1. Mathematical biology From individual cell behavior to biological growth and form Lecture 3: To Turing or not to Turing?
Roeland Merks Lisanne Rens (1,2)
(1) Centrum Wiskunde & Informatica, Amsterdam
(2) Mathematical Institute, Leiden University

2. Previous lecture: Turing patterns
Take two chemicals, X and Y; X and Y are called “morphogens”: a “form producer”
Short range activator longe range inhibitor
video 2 2

3. Zebrafish Nakamasu et al. PNAS 2009
Long-range and short range interactions between pigment cells
cell1 promotes differentiation cell2 (long range)
cell1 promotes survival cell2 (long range)
Cell2 limits diff & surv of cell1 (short range) 3 3

4. Zebrafish Nakamasu et al. PNAS 20094 4

5. Zebrafish Nakamasu et al. PNAS 20095 5

6. Continuing Last week's cliffhanger....
Stripes and spots are “easy” (Turing) So: can we understand all stripes now?
Continuing Last week's cliffhanger....
Even-skipped (blue) - 6

7. Stripe formation during development of the fruitfly, Drosophila melanogaster
Each stripe of even skipped is controlled by a different section of the promotor!
It’s controlled by different combinations of morphogens.
Is this consistent with the Turing hypothesis??
Orange: expression of gene “even-skipped”
Blue: marker gene under control of parts of promotor 7 7

8. Gradients in developmental biology
E.g. Bicoid gradient in Drosophila
(See Lecture 1)
video
SDD model
(Driever and Nüsslein-Volhardt)

9. How is gradient translated to pattern?
Lewis Wolpert (1969):
Positional information: Cells interpret morphogen gradients
Wolpert et al. Principles of Development, 2006

10. Cascade of subdivisions
Wolpert et al. Principles of Development, 2006 10 10

11. Problem: diffusion makes things “messy”
Irregular spots and stripes
No sharp boundaries, so
Cells would have to detect small concentration difference
video 11 11

12. How do morphogen gradients instruct sharp gene expression boundaries?
Reinitz and Sharp (1995): “reverse engineering” a fly
Derive model from gene expression data 12

13. Reverse-engineering eve patterning Reinitz and Sharp (1995)
After 14 divisions, nuclei are in syncytium
no cell membranes between nuclei
Gene products regulate one another, they diffuse and degrade
Lewis (2008) Science
Effect of Bicoid
Sigmoid function 13 13
Network of gene interactions (the unknown!)

14. ? Effect of Bicoid Sigmoid function14 14
Network of gene interactions (the unknown!) ?

15. Simulated annealing
To find a gene network that reproduces eve patterns
Choose a random set of interactions between genes
Simulate the system, and compare the output with data
Summed squared deviations from data:
C: Cost function
Randomly change one entry in gene regulatory network, , then simulate again
If C decreases: accept, and make next change
If C increases: accept with probability inversely proportional to increase; else reject
Acceptance probability drops over time
Why is this called “annealing”? Heating, then cooling metals improves crystal structure

16. Six gene products in simulation
Interactions between three gap genes:
Krüppel, Knirps, Giant
Maternal gene products: Bicoid, and Hunchback
Pair-rule gene: even-skipped
Wikipedia.org 16 16
Wikipedia.org

17. Data set Reinitz and Sharp (1995)
{Reinitz:1995ws} 17 17

18. Reverse-engineered network
Autocatalysis (diagonal elements)
Gap genes mutually repress one another
Gap genes repress eve
Bcd upregulates all gap genes, except knirps
(Some elements are zero) 18

19. Eve stripes in silico/in vivo
Reinitz and Sharp (1995) 19 19

20. Follow-up work (selection)
In further work this method was optimized, producing refined networks
Jaeger et al. Genetics 2004; Nature 2004
Reverse-engineering can produce alternative circuits
Studies of robustness to perturbations help identify most plausible alternative:
Fomekong-Nanfack et al. BMC Systems Biology 2009

