Thermodynamics and kinetics in single-molecule pulling, nanotubes, and proton pumps

Gerhard Hummer

Laboratory of Chemical PhysicsNIDDKNational Institutes of HealthBethesda, MD 20892-0520Gerhard.Hummer@NIH.GOVDepartment of Health and Human Services

∆x(t)=F(t)/κ

Premise and Goals

• Thermodynamics and kinetics are the key to understanding molecular processes– Peptide dynamics and protein folding

– Mechanical manipulation of single molecules

– Transport of water through membranes

– “Maxwell demons” in biology: proton pumps in mitochondria

Outline

• “Feynman-Kac” theorem: S(t)=<exp(-∫0

tk(t’)dt’)>– Kinetics of end-to-end contact formation in peptides

– Kinetics from single-molecule pulling: protein unfolding under mechanical force

– Equilibrium thermodynamics from non-equilibrium pulling via an extension of Jarzynski’s identity

• Histogram methods: P(x)=<δ(x(t)-x)>– Water filling of nanotubes

– Kinetics of water and proton transport through nanotubes and membranes

– Biological proton pumps

• “Derivation”

• Applications– Peptide loop closure

– Single-molecule pulling• Kinetics

• Thermodynamics

PART I: “Feynman-Kac”

Kinetics from Equilibrium Trajectories

ettpt dttk

ii∫−= 0 ')'()()( δ

Q1

0k0(t)

k1(t)

time t

state• Simulation using first-

passage times– During time dt, either stay,

switch state, or quench

• Equilibrium simulation– Switch between states and

weight trajectory with survival probability:

time t

ln Si(t)= -∫0

tk(t’)dt’

1

0.1

et dttk

tQtS ∫−=−= 0 ')'()(1)(

Feynman-Kac Theorem

• Dynamics without quenching (with rate matrix L)

∂p(x,t)/∂t = Lp

• Dynamics with quenching (with k(x,t) state and time dependent quenching rate )

∂p(x,t)/∂t = Lp – k(x,t) p(x,t)

• Feynman-Kac theorem:

• Survival probability:

1 2 3 4 5 6 7 8 9 10 11 12

k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12

Q

et dtttxk

tQtS ∫−=−= 0 ']'),'([)(1)(

et dtttxk

xtxtxp ∫−−= 0 ']'),'([))((),( δ

Application 1: Amino-Acid Contact Formation in Peptides

(Yeh & Hummer, J. Am. Chem. Soc. 124, 6563, 2002)

• Contact formation defines a ‘speed limit’ of protein folding

• ~10 ns-1 µs (Hagen et al., PNAS 1996; Bieri et al., PNAS 1999;Lapidus et al., PNAS 2000; Hudgins et al., JACS 2002)

Lapidus, Eaton and Hofrichter, PNAS 97, 7220, 2000Cys-(Ala-Gly-Gln)n-Trp (n=1,…,6): ~1/(100 ns)

Cys Cys

Trp Trp*quenching

Approach

• Define distance-dependent tryptophanquenching rate κ(r) [e.g., ~exp(-r/r0)]

• Use molecular dynamics to generate an ensemble of equilibrium trajectories with end-to-end distance r(t)

• Analyze trajectories with “Feynman-Kac” to estimate kinetics from fraction of “surviving” Trp triplets

et dttr

tS ∫−= 0 ')]'([)(

κ

r(t)

Molecular Dynamics Simulations

• All-atom MD with solvent

• Cys-(Ala-Gly-Gln)1,2-Trp

• 526 & 1064 TIP3P water

• Particle-Mesh Ewald

• AMBER and CHARMM

• ~150 runs of ~20 ns with total 2.8 µs

• Initial configurations from high-T run, equilibrated for ~100 ps (or 10 ns)

Molecular Dynamics with CHARMM

Diffusion-Limited Contact Formation

Time t (ns)

• Quenching rate q→∞ for r<4Å

• Contact formation rate from integration of S(t)

• Corrected for TIP3P viscosity

908.2±1.16.5±0.9C(AGQ)2W

734.6±0.34.0±0.3CAGQW

expCHARMMAMBER1/kc [ns]

Kinetics for Finite Trp Quenching Rate

Model 1:

κ(r)=0 for r>4Åκ(r)=0.8/ns for r<4Å(Lapidus et al, PNAS

97, 7220, 2000)

Model 2:

κ(r)∝e-r/r0

(Lapidus et al, PRL 87, 258101, 2001)

Triplet Decay Rate

90143±26103±29C(AGQ)2W

73127±671±6CAGQW

expCHARMMAMBER1/k [ns]

Time t (ns)

(scaled by TIP3P viscosity)

• Excellent agrement between two force fields and experiments

• BUT: structure!!!

