V7: Diffusional association of proteins and Brownian dynamics simulations
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Transcript of V7: Diffusional association of proteins and Brownian dynamics simulations
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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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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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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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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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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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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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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