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Constructive effects of stochasticity in biochemical pathways

Constructive effects of stochasticity in biochemical pathways. Yao-Chen Hung （洪耀正）. Department of Physics, National Chung Cheng University, TAIWAN. Outline. 契子： Biological Mathematics Introduction Noise in biochemical pathways Noise induced phenomena p53 regulatory network

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Constructive effects of stochasticity in biochemical pathways

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  1. Constructive effects of stochasticity in biochemical pathways Yao-Chen Hung （洪耀正） Department of Physics, National Chung Cheng University, TAIWAN

  2. Outline • 契子：Biological Mathematics • Introduction Noise in biochemical pathways Noise induced phenomena • p53 regulatory network • Noise induced bifurcation • Discussion and Conclusion

  3. Biological Mathematics • Introduction • p53 regulatory network • Noise induced bifurcation • Discussion and Conclusion

  4. Fight! Why should war favor shark? Biological Mathematics • 生物動力學的起源: 第一次世界大戰期間，亞得里亞海域（Adriatic sea）的漁業活動大為衰減。幾年過後，義大利生物學家D’Ancona分析魚貨市場的魚獲量。他發現經歷戰爭後，掠奪性魚類的總數大為增多。

  5. Biological Mathematics • Volterra原理: 令 與 分別代表獵物與掠食者的數量。假設獵物以 的速率繁殖，並存在 的機率被掠食者捕獲。掠食者的繁殖速度由食物與掠食者數量所共同決定，其為 。同時，掠食者存在 的機率死亡。

  6. Biological Mathematics • Steady state 掠食者對獵物比例

  7. Biological Mathematics 原野上有一個牧羊人以及放養的一群羊。一個陌生男子走近牧羊人並問他說：「如果我猜中你羊的總數，你是不是可以送我一頭羊呢？」牧羊人回答：「沒問題。」那男子沈吟了一下，說道：「83頭。」牧羊人很是訝異，因為這正是確切的數目。於是陌生男子從羊群中挑了一頭，轉身準備離去。此時牧羊人突然叫道：「站住！如果我猜中你的職業，你是不是可以把它還給我？」男子回答：「沒問題。」牧羊人說：「你是一名數學生物學家。」男子張大了嘴很是訝異，問道：「你怎麼知道？」牧羊人笑笑…

  8. Biological Mathematics 原野上有一個牧羊人以及放養的一群羊。一個陌生男子走近牧羊人並問他說：「如果我猜中你羊的總數，你是不是可以送我一頭羊呢？」牧羊人回答：「沒問題。」那男子沈吟了一下，說道：「83頭。」牧羊人很是訝異，因為這正是確切的數目。於是陌生男子從羊群中挑了一頭，轉身準備離去。此時牧羊人突然叫道：「站住！如果我猜中你的職業，你是不是可以把它還給我？」男子回答：「沒問題。」牧羊人說：「你是一名數學生物學家。」男子張大了嘴很是訝異，問道：「你怎麼知道？」牧羊人笑笑… 「因為，你挑中的是我的牧羊犬。」

  9. Biological Mathematics • 理論生物學家在過去數十年一直處在弱勢的地位；甚至在沒有惡意的前提下被謔稱為「搖椅生物學家」。 Reason 1: 偏好純粹、乾淨的數學，而忽略與實驗學家 進行交流合作才有機會解決真正的問題。 Reason 2: 生物系統的高度複雜性。

  10. Biological Mathematics Time series of the lynx population (1821–1934) from six regions in Canada.

  11. Biological Mathematics

  12. Biological Mathematics

  13. Biological Mathematics 「只有理論才能告訴我們必須量測什麼，以及如何 去解釋測量的結果。」 ~愛因斯坦

  14. Biological Mathematics • Introduction • P53 regulatory network • Noise induced bifurcation • Discussion and Conclusion

  15. Introduction • Genetically identical cells exposed to the same environmental conditions can show significant variation in molecular content and marked differences in phenotypical characteristics.  arising from the stochasticity in gene regulation. • Such effects play crucial roles in biological processes, such as development, which would finally determine cell fates. J. L. Spudich et al., Nature 262, 467 (1976). J.M. Raser and E.K. O‘shea, Science 309, 2010 (2005).

  16. Introduction • The noise in biochemical pathways originates from two different mechanisms: a. Intrinsic noise: arising from the discrete nature of the biochemical process of gene expression; . M. Kaern et al., Nature Reviews 6, 451 (2005).

  17. Introduction b. Extrinsic noise: generated by the interactions of the system with other stochastic systems in the cell or its environment; . • The two different sources of stochasticity now can be identified and quantified experimentally. M. B. Elowitz et al., Science 297, 1183 (2002)

  18. Introduction • Noise is generally regarded as a destructive factor because fluctuations in protein level can inevitably obscure the transduction signals and negatively affect the cellular regulation. • Most previous works have focused on the design of biochemical networks which could reduce or resist these existing fluctuations. O. Brandman, J. E. Ferrell Jr., R. Li, and T. Meyer, Science 310, 496 (2005).

