1 / 30

Applications of Transition State in System Biology

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).

nuru
Télécharger la présentation

Applications of Transition State in System Biology

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. Outline • Introduction • Noise drives boundary sharpening in zebrafish hindbrain • Neuroblast delamination in Drosophila • Summary

  3. 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) Saddle Point (Wikipedia)

  4. Transition state in biology?

  5. 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

  6. Zebrafish Hindbrain Krox20 gene

  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. Morphogengradient Gene expression

  8. 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)

  9. Noises in biological systems Noise in gene expression Noise in morphogen gradient Arthur Lander, UC Irvine 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?

  10. 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

  11. Morphogen Model (10-4sec-1) (18 um2/sec) Synthesis rate at position x Permeability coefficient Diffusion coefficient Allows flux rate out to be higher than rate in Regulated degradation shapes the gradient Location along Fgf gradient where [Fgf] = f0 n=2 (indicates modest cooperativity in signaling) : extracellular RA concentrations, : intracellular RA concentrations. (R. White, Q. Nie, A. Lander, T. Schilling PLoS Biology (2007) 5-11)

  12. Gene Model : hoxb1 gene, : krox20 gene, Autoregulation Degradation rate of genes Mutual inhibition Sensitivity to RA feedback

  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? Krox20 Hoxb1 r3 r4 r5 Dr Schilling’s lab

  14. Initial level of Hoxb1 A model for chick hindbrain patterning, Giudicelli et al, 2001. 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.

  15. Krox20 Hoxb1 Hoxb1 Hoxb1 Krox20 Krox20 Mutual inhibitions are necessary

  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.

  17. Minimum Action Path ( are the two steady states). With the constraint that and • A random dynamic system: • Wentzell-Freidlin theory of large deviations gives an estimate of the probability distribution over any fixed time interval • The most probable path from one stable steady state to another stable steady state is Minimum Action Path (MAP) (Freidlin and Wentzell. 1998) • Numerical method:Minimum action method to find the MAP for a given switching time ( ) ( E, Ren,Vanden-Eijnden, 2004; Zhou, Ren, E, 2008)

  18. 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).

  19. Switching Probability • Find Minimum Action Path: connecting Hoxb1 with Krox20 through a saddle point . • Distances andserver 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

  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.

  21. T=1 T=25 T=50 Morphogen noise • Self-degradation enzyme Cyp26 is able to absorb the most extracellular noise. • Both extra- and intra-cellular noise on RA gradient. 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.

  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. (L. Zhang et al, Nature Molecular Systems Biology, 2012) • a novel noise attenuation mechanism that intracellular noise induces switching and coordinate cellular decisions

  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.

  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 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)

  25. Gene noise amplitude a is noise amplitude Sharpness Index Gene noise frequency ratio:

  26. Other noise attenuation mechanism? • Cell sorting (movement)discrete stochastic model • Effect of growing domain • Time delay • Noise in gene expression is critical for boundary sharpening.

  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 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

  28. Thank You !

More Related