1. (the end of the world?) 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 Microtubule CATASTROPHE!!!

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/vectorialSpontaneous/stochastic –Catastrophe

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

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) 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 Probability cap has length x at time t Boundary condition at x = 0 (no cap = catastrophe)

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

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. 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. Results 2: Dilution/spontaneous Time of catastrophe indeed independent from initial cap lengths in dilution experiment, except at very short cap lengths 00.20.40.60.81 0 0.5 1 1.5 2 2.5 3 Initial cap length Average delay before catastrophe Spontaneous hydrolysis (varying initial conditions, rates fixed)

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., 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