Transcript Modeling of Neuroblast Delamination

Applications of Transition State in
System Biology
Lei Zhang （张磊）
Beijing International Center for Mathematical Research,
Peking University
Joint with
Qing Nie (Math, UC Irvine), Tom Schilling (Dev. & Cell Bio, UC Irvine),
Yan Yan (Life Science, HKUST)
Workshop on Modeling Rare Events in Complex Physical Systems, IMS,
Singapore, Nov. 5-8, 2013
Outline
 Introduction
 Noise drives boundary sharpening in zebrafish hindbrain
 Neuroblast delamination in Drosophila
 Summary
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What is transition state?
 Transition state is a particular configuration
corresponding to the highest energy along
the minimum energy path.
 Transition state is a saddle point and
transition is often driven by very small
thermal noise.
Saddle Point
(Wikipedia)
 Transition state (Rare events) are of general interests:
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Nucleation in materials (Zhang-Chen-Du, PRL 2007, CiCP 2010; Cheng-Lin-
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E-P.W. Zhang-Shi, PRL 2010; Li-Zhang-Zhang MMS 2013 )
Chemical reactions (E-Ren-Vanden-Eijnden, Annu. Rev. Phys. Chem. 2010)
Conformational changes of biomolecules (Bolhuis, PNAS 2003)
Data sciences (E-Lu-Yao, Methods Appl. Anal. 2013)
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Transition state in biology?
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Numerical methods for saddle point
 Numerical methods for saddle point and transition pathway
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Minimax method: Rabinowitz (1986); Li, Zhou (2001),
Zhang, Chen, Du (2007), Chen, Zhou (2010)
String method: E, Ren, Vanden-Eijnden (2002, 2007), Cameron,
Kohn, Vanden-Eijnden (2009), Du, Zhang (2009, 2010)
Nudged Elastic Band method: Henkelman, Jonsson (2000),
Henkelman, Uberuaga, Jonsson (2000), Sheppard, Terrell,
Henkelman (2008),
Dimer method: Henkelman, Jonsson (1999)
Shrinking Dimer Dynamics: J.Y. Zhang, Du (2012)
Minimum Action method: E, Ren,Vanden-Eijnden (2004); Zhou, Ren,
E (2008)
Gentlest Ascent Dynamics: E, Zhou (2011)
Eigenvector-following method, activation-relaxation technique,
trajectory-following algorithm, step and slide method, etc
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Zebrafish Hindbrain
Krox20 gene
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Roles of Retinoic Acid (RA)
 A vitamin A derivative and a signal that patterns
the nervous system.
 Also involved in development of many organs
(eye, ear, limbs, heart, pancreas, gonads,
kidney, and lungs).
Morphogen
gradient
Gene expression
 Disrupted in many neurological diseases (e.g.
Parkinson’s, schizophrenia) and cancer (acute
promyelocytic leukemia).
 Neurons in the hindbrain know their positions
along the body axis based on levels of RA.
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Boundary Sharpening during
Segment Development
Transient process of boundary sharpening of krox20 stripes in r3 and r5
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
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Noises in biological systems
Noise in morphogen gradient
Arthur Lander, UC Irvine
Noise in gene expression
Michael Elowitz, CalTech
• Effect of noise in gene expression
- Regulation of noise in biological switches (Hasty et al, 2000 )
- Noise attenuation in an ultrasensitive signal (Thattai et al, 2002 )
- Gene expression noise in Drosophila segmentation (Holloway et al, 2011 )
• Study of noise in a single cell.
- Stochastic gene expression in a single cell (Elowitz et al, 2002 )
- Spontaneous switch system generated by noise (To and Maheshri, 2010 )
- Bistability and bimodal population (Ferrell et al, 2002; Lopes et al, 2008)
• Little is known how the coupling between the spatial extracellular and intracellular
components, both of which contain noise, regulate the spatial gene patterning?
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Multiscale Model
• RA gradient specifies the fates of rhombomere segments by activating
different genes in the hindbrain.
• Hoxb1 and Krox20 genes: auto-regulation and mutual inhibition.
Noise
Noise
Noise
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Morphogen Model
(18
um2/sec)
Diffusion coefficient
(10-4sec-1)
Permeability
coefficient
Synthesis rate
at position x
[RA]out
 DRA [RA]out  VRA (x,t)  (1   )kA [RA]out  kA [RA]in .
t
Allows flux rate out to be
[RA]in
higher than rate in
 kA [RA]out  (kA  [Cyp])[RA]in
t
Regulated degradation
shapes the gradient
 RAsignal

