Applications of Transition State in System Biology Lei Zhang （张磊） Beijing International Center for Mathematical Research, Peking University Joint with.

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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