1. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding1 V7: Diffusional association of proteins and Brownian dynamics simulations Brownian motion The particle movement was discovered by Robert Brown in 1827 and was interpreted correctly first by W. Ramsay in 1876. Exact proofs by Albert Einstein and M. von Smoluchowski in the years 1905/06. http://www.deutsches-museum.de/ausstell/dauer/physik/e_brown.htm

2. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding2 Diffusion - Brownian dynamics 0.5  m t = 0 s t = 24 s Diffusion of 2  m particles in water and DNA solution http://www.deas.harvard.edu/projects/weitzlab/research/micrheo.html Diffusion of 0.5  m particles in water

3. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding3 Langevin equation Theory of stochastic processes: -> colloidal suspensions (particles in a liquid) more collisions in the front than in the back => force in opposite direction and proportional to velocity: Hydrodynamics:  is the friction constant  : viscosity a : radius of the particle m : mass of the particle : stochastic force Statistical calculations: Einstein relation

4. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding4 Smoluchowski equation Flux of particles in 1D: in 3D: Fick’s 1st law Fick’s 1st law + conservation of particles -> Diffusion equation in 1D: in 3D: (Fick’s 2nd law) … considering the friction force Smoluchowski equation

5. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding5 Kramer’s Theory Transition state theory assumptions: - thermodynamic equilibrium in the entire system - transition from reactant state which crosses the transition state will end in the product state Kramers (1940): escape rate for strong (over-damped) friction (large  ) ab r U(r)

6. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding6 Protein-protein association Protein-protein association is crucial in cellular processes like signal transduction, immune response, etc. Diffusive association of particles to a sphere steady state: Diffusion equation: (without friction) after integrating: particle flux: number of collisions per second at r = a : association rate: ~ 10 9 M -1 s -1

7. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding7 Protein-protein association II a more realistic scenario … typical association rates ~ 10 3 - 10 9 M -1 s -1 barnase / barstar

8. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding8 Forces between the proteins Long range interactions: electrostatic forces desolvation forces hydrodynamic interactions Entropic effects: (restriction of the degrees of freedom) translational entropy rotational entropy side chain entropy Short range interactions: van der Waals forces hydrophobic interactions formation of atomic contacts structure of water molecules

9. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding9 The association pathway Steps involved in protein-protein association: random diffusion electrostatic steering formation of encounter complex dissociation or formation of final complex Association pathway depends on: forces between the proteins solvent properties like temperature, ionic strength MDBD

10. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding10 Brownian dynamics simulations Diffusional motion of a particle Translational / rotational diffusion coefficients D / D R Translational displacement during each time step: with and Rotational displacement during each time step : with and Ermak-McCammon-Algorithm:

11. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding11 SDA Simulation of Diffusional Association of proteins Gabdoulline and Wade, (1998) Methods, 14, 329-341

12. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding12 Example trajectory barstar barnase

13. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding13 Example system: barnase / barstar barnase: a ribonuclease that acts extracellularly barstar: its intracellular inhibitor diameters of both ~ 30 Å provides well-characterized model system of electrostatically steered diffusional encounter between proteins interaction between barnase and barstar is among the strongest known interactions between proteins very fast association rate: 10 8 – 10 9 M -1 s -1 at 50 mM ionic strength simulated rates are in good agreement with experimental results barnase barstar - 7 kT/e + 7 kT/e

14. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding14 Computation of the occupancy landscape d 1-2 position:  orientation: d 1-2 30° 60° 90° nn 30° 60° 90°

15. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding15 Results: Occupancy landscape bound complex: d 1-2 = 23.8 Å

16. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding16 Choice of the distance axis detailed view global view center-center distance d 1-2 : minimum distance between contact pairs cd min : distance between geometric centers of contact surfaces cd center : average distance between contact pairs cd avg :

17. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding17 Results: Occupancy landscape II bound complex: cd avg = 3.56 Å

18. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding18 Entropy from occupancy maps Occupancy maps can be interpreted as probability distributions for the computation of an entropy landscape Proteins can only explore the surrounding region entropy for each grid point is calculated from the probability distribution within accessible volumes V and Y Take V as sphere with radius , Y as sphere with radius  around protein position and orientation Average displacement within BD time step of  t ~ 1 ps: In the simulations:  = 3 Å,  = 3°

19. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding19 Entropy from occupancy maps II Entropy of a system with N states: probability for each state if all states are equally probable, P n = 1/N: Entropy in protein-protein encounter: Basic entropy formula applied for all states within V and Y: 

20. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding20 Free energy landscape bound complex: d 1-2 = 23.8 Å

21. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding21 Results: Occupancy landscape II bound complex: d 1-2 = 23.8 Å

22. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding22 Energy profiles encounter state -T  S GG  E el  E ds

23. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding23 Encounter complex free energy:  G = -4.053 kcal/mol

24. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding24 Encounter complex II free energy:  G = -4.0 kcal/mol volume of encounter region: V enc = 14.4 Å 3 lifetime:  t = 2.1 ps

25. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding25 Encounter complex III free energy:  G = -3.5 kcal/mol volume of encounter region: V enc = 1492 Å 3 lifetime:  t = 11.5 ps

26. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding26 Encounter complex IV free energy:  G = -3.0 kcal/mol volume of encounter region: V enc = 5338 Å 3 lifetime:  t = 20.1 ps

27. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding27 Encounter complex V free energy:  G = -2.8 kcal/mol volume of encounter region: V enc = 8377 Å 3 lifetime:  t = 18.5 ps

28. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding28 Encounter regions: comparison regions for energetically favourable regions for each protein from BD simulations:  G ≤ -3 kcal/mol from a Boltzmann factor analysis Gabdoulline & Wade: JMB (2001) 306:1139

29. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding29 Encounter regions: comparison II regions for energetically favourable regions for each protein from BD simulations:  G ≤ -2.5 kcal/mol Gabdoulline & Wade: JMB (2001) 306:1139 from a Boltzmann factor analysis

30. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding30 Association pathways paths of highest occupancy vs. paths of lowest free energy

31. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding31 Coupling of translation and orientation

32. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding32 Mutant effects 60 59 27 83 87 Energy Profiles:  E el  E ds -T  S GG ---- WT ---- E60A ---- K27A ---- R59A ---- R83Q ---- R87A

33. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding33 Mutant effects II 60 59 27 83 87 Encounter Regions:  G ¡Â  G min + 0.5 kcal/mol WT E60A K27A R59A R83Q R87A GG ---- WT ---- E60A ---- K27A ---- R59A ---- R83Q ---- R87A

34. 7. Lecture SS 2005Optimization, Energy Landscapes, Protein Folding34 Summary Brownian motion: Particles in solutions move according to a random force average displacement: Association: association of spherical particles -> ‘diffusion limit’, protein association is steered along the free energy funnel Interactions: long-range association can be modeled by BD simulation, short-range association by MD simulation BD simulations allow the calculation of association rates analysis of association paths identification of the encounter complex