Peptide Contact Formation -Conclusions

• Multiple independent simulations (~3 µs) efficiently sample conformation space

• Contacts between ends of short peptides form rapidly (~10 ns)

• Simulations in good agreement with experimental Trp quenching rates (~100 ns)

• Short lifetime of Trp/Cys contacts → reaction-dominated Trp quenching

• Unfolded peptides, unlike folded proteins, sensitive to force-field details (kT effects):

– Two force fields give same rate and equilibrium constant for contacts, yet very different structures

Application 2: Kinetics from Single-Molecule Pulling

Hummer & Szabo, Biophys. J. 85, 5,2003

Kinetics

• Pulling on a molecule or molecular assembly drives the system out of equilibrium– External force accelerates

rare “mechanical” transitions (such as protein unfolding or complex dissociation)

– Use statistics of rupture force as a function of pulling speed to extract intrinsic rate constant

Pulling on tenascin(Oberhauser et al., Nature393, 181, 1998)

Simulated Pulling Experiment

• Cantilever spring constant k = 20.6 pN/nm

• Pulling velocity v = 1 µm/s

• Brownian dynamics with D = 10-7 cm2/s

• Equilibrium ‘unfolding’ rate ~10-6 s-1

free energy G(z) [molecule]G(z) + k(z–vt)2/2 [+spring]z = vt

Approach

• Theory– Simplify “position-dependent” rupture kinetics with

adiabatic approximation

– Describe kinetics of rupture using Kramers theory

vt

x

G(x)• Model

– Molecular coordinate x moves on harmonic free energy surface

– Irreversible rupture (unfolding, dissociation etc.) occurs when xreaches extension x*

– Harmonic pulling spring

x*

Kinetics from Phenomenological Description

• Assumption: time-dependent generalization of Bell’s formula (Bell, Science 200, 618, 1978)

k0 ….. intrinsic rate constantF(t) …external force with explicit time dependencex*……location of transition state

• Widely used in ‘Monte Carlo’ simulations of the kinetics of rupture– e.g., F(t) given by the force extension-curve of a

worm-like chain

ektk xtF *)(0)( β=

Comparison of PhenomenologicalDescription to Experiment & Simulation

• x* = 4.2 Å; k0 = 10-4 s-1;

G(x*) = 11.5 kcal/mol; κ/β = 41pN/nm

• Brownian dynamics

• Fit: k=140 k0; x* = 1.8 Å

κvt

x

x*βG(x)=κmx2/2

Titin I27 data from Carrion-Vazquez et al,Proc Natl Acad Sci USA 96, 3694, 1999

• simple microscopic model with realistic parameters

• Before rupture, pulling trajectories fluctuate about mean x(t)

• Rupture from quasi-equilibrium around x(t):

• Fraction of intact molecules

• Kramers rate

Microscopic Theory using Adiabatic Approximation

)()()( tStktS −=•

e xxDxk2/

2/3 2

2)(

κ

πκ −≈

[ ] [ ]eetSt

Kramers

tdttxxkdttxk ∫≈∫= −−−

000 ')'(*')'()(

x* x(t) x(t)

Distribution of Rupture Forces

• Fraction of intact molecules

• Cumulative distribution of rupture forces

Microscopic Theory of Rupture Reproduces Experiment and Simulation

• x* = 4.2 Å

• k0 = 10-4 s-1

• G(x*) = 11.5 kcal/mol Titin I27 data from Carrion-Vazquez et al,Proc Natl Acad Sci USA 96, 3694, 1999

