Membranes under transmebrane voltages
Werner TreptowWerner [email protected]
Laboratório de Biologia Teórica e ComputacionalUniversidade de Brasília UnB
Molecular Modeling of Ion ChannelsMolecular Modeling of Ion Channels
VII Escola de Modelagem Molecular em Sistemas BiológicosVII Escola de Modelagem Molecular em Sistemas Biológicos
Membranes under transmebrane voltages
Biological Membranes
Ionic Currents and Channels
Ion flow
(#) High free energy barrier
Ion Channel
Eletrochemical Gradient
dW = V dqV= voltageq = charge
dW = µA dnA
µA = µi = [∂/∂ni(G)]T,P,nj≠i
nA mol number of A in the system
sy
ste
m:
q e
nA
KCl
Energy source for cell transport and signaling!
Ion channels
Energetics
Chemical, Electrical and Mechanical Stimuli Modulate Ion Channels and Conduction
Reaction Coordinate
C ← → O
Mo
du
lati on
Membranes under transmebrane voltages
Ion transport triggered by a TM potential Voltage gradient from explicit ion dynamicsChannels versus Microscopic ProcessesX
-Ray
Cry
stal
S
tru
ctu
res KcsA, a pH-dependent K+ channel MthK, Ca+2-gated K+ channel
Kv1.2, a voltage-gated K+ channel
Structural Models
Homology Modeling and ab initio modeling
Palovcak et al., Sequence-structure relationships in the TRP and Kv channel superfamilies (2014)
MI: Mutual Information (Gobel et al., Proteins 1994)
MI ij=∑A ,B
f ij( A ,B )lnf ij (A , B)
fi (A ) fj(B)
MSA
i j … L
P (A1 , ... , A L)=1Z
exp[−∑i < j
eij( Ai , A j )+∑i
h i( Ai) ] Potts Model: Max. Shannon Entropy
DCA: Direct Coupling Analysis (Morcos et al., PNAS 2011)
Membranes under transmebrane voltages
Ion transport triggered by a TM potential Voltage gradient from explicit ion dynamicsWatching water, ions, lipids and channel move
Kv channels and excitable membranesAtomistic MD Simulations
Channels versus Microscopic Processes
Treptow & Tarek (Treptow & Tarek (Biophys. J. 2006aBiophys. J. 2006a)) FORCE FIELD: water TIP3 model CHARMM36-CMAP POPC lipids
SYSTEM SIZE ~ 105 atoms
DYNAMICS: NPT PBC time step = 2.0 fs (multiple) langevin dynamics (300 K) langevin piston (1 atm) PME
SY
ST
EM
OF
N
PA
RT
ICL
ES
i
NN
iUf r
rr )()(FORCES ON
PARTICLE i
)( Niii fm rr SOLVE THE
EQUATIONS OF MOTION
)()...,,( 21N
N UU rrrr
Standard Simulations
Equilibrium properties over short timescales ~ 100 ns – 10 us: Sampling a single state “A”!
Membranes under transmebrane voltages
Ion transport triggered by a TM potential Voltage gradient from explicit ion dynamics
MD Simulations
Advanced Simulations: Sampling State “B” Channels versus Microscopic ProcessesS
imu
lati
on
s u
nd
er a
pp
lied
vo
lta
ges
F z=qE z Additive force in the equations of motion
E z=VLz
Ez drops across the entire simulation box Lz
i) Homogeneous Electrical Field Ez across the membrane
ii) Explicit electrolyte imposing V across the membrane:
ΔV=QC
Delemotte, Dehez, Treptow & Tarek JPCB 2008
Ess
enti
al D
ynam
ics
C=⟨ p (t) p (t )T ⟩Covariance:
p (t)= x( t)−⟨ x ⟩Atomic motion:
q (t)=T T p (t)
C=T DTTEigenvectors and eigenvalues of C:
in which
each eigenvector contributes with
i to the total atomic motion !
⟨ q(t )T q( t)⟩=∑i
λ i
Treptow et al. (Biophys. J. 2004)
Sampling – up to ns!
M ' q(t )=M x( t)
Motion in the Essential Space:
In which:
M '=TT MTis the transformed mass tensor!
Effective Sampling up to µs!
