species (simulate differential adhesion hypothesis?) Extend model to 2 spatial dimensions. Explore possibility of implementing velocity jump models on growing domain.

33 Thanks... Dr Ruth Baker (Centre for Mathematical Biology, University of Oxford). Christian Yates (Centre for Mathematical Biology, University of Oxford). Professor Endre Suli (Worcester College, University of Oxford). Centre for Mathematical Biology, University of Oxford. The Nuffield Foundation. The Oxford University SIAM Student Chapter.