International Workshop on Mathematical Biology:

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 International Workshop on Mathematical Biology
Modeling and Analysis Taipei, 1000-1050,
December 16, 2008
Modeling morphogenesis in development.
Yoh Iwasa Department of Biology, Kyushu University
collaborators T. Hirashima, Y. Morishita, K.
Uriu
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Multicellular organisms start their life from a
"fertilized egg", and develop complex structures.
Development
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Todays talk
Modeling for organ growth
1. Limb bud formation
2. Ureteric bud branching in kidney
Traveling wave of gene expression in vertebrate
somitogenesis
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Todays talk
Modeling for organ growth
1. Limb bud formation
2. Ureteric bud branching in kidney
Traveling wave of gene expression in vertebrate
somitogenesis
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Molecular biology revealed chemical aspects of
development

• responsible genes for morphologies

• gene regulatory networks

• spatio-temporal expression patterns

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Mathematical models for pattern formation
Approach from mathematics

• chemical network dynamics

• network estimation by reverse engineering

• pattern formation (e.g. Turing)

Perkins et al. (2006)
Kondo and Asai (1995)
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Mathematical models for mechanical aspects needs
to develop.
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Limb-bud formation and elongation
AER
FGF diffusive protein inhibits
apoptosis inhibits differentiation promotes
cell proliferation
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Previous studies
fluid dynamics model
Dillon and Othmer (1999)
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Our model
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Modeling of epithelial and mesenchymal tissues

• Network of nodes

• Two tissues

Mesenchyme
movable, deformable
Ectoderm
tight-junction, layer
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potential for distance between nodes
There exists the equilibrium inter-nodal distance.
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?
Nodes move to reduce the potential energy.
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Division probability ? FGF concentration
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Morphogenetic mechanism
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(No Transcript)
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Comparisons with Experiments (1)
Inhibition of FGF-receptor expression
(Verheyden et al. (2005) Development)
Normal (model)
Mutant (model)
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Comparisons with Experiments (2)
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Chemical signals specify the spatio-temporal
pattern of volume growth, which automatically
guides organ morphogenesis.
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Initially uniform limb bud becomes differentiated
into distinct regions
Autopod
Stylopod
Zeugopod
marker genes for Stylopod and Zeugopod
Yashiro et al. (2004)
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Initially uniform limb bud becomes differentiated
into distinct regions
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Todays talk
Modeling for organ growth
1. Limb bud formation
2. Ureteric bud branching in kidney
Traveling wave of gene expression in vertebrate
somitogenesis
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Branching morphogenesis in early kidney
development
provided by Prof. Costantini (Columbia Univ.)
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Factors controlling kidney branching
BSA beads (control)
GDNF beads

• Chemotaxis
• (Tang et al., 1998 Kim and Dressler, 2007)

