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 Lei Zhang (PKU)1

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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) Lei Zhang (PKU)4 Saddle Point

Lei Zhang (PKU)5 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 Lei Zhang (PKU)6

Krox20 gene Zebrafish Hindbrain Lei Zhang (PKU)7

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. Lei Zhang (PKU)8 Morphogen gradient 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 Lei Zhang (PKU)9

Noises in biological systems Noise in gene expression Michael Elowitz, CalTech Arthur Lander, UC Irvine Noise in morphogen gradient Lei Zhang (PKU)10 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 Lei Zhang (PKU)11

Diffusion coefficient Synthesis rate at position x Permeability coefficient Allows flux rate out to be higher than rate in (18 um 2 /sec) n=2 (indicates modest cooperativity in signaling) (10 -4 sec -1 ) Regulated degradation shapes the gradient Location along Fgf gradient where [Fgf] = f 0 Morphogen Model : extracellular RA concentrations,: intracellular RA concentrations. (R. White, Q. Nie, A. Lander, T. Schilling PLoS Biology (2007) 5-11) Lei Zhang (PKU)12

Autoregulation Degradation rate of genes Gene Model : hoxb1 gene,: krox20 gene, Sensitivity to RA feedback Mutual inhibition Lei Zhang (PKU)13

Question 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 Krox20Hoxb1 Dr Schilling’s lab Lei Zhang (PKU)14

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, r3 r4 r5 Initial level of Hoxb1 (L. Zhang et al, Nature Molecular Systems Biology, 2012) 1D 2D Lei Zhang (PKU)15

Mutual inhibitions are necessary Hoxb1Krox20 Hoxb1 Krox20 Lei Zhang (PKU)16

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?  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. Lei Zhang (PKU)17

Minimum Action Path 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 thatand are the two steady states).( Wentzell-Freidlin theory of large deviations gives an estimate of the probability distribution over any fixed time interval  Numerical method: Minimum action method to find the MAP for a given switching time ( ) ( E, Ren,Vanden-Eijnden, 2004; Zhou, Ren, E, 2008)  A random dynamic system: Lei Zhang (PKU)18

Results II Number of gene states is 5 (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) Lei Zhang (PKU)19

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. and Lei Zhang (PKU)20

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. White noise and color (spatial- & temporal-correlated) noise. Lei Zhang (PKU)21

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=1T=25T=50 Lei Zhang (PKU)22

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) Lei Zhang (PKU)23

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. Lei Zhang (PKU)24

Gene switching in vivo  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 krox20 co-expression cells (L. Zhang et al, Nature Molecular Systems Biology, 2012) Lei Zhang (PKU)25

Gene noise amplitude Sharpness Index a is noise amplitude Gene noise frequency ratio: Lei Zhang (PKU)26

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)27

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 Simultaneous Prediction of Morphologies of a Critical Nucleus and an Equilibrium Precipitate in Solids, Zhang-Chen-Du, CiCP, 2010, JCP, Heterogeneous nucleation in solid, Zhang-Zhang-Du, submitted, Incorporating diffuse-interface nuclei in phase-field simulations. Heo-Zhang-Du-Chen, Scr. Mater., 2010; Li-Hu-Zhang-Sun, submitted, 2013 Lei Zhang (PKU)28

Thank You ! Lei Zhang (PKU)29