Membranes under transmebrane voltages

Werner TreptowWerner Treptowtreptow@unb.brwww.lbtc.unb.br

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