Lewis DDE Model of Somite
Formation in Zebrafish
Jason James
Tanya Kazakova
Oleg Kim
Alexander Vorobiev
May 8, 2006
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Outline
▀ About the Model
▀ Biological background
►What are zebrafish?
►Why should we care about somitogenesis?
►Temporal and spatial coupling
Jason James - JJ
Tanya Kazakova - TK
Oleg Kim - OK
Alexander Vorobiev - AV
▀ Model set up
►About gene expression
►Notch pathway
►Modeling Oscillatory Mechanism ►
JJ
▀ Mathematical Model
►ODE and DDE Models
►Motivation and mechanism of delays
►Numerical results
▀ Conclusions and applications
▀ Movie time!
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TK
About
“Autoinhibition with
Transcriptional Delay: A
Simple Mechanism for the
Zebrafish Somitogenesis
Oscillator” by Julian
Lewis
Julian Lewis
Vertebrate Development Laboratory
Cancer Research UK London Research Institute
Current Biology, Vol. 13,
1398–1408, August 19,
2003
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The Zebrafish
• The Zebrafish, also called
Zebra Danio, is a tropical fish
belonging to the minnow
family.
• The fish is known for its
distinctive horizontal stripes
extending across the length of
its entire body.
• Zebrafish have few
commercial or food purposes,
but are ideal for keeping in an
aquarium because of its
popularity, low price, and low
maintenance physical
constitution.
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The Zebrafish
• The Zebrafish originates
from eastern India and
inhabits bodies of fresh
water such as streams,
ponds, and canals.
• The Zebrafish can grow
to be six centimeters
long and can live for up
to five years.
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The Zebrafish
• The Zebrafish holds great
value as a model organism in
the studies of vertebrate
development and genetics.
• A Zebrafish egg becomes a
larvae in three days.
• This speedy development
facilitates research. In Lewis’
case, a Zebrafish’s period of
oscillation takes 30 minutes
compared to 90-120 minutes
for a mouse or chick.
• Another consequence of the
Zebrafish’s fast development
is a simpler, more easily
observed chain of
transcriptional regulation.
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Fun facts about the Zebrafish
• A Zebrafish embryo is
transparent, allowing researchers
to see results of genetic
experiments in less than three
days.
• The Zebrafish is one of the a
select group of animals to have
ventured into outer space.
• The Zebrafish has been
genetically engineered into the
Glofish, a brightly fluorescent
fish.
• A melanin-producing gene in
Zebrafish has also been found in
humans and segregates strongly
between light skin and dark skin.
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Somites in vertebrate embryos
• Somites lie on either side
of the neural tube in
vertebrate embryos,
including humans, in
head to tail succession.
• The somites develop into
dermis, skeletal muscle,
and the vertebrae.
• The segmented pattern
in somites is determined
by cyclic changes in gene
expression in the
presomitic mesoderm.
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Segmentation and somitogenesis
• During segmentation, the embryo of an
organism becomes subdivided into
similarly-appearing tissue modules.
• During somitogenesis, the presomitic
mesoderm of a vertebrate segments into
parallel blocks of tissue.
• Some blocks of tissue retain a segmented
appearance, such as the spinal cord,
while others do not, such as skin.
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Segmentation and somitogenesis
• Somite segmentation is dictated by a kind of
molecular clock in all vertebrate embryos.
• This ‘clock’ takes the form of a transcriptional
oscillator in the presomitic mesoderm. Most
genes that participate in this oscillator belong
to a certain set called the Notch pathway.
• In particular, Lewis has examined the her1 and
her7 genes in the Zebrafish.
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Mathematical model
• Lewis has developed a model for the
mechanism of expression of the her1 and her7
genes which serves to explain certain
observations about somitogenesis in Zebrafish.
• According to the model, the period of
oscillation in the molecular clock amounts to a
system with delay – the delays of transcription
and translation, i.e., the amount of time
needed to construct the mRNA molecule and
the protein molecule.
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Temporal + spatial
• Dynamic behavior of gene expression
• Clock and wavefront mechanism
• The cell oscillation components belongs to the Notch
Signaling Pathway
• PSM cells may continue oscillate without their
neighbors
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Gene expression and mRNA
• Gene expression, (protein
expression or simply
expression) is the process by
which a gene's DNA
sequence is converted into
the structures and functions
of a cell.
