Morphogen Gradient Formation through Interactions with HSPG: An Anomalous Diffusion Model

1. Morphogen Gradient Formation through Interactions with HSPG: An Anomalous Diffusion Model Gil Hornung Supervisors Brian Berkowitz, Naama Barkai

2. Cth1 Cth2 Source Cth3 Cth4 Receptors Morphogens • Coordinate the cell growth and differentiation. • Formation of a long-rangeconcentration profile. • Cellular response is concentration dependent.

3. Drosophila Wing Imaginal Disc as a Model System Belenkaya et al. 2004

4. Degrade: p Constant production x=0 Protected while waiting f (Dx) y(Dt) Transition time distribution Transition length distribution Dt Dx Diffusion can be Modeled by a Random Walk Process Wait Dt y(Dt)

5. mean tm Fickian diffusion coefficient f (Dx) Constant degradation rate k y(Dt) variance l2 Dt Dx Accepted Model: Fickian Diffusion Fickian Diffusion & Reaction DiffusionEquation Transition time distribution Transition length distribution Widely accepted model, but is it applicable?

6. Possible Morphogen-HSPG interactions • Methods of Analysis • Analysis of Gradients Formed by Anomalous Diffusion • Model Implications for Robustness • Theoretical Predictions and Methods of Experimental Verification

7. Crucial for morphogen transport Morphogens Move through Heparan Sulfate Proteoglycans Protein • HSPG are membrane-tethered proteins, modified by saccharide chains Disaccharide chain • Highly abundant – much more than receptors • Morphogens interact with the chains

8. What Could be the Effect of HSPG? • Distribution of Affinities • Many Transport Mechanisms • Entangled Domains HSPG may lead to broad transition time distributions!

9. Power Law Tailed Distributions Probabilities for encountering long transition times that do not diminish even for very long transition times y(Dt) ~ Dt -1-b y(Dt) 0 < b < 1 – Infinite mean b > 1 – Finite mean Dt

10. AnomalousDiffusion Fickian Diffusion(Reaction-diffusion equation) Observable Scale Observable Scale Molecular Scale Molecular Scale HSPG may Lead to Anomalous Transport Anomalous: non-Fickian

11. Possible Morphogen-HSPG interactions • Methods of Analysis • Analysis of Gradients Formed by Anomalous Diffusion • Model Implications for Robustness • Theoretical Predictions and Methods of Experimental Verification

12. Computer Simulations

13. Anomalous Transport can be Studied by Utilizing the Method of Continuous Time Random Walks • Statistical approach – random walks. • All properties can be inferred from the transition time distribution y(Dt). • CTRW was extended in this work to include degradation. R(r,t)

14. Possible Morphogen-HSPG interactions • Methods of Analysis • Analysis of Concentration Profiles Formed by Anomalous Diffusion • Model Implications for Robustness • Theoretical Predictions and Methods of Experimental Verification

15. Phenomenological parameter. Determined by the morphogen-HSPG interactions. As b approaches 1 the diffusion becomes closer to Fickian. b Analogous to the Fickian diffusion coeffcient. Determines the internal length scale. Db kb Related to the degradation probability / analogous to degradation rate. Important Quantities

16. Steady state profile: Pre-steady state: Gaussian to exponential profile Profiles Resulting from the Reaction-Diffusion Equation b>1 : Converges to the reaction diffusion equation Profile at Different Times Concentration Source Position

17. No Steady State for Anomalous Steady State for Fickian kbtb<< 1 kbtb>> 1 No Effect of Degradation Anomalous Diffusion Show Two Distinct Time Regimes Total Number of Particles Time

18. The Profile at the Early Time Regime is a Time-Dependent Exponential kbtb<< 1 (equivalent to pre-steady state) Concentration Source Position

19. Exponential profile Exponential profile (at steady state) Degradation is not required Degradation is important (at steady state) HSPG properties (b) determine the profile Effect of HSPG is unclear Time dependent Time independent (at steady state) Novel Mode for Morphogen Gradient Formation Anomalous Diffusion Model Reaction-Diffusion Equation

20. Possible Morphogen-HSPG interactions • Methods of Analysis • Analysis of Gradients Formed by Anomalous Diffusion • Model Implications for Robustness • Theoretical Predictions and Methods of Experimental Verification Skip

21. Cth Wild Type Perturbed Robustness • Fluctuations due to: • Genotype: heterozygous mutant, overexpression • Developmental noise • Environmental effects: changes in temperature, nutrition... • Robustness of what and to what • Robustness is a comparative term

22. Shift in Position Due to Doubling Degradation Rate Shift in Position Original Position Robustness to Degradation Comparison to Reaction-Diffusion at Steady State Concentration Profiles Due to Doubling Degradation Rate Concentration (Normalized to Max) Position

23. Shift in Position Due to Doubling Development Time Shift in Position Original Position Robustness to Development Time Comparison to Reaction-Diffusion at Pre-Steady State Concentration Profiles Due to Doubling Development Time Concentration (Normalized to Max) Position

24. Robustness to Morphogen Production Rate Shift in position is constant when the concentration profile is exponential. Concentration (Normalized to Max) Shift in Position Source Position Original Position Opposing requirements between the need for robustness and need for high spread.

25. Morphogen Concentration HSPGproperties (b) Feedback – Combining the Two Length Scales Small b Steep slope Cth Concentration (log scale) Large b Moderate slope Source Position

26. Morphogen Concentration Concentration Profiles Due to Doubling Production Rate Shift in Position Due to Doubling Production Rate HSPGproperties (b) Concentration (Normalized to Max) Shift in Position Position Original Position Robustness to Morphogen Production Rate Comparison to Reaction-Diffusion atSteady State; Inclusion of Feedback

27. Possible Morphogen-HSPG interactions • Methods of Analysis • Analysis of Gradients Formed by Anomalous Diffusion • Model Implications for Robustness • Theoretical Predictions and Methods of Experimental Verification

28. Bleach Label Follow recovery Experimental Predictions – Use of Fluorescent Recovery After Bleaching Is morphogen transport Fickian or anomalous? Fluorescent recovery after photo-bleaching (FRAP) has been used to study diffusion in various biological systems (cellular level)

29. Experimental Predictions – Simulation Results Recovery of: • Fickian t1/2 • Anomalous t1-b/2(Power-law exponent larger than 0.5) Number of Particles (Normalized to First Point) • Previous predictions tb/2(Power-law exponent lower than 0.5) • Reason: particle “memory” Time Since Bleach

30. 25 mm 25 mm 25 mm Prebleach t = 0 t = 278 s Experimental Testing of the Model Dpp tagged with GFP. Inconclusive results.

31. Conclusions • Reasonable that morphogen diffusion is anomalous. Reaction-diffusion equation is no longer valid. • HSPG interactions (b) and time determine concentration profile (rather than degradation). • Robustness to degradation, less sensitive to time (than pre-steady state) and robust to morphogen production rate (with feedback on HSPG properties). • FRAP can be used to test model assumptions.

32. Naama Barkai Brian Berkowitz y f(k) a G(x,u) ò x e Harvey Scher Gennady Margolin Avigdor Eldar Efrat Assa-Kunik Eyal Schejter Benny Shilo’s group Offer Gerlitz Hebrew University of Jerusalem Ran Kafri Acknowledgments