Applications of Transition State in Applications of Transition State in System BiologySystem 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.
Transition state (Rare events) are of general interests: Nucleation in materials (Zhang-Chen-Du, PRL 2007, CiCP 2010; Cheng-Lin-
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)
(Wikipedia)
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Saddle Point
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Transition state in biology?
Numerical methods for saddle point Numerical methods for saddle point and transition pathway
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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Krox20 gene
Zebrafish Hindbrain
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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).
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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Morphogengradient
Gene expression
Transient process of boundary sharpening of krox20 stripes in r3 and r5 (L. Zhang et al, Nature Molecular Systems Biology, 2012)
Boundary Sharpening during Segment Development
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Noises in biological systemsNoise in gene expression
Michael Elowitz, CalTechArthur Lander, UC Irvine
Noise in morphogen gradient
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• 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?
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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Diffusion coefficientSynthesis rate at position x
Permeability coefficient
Allows flux rate out to behigher than rate in
(18 um2/sec)
n=2 (indicates modest cooperativity in signaling)
(10-4sec-1)
Regulated degradationshapes the gradient
[RA]outt
DRA[RA]out VRA (x, t) (1 )kA[RA]out kA[RA]in .
[RA]int
kA[RA]out (kA [Cyp])[RA]in
[Cyp] kdeg
RAsignal1RAsignal f0e
(r7 x ),0 x x f 40
kmax , x 0 or x f 40 x x f
RAsignal ([RA]in )n . Location along Fgf gradient
where [Fgf] = f0
Morphogen Model
[RA]out : extracellular RA concentrations, [RA]in : intracellular RA concentrations.
(R. White, Q. Nie, A. Lander, T. Schilling PLoS Biology (2007) 5-11)
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Autoregulation
Degradation rate of genes
Gene Model
dghdt
chgh
nh ( h[RA]in )m
1 chghnh ckgk
nk ( h[RA]in )m dhgh ,
dgkdt
ckgk
nk ( k[RA]in )m
1 chghnh ckgk
nk ( k[RA]in )m dkgk .
gh : hoxb1 gene, gk : krox20 gene,
Sensitivity to RA feedback
Mutual inhibition
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Question IQuestion I
In the deterministic model: How to generate a three-segment alternating striped expression of two
genes activated by a smooth RA gradient?
r3 r4 r5
Krox20 Hoxb1
Dr Schilling’s lab
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r3 r4 r5
Results I In the absence of noise, the initial level of Hoxb1 and mutual
inhibition are essential for the normal gene patterning.
•Activation of hoxb1 and krox20 is determined by the initial level of hoxb1 and RA gradient.
A model for chick hindbrain patterning, Giudicelli et al, 2001.
r3 r4 r5
r3 r4 r5
Initi
al le
vel o
f Hox
b1
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
1D
2D
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Mutual inhibitions are necessary
Hoxb1 Krox20 Krox20 Hoxb1Hoxb1 Krox20
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Question IIQuestion IIDuring the segment development, What kind of noise induces the initial ragged boundary during the segment development?
--- Extracellular or intracellular noise,
--- Morphogen noise, gene noise?
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
dXt b(Xt )dt dWt ,
• The most probable path from one stable steady state to another stable steady state is Minimum Action Path (MAP) (Freidlin and Wentzell. 1998)
With the constraint that (T1) a1(T2 ) a2and are the two steady states).a1,a2(
P{(X, ) } exp(1
ST1 ,T2
[ ])
• Wentzell-Freidlin theory of large deviations gives an estimate of the probability distribution over any fixed time interval [T1,T2 ]
Numerical method:Minimum action method to find the MAP for a given switching time ( ) ( E, Ren,Vanden-Eijnden, 2004; Zhou, Ren, E, 2008)T2 T1
*
A random dynamic system:
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Results II
Number of gene states is 5 (RA<0.22), 3 (0.22<RA<0.85), 1 (RA>0.85).
Gene state bifurcation and their Minimium Action Paths determine the capability of gene switch between different states.
Hoxb1 on
MAP
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
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Switching Probability Find Minimum Action Path: connecting Hoxb1
with Krox20 through a saddle point . Distances and
server as a minimal barrier to overcome for switching. Estimate gene switching probability
within a time interval [0, T ]:
Monte Carlo simulation is also carried out to compute the switching probability at the same time interval.
HX*KX CX
| *( ) *( ) |H CX X | *( ) *( ) |K CX X
exp( | *( ) *( ) | )H K
nX X H CP X X
exp( | *( ) *( ) | )K H
nX X K CP X X and
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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.
dghdt
chgh
nh ( h[RA]in )m
1 chghnh ckgk
nk ( h[RA]in )m
dhgh h ghd(t)
dt,
dgkdt
ckgk
nk ( k[RA]in )m
1 chghnh ckgk
nk ( k[RA]in )m
dkgk k gkd (t)
dt.
[RA]outt
DRA
2[RA]outx2
VRA (x,t) (1 )kA[RA]out kA[RA]in out [RA]out2Wout (t, x)
tx,
[RA]int
kA[RA]out (kA [Cyp])[RA]in in [RA]in2Win (t, x)
tx,
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.
If the noise exists in extra/intracellular RA gradient, initial ragged boundary is established and do not become sharp over time.
Dynamics of gene distributions
T=1 T=25 T=50
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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.
a novel noise attenuation mechanism that intracellular noise induces switching and coordinate cellular decisions
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
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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 Gene switching in vivovivo Co-expression of two genes and mis-expressing cells along the r4/5
boundary
Confocal projections of two color FISH for hoxb1a and krox20
Sample distributions of mis-expressing cells along the r4/5 boundary.
hoxb1a krox20co-expression cells
(L. Zhang et al, Nature Molecular Systems Biology, 2012)
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Gene noise amplitudeS
har
pn
ess
Ind
exa is noise amplitude
Gene noise frequency ratio: RA
gene
freq
freq
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Other noise attenuation mechanism? Effect of growing domain
Time delay
Cell sorting (movement)discrete stochastic model
Noise in gene expression is critical for boundary sharpening. Lei Zhang (PKU) 28
Summary Computational biology involves all kinds of mathematics: modeling,
theoretical analysis, numerical methods, etc. Transition state plays a big role in complex biological systems.
A novel noise attenuation mechanism for boundary sharpening in zebrafish hindbrain.
Myosin signaling drives neuroblast delamination in Drosophila.
Some other applications in materials: 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 !Thank You !
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