Microscopic manipulation of the channel:
Moving Harmonic Constraints
Dellemotte, Tarek, Klein, Amaral, Treptow PNAS 2011Amaral, Carnevale, Klein, Treptow PNAS 2012Jensen et al., Science 2012
Simulation time ~ 2.25 μs JADE (SGI / 910 cores)
ξ
F
Membranes under transmebrane voltages
Voltage gradient from explicit ion dynamics
Ion channels and conduction
Conduction mechanism
1 2 3 4
5 6 7 8
Knock-on process: ions and water move in concert in the SF adopting a single-file configuration
Up-Down Cycle
V
ξ
Steered MD and Irreversible work
thxt,x 0
2
2tt,x
kth
vdtttxktW ,)(
vtt 0
Hamiltonian:External Potential:Potential position:External work:
-ξ:
ξ forward
backward
Treptow & Tarek Biophys. J. (2006b) - Free energy of the conduction cycle
PMF at 0 mV
Up-Down throughput cycle triggered by knock-on
(Åqvist & Coworkers; Roux & coworkers; Sansom & coworkers; Carloni &
coworkers; Schulten & coworkers; Treptow & Tarek, 2006)
General mechanism
in which, mean <W> and the variance s over M values
Fourth order formula for Jarzynki's identity
Jarzynski’s identity relating the free-energy variation (F) to the irreversible work (W)
Channels versus Microscopic Processes
Membranes under transmebrane voltages
Computing Binding Affinities
Ion transport triggered by a TM potential Voltage gradient from explicit ion dynamics
Ion channels and conduction
Ligand BindingChannels versus Microscopic Processes
ΔG=∑i=1
N −1
ΔGi , i+1=−1β∑i=1
N−1
ln ⟨exp−βΔU i , i+1 ⟩i
Free energy Perturbation
ΔG=1β
lnV
V oLoss of translational entropy
2) Free Energy Calculations: Free-Energy Perturbation
ΔGbinding=ΔGgas−R−ΔGsolv−ΔGrestraint
L+R←→ L⋯R
nothing+R←→ nothing⋯R
ΔGsolv↑↓ ↑↓ΔGgas− R
ΔG=0
ΔGbinding
ΔGrestraint
Bin
din
g A
ffin
itie
s
3) Free Energy Calculations: LIE
ΔG=α(⟨U LJLR ⟩−⟨U LJ
L ⟩)+β( ⟨U ElectLR ⟩−⟨U Elect
L ⟩ )
Kraszewski et al. ACS Nano (2010)
Simple Overlap Sampling method
ΔG=−1β∑i=1
N
lnxi( f)
x i(b )
ΔG=−1β ln (K∗C
o)
Binding Constant K
Barber et al. Biophys. J. (2011)
1) Ligand Force Field: CGenFF – CHARMM General Force Field
QM (Gaussian)Optimization and validation
of parameters MM (NAMD)
FF related to CHARMM.
Iterative Procedure: to minimize differences between QM and MM
QM optimization (g03 runs). Geometry MP2/6-31G(d). Partial charges: Merz Kollman (MK) method. Interaction energy: HF/6-31G(d). Bond, angle and torsion parameters: MP2/6-31G(d).
Chemical Free energy
● Reaction Coordinate RMSD + Enhanced Sampling Techniques (ABF, Metadynamics)
Huge CPU Cost!
Chemical, Electrical and Mechanical Stimuli Modulate Ion Channels: Challenges
F (X .V , pH , ligand ...) = F(X ,0 ,0. ..) + Δ F(X ,V , pH , ligand ...)
Reaction Coordinate
C ← → O
+
Chemical [F(X, 0, 0...)]
Excess [F(V, pH, ligand...)]
Excess Free Energy (Voltage)
Δ F (X ,V )=V Q(X)
Effective ChargeQ(X)=∑i
qiφX(r i)
Excess Free Energy
∇⋅[ϵ(r)∇ φ(r )]−κ2(r) [φ(r)−V Θ(r)]=0
Linearized Poisson Boltzmann: PB-V
Electrical distanceφ(r)≡ ∂∂V
Φ(r ,V ) |V=0
∇ 2Φ(r ,V )=−4π∑i
ρi(r ,V ) Poisson Equation
Membranes under transmebrane voltages
Acknowledgements
Ion channels and conduction
Graduate Students:
Letícia Stock (LBTC, UnB BR)Caio Souza (LBTC, UnB BR)Juliana Hosoume (LBTC, UnB BR)Camila Pontes (LBTC, UnB BR)
Researchers:
Dr Mounir Tarek (eDAM, UHP France)Dr Michael Klein (ICMS, Temple US)Dr Vincenzo Carnevale (ICMS Temple US)
Post-Docs:
Dr Lucie Delemotte (ICMS, Temple US)Dr Cristiano Amaral (LBTC, UnB BR)Dr. Alessandra Kiamentis (LBTC, UnB, BR)
UnderGraduate Students:
Matheus CostaLeonardo Cirqueira
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