Phenomenological Model as a Limiting Case of Microscopic Theory

Pulling on a Multimodule Construct with Anharmonic Linkers

Titin I27 data from Carrion-Vazquez et al,Proc Natl Acad SciUSA 96, 3694, 1999

Simulation of microscopic model

Multimodule Structure and AnharmonicLinkers Require only Minimal Modifications

• Effective spring constant κs– Describes combined response of AFM

cantilever, molecular linkers, and already ruptured units

– Slope of n-th event in force extension curve: βF(x) ≈ κs x

• Combinatorial factor– Different modules assumed to unfold

independently– Apparent rate of rupture is nk0, with n the

number of remaining intact modules

n

Comparison of Theory and Simulationfor Anharmonic Multimodule Construct

• 10 subunits that convert to worm-like chain polymers upon rupture

• Worm-like chain linker between protein and cantilever

• Spring constants fitted to slopes of force-extension curves and averaged over multiple rupture events

Extracting Unfolding Rates from Pulling on I278 Titin Constructs

Figure 3 from Carrion-Vazquez et al,Proc Natl Acad Sci USA 96, 3694, 1999): mean force and variance averaged over all rupture events

Results for I27 Titin

• Broad parameter range consistent with unfolding forces (mean and variance)– 10-5 < k0 < 10-2 s-1 (including correction for the number of

subunits)– 12 < ∆G*u < 16 kcal/mol (free energy barrier to unfolding)– 3 < x* < 5 Å (location of transition state)

• Rate roughly consistent with chemical denaturation: k0~5x10-4 s-1

• Free energy barrier to folding, ∆G*f = ∆G*

u − ∆G ≈ 4.5 to 8.5 kcal/mol similar to that estimated for CspTmfrom single-molecule fluorescence spectroscopy (2.4 to

6.6 kcal/mol; Schuler, Lipman and Eaton, Nature 419, 743, 2002)

• Unfolding from intermediate? (Marszalek et al., Nature 402, 100, 1999; Fowler et al., J. Mol. Biol. 322, 841, 2002)

Application 3: Free Energy from Single-Molecule Pulling

(Hummer & Szabo, Proc. Natl. Acad. Sci. USA 98, 3658, 2001)

• Goal: obtain free energy of binding, unfolding, etc., from single-molecule pulling

• Approach: Use Feynman-Kac to derive and extend Jarzynski’s identity for time-dependent perturbations (Jarzynski, Phys.Rev.Lett.78, 2690, 1997)

• Jarzynski’s identity produces free energies from repeated time-dependent perturbations

<exp(-∆W/kT)> = exp(-∆G/kT)

with work ∆W being the integrated change in energy

∆W = ∫(∂H/ ∂t)dt

Pulling as a Time-Dependent Perturbation

• Hamiltonian

H(p,q,t) = H0(p,q) + k(z–vt)2/2k ……. spring constant z(q) … extension

v ….… pulling speed

t ….… time

• Free energy G(z) along pulling coordinate?

δzt

z=vt

Reconstructing Equilibrium Phase-Space Density from Non-Equilibrium Pulling

• Phase-space density f(x,t) evolves under Liouville operator L with Boltzmann distribution stationary∂f/∂t = Lf with Le-βH(x)/kT = 0

(for diffusion: L = ∇ De-βV∇ e+βV)

• p(x,t)=e-βH(x,t)/∫e-βH(x’,0)dx’ is solution to a ‘sink-equation’∂p/∂t = Lp–kp with k(x,t) = β∂H(x,t)/∂t and p(x,0) = e-βH(x,0)

• Feynman-Kac theorem:

• Integration over x gives Jarzynski’s identity<exp(-∆W/kT)> = exp(-∆G/kT)

∫ −

−∫− =−=

'))((),(

)0,'(

),(']'),'([0

dxe

extxtxpxH

txHdtttxk

et

β

β

δ

Free Energy from Pulling

• Free energy profile G(z) along pulling coordinate can be determined from histograms of extensions at fixed time texp[-βG(z)] = <δ(z-z(t))e-βw(t)> with

• Work w(t) from integration over forces has a correction for the biased choice of the initial state

