V7: Diffusional association of poteins and Bownian dynamics simulations Bownian motion The paticle movement was discoveed by Robet Bown in 1827 and was intepeted coectly fist by W. Ramsay in 1876. Exact poofs by Albet Einstein and M. von Smoluchowski in the yeas 1905/06. http://www.deutsches-museum.de/ausstell/daue/physik/e_bown.htm 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 1
Diffusion - Bownian dynamics t = 24 s t = 0 s Diffusion of 2 µm paticles in wate and DNA solution 0.5 µm Diffusion of 0.5 µm paticles in wate http://www.deas.havad.edu/pojects/weitzlab/eseach/micheo.html 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 2
Langevin equation Theoy of stochastic pocesses: -> colloidal suspensions (paticles in a liquid) moe collisions in the font than in the back => foce in opposite diection and popotional to velocity: d dt F dv = v, m = m v dt : stochastic foce Hydodynamics: is the fiction constant Statistical calculations: = 6a / Einstein elation D m F( t) = 0, = + F = 0, k BT 6a ( ( t), : viscosity a: adius of the paticle m: mass of the paticle 6k BT (0) F( t) = ( t) m 2 6k BT 0) = t m 2 ( ( t) 0) = 6Dt 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 3 F
Smoluchowski equation Flux of paticles in 1D: in 3D: J J = c dx dt = Dc dc dx = cv, J = D Fick s 1st law Fick s 1st law + consevation of paticles -> Diffusion equation in 1D: in 3D: (Fick s 2nd law) c t c t J c = = D 2 x x = J = D 2 2 c consideing the fiction foce c f J = Dc + f = v c t = D 2 c f c Smoluchowski equation 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 4
Kame s Theoy Tansition state theoy assumptions: - themodynamic equilibium in the entie system - tansition fom eactant state which cosses the tansition state will end in the poduct state k exp[ U ] Kames (1940): escape ate fo stong (ove-damped) fiction (lage ) U() k esc = 1 U"( a) U"( b) exp[ U ] 2 a b 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 5
Potein-potein association Potein-potein association is cucial in cellula pocesses like signal tansduction, immune esponse, etc. Diffusive association of paticles to a sphee steady state: Diffusion equation: (without fiction) c / t = 0 2 c = 1 2 d ( c) 2 d ( a ) afte integating: c( ) = c 1 Dc a paticle flux: ( ) D dc J = = 2 d numbe of collisions pe second at = a: association ate: k a = 0 I( a) = J ( a) 4 a = 4 Dc = I( a) / c = 4 Da ~ 10 9 M -1 s -1 2 a 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 6
a moe ealistic scenaio Potein-potein association II typical association ates ~ 10 3-10 9 M -1 s -1 banase / basta 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 7
Foces between the poteins Long ange inteactions: electostatic foces desolvation foces hydodynamic inteactions Shot ange inteactions: van de Waals foces hydophobic inteactions fomation of atomic contacts stuctue of wate molecules Entopic effects: (estiction of the degees of feedom) tanslational entopy otational entopy side chain entopy 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 8
The association pathway Steps involved in potein-potein association: andom diffusion electostatic steeing fomation of encounte complex dissociation o fomation of final complex MD BD Association pathway depends on: foces between the poteins solvent popeties like tempeatue, ionic stength 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 9
Bownian dynamics simulations Emak-McCammon-Algoithm: Diffusional motion of a paticle Tanslational / otational diffusion coefficients D / D R Tanslational displacement duing each time step: with and Rotational displacement duing each time step : with and 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 10
SDA Simulation of Diffusional Association of poteins Gabdoulline and Wade, (1998) Methods, 14, 329-341 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 11
Example tajectoy basta banase 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 12
Example system: banase / basta banase basta banase: a ibonuclease that acts extacellulaly basta: its intacellula inhibito diametes of both ~ 30 Å povides well-chaacteized model system of electostatically steeed diffusional encounte between poteins inteaction between banase and basta is among the stongest known inteactions between poteins vey fast association ate: 10 8 10 9 M -1 s -1 at 50 mm ionic stength simulated ates ae in good ageement with expeimental esults - 7 kt/e + 7 kt/e 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 13