k
,0  x  x f  40
 deg 1   RA
  (r7  x)
[Cyp]  
signal  f0 e
k ,
x  0 or x f  40  x  x f
 max
RAsignal  ([RA]in )n .
n=2 (indicates modest
cooperativity in signaling)
[RA]out : extracellular RA concentrations, [RA]in
Location along Fgf gradient
where [Fgf] = f0
: intracellular RA concentrations.
(R. White, Q. Nie, A. Lander, T. Schilling PLoS Biology (2007) 5-11)
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Gene Model
Autoregulation
dgh
ch ghnh  ( h [RA]in )m

 dh gh ,
nh
nk
m
dt 1  ch gh  ck gk  ( h [RA]in )
dgk
ck gknk  ( k [RA]in )m

 d k gk .
nh
nk
m
dt 1  ch gh  ck gk  ( k [RA]in )
Degradation rate of genes
Mutual inhibition
Sensitivity to
RA feedback
gh : hoxb1 gene,
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gk : krox20 gene,
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Question I
In the deterministic model:
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How to generate a three-segment alternating striped expression of two
genes activated by a smooth RA gradient?
Krox20
Hoxb1
r3 r4 r5
Dr Schilling’s lab
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Results I
 In the absence of noise, the initial level of Hoxb1 and mutual
inhibition are essential for the normal gene patterning.
1D
r3 r4 r5
r3 r4 r5
r3 r4 r5
2D
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
•Activation of hoxb1 and krox20 is determined by the
initial level of hoxb1 and RA gradient.
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A model for chick hindbrain
patterning, Giudicelli et al, 2001.
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Mutual inhibitions are necessary
Hoxb1
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Krox20
Hoxb1
Krox20
Krox20
Hoxb1
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Question II
During the segment development,
 What kind of noise induces the initial ragged boundary during the
segment development?
--- Extracellular or intracellular noise,
--- Morphogen noise, gene noise?
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How can the ragged boundary become sharp?
--- Regulation of morphogen?
--- Still noise?
Our approach:
 Theoretical analysis:
Rare events: Minimum Action Path - Gene switching probability
 Numerical simulations for boundary sharpening
(a) Stochastic PDE, (b) Stochastic Simulation Algorithm.
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Minimum Action Path

A random dynamic system:
dXt  b(Xt )dt   dWt ,
• Wentzell-Freidlin theory of large deviations gives an estimate of the
probability distribution over any fixed time interval [T1,T2 ]
1
P{(X,  )   }  exp( ST1 ,T2 [ ])

• The most probable path from one stable steady state to another stable
steady state is Minimum Action Path (MAP) (Freidlin and Wentzell. 1998)
1 T2
MAP = min{ST ,T [j ]} = min{ ò | j (t) - b(j )2 |2 dt}.
1 2
j
j
2 T1
With the constraint that  (T1 )  a1 and  (T2 )  a2 ( a1 , a2 are the two steady states).
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Numerical method:
Minimum action method to find the MAP  * for a given switching time
( T2  T1 ) ( E, Ren,Vanden-Eijnden, 2004; Zhou, Ren, E, 2008)
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Results II
 Gene state bifurcation and their Minimium Action Paths determine
the capability of gene switch between different states.
MAP
Hoxb1 on
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
Number of gene states is 5 (RA<0.22), 3 (0.22<RA<0.85), 1 (RA>0.85).
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Switching Probability
 Find Minimum Action Path:  * connecting Hoxb1 X H with
Krox20 X K through a saddle point X C .
 Distances |  *( X H )   *( X C ) | and|  *( X K )   *( X C ) |
server as a minimal barrier to overcome for switching.
 Estimate gene switching probability
within a time interval [0, T ]:
PX H X K  exp( |  *( X H )   *( XC ) |n )
and
PX K X H  exp( |  *( X K )   *( X C ) |n )
 Monte Carlo simulation is also carried out to compute the
switching probability at the same time interval.
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Stochastic Modeling
 Theoretical analysis of MAP suggests that gene switching may
regulate the gene patterning.
 Stochastic model of both extracellular noise and intracellular
noise on RA gradient and genes.
dgh
ch ghnh  ( h [RA]in )m
d (t)