• Data weighted by a Boltzmann factor of the workinstead of the energy!

zk

dztzFdtt

Htw

C

t

δ 20

0 2),()( ∫∫ −=

∂∂

=

Free Energy Reconstruction from 10 Pulling Simulations

• Robust at low pulling speeds with variance of work ≈ (kBT)2

• Strong bias in <δ(x-xt)e-∫kdt > if variance >> kBT

reconstructed G(z)

Unfolding of the P5abc RNA Domain of the tetrahymena thermophila Group 1 Intron

(Liphardt et al.., Science 296, 1832, 2002)

• Pulling on P5abc RNA at different rates (2-50 pN/s)

• Approximate integration of work (neglecting initial bias) up to given extension

• Free energies at fast pulling rates consistent with reversible pulling at slowest rates

Figure 2 of Liphardt et al.., Science 296, 1832, 2002: force-extension curves at different switching rates

Single-Molecule Pulling - Conclusions

• Rigorous theory for extracting thermodynamic information from nonequilibrium pulling experiments

• Phenomenological description (k0eβx*F(t)) used widely in ‘Monte Carlo’ simulations of the kinetics is inadequate

• Microscopic theory describes rupturing accurately over whole velocity range

• Unfolding rate and free energy barriers for titin consistent with estimates from other experiments

• Accurate fit requires rupture force distributions for broad range of pulling velocities

• Test-particle histogram methods to calculate free energies

• Thermodynamics and kinetics of filling nanotubes with water

• Mechanisms of biological proton pumps

PART II: Water in Nanotubes and Biological Proton Pumps

Test-Particle Histogram Method(Bennett, J. Comput. Phys. 22, 245, 1976)

• Define a volume V from which particles are virtually removed and added (typically a simulation cell)

• At regular intervals, identify particles inside V, and calculate the difference ∆uin potential energy with and without a given particle

• Randomly insert a virtual new particle inside V and calculate the difference ∆u in the potential energy with that particle and without

• Probability histograms of insertion and removal energies give free energy: exp(β∆u-β∆µ)=pins(∆u)/prem(∆u)

inse

rtio

n removal

∆u∆u

Test-Particle Insertion(Widom, J. Chem. Phys. 39, 2808, 1963; Bennett, J. Comput. Phys. 22, 245, 1976)

µex = Aex(N+1,V,T) - Aex(N,V,T) = -kT ln [ZN+1 /VZN]= - kT ln [∫exp(-UN+1/kT)dr1…drN+1 / V∫exp(-UN /kT) dr1…drN]= - kT ln <exp(-∆u /kT)>N [particle insertion]= kT ln <exp(+∆u /kT)>N+1 [particle removal]

Insertion: pins(∆u) = ∫δ(UN+1- UN- ∆u)exp(-UN /kT) dr1…drN+1

x [∫exp(-UN /kT) dr1…drN+1]-1

Removal: prem(∆u) = ∫δ(UN+1- UN- ∆u)exp(-UN+1 /kT) dr1…drN+1

x [∫exp(-UN+1 /kT) dr1…drN+1]-1

pins(∆u)/prem(∆u) = exp(∆u/kT) <exp(-∆u /kT)>N

= exp(∆u/kT -µex/kT)

from δ functions from normalizations

Test of Equilibrium

pins(∆u)/prem(∆u) = exp(∆u/kT -∆µ/kT)

• Thermal equilibrium– Plot of ln pins(∆u)/prem(∆u) versus ∆u has slope 1/kT

• Chemical equilibrium– Average number of particles inside volume V is given by

<N> = ρV exp(-∆µ/kT) [follows after a more careful grand-canonical treatment]

Biological “Fuel Cell”: The “Proton Pump”Cytochrome c Oxidase

H+

ADPATP

+ + + + +

- - - - -

O2 + 8H+ + 4e- → 2H2O + 4H+(pumped)