2. Proliferation on tips
(Tang, M. et al., 1998)
Gdnf is localized
(Michael and Davies, 2004)
Chemoattractant
GDNF
Growth factor
Gdnf expression is maintained in mesenchyme
through epithelium-mesenchyme interaction.
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Cellular Potts Model
H Hadhesion Hvolume Helasticity Hsurface
Hchemotaxis
aNB, anon-NB, aLIQ, aMES,
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Cellular Potts Model
Model includes (1)Cell-cell adhesion (2)Elastici
ty of epitherial cells (3)Chemotaxis (4)Cell
growth and division
morphogenesis
time
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Control
Inhibiting chemotaxis
Inhibiting cell proliferation
(Kim and Dressler, 2007)
(Michael L. et al., 2005)
Bloated pattern
Kinked pattern
Y-shaped pattern
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Phase diagram
(2007, Kim and Dressler)
(Watanabe and Costantini, 2004)
Kinked
Y-shape
(2005, Michael L., et al.)
K
Y
Cell proliferation rate
Bloated
B
Intensity of chemotaxis
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Branching of ureteric bud requires proper
balance between intensity of chemotaxis and
cell proliferation rate.
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Todays talk
Modeling for organ growth
1. Limb bud formation
2. Ureteric bud branching in kidney
Traveling wave of gene expression in vertebrate
somitogenesis
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Somites are produced through one-by-one addition
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Somite formation in zebrafish
traveling wave of her gene expression
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1. condition for synchronization
2. traveling wave formation
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Theoretical studies
Cooke and Zeeman, 1976
Meinhardt, 1986
spatio-temporal periodicity in somite formation
Collier et al., 2000
Kerszberg and Wolpert, 2000
Baker et al., 2006
Lewis, 2003
oscillation of segmentation clock
synchronization
Horikawa et al. 2006
Zeiser et al., 2006, 2007, 2008
Tiedemann et al., 2007
Cinquin, 2007
Kærn et al., 2000
traveling wave
Jaeger and Goodwin, 2001
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Lewis (2003)
Delta
Notch
Her protein
her mRNA
with time delay
1.This negative feedback generates oscillation
delta mRNA
Delta protein
2.Notch-Delta signal couples her expression
between cells
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Lewis (2003)
effect of Delta protein from the neighboring cell
her mRNA
Her protein
delta mRNA
Delta protein
needed for oscillation
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WithOUT time delay
Cell b
Cell a
Her protein
Her protein
in cytoplasm
in cytoplasm
her mRNA
her mRNA
Her protein
Her protein
in nucleus
in nucleus
Delta protein
Delta protein
Uriu et al.
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Model for Cell a
her mRNA
Her protein
in cytoplasm
Her protein
in nucleus
Delta protein
no explicit time delay is needed for oscillation
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Cell a
Cell b
Synchronized oscillation
a limit cycle with
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limit cycle with cells perfectly synchronized
cell a
cell b
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stability of the limit cycle
stable
unstable
maintained
broken
The limit cycle must be stable for
synchronization to be achieved
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limit cycle
small deviation
after one period of the cycle
F(T) fundamental solution matrix
Eigenvalues of F(T) determine the stability of
the limit cycle.
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synchronized oscillation is
unstable
absent
stable
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larger values promote synchronization
smaller values promote synchronization
Her protein in nucleus
delta
her
Delta protein
her mRNA
Her protein in cytoplasm
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Traveling wave
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traveling wave of her gene expression
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pre-somitic mesoderm
determined region
empty
tail
anterior
posterior
position
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(No Transcript)
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1
k-1
k
kL-1
N
1
. . .
tail of embryo
M
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1
k-1
k
kL-1
N
1
. . .
tail of embryo
M
1
. . .
tail of embryo
elongation
determination
M
copy
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her mRNA
in cytoplasm
Her protein
in nucleus
Delta protein
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the gradient of a reaction parameter
elongation
a reaction parameter
determination
length L
length L
pre-somitic mesoderm
pre-somitic mesoderm
determined region
1
k
kL-1
k1
kL
N
tail
tail
cell position
anterior
posterior
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pre-somitic mesoderm
determined region
empty
tail
anterior
posterior
position
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gradient of her mRNA degradation rate ??
a conclusion similar to Tiedeman et al. (2007)
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There are many other ways to generate .....
translation rate of Her protein
her mRNA transcription rate
degradation rate of Her protein in nucleus
transportation rate of Her protein
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To generate the observed traveling wave
long
short
Period of oscillation
pre-somitic mesoderm
anterior end
tail
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Sensitivity of the period to rate parameters
Larger values
lengthen the period
shorten the period
delta
in nucleus
her
Delta protein
her mRNA
Her protein in cytoplasm
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All of these can generate similar traveling waves.
parameter
period

• her mRNA degradation rate

• her mRNA transcription rate

• translation rate of Her protein

• degradation rate of Her protein

• transportation rate

• synthesis rate of Delta protein

• degradation rate of Delta protein

• activation rate by Notch signal

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density plot of the waves
white

high
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analytical formula for traveling wave
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t t1
L
t time
L length of pre-somitic mesoderm
u speed of the elongation of an embryo
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t t1
L
t t2 (gt t1)
x ut2
x ut2 L
pre-somitic mesoderm
determined
x
L
t time
L length of pre-somitic mesoderm
u speed of the elongation of an embryo
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State of each cell is described by phase ??t?
?
?
0
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1. The speed of phase changes linearly.
speed of phase
period
x
ut
utL
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2. New cells in the tail are produced by copying
the existing cells.
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solution
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nonlinear dynamics
period of oscillation
estimated at two ends
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nonlinear model
formula
t
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wave speed
spatial distance between adjacent peaks
time interval between expression peaks
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Gradient of a single reaction parameter produces
traveling wave.
Period of oscillation
posterior
anterior
shorter
longer
A mathematical formula is derived for the
trajectory of traveling wave.
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Thank you !