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(the end of the world?)

Microtubule. (the end of the world?). CATASTROPHE!!!. Project L, Assigned by Alex Mogilner, UC-Davis Omer Dushek, University of British Columbia Zhiyuan Jia, Michigan State University Maureen M. Morton, Michigan State University Mentor: Xiao Yu Luo, University of Glasgow.

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(the end of the world?)

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  1. Microtubule (the end of the world?) CATASTROPHE!!! Project L, Assigned by Alex Mogilner, UC-Davis Omer Dushek, University of British Columbia Zhiyuan Jia, Michigan State University Maureen M. Morton, Michigan State University Mentor: Xiao Yu Luo, University of Glasgow

  2. Biological Background • Microtubule-what is it? • Dynamic instability: back and forth between growth/shrinking • One biological model: • Cap • Polymerization adds GTP-tubulin • Hydrolysis of GTP to GDP • Induced/vectorial • Spontaneous/stochastic • Catastrophe

  3. The Situation GDP-bound tubulin GTP-bound tubulin Catastrophe Microtubule Growth Hydrolysis faster than addition GTP-tubulin addition Induced GTP hydrolysis Rapid dissociation of GDP-tubulin Cap Spontaneous GTP hydrolysis en.wikipedia.org/wiki/Microtubule

  4. Experimental Results 1 • Greater concentration of tubulin causes higher growth velocity causes longer time before catastrophe occurs. Flyvbjerg et al 1994

  5. Experimental Results 2 • Grow MT at high tubulin concentration then dilute. Upon dilution, time to catastrophe is independent of initial cap length or growth velocity before dilution. Pre-dilution Length (m) Walker et al 1991

  6. A Stochastic Model(Flyvbjerg, et al, 1994) p(x,t) Probability cap has length x at time t Average growth of the cap’s length between spontaneous shortening in its interior Fluctuations, i.e. random walk superposed on this average growth Rate at which caps of length x are spontaneously shortened Rate at which caps longer than x are shortened to length x Boundary condition at x = 0 (no cap = catastrophe)

  7. Solutions to Flyvbjerg et al’s Stochastic Model Post-dilution catastrophe Delay before catastrophe (s) Pre-dilution vg (m/min) Catastrophe frequency as dependent on rate Average catastrophe time after dilution Flyvbjerg et al 1994

  8. What We Did: Monte Carlo Simulations • Testing increased growth rate => increased delay before catastrophe • Changed growth rate, kept induced hydrolysis rate constant, no spontaneous hydrolysis • 500—5000 times • Testing post-dilution catastrophe independence from initial conditions • Changed initial length of cap, kept constant probabilities of induced and spontaneous hydrolysis and kept probability of growth at 0 • 5000 times

  9. Induced hydrolysis (varying polymerization rate, fixed start time) 600 Highest(1) (2) 500 (3) (4) Lowest(5) 400 Number of events 300 200 100 0 0 10 20 30 40 50 60 70 80 90 100 Delay before catastrophe Results 1: Changing Growth Rate • Number of catastrophes at certain times follows a not purely exponential curve (which might be mildly interesting) • Greater growth => longer catastrophe time Frequency of catastrophe related to growth rate

  10. Spontaneous hydrolysis (varying initial conditions, rates fixed) 3 Spontaneous hydrolysis (varying initial conditions, rates fixed) 2.5 140 2 120 Average delay beforecatastrophe 1.5 Short cap(1) 100 (2) 1 (3) 80 (4) Number of events 0.5 Long cap(5) 60 0 0 0.2 0.4 0.6 0.8 1 Initial cap length 40 20 0 0 1 2 3 4 5 6 7 8 9 10 Delay before catastrophe Results 2: Dilution/spontaneous • Time of catastrophe indeed independent from initial cap lengths in dilution experiment, except at very short cap lengths

  11. Concluding Remarks • Our Monte Carlo simulations agree qualitatively well with the experiments and the stochastic model by Flyvbjerg et al (1994) • This suggests that a simple Monte Carlo modeling approach can explain the biological situation and verify a complicated equation (polymerization and two kinds of hydrolysis) • More detailed research is required to obtain a quantitative match with experimental data (as also admitted by Flyvbjerg et al 1996)

  12. References • Drechsel, D. N., A. A. Hyman, M. H. Cobb, and M. W. Kirschner. 1992. Modulation of the dynamic instability of tubulin assembly by the microtubule-associated protein tau. Mol. Biol. Cell. 3: 1141-1154. • Flyvbjerg, H., T. E. Holy, and S. Leibler. 1996. Microtubule dynamics: caps, catastrophes, and coupled hydrolysis. Phys. Rev. E. 54: 5538-5560. • Flyvbjerg, H., T. E. Holy, and S. Leibler. 1994. Stochastic dynamics of Microtubules: a model for caps and catastrophes. Phys. Rev. Let. 73: 2372-2375. • Matlab 7. 2004. The MathWorks, Inc. • Mogilner, A. (Web site, notes, personal correspondence). http://www.math.ucdavis.edu/~mogilner/ParkCity.html • Voter, W. A., E. T. O’Brien, and H. P. Erickson. 1991. Dilution-induced disassembly of microtubules: relation to dynamic instability and the GTP cap. Cell Motil. Cytoskeleton. 18: 55. • Walker, R. A., N. K. Pryer, and E. D. Salmon. 1991. Dilution of individual microtubules observed in real time in vitro: evidence that cap size is small and independent of elongation rate. J. Cell Biol. 114: 73-81. • Weisstein, E. W. Airy Differential Equation. From Mathworld—A Wolfram Web Resource. http://mathworld.wolfram.com/AiryDifferentialEquation.html • Weisstein, E. W. Airy Functions. From Mathworld—A Wolfram Web Resource. http://mathworld.wolfram.com/AiryFunctions.html • Weisstein, E. W., et al. Asymptotic Series. From Mathworld—A Wolfram Web Resource. http://mathworld.wolfram.com/AsymptoticSeries.html • Weisstein, E. W. Gamma Function. From Mathworld—A Wolfram Web Resource. http://mathworld.wolfram.com/GammaFunction.html

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