Active-site cavity

H+H+

e-

O2

H2O

Water Pores: Aquaporin 1

Approach: Study Water in Confinement

• Does water fill narrow hydrophobic channels? – Thermodynamics

– Kinetics

• What is the functional role of hydrophobic channels?– Water conduction

– Proton transfer

– Proton pumping

Carbon Nanotube as Simplest Molecular Channel

• Fullerene-type cylindrical molecules

• sp2 carbons in ‘honeycomb’ lattice

• Open or closed ends

• Single or multi-wall structure

• Diameters of ~1 nm and larger

• Chemically functionalizable

Carbon Nanotube in Water(Hummer, Rasaiah & Noworyta, Nature 414, 188, 2001; Waghe, Rasaiah

& Hummer, J. Chem. Phys. 117, 10789, 2002)

• Classical molecular dynamics simulation– Flexible (6,6)-nanotube

(8Å diameter, 13.5Å length)

– Graphite parameters

– ~1000 TIP3P water molecules

– AMBER 6.0

– Particle-mesh Ewald

– 66 ns

Water Occupancy

• Nanotube fills within picoseconds and remains filled for 66 ns

Test for Thermodynamic Equilibrium

• Water inside nanotube should be in thermalequilibrium with outside– Boltzmann distribution of energies

• Water inside nanotube should be in chemicalequilibrium with outside– Occupancy given by difference in excess chemical

potential relative to bulk phase

Thermal and Chemical Equilibrium between Water in Nanotube and Bulk Phase

• Excess chemical potentials from histogram analysis− µex

w = -6.05 ± 0.02 kcal/mol (bulk TIP3P water)

− µexnt = -6.87 ± 0.07 kcal/mol (nanotube)

− -kT ln(<N>/ρ∆V) = -0.87 kcal/mol ~ µnt - µw

chemicalequilibrium}

slope 1/kT:thermal

equilibrium

intercept:chemicalpotential

Thermodynamics from Binding Energies: High-Energy Tail Determines Vapor Pressure

=ββ(µµ-u)

Weakly boundstates determine

chemical potential

duupee remuex

)(∫= βµβ• Channel shields from fluctuations

• High-energy states weakly populated

water binding energyu (kcal/mol)

~15% of neighboring watermolecules have unfavorablepair interaction energies

Effects of Interaction Potentials and Solvent Conditions: Modified Carbon-Water Attractions

• Modified carbon parameters§ εCO = 0.065 (0.114) kcal/mol

§ σCO = 3.41 (3.28) A

kBT

original

new

Emptying Transition

Filling/Emptying Transitions

Average density in tube~ bulk density

Change in parameters results in fluctuations between filled and empty states

Bimodal distribution

µex

µexbulk

µexnt

Average occupancyN~exp(-β∆µex)corresponds to ‘unstable’ fragmented chain→ fluctuationsbetween filled and empty states

redu

ced

vdW

attr

actio

n

Water Transport

• >1000 water molecules transported through nanotube

Single-File Transport as Continuous-Time Random Walk (Berezhkovskii & Hummer, Phys. Rev. Lett. 89, 064503, 2002)

• Ptr = (N+1)-1 ~ L-1

Water Flow Through Nanotube Membranes under Osmotic Gradient

(Kalra, Garde & Hummer, Proc. Natl. Acad. Sci. USA, in press, 2003)

NaCl + H2O

H2O

NaCl + H2O

• ~5 mol/l NaCl generates net flow of ~5 H2O/ns per tube (~independent of tube length up to at least 5 nm)

• Flow rate as in aquaporin-1

Water Flow and Water Monolayers

• Metastable water monolayer sandwiched for ~60 ns between porous and nonpolar nanotube membranes

Proton Transport(Dellago, Naor & Hummer, Phys. Rev. Lett. 90, 105902, 2003)

• Molecular dynamics simulations of water and excess proton in nanotube– Car-Parrinello dynamics (DFT/BLYP)

– Empirical-valence-bond model (Schmitt and Voth, J. Phys. Chem. B 102, 5547, 1999)

CPMD

EVB

Proton Transport Coupled to Defect Motion

• Proton diffusion approximately 40 times faster than in bulk water: D(H+)≈170x10-5 cm2s-1

• Strong 1/r-coupling to H-bond (D) defect in periodic tube: 10-fold reduction of apparent diffusion constant