• The expression of particular
genes may be estimated by
measurement of the cellular
concentration of messenger
RNAs (mRNA).
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AV
Notch pathway - cell
communication mechanism.
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Time scales - transcriptional and the translational delays
• Stages: transcription of DNA, into messenger RNA, post transcriptional
modification, translation, folding, post-translational modification and
targeting.
• Degradation: After a certain amount of time the message degrades into
its component nucleotides.
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AV
Notch pathway
• The Notch pathway is a gene regulatory
pathway involved in multiple differentiation
processes.
• The Notch protein sits like a trigger penetrating
the cell membrane, with part of it inside and
part outside. Proteins when binding to the
exterior part cause the interior part to release
other proteins, which typically then make their
way to the cell nucleus to alter gene
expression.
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OK
Zebrafish Somitogenesis Oscillator
The pattern of somites is traced out by
a mechanism involving oscillating gene
expression at the tail end of the
embryo. In zebrafish two linked
oscillating genes, her1 and her7, coding
for inhibitory gene regulatory proteins,
are especially implicated in genesis of
the oscillations, while Notch signaling
appears necessary for synchronization
of adjacent cells.
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Stages of embrionic development of the zebrafish
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Mathematical Model: ODE
Variables:
m – the number of mRNA
molecules in the cell at any
instant
p - be the number of molecules
of the corresponding protein
Constants:
b,c – decay rates (inverse
lifetimes)
k,p0 – constants representing the
action of an inhibitory protein
dp
 am  bp
dt
dm
 f ( p)  cm
dt
k
f ( p) 
p2
1 2
p0
f(p) – rate of production of
new mRNA molecules
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Plausible Parameter Values
• a = 4.5 protein molecules per mRNA molecule
per minute
• b = c = 0.23 molecules per minute,
corresponding to protein and mRNA half-lives of
3 min
• k = 33 mRNA molecules per diploid cell per
minute corresponding to 1000 transcripts per
hour per gene copy in the absence of inhibition
• p0 = 40 molecules, corresponding to a critical
concentration of ~10-9 M within a 5 mdiameter cell nucleus
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Mathematical Model: DDE
Constants:
Tm – time that elapses between
the initiation of transcription
and the arrival of the mature
mRNA molecule in cytoplasm
Tp – delay between the initiation
of translation and the
emergence of a complete
functional protein molecule
dp (t )
 am(t  T p )  bp (t )
dt
dm
 f ( p (t  Tm ))  cm(t )
dt
k
f ( p) 
p2
1 2
p0
Values:
10.2 < Tmher1 < 31.5 min
Tphe1 ~ 2.8 min
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Oscillator: Illustrating Basic Principles
The Period is determined by the delay in the feedback loop.
A) On/of switch. Transition to the “on” state creates a delayed
negative feedback signal, causing transition to the “off” state
after time T, and vice versa; the time for one complete cycle
(on->off->on) is 2T.
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Oscillator: Illustrating Basic Principles
B) System described by a continuous variable, x(t), autoregulated by
negative feedback with delay T such that x(t) = f (x(t - T)), where f is a
monotonically decreasing function. If, for example, f(x) = K/(1+x2),
oscillations will be sustained and will have period 2T whenever K >2. (This is
the condition under which the pair of equations x1= f[x2] and x2 = f[x1] has
solutions x1≠x2; the system then alternates between state x = x1 and state x
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= x2).
OK
Oscillator: Illustrating Basic Principles
C) A chemical system in which the rate of synthesis of
substance x is regulated by delayed negative feedback and
degradation occurs at rate cx(t):
dx
 f ( x(t  T ))  cx (t )
dt
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Model without noise
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Including noise
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Conclusions and applications
• The systems investigated by Lewis, despite their
simplicity in Zebrafish, often exhibit surprising
behaviors.
• The mathematical model offered by Lewis attempts to
more completely understand somitogenesis and
suggests a method to describe other oscillations in
vertebrate embryos which currently go undetected by
standard means.
• A more complete understanding of the segmentation
clock within an embryo will greatly contribute to the
overall understanding of the very earliest stages of
human and animal life.
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Movie
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