Bhavin Khatri and Tom McLeish Polymer & Complex Fluids Group School of Physics & Astronomy

1. Come on – Feel the NoiseorViscoelastic Force Spectra of Single Biomolecules:Mapping the Energy Landscape of Conformational Transitions Bhavin Khatri and Tom McLeish Polymer & Complex Fluids Group School of Physics & Astronomy

2. Protein Concatamers Polysaccharides Chair  Boat transition A physicist’s definition of a biopolymer? • Biopolymer monomers can adopt different conformations • Different conformations have different energies and sizes • e.g. Dextran: Chair Boat: extra length, but cost in energy • Fluctuations on energy landscape  Viscoelasticity of transitions??

3. FJC Something interesting Probing Biopolymers:Conventional Force Spectroscopy Features in force extension trace indicate conformational transitions

4. 200 150 100 Force [pN] 50 0 0 20 40 60 80 100 120 140 160 Extension [nm] [Data:M. Kawakami] Spectroscopy of Proteins • Relaxation time of monomers slower than experimental time=> pulling at different speeds can give global dynamical information; e.g. rate of unfolding of protein • Cannot easily access local dynamical information; e.g. rubbing of helices and sheets

5. Recall Polymer Rheology Non-linear experiments came before ….   .. linear ones

6. A Photo Detector LASER Diode Set Point Cantilever Polysaccharide chain PID Piezoelectric Stage Force Clamp Thermal Noise Spectroscopy

7. Free Cantilever 320pN 620pN -4 8x10 920pN 6 PSD [nm2/kHz] 4 2 5 10 15 20 25 30 35 40 Frequency [kHz] Power Spectral Density Example:real PSD of cellulose Data: M. Kawakami Need model for power spectra…

8. Like Rouse but mode friction: => Internal friction important for short chains: Rouse with Internal Friction: RIF Model D. McInnes (1977) Polymer,18,505 de Gennes, Scaling Concepts

9. Brownian response due to • Populations obey • Linear response solution Brownian Response of a 2-state monomer

10. Frictional Freely Jointed Chain • Joints with constant friction and at high stretch Torque Restoring Force From statistical mechanics

11. Detailed Balance is obeyed Dextran Cellulose Force [pN] Fits to Internal Friction Spectrum • Friction is underdetermined for dextran • Minima in elasticity & friction • Geometry of landscape • Hence:

12. Reconstruction of Energy Landscape

13. Displacement Force time Input: Brownian ‘kicks’ Output: Sum over responses time Friction dominates Power spectrum: frequency distribution of fluctuations Elasticity dominates Dissipation and Dynamics • Brownian Fluctuations give inherent viscoelastic information • Example: Overdamped Spring & Dashpot

14. Idea • Probe local dissipation and dynamics • Noise in AFM experiments usually detrimental • watch single molecule fluctuations under controlled force • Fingerprint of dynamics on energy landscape • Analogy: macroscopic rheology of complex fluids

15. Overview of rest of talk • Force Clamp Thermal Noise Spectroscopy • Coarse-grained biopolymer models • Molecular scale models • Comparison to experiment • Reconstruction of dextran energy landscape

16. Dextran Cellulose Force Force Extension Extension Spectroscopy of Polysaccharides • Polymers with ringed monomers (e.g. glucose) No hystereris observed => relaxation time of chain faster than experimental time Cannot probe dynamics of monomers; only eqm elastic info [Data:M. Kawakami]

17. Solvent Friction Diffusion Equation Rouse model • Simplest model Spring and dashpot • But biomolecules are actually polymers • Spectrum of relaxation times

18. AFM experiments End-End Vector Important Rouse Spring & Dashpot Frequency Response of End to End Vector • Fori >> R spring & dashpot (-1)   • ForR >> i Rouse behaviour (-1/2) • up to c ~1/iwhereinternal friction of high modes dominate