H+ D H+ D H+ D

D defect

L defect

H+

H+/D

Implications for Proton Transfer in Proteins

• Proton wires ‘on demand’– Increase in local polarity can trigger water’ influx to

establish protonic connectivity

• High ‘delivery speed’– High mobility of a single proton along ordered

water chain inside hydrophobic pore

• Unidirectional wires (‘diode’)– Hydrogen-bond orientation controlled by

electrostatics

• High ‘fidelity’– Only single proton delivered without reorientation

Light-driven proton pump: Water Filling/Emptying in Bacteriorhodopsin

(Hummer, Rasaiah, Noworyta, ICCN Proceedings, 2002)

• Reprotonation of Asp-96 from solvent

N

O

O

H

H+

N

O

O- -O

O

H+

N

-

H+

H+

N

O

O

H

Proton Pumping in Cytochrome C Oxidase(Wikström, Verkhovsky and Hummer, Biochim. Biophys. Acta-Bioenerget. 1604, 61, 2003)

• Gating of ‘chemistry’ and ‘pumping’– 2 alternative water-mediated proton

pathways through nonpolar cavity for ‘chemistry’ and ‘pumping’

– Water in cavity orientated by electric field between hemes a and a3/CuB

– Switching between H+ paths by water orientation (‘diode’)

• Coupling of O2 reduction to pumping– H+ transfer from E286 to heme-a3

propionate provides gate (‘solvent fluctuation’) for electron transfer fromheme a to heme-a3/CuB

– ‘Pumped’ H+ trapped by electron transfer and reprotonation of E286

P

N

Water-Gated Proton Pump

Heme-a/Heme-a3/CuB Charge Distribution Switches Water-Chain Orientation

• Heme-a reduced

• Heme-a3/CuB oxidized

• Heme-a oxidized

• Heme-a3/CuB reduced

• Reorientation within ~ 1 ps of electron transfer in molecular dynamics simulations

E E

Water & Proton Transport -Conclusions

• Water-filled nonpolar channels form efficient wires for water and proton transport

• Gated by filling/emptying and electric field

• Proton pumping is ‘solvent fluctuation’ required for subsequent redox chemistry

Acknowledgments

• Peptide dynamics– In-Chul Yeh (NIH)

• Single-molecule pulling– Attila Szabo (NIH)

• Nanotube in water– J. C. Rasaiah (U. Maine)

– P. J. Noworyta (U. Maine)

– A. Waghe (U. Maine)

– S. Vaitheeswaran (NIH, U. Maine)

• Water transport– A. Berezhkovskii (NIH)– A. Kalra (NIH, RPI)– S. Garde (RPI)

• Proton transport– C. Dellago (U. Rochester;

U. Vienna)– M. Naor (U. Rochester)

• Proton pumping in oxidase– M. Wikström (U. Helsinki)– M. Verkhovsky (U. Helsinki)

• Proton transfer in proteins– S. Taraphder (NIH, IIT

Kharagpur)

References

• Peptide dynamics

– Yeh & Hummer, J. Am. Chem. Soc. 124, 6563, 2002

• Single-molecule pulling

– Hummer & Szabo, Proc. Natl. Acad. Sci. USA 98, 3658, 2001

– Hummer & Szabo, Biophys. J. 85, 5,2003

• Nanotubes

– Hummer, Rasaiah & Noworyta, Nature 414, 188, 2001

– Waghe, Rasaiah & Hummer, J. Chem. Phys. 117, 10789, 2002

– Berezhkovskii & Hummer, Phys. Rev. Lett. 89, 064503, 2002

– Kalra, Garde & Hummer, Proc. Natl. Acad. Sci. USA, in press, 2003 (http://www.pnas.org/cgi/content/abstract/1633354100v1)

– Dellago, Naor & Hummer, Phys. Rev. Lett. 90, 105902, 2003

• Mechanism of proton pumping in cytochrome c oxidase

– Wikström, Verkhovsky & Hummer, Biochim. Biophys. Acta-Bioen. 1604, 61, 2003

• Network model of proton transfer in proteins

– Taraphder & Hummer, J. Am. Chem. Soc. 125, 3931, 2003