d
g


g
,
h h
h h
dt 1  ch ghnh  ck gknk  ( h [RA]in )m
dt
dgk
ck gknk  ( k [RA]in )m
d (t)


d
g


g
.
k
k
k
k
dt 1  ch ghnh  ck gknk  ( k [RA]in )m
dt
[RA]out
2 [RA]out
2Wout (t, x)
 DRA

V
(x,t)

(1


)k
[RA]

k
[RA]


[RA]
,
RA
A
out
A
in
out
out
t
x 2
tx
[RA]in
2Win (t, x)
 kA [RA]out  (kA  [Cyp])[RA]in   in [RA]in
,
t
tx
White noise and color (spatial- & temporal-correlated) noise.
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Morphogen noise
 Self-degradation enzyme Cyp26 is able to absorb the most extracellular
noise.
 Both extra- and intra-cellular noise on RA gradient.
T=1
T=25
T=50
Dynamics of gene distributions
 If the noise exists in extra/intracellular RA gradient, initial ragged
boundary is established and do not become sharp over time.
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Morphogen noise
+ Gene expression noise=Less noise
 Noise in morphogen gradient induces initial noisy boundary, but
noise persists.
 Noise in gene expression could be a secret ingredient for the noise.
attenuation.
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
 a novel noise attenuation mechanism that intracellular noise
induces switching and coordinate cellular decisions
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Measure the boundary sharpening
 Define a quantity to measure the noise:
1. A sharp boundary is defined as the intersection where both gene
distributions are 50%,
2. The sample standard deviation is defined as “Sharpness Index”.
 A decreasing of the Sharpness Index over time indicates the
noise attenuation during development.
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Gene switching in vivo
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Co-expression of two genes and mis-expressing cells along the r4/5
boundary
Confocal projections of two color FISH for hoxb1a and krox20
hoxb1a
krox20
co-expression cells
Sample distributions of mis-expressing cells along the r4/5 boundary.
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
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Gene noise amplitude
a is noise amplitude
freqRA
Gene noise frequency ratio:  
freqgene
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Other noise attenuation mechanism?
 Effect of growing domain
 Cell sorting (movement)
discrete stochastic model
 Time delay
 Noise in gene expression is critical for boundary sharpening.
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Summary
 Computational biology involves all kinds of mathematics: modeling,
theoretical analysis, numerical methods, etc.
 Transition state plays a big role in complex biological systems.
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A novel noise attenuation mechanism for boundary sharpening in zebrafish
hindbrain.
Myosin signaling drives neuroblast delamination in Drosophila.
 Some other applications in materials:
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Finding morphology of critical nucleus in solid-state phase transformation,
Zhang-Chen-Du, PRL, 2007, Acta Mater. 2008, JSC 2008.
Simultaneous Prediction of Morphologies of a Critical Nucleus and an
Equilibrium Precipitate in Solids, Zhang-Chen-Du, CiCP, 2010, JCP, 2010.
Heterogeneous nucleation in solid, Zhang-Zhang-Du, submitted, 2013.
Incorporating diffuse-interface nuclei in phase-field simulations.
Heo-Zhang-Du-Chen, Scr. Mater., 2010; Li-Hu-Zhang-Sun, submitted, 2013
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Thank You !
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