7. Lecture SS 2005

Optimization, Energy Landscapes, Protein Folding 1

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

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Optimization, Energy Landscapes, Protein Folding 2

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

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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

ma /6 : viscositya: radius of the particlem: mass of the particle

F

Fvmmv dtvd

dtrd

,

: stochastic force

Statistical calculations:

DtrtrD

trtrr

ttFFtF

aTk

mTk

mTk

B

B

B

6))((,

))((,0

)()()0(,0)(

206

620

6

Einstein relation

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Optimization, Energy Landscapes, Protein Folding 4

Smoluchowski equation

Flux of particles in 1D:

in 3D: cDJ

DJcvcJdxdc

dtdx

, Fick’s 1st law

Fick’s 1st law + conservation of particles -> Diffusion equation in 1D:

in 3D: (Fick’s 2nd law) cDJ

D

tc

xc

xJ

tc

2

2

2

ccDcDJf

tcfc

2

… considering the friction force vf

Smoluchowski equation

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Optimization, Energy Landscapes, Protein Folding 5

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

]exp[ Uk

Kramers (1940): escape rate for strong (over-damped) friction (large )

a br

U(r)

]exp[)(")("2

1 UbUaUkesc

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Optimization, Energy Landscapes, Protein Folding 6

Protein-protein association

Protein-protein association is crucial in cellular processeslike signal transduction, immune response, etc.

Diffusive association of particles to a sphere

0/ tcsteady state:

02

2 )(12 dr

rcdr

cDiffusion equation:(without friction)

racrc 1)(after integrating:

particle flux: 2)(r

aDcdrdcDrJ

number of collisions per second at r = a: aDcaaJaI 44)()( 2

association rate: DacaIka 4/)( ~ 109 M-1s-1

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Optimization, Energy Landscapes, Protein Folding 7

Protein-protein association II

a more realistic scenario …

typical association rates ~ 103 - 109 M-1s-1

barnase / barstar

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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

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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

MD BD

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Brownian dynamics simulations

Diffusional motion of a particle

Translational / rotational diffusion coefficients D / DR

Translational displacement during each time step:

with and

Rotational displacement during each time step :

with and

Ermak-McCammon-Algorithm:

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SDA

Simulation of Diffusional Association of proteinsGabdoulline and Wade, (1998) Methods, 14, 329-341

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Example trajectory

barstar

barnase

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Example system: barnase / barstar

• barnase: a ribonuclease that acts extracellularlybarstar: its intracellular inhibitordiameters 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: 108 – 109 M-1s-1 at 50 mM ionic strength

• simulated rates are in good agreement with experimental results

barnase

barstar

- 7 kT/e + 7 kT/e

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Optimization, Energy Landscapes, Protein Folding 14

Computation of the occupancy landscape

d1-2position:

orientation: d1-2

30°

60°90°

n

30°

60°90°

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Results: Occupancy landscapebound complex:

d1-2 = 23.8 Å

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Choice of the distance axis

detailedview

globalview

• center-center distance d1-2:

• minimum distance between contact pairs cdmin:

• distance between geometric centers of contact surfaces cdcenter:

• average distance between contact pairs cdavg:

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Results: Occupancy landscape IIbound complex:

cdavg = 3.56 Å

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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:

9.06

Å4.06

tD

tD

rot

trans

In the simulations: = 3 Å, = 3°

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Entropy from occupancy maps II

Entropy of a system with N states:

:ln1

n

N

nnnB PPPkS

probability for each state

if all states are equally probable, Pn = 1/N:

NkS B ln

Entropy in protein-protein encounter:

rottrans SSS

Basic entropy formula applied for all states within V and Y:

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Free energy landscapebound complex:

d1-2 = 23.8 Å

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Results: Occupancy landscape IIbound complex:

d1-2 = 23.8 Å

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Energy profiles

encounter state

-TS

G

Eel

Eds

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Encounter complex

free energy: G = -4.053 kcal/mol

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Encounter complex II

free energy: G = -4.0 kcal/mol

volume of encounter region: Venc = 14.4 Å3

lifetime: t = 2.1 ps

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Encounter complex III

free energy: G = -3.5 kcal/mol

volume of encounter region: Venc = 1492 Å3

lifetime: t = 11.5 ps

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Encounter complex IV

free energy: G = -3.0 kcal/mol

volume of encounter region: Venc = 5338 Å3

lifetime: t = 20.1 ps

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Encounter complex V

free energy: G = -2.8 kcal/molvolume of encounter region: Venc = 8377 Å3

lifetime: t = 18.5 ps

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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

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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

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Association pathways

paths of highest occupancyvs.

paths of lowest free energy

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Coupling of translation and orientation

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Mutant effects

60

59

2783

87

Energy Profiles:

Eel

Eds

-TS

G

---- WT---- E60A---- K27A---- R59A---- R83Q---- R87A

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Mutant effects II

60

59

2783

87

Encounter Regions: G ¡Â Gmin + 0.5 kcal/mol

WTE60AK27AR59AR83QR87A

G

---- WT---- E60A---- K27A---- R59A---- R83Q---- R87A

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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

Dttrr 6)(,0 2