Modeling Length Regulation of Stereocilia

1. Modeling Length Regulation of Stereocilia Alexandra Jilkine, Karin Leiderman and Attila Toth

2. Background • Stereocilia are actin-based cellular protrusions found on the surface of hair cells in the inner ear • Needed to detect sound waves • Persist for a lifetime • Composed of several hundred parallel actin filaments, which are cross-linked by the proteins espinandfimbrin • Organized into bundles of precisely specified rows of increasing heights • Length is maintained to 0.01% accuracy

3. Stereocilia Staircase From Frolenkov et al. (2004)

4. Model of Mature Stereocilia Renewal (+) • Stereocilia continuously self-renew via an actin molecular treadmill • Actin monomers are added to the plus end and removed from the minus end, while constant length is maintained • Espin is also incorporated at the tip of the stereocilia and progresses down at the same rate as actin • There is a stable steady state for various stereociliar lengths in the staircase • How actin flux rates are regulated in the stereocilia is an important unresolved issue (-) From Lin et al. (2005)

5. Actin Treadmill Rate • Actin is incorporated into stereocilia and treadmills down at the rate of 0.004-0.04 actin subunits (per actin filament) per second • The treadmill rate is proportional to stereocilia length • Actin overexpression does not change the treadmill rate or stereociliar length • Entire bundle is turned over synchronously From Rzadzinska et al. (2004)

6. Depolymerization Rate • Stereocilia shorten linearly when actin polymerization is blocked with Cytochalasin D • Rate of shortening is scaled to the steady state length of a stereocilium • Experiment suggest there is no direct link between polymerization and depolymerization machinery From Rzadzinska et al. (2004)

7. Role of Myosin XVa • Myosin XVa is localized to the tips; amount of myosin XVa is scaled to the length of the stereocilia • Mutations in myosin XVa result in abnormally short stereocilia • Hence, myosin XVa has a role in regulating the steady state length of stereocilia

8. How is Stereociliar Length Regulated? Hypothesis • The concentration of Myosin XVa is proportional to the length of the stereocilium • Actin incorporation rates are proportional to myosin XVa concentration • The espin incorporation rates are independent of the actin flux; longer stereocilia have a smaller espin/actin ratio • The depolymerization rate directly correlates to the espin/actin ratio

9. Model Formulation Stereociliium length is at a steady state: Hence, the monomer profile is static. Assume actin monomers are delivered to the tip by diffusion alone: Monomer concentration at the base Monomer flux at the tip

10. Boundary conditions inside the cell x=0 x=L • depolymerization at x=0 and polymerization at x=L • If we integrate the steady state equation up once: • the net flux is constant! • At x=L, we need the constant flux to balance with the degradation of the actin monomers (polymerization) to avoid build up of monomers • Thus, # of monomers/sec assembling at the tip φ=conversion factor between length and concentration

11. Boundary conditions continued… inside the cell x=0 x=L • At x=0, what is going on? • There is diffusion of monomers back into the cell as well as the depolymerization of the filament creating free monomers • Is the depolymerization fast enough relative to the diffusion to create a build up of monomers? • How can we compare the measured depolymerization rate with the rate of diffusion?

12. Boundary conditions continued… • Peclet numbercompares the relative effect of convection versus diffusion. • Here the shortening rate of stereocilia plays the role of the convection rate. We use the Peclet number to compare the effect of depolymerization and diffusion. From Rzadzinska et al., we derive: • Diffusion wins! • Since the monomers at the base can diffuse away faster than any rate of depolymerization, we can now assume that the concentration of actin monomers at x=0 is constant (M0).

13. Monomer Profile Solution to is given by

14. Can Diffusion Supply the Necessary Monomer Flux? The flux is given by Experimentally measured parameters: But measured actin flux is increasing with length!

15. What happens if we don’t use a constant kon? Diffusion does not give the right flux!

16. How does active transport affect the monomer flux? Actin monomer concentration equation: Diffusion termConvection term (active transport)

17. Monomer Profile The steady state solution is given by: Monomer concentration at the tip is:

18. Experimentally Measured Flux Based on experimental data, we obtain J=0.0107L . We use this relationship together with the flux obtained from the analytical solution to solve for U numerically for a given M0.

19. Effect of Monomer Concentration at the Base on Transport Speed and Tip Monomer Concentration

20. Necessary Active Transport Rates [M] at the base vs. transport rate, L = 4, 12, 20 Uμm/sec use M(0) = 0.01 to investigate the L dependence of U L = 20 L = 12 L = 4

21. Dependence of Transport Rate of Stereocilium Length

22. Analytic Expression for Active Transport Rate Parameters used: D=5μm2/sec, φ=18 (μM μm)-1, N=200, kon=10 (μM sec)-1

23. Conclusions • Pure diffusion model doesn’t give the flux as an increasing function of L • Indicates an addition to the model is needed  transport • For a given base concentration we can calculate the necessary transport rates for any length stereocilia to maintain steady state monomer concentration • The flux is not sensitive to parameter kon • Hence, variability in kondoes not explain greater flux in longer stereocilia if monomer transport to the tip is achieved by diffusion and active transport of monomers

24. Biological Consequences • Our results indicate that unless monomer concentration at the base is small, active transport will take monomers from the tip to the base • We do not believe this is realistic as it is not energy-efficient for the cell • We conclude that either monomer concentration at the base is very low or a additional factors must be taken into account

25. References • H. Lin, M. Scneider and B. Kachar. (2005) When size matters: the dynamic regulation of stereocilia lengths. Current Opinion in Cell Biology. 17:55-61. • G. Frolenkov, I. Belyantseva, T. Friedman and A. Griffith. (2004) Genetic insight into the morphogenesis of inner ear hair cells. Nature Reviews Genetics. 5:489-498. • A. Rzadzinska, M. Scneider, C. Davis, G. Riordan and B. Kachar. (2004) An actin molecular treadmill and myosins maintain stereocilia functional architecture and self-renewal. The Journal of Cell Biology. 164: 887-897. • L. Tilney, M. Tilney and D. Cotanche. (1988) Actin filaments, stereocilia, and hair cells of the bird cochlea V. How the staircase pattern of stereociliary lengths is generated. The Journal of Cell Biology. 106: 355-365.

26. Acknowledgements We would like to thank Nicholas HillLeah Edelstein-KeshetandAlex Mogilner for their advice and help!