  19. Introduction Question: If the stochasticity can provide any advantages for living cells to sustain their functions?

  20. Biological Mathematics • Introduction • p53 regulatory network • Noise induced bifurcation • Discussion and Conclusion

  21. p53 Regulatory Network • In this report, we study the effect of noise on a specific crosstalk of signal transduction pathways, the p53 regulatory network. • The tumor suppressor protein, p53, is a major regulator in preserving genomic integrity in mammalian and therefore plays a crucial role in the origins of cancer.

  22. p53 Regulatory Network G. Lahav, et al., Nature Genetics 36, 147 (2004)

  23. p53 Regulatory Network B. Vogelstein, D. Lane, and A. J. Levine, Nature 408, 307 (2000).

  24. p53 Regulatory Network Noise In a DNA-damaged cell, damage-dependent kinases (e.x. ATM) would phosphorylate p53 and then stabilize it. kd0=0.056 f1(t)=-0.008 or 0 DNA Damage f2(t)=-0.006 or 0 Y.-C. Hung and C.-K. Hu, (2009), submitted.

  25. Results and Discussion • In the absence of noise The resonant switches of p53 between the surviving and apoptotic states in response to a stimulus provide a control mechanism to kill thecancerous/stressed cells and keep normal/recovered cells alive. Apoptosis-Survival Bistability

  26. Results and Discussion • Noise Effect

  27. Results and Discussion Question: What happened in the new emerged region?

  28. Noise

  29. Results and Discussion • The mechanism of the results can be understood as follows: the noise enhances the possibility of threshold crossing and thereby improves the network's resonant response to a subthreshold stimulus  stochastic resonance (SR)..

  30. Results and Discussion

  31. Results and Discussion • Biochemical circuits are not fine-tuned to exercise their functions only for precise values of their biochemical parameters  they must be robust to parameter change. • The inherent noise is of benefits to maintain the basal function of the regulatory network over a broader range of parameter values. • The enhanced robustness arises from the aforementioned SR mechanism. Y.-C. Hung and C.-K. Hu, Comput. Phys. Commun. 182, 249 (2011).

  32. Biological Mathematics • Introduction • p53 regulatory network • Noise induced bifurcation • Discussion and Conclusion

  33. Noise induced bifurcation • Recently, an interesting phenomena induced by the propagation of noise was found in a gene cascade network. W. J. Blake, M. Kaern, C. R. Cantor, and J. J. Colins, Nature 422, 633 (2003).

  34. Noise induced bifurcation Noise induced bistability (bifurcation) Such an phenomenon is important for the origin of phenotypic variation and cellular differentiation. N. Maheshri and E. K. O'Shea, Annu. Rev. Biophys. Biomol. Struct. 36, 413 (2007) .

  35. Noise induced bifurcation • Several questions remain regarding the new bistable behavior. a. How does an external noise affect the dynamics of systems which can present typical bistable behavior deterministically? b. If it is possible to give a clear picture to explain such a noise induced bifurcation? c. What are the potential applications of this phenomenon?

  36. Noise induced bifurcation • We adopt the Schlögl model as our working system for its simplicity and analyzability. The model is an autocatalytic, trimolecular reaction scheme, proposed as a prototype exhibiting bistability. F. Schlögl, Z. Physik. 253, 147 (1972).

  37. Noise induced bifurcation • Mean-Field Equation: • At steady state concentration of X

  38. Noise induced bifurcation • We consider the noise originated from the chemical reaction rate. • In biological systems, the system control via single reaction rate is realizable. The modulation process often accompanies with additional fluctuations dominant over other noise source. ex. a temperature-sensitive protein cI J. Hasty, D. McMillen, F. Isaacs and J. J. Collins, Nat. Rev. Genet. 2, 268 (2001).

  39. Noise induced bifurcation b. A single biochemical reaction equation is likely to represent a complex sequence of reactions. It is therefore nature to assume that the reaction rate is affected by fluctuations caused by many internal or external factors.

  40. Noise induced bifurcation The equivalent stochastic differential equation (Langevin equation):

  41. Noise induced bifurcation • Numerical simulations are performed based on the simple forward Euler algorithm: J. M. Sancho, et al., Phys. Rev. A 26, 1589 (1982).

  42. a=2.1, b=0.2, k=1.2 Blue: D=0.01Red: D=0.1 Noise induced bifurcation The stable fixed points in the deterministic case (D = 0) locate at x = 0.31 and x = 1.29.

  43. Blue: D=0.01Red: D=0.1 Noise induced bifurcation The external noise causes the transitions from bistability to monostability and this is the phenomenon of noise induced bifurcation (or stochastic bifurcation)

  44. a=2.3, b=0.2, k=1.2 Blue: D=0.01Red: D=0.03Green: D=0.2 Noise induced bifurcation The single stable fixed point in the deterministic case locates at x = 1.64.

  45. Noise induced bifurcation • To analytically understand the underlying mechanism of the aforementioned phenomena, we transform the original formulation to an equivalent Fokker-Plank equation:

  46. Noise induced bifurcation Stationary probability distribution

  47. Noise induced bifurcation in which

  48. Noise induced bifurcation a=2.1, b=0.2, k=1.2 Blue: α=0.05 (D=0.01) Red: α=0.5 (D=0.1)

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