19. PSD of Cantilever and RIF polymer • Cantilever response • Combined parallel response • Fluctuation Dissipation Theorem

20. Folding Funnel (Onuchic, Wolynes,..) Or Hyperspace?

21. Extract from experiments: Fitting to RIF Model to Experimental PSD [Data:M. Kawakami]

22. Monomer Internal Friction [gkHz] Monomer Elasticity [pN/nm] Cellulose Cellulose Solid lines are gradient of Force-Extension experiment Dextran Dextran Force [pN] Force [pN] End-End Solvent Friction [gkHz] Expect Force [pN] Results:Viscoelastic Force Spectra [Data: M. Kawakami & K. Byrne] These biopolymers are ‘short’

23. Frequency response of monomer length ~ spring and dashpot Viscoelasticity on a 2-state landscape • Identify Elasticity and Friction in terms of microscopic parameters Depends only on eqm populations and obeys equipartition Depends only on hopping time Like an asymmetric 1D lattice diffusion process

24. Overview of rest of talk • Force Clamp Thermal Noise Spectroscopy • Coarse-grained biopolymer models • Molecular scale models • Comparison to experiment • Reconstruction of dextran energy landscape

25. Fluctuations on a 2-state landscape • Chain of 2-state monomers (1 dimensional) • Average length  Equilibrium populations: • Fluctuations of length  hopping:

26. Physical Interpretation: Elasticity spectrum • Hopping elasticity entropic in origin  applying force changes effective size of ‘box’ • Can measure zero force ΔG and Δx

27. Physical Interpretation:Internal Friction Spectra • Force controls barriers heights and thus friction • In principle can measure:

28. Monomer Internal Friction [gkHz] Monomer Elasticity [pN/nm] Force [pN] Force [pN] Experiments again.. • Microscopic model explains minima in elasticity and friction • Can explain elasticity spectrum qualitatively • Low force friction? • Bond friction? ? ?

29. Modelling Entire Force Regime • FJC • Conformational hopping • Bond stretching • All independently additive to total extension

30. Dextran Cellulose Force [pN] Fit to Elasticity Spectrum • Very good agreement between experiment & model • Agree with literature values • Minima in elasticity: entropically favourable to elongate

31. ? Dynamics controlled by barrier -> Internal Friction Force Spectrum Reconstruction of Landscape

32. Implications.. • Bond & joint friction 6-7 orders of magnitude larger than expected solvent friction • One explanation:Diffusion in rough potential (R.Zwanzig (1988), PNAS, 85, 2029) • Barrier curvature • Discrete Kramer’s prefactor Very sharp!

33. 200 150 100 Force [pN] 50 1 2 0 3 0 20 40 60 80 100 120 140 160 Extension [nm] Protein experiments A • Future: controlled viscoelastic force spectra of refolding proteins B [Data: M. Kawakami]

34. Summary • Brownian Noise can give detailed viscoelastic information of single molecules • RIF model: a generic coarse-grained model of for biopolymer with internal transitions • 2-state model provides insight to viscoelasticity of conformational transitions • Viscoelastic force spectra reveal the statics and dynamics on the conformational energy landscape of biomolecules • Reveals a dominance of internal friction in the nanoworld • Future experiments on proteins may probe dynamics of secondary and tertiary structure formation in protein folding • Thanks to Masaru Kawakami, Katherine Byrne and Alastair Smith for doing the AFM experiments! • Thanks to EPSRC for funding. References, Experiment: Langmuir, 401,400 (2004) ; Theory: submitted to Nature Physics

35. Modelling Entire Force Regime • Each process spring & dashpot in nature, and independently additive to total extension • Hence, at low frequency

36. Mechanical Oscillation Experiments Dextran Humphris, Tamayo & Miles Langmuir, Vol. 16, No. 21, 2000

37. 3 10 2 10 1 10 0 10 -1 10 -2 10 0 1 2 10 10 10 Relaxation in RIF Model RIF Rouse