21. Biological development DNA + fertilized egg cell -> 3D shape + function
Wikipedia
Zebrafish
Wikipedia
video
Movie zebrafish development: P.Z. Myers, University of Minnesota (YouTube)

22. From single cell behavior to tissue scale phenomena
“Simple” problem:
Growth of avascular tumors in vitro
Proliferative cells
Necrotic core
Folkmann, J. Exp. Med. 1973

23. Tumor growth dynamics Unregulated cell division: dN/dt = rN
N(t) = N0*ert : exponential growth?
Typical growth curve of tumor: not exponential
Radius R(t) for exponential growth:
In vitro tumors: R(t) ~ t
In vitro brain tumor (rat C6 astrocyte glioma cell line)
Brú et al. Phys. Rev. Lett. 1998

24. Why don’t tumors grow exponentially?
Unregulated cell division limited by food supply?
logistic growth? dN/dt=rN(1-N/K)
K = “carrying capacity” - growth to maximum population size
So, tumors do not follow logistic growth

25. Why space matters: Eden Growth
Spatial model of colony growth (Eden, 1961)
Stochastic cellular automata model
Cells divide into adjacent site if space is available

27. Eden Growth
video

28. Eden growth: radius grows linearly
Brú et al. Phys. Rev. Lett. 1998

29. Eden Growth Eden growth produces “rough” (fractal) boundaries
Faster invasion and more access to nutrients than smooth boundaries
In vitro “glioma” also has fractal boundary

30. Deterministic cellular automata
Lattice of finite state automata (FSA)
Adjacent sites (“cells”) are coupled to one another
updates a fixed clock ticks
Neighborhood: what lattice sites are adjacent?
Von Neumann Moore 30

31. Typical CA rules Conway’s Game of Life
Moore Neighborhood
“Game of Life”
For a space that is 'populated':
Each cell with one or no neighbors dies, as if by solitude.
Each cell with four or more neighbors dies, as if by overpopulation.
Each cell with two or three neighbors survives. For a space that is 'empty' or 'unpopulated' Each cell with three neighbors becomes populated. 31

32. Conway’s Game of Life

33. Typical CA rules

34. Typical CA rules

35. Typical CA rules

36. Typical CA rules

37. Classification of cellular automata Wolfram, 1981
1D cellular automata, left and right neighbors
How many possible rules?
e.g., {0,0,0}->0, {0,1,0}->1, {1,1,0}->0 37

38. Rules 38

39. 39

40. 40

41. 41

42. 42

43. “Wolfram classes” Class 1: Converges onto uniform state
Class 2: Converges onto repetitive or stable state
Class 3: Converges onto “random” state
Class 4: .... 43

44. Probabilistic cellular automata
Synchronous updating
Transition rules are probabilistic
Example:
Von Neumann Neighborhood 44

45. Lab session: cellular automata
Experiment with 2D cellular automata (synchronous)
Can you produce class 1, 2, 3 CA rules?
What class of CAs is Game of Life?
Next time:
Diffusion and reaction (Turing) in cellular automata 45

46. Eden growth is highly simplified model for tumor growth
Real cells move
Real cells can push each other aside
Real cells communicate
Real cell growth depends on food and oxygen
Real cells can die or become quiescent
Etc. etc.
Cell-based models allow for much more detail

47. Cell-based modeling (Merks and Glazier, Phys. A 2005)
Input at microscale: behavior of cells
Output at macroscale:
Emergent tissue shapes and patterns
Change in cell behavior alters tissue shape
Cell-based models help analyze:
How cells build tissues and organisms
How tissue structure feeds back on cell behavior

48. Cell-based simulation methods
Membrane movements and cell shape are crucial
Multi-particle method required

49. Cellular Potts Model Pick random site Pick random neighbor H < 0 ?
Consider energy change
H if we accepted this copy
H < 0 ?
Accept always
H > 0 ? Accept with
Repeat
Cell adhesion
Volume conservation

50. Cell proliferation Cells growth at fixed rate
Divide once volume has doubled
video
video
High cell-cell adhesion
No cell-cell adhesion

