Mathematical biology From individual cell behavior to biological growth and form

Transcription

1 Mathematical biology From individual cell behavior to biological growth and form Lecture 6: Branching morphogenesis Roeland Merks (1,2) (1) Centrum Wiskunde & Informatica, Amsterdam (2) Mathematical Institute, Leiden University

2 Branched structures are common in biology Lung coral: Kidney (Qiao et al. PNAS 1999) Trees (sugarmntfarm.com)

3 What is this? Source:

4 How can branching structures form? Diffusion limited aggregation (Witten and Sander, 1981) Seed Seed Random Walker Outcome of this procedure?

5 Diffusion in Cellular Automata Can you propose an algorithm?

6 Margolus diffusion Problem: local particle conservation particle collisions Solution: Rotate two-by-two blocks randomly CW or CCW Then shift two-by-two blocks for next step Margolus neighborhood

7 100x100 lattice, 1,000 particles

8 500x500 lattice, 25,000 particles

9 Characterizing branching structures Fractal dimension: scaling dimension (source: Wikipedia) N ~ ε D

10 Box counting dimension N ε /2

11 Box counting dimension N ε /2 16 1/8

12 Box counting dimension N ε /2 16 1/4 53 (!) 1/8

13 Box counting dimension N ε /2 16 1/4 53 (!) 1/8... 1/16

14 (Uitkomst metingen) "./boxcount_13000" u 2: N(eps) D epsilon

15 Alternative calculation of (Witten and Sander, 1981) D f Autocorrelation function C(r) N 1 ρ( r )ρ( r + r) ~ r D f d r

16 Curvature effects Example: growth of blood vessels and organs Bud branching in the embryonic kidney Watanabe and Constantini, Dev. Biol HUVEC cells (blood vessels)

17 Contact-inhibited motility Cell contact suppresses response to chemoattractant? Context-dependent effect of VEGF (VEGF = growth factor stimulating blood vessel growth) Dejana, Nat. Rev. MCB, 2004

18 VE-cadherin knock-outs? VE-cadherin knock-out mice (Gory-Fauré et al., Development 1999) Blood-islands in E10.5 yolk sacs Wild-type VE-cadherin -/-

19 Contact-inhibited motility with Erica Perryn, Abbas Shirinifard and James Glazier Indiana University Bloomington and Kansas University Medical Center Merks, Perryn, Shirinifard and Glazier, PLoS Comp. Biol., 2008 Merks & Glazier (2006). Nonlinearity 19, C1-C10

20 Vasculogenesis and sprouting: two sides of the same coin? Same mechanism also drives sprouting Merks & Glazier (2006). Nonlinearity 19, C1-C10 Merks, Perryn, Shirinifard and Glazier, PLoS Comp. Biol., 2008, e

21 Contact-inhibited chemotaxis Buckling instability? A. B. C. A. Only peripheral cells chemotact to center B. Invading surface cells displace interior cells Only possible with contact inhibition C. Resulting pressure pushes peripheral cells further outwards

22 Chemotaxis at cell-cell interfaces Chemotaxis at cell-matrix interfaces: Vary chemotaxis at cell-cell interfaces: Calculate compactness of objects

23 Chemotaxis at cell-cell interfaces

24 Chemotactic pushing required for buckling instability? Eliminate pushing: extension-only chemotaxis µ=100 c(x )=0.5 c(x)=1 H -= 50

25 Chemotactic pushing required for buckling instability? Eliminate pushing: extension-only chemotaxis µ=100 c(x )=0.5 c(x)=1 H -= 50

26 Chemotactic pushing required for buckling instability? Eliminate pushing: extension-only chemotaxis µ=100 c(x )=0.5 c(x)=1 H -= 50

27 Extension-only chemotaxis Pseudopod retraction energetically neutral µ=100 c(x)=1.0 c(x )=0.5 H -= 0

28 Extension-only chemotaxis Pseudopod retraction energetically neutral µ=100 c(x)=1.0 c(x )=0.5 H -= 0

29 Extension-only chemotaxis Pseudopod retraction energetically neutral µ=100 c(x)=1.0 c(x )=0.5 H -= 0

30 No sprouting T=50

31 Random motility drives sprouting T=200

32 Chemoattractant inhibits pseudopod extension most strongly at concavities At high motility (T): many pseudopod extensions, which the chemical gradients counteracts Gradient is most shallow at convexities Cells at sprout tips move faster than those between branches Note: chemoattractant here also acts as an inhibitor of cell motility! Merks et al. PLoS Comp. Biol., 2008, e

33 Random motility drives sprouting Extension only chemotaxis Extension retraction chemotaxis 0.8 Compactness (C) Cell Motility (T) Merks et al. PLoS Comp. Biol., 2008, e

34 endothermic mechanism 2e+08 1e+08 Extension retraction chemotaxis ( T =50) Extension retraction chemotaxis ( T =200) Extension only chemotaxis ( T =200) (H H ) 0 Cumulative Energy Change 0 1e+08 2e+08 3e+08 4e+08 5e Time (MCS)

35 Branching morphogenesis in kidney or gland Bud branching in the embryonic kidney Watanabe and Constantini, Dev. Biol. 2004

36 Curvature effects in epithelial tissues Secreted morphogen: Nelson et al. Science 2006 Dynamics of morphogen: c t = c(x,t) + D c(x,t) + (1 δ (σ x,0))s

37 CPM model: Morphogen (TGF-β) inhibits protrusion of pseudopods ΔH * = ΔH χ inhibition c( x), so P protrusion = f (c( x))

38 CPM model: Morphogen (TGF-β) inhibits protrusion of pseudopods ΔH * = ΔH χ inhibition c( x), so P protrusion = f (c( x))

39 CPM model: Morphogen (TGF-β) inhibits protrusion of pseudopods ΔH * = ΔH χ inhibition c( x), so P protrusion = f (c( x))

40 CPM simulation All cells secrete TGF-β: c( x,t) t = D 2 c( x,t) εc( x,t) + (1 δ(σ( x),0))s term becomes 0 in medium Extension probability depends on local concentration of TGF-β ΔH = ΔH + χ c ( x)δ(σ( x),0)(1- δ(σ( x '),0)) new old τ,i i contact inhibition I.e. TGF-β inhibits cellular extension

41 Iraes Rabbers, Andras Szábo

42 Iraes Rabbers, Andras Szábo

43 Kidney cells (renal duct) Kieran Bannerman, Jamie Davies (University of Edinburgh) Patterned substrate: Huabing Yin (Glasgow) Simulation Experiment Andras Szábo

44 Curvature dependence makes blob transform into a branched structure - Volume is conserved -

45 Cell motility is key

46 Interaction of kidney models

47 Branching growth and curvature effect Convex parts of growing surface more contact with environment than concave sites DLA: tips stick into fluid with highest solute concentration Particles cannot diffuse to Result: convex surfaces grow faster than flat or concave curved surfaces -> Instability (Mullins-Sekerka) In physics: e.g., Diffusion-limited aggregation Biology: e.g. epithelial branching, coral growth

48 L-systems Descriptive models of branching growth Rewriting system to describe, e.g., plant growth Lindenmayer (1968), further developed e.g. by Prusinkiewicz Parallel rewriting system: G = {V,ω,P} Alphabet, V Production rules, ω Axioma, P Describe regular placement of heterocysts (gonads) Example: Anabaena catenula: filamentous cyanobacterium ( blauwalg ) V = {A, B}, ω = {A AB, B A}, P = A Result?

49 L-system of branching structure V = {F, X,[,]}, ω = {F FF, X F[+X]F[+] X}, P = X Iteration: 0: X 1: X[+X]F[+X]-X 2: X[+X]F[+X]-X[+X[+X]F[+X]-X]FF[+X[+X]F[+X]-X]- X[+X]F[+X]-X 3: X[+X]F[+X]-X[+X[+X]F[+X]-X]FF[+X[+X]F[+X]-X]-X[+X]F[+X]- X[+X[+X]F[+X]-X[+X[+X]F[+X]-X]FF[+X[+X]F[+X]-X]-X[+X]F[+X]- X]FFFF[+X[+X]F[+X]-X[+X[+X]F[+X]-X]FF[+X[+X]F[+X]-X]-X[+X]F[+X]-X]- X[+X]F[+X]-X[+X[+X]F[+X]-X]FF[+X[+X]F[+X]-X]-X[+X]F[+X]-X [...] is side branch What to do with this? Turtle interpretation : F: Draw line forward; X: Do Nothing; +: Turn right; -: Turn left [: Put position on stack; ]: pull position from stack

50 Turtle interpretation Example: iteration 1: X[+X]F[+X]-X (Bord) Iteration 5: Note: additional data is required for turtle interpretation

51 Turtle interpretation Example: iteration 1: X[+X]F[+X]-X (Bord) Iteration 5: Note: additional data is required for turtle interpretation

52 L-systems used for computer graphics tekst

53 Next week: guest speakers Fred Vermolen (TUDelft) Application Of Semi Stochastic Cell Colony Models: Infectious Wound Healing And Tumor Growth Sander Hille (MI, Leiden University) Mathematical modeling of polar auxin transport in Arabidopsis inflorescence stems

54 Paper seminars Three papers of a topic of your choice Prepare seminar of ±30 mins, plus 15 minutes discussion What question do the papers study? What was the hypothesis? Did the papers propose alternative ideas? Did they work equally well? How was the hypothesis translated into a mathematical model?

55 Possible topics Mechanisms of phyllotaxis (upstream auxin transport vs. canalization) Gastrulation Blood vessel growth (angiogenesis) Reaction-diffusion models E.g. of finger prints Spiral waves in Dictyostelium discoideum Theory of cellular automata Morphogenesis due to differential growth

56 Mini-projects Small research project Typically based on a simulation you have seen during the computer labs Two guided afternoons and individual work Final presentation ±30 mins Introduce the problem, existing results, your research question, and your new results. Discuss the biological and mathematical relevance. What can biologists learn from your model?

57 Mini-projects Small research project (II) Final report: In the form of a paper: Introduction/Methods/Results/Discussion/Future work Figures, analysis of your model Size: around 8 pages Deadline: 31 January 2013 It s okay to choose the same topic for your seminar and research project

58 Potential topics Pattern formation (e.g. leaf venation, fur patterns, shell patterns) Theory of 1D/2D cellular automata Plant morphogenesis: interaction between reactiondiffusion and auxin pumping Blood vessel growth: extend models with additional cell types; angiogenesis in stromal tissues Branching growth (DLA): modify models with additional particle types, ballistic particle motion, aggregation probability: effect on? Models of tumor invasion Theory of cell sorting D f