Computation of the occupancy landscape position: d 1-2 oientation: d 1-2 n 30 60 90 30 60 90 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 14
Results: Occupancy landscape bound complex: d 1-2 = 23.8 Å 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 15
Choice of the distance axis cente-cente distance d 1-2 : global view minimum distance between contact pais cd min : distance between geometic centes of contact sufaces cd cente : aveage distance between contact pais cd avg : detailed view 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 16
Results: Occupancy landscape II bound complex: cd avg = 3.56 Å 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 17
Entopy fom occupancy maps Occupancy maps can be intepeted as pobability distibutions fo the computation of an entopy landscape Poteins can only exploe the suounding egion entopy fo each gid point is calculated fom the pobability distibution within accessible volumes V and Y Take V as sphee with adius, Y as sphee with adius aound potein position and oientation Aveage displacement within BD time step of t ~ 1 ps: 6D 6D tans t t In the simulations: = 3 Å, = 3 ot 0.4Å 0.9 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 18
Entopy fom occupancy maps II Entopy of a system with N states: S N = k P ln P P : pobability fo each state B n= 1 n if all states ae equally pobable, P n = 1/N: S = k ln B N n Entopy in potein-potein encounte: S = S tans + S ot Basic entopy fomula applied fo all states within V and Y: n 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 19
Fee enegy landscape bound complex: d 1-2 = 23.8 Å 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 20
Results: Occupancy landscape II bound complex: d 1-2 = 23.8 Å 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 21
Enegy pofiles E el -TS E ds G encounte state 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 22
Encounte complex fee enegy: G = -4.053 kcal/mol 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 23
Encounte complex II fee enegy: G = -4.0 kcal/mol volume of encounte egion: V enc = 14.4 Å 3 lifetime: t = 2.1 ps 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 24
Encounte complex III fee enegy: G = -3.5 kcal/mol volume of encounte egion: V enc = 1492 Å 3 lifetime: t = 11.5 ps 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 25
Encounte complex IV fee enegy: G = -3.0 kcal/mol volume of encounte egion: V enc = 5338 Å 3 lifetime: t = 20.1 ps 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 26
Encounte complex V fee enegy: G = -2.8 kcal/mol volume of encounte egion: V enc = 8377 Å 3 lifetime: t = 18.5 ps 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 27
Encounte egions: compaison egions fo enegetically favouable egions fo each potein fom BD simulations: G K -3 kcal/mol fom a Boltzmann facto analysis Gabdoulline & Wade: JMB (2001) 306:1139 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 28
Encounte egions: compaison II egions fo enegetically favouable egions fo each potein fom BD simulations: G K -2.5 kcal/mol fom a Boltzmann facto analysis Gabdoulline & Wade: JMB (2001) 306:1139 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 29
Association pathways paths of highest occupancy vs. paths of lowest fee enegy 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 30
Coupling of tanslation and oientation 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 31
Mutant effects Enegy Pofiles: 87 83 27 E el -TS ---- WT ---- E60A ---- K27A ---- R59A ---- R83Q ---- R87A 60 59 E ds G 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 32
Mutant effects II Encounte Regions: G Â Gmin + 0.5 kcal/mol WT E60A K27A R59A R83Q R87A 83 27 87 59 60 G ------------------7. Lectue SS 2005 WT E60A K27A R59A R83Q R87A Optimization, Enegy Landscapes, Potein Folding 33
Summay Bownian motion: Paticles in solutions move accoding to a andom foce 2 aveage displacement: = 0, ( t) = 6Dt Association: association of spheical paticles -> diffusion limit, potein association is steeed along the fee enegy funnel Inteactions: long-ange association can be modeled by BD simulation, shot-ange association by MD simulation BD simulations allow the calculation of association ates analysis of association paths identification of the encounte complex 7. Lectue SS 2005 Optimization, Enegy Landscapes, Potein Folding 34