51. Exponential growth
Growing cells “push” surrounding cells away
Growth:

52. Cell sorting
From Gierer, Int. J. Dev. Biol. 2012

53. Cell sorting
Cell sorting in vitro: zebrafish endoderm ectoderm and mesoderm
Krieg et al. Nature Cell Biology, 2008

54. Steinberg (Science 1963) Differential Adhesion Hypothesis
Cell sort out due to differences in adhesion between cells
different interfacial tensions between different pairs of cells
Cells make and break connections to one another
Strongest connections are kept with highest probability

55. video
ecto
meso

56. Initial configuration
Sorting
Jyellow, yellow = Jred,red < Jyellow, red
Mixing
Jyellow, red > Jyellow, yellow = Jred,red
Engulfment
Jyellow, red < Jyellow, medium
Jred,medium < Jyellow, medium
No cell-cell adhesion
Jcell, cell > 2 Jcell, medium

57. Experimental validation Krieg et al. Nature Cell Biol. 2008
Measure cell-cell adhesions of progenitor cells in zebrafish embryos

58. Measurements of adhesion
Homotypic adhesion:
So:
and:

59. Model prediction? Which cells will go to the center in simulation?
(Hint: In what place do cells have most contact with one another...?)
# Cellular Potts parametersT = 100target_area = 50target_length = 0lambda = 50lambda2 = 0Jtable = Jkrieg.datconn_diss = 0vecadherinknockout = truechemotaxis = 0border_energy = 100# note: do not change the following parameters for "long" cells (lambda2>0)neighbours = 3periodic_boundaries = false# PDE parameters (irrelevant for this simulation)n_chem = 1diff_coeff = 1e-13decay_rate = 1.8e-4secr_rate = 1.8e-4saturation = 0dt = 2.0dx = 2.0e-6pde_its = 15# initial conditions (create a "blob" of cells in the middle)n_init_cells = 1size_init_cells = 50sizex = 200sizey = 200divisions = 7mcs = rseed = -1subfield = 1.0relaxation = 0# outputstorage_stride = 100graphics = falsestore = truedatadir = krieg2MacBook-Pro-van-Roeland:src roel$ more Jkrieg.dat
Experimental observation
Model simulation
10x speed up
video
video
ecto
ecto
meso
meso

60. Surface tensions Other sources of surface tension
Cell differ in cortical tension - actin
Including tension in Hamiltonian “fixes” the simulations

61. Sources of surface tension
Including cortical tension in simulations “fixes” model predictions (Krieg et al. Nat. Cell Biol )

62. Conclusion Understand how cell behavior affect tissue scale
Cell based model can help
Seperate system components
Track and measure everything
Understand the dynamics
Start with a simple model and add dynamics if necessary
Generate hypothesis that can be tested
Feedback of biologists and is needed to further develop the model

63. Application: angiogenesis
Growth of new blood vessels
Wound healing, tumor growth, etc.
Source: Genentech

64. Simplified experimental system Human umbilical vein endothelial cells (HUVEC) in Matrigel
video
Movie: courtesy Luigi Preziosi, Politecnico di Torino, Italy

65. Hypothesis: Chemotaxis (Gamba et al. 2003; Serini et al., 2003)
Observation: cells migrate to higher concentrations of cells

66. Cells secrete signal that degrades in the matrixc ( x , t ) = D Ñ 2 - e + 1 d s r m b o i n u
Chemotaxis D H n e w = o l d - c t , i ( x ) '

67. Chemotaxis Pseudopods extend and retract more likely up chemical gradients
µ=100
c(x’)=0.5
c(x)=1
∆H -= 50
Savill and Hogeweg, J Theor Biol 1997

68. video

69. Close-up: cells are elongated
video
Movie: courtesy Luigi Preziosi, Politecnico di Torino, Italy

70. Collective behavior of elongated cells
video
L ≈ 100 µm
A ≈ 400 µm2
Merks et al. Dev. Biol. 2006;
Palm and Merks. Phys. Rev. E 2013 (arXiv: )

71. Pattern analysis Branching points Lacunae Morphological skeleton
(Method: Guidolin et al. Microvasc. Res. 67 (2004) 117)

72. Remodeling of vascular networks
Simulation
Experiments

73. Control +soluble VEGFR-1
Figure courtesy of Charles Little (from Drake et al. 2000)

74. Many models ignored key component: Extracellular matrix (ECM)
Extracellular matrix: materials that cells produce
Mechanical support
Signaling
Signals in ECM can be long-lived and long-distance
ECM binds chemical signals
Cells respond to strains in ECM
Thus: ECM is key to tissue morphogenesis
Mol. Biol. Cell
Van Oers, Rens, et al. PLoS Comp Biol. 2014

75. Matrix mechanics matters
Substrate thickness (Matrigel)
Vernon et al. Lab Invest. 1992 Pa Pa
Soft matrix Pa Pa Pa
Stiff matrix
Califano and Reinhart-King, 2008
Polyacrylamide gels

76. Cells respond to ECM mechanics
ECM supports tissue cells adhere to ECM ECM guide cell migration
Figure: Copied from
cells:
migrate to stiffer areas
spread more on stiff substrates
more stable focal adhesions on stiff substrates
Figure: Adapted from [Plotnikov et al., 2012]

77. Cells deform the ECM
(a) Cell traction forces (b) Wrinkles in the substrate [Califano and Reinhart-King, 2010] [Lemmon and Romer, 1997]

78. [van Oers, Rens et al. PLoS Comp. Biol. 2014]
Model overview
[van Oers, Rens et al. PLoS Comp. Biol. 2014]

79. Traction forces and mechanotaxis
Cell traction forces: cell nodes pull on cell nodes
Fi = µ .j dij [Lemmon and Romer, 2010] SubstrateLinear elastic, isotropic, infinitesimal strain
Ku = f , s = (sxx , syy , 2sxy ) = ( ∂ux , ∂uy , ∂ux + ∂uy )
∂x ∂y ∂x ∂y
MechanotaxisCells prefer to adhere to higher strained areas and in
the strain orientation. [van Oers, Rens et al. PLoS Comp. Biol. 2014]
(a) Traction forces
(b) Resulting strains

80. Single cell pulls on substrate and responds to strain (movie)
[van Oers, Rens et al. PLoS Comp. Biol. 2014]
video

81. (b) Copied from [Winer et al., 2011]
Substrate stiffness influences cell length and area [van Oers, Rens et al. PLoS Comp. Biol. 2014]
8 kPa
12 kPa
14 kPa
(a) Model results
16 kPa
(b) Copied from [Winer et al., 2011]

82. Mechanical cell-ECM feedback enables network formation
[van Oers, Rens et al. PLoS Comp. Biol. 2014]
video

83. Network formation is best on substrates of intermediate stiffness
[van Oers, Rens et al. PLoS Comp. Biol. 2014]

84. Bridging events across lacunae
[van Oers, Rens et al. PLoS Comp. Biol. 2014]

85. Sprouting from a blob (movie)
[van Oers, Rens et al. PLoS Comp. Biol. 2014]
video

86. Conclusion Models can give clues of how morphogenesis works
Different hypothesis for vascular network formation:
Chemotaxis + elongation
Strains
More....
Need biologists to verify/differentiate!

87. Acknowledgments CWI Sonja Boas (Josephine Daub)
(René van Oers)
(Margriet Palm)
(Iraes Rabbers)
Lisanne Rens
(András Szabó)
VUMC, Amsterdam
Pieter Koolwijk
Victor van Hinsbergh
Cornell University
Danielle LaValley Cynthia Reinhart-King
Indiana University Bloomington
James Glazier
Abbas Shirinifard
New York Medical College
Sergey Brodsky
Stuart Newman
Michael Goligorsky
Amsterdam Medical Center
Marchien Dallinga
Ingeborg Klaassen
Reinier Schlingemann
Funding: