Transport of proteins acrossnanopores: a physicist's perspective

F. Cecconi

CNR-ISC Istituto dei Sistemi Complessi (Roma)

A. Ammenti (Univ. di Perugia, INFN Italy)

U. M.-B.-Marconi (Univ. di Camerino, Italy)

A. Vulpiani (Univ. �“Sapienza�” Roma, INFN, Italy)

M. Chinappi (Univ. �“Sapienza�” Roma, Italy)

M.C. Casciola (Univ. �“Sapienza�” Roma, Italy)

Anomalous Transport: from Billiards to Nanosystems (Sperlonga 2010)

Sperlonga 2010

Overview

�• Biology of translocation (under physics view)

�• Voltage-driven translocation experiments (clean data)

motivation

�• Computer modeling (simulation results)

�• Math. Theory: First Passage Time (interpretation ?)

�• Conclusions

Sperlonga 2010

main �“characters of the tale�” are cellularmembranes and pores which constituteselective biological gates = transport proteins

Translocation (molecular transport)

Proteins (complex biopolymers) areneeded in a variety of locationsinside/outside the cellefficient transport mechanisms

Cell: non isolated chemical lab (continuousexchange of chemical compounds)

Sperlonga 2010

Cell membrane

Barrier separating material inside the cellfrom environment with a complex structure

Structural elements: Lipid bilayer (fatty molecules) +

proteins (receptors,catalyst, mechanical�…�…

transport proteins (pores) allow the exchange of signals, chemicals, genetic information

Sperlonga 2010

Phospholipid bi-layer

Gases (CO2,N2,O2)

Q=0 Small Polar Mol. (Ethanol)

Water & Urea

Q=0 Large Polar Mol. (Glucose)

Ions (K+,Ca2+,Mg2+)

Q 0 Polar Mol.

(Am-Ac,ATP)

Biological relevance of pores (= transport proteins)

Without gates fewmolecules onlywould cross thecell membranes

trappin

g

Sperlonga 2010

-Hemolysin ion channel ( HL)

Transport protein �“mushroom shape�”

2) Can be integrated into lipid bilayers or solid-state substrates to form conducting and transport devices �“nanopore systems�”

Structure resolved bySong et al. Science (1996)

Refined by

Gonaux J.Str.Biol (1998)

Eptameric structure

1) Stable in vitro

Recent interest

with potential technological/biomedical appl.)

Sperlonga 2010

Voltage driven translocation

ion-current drop indicates clogging of the channel single molecule passage

Direct detection of �“translocation�” events

Kasianowicz et al. PNAS (1995ff), Meller et al. PRL (2001)

Experimental setup is a circuit

Sperlonga 2010

Sequencing & mass spectroscopy

the analysis of current signals allows translocatingmolecules to be identified and sequenced, eachmolecule has its own signature in the plane (I,t) ?patterns are affected by unpredictable details of thedynamics (importance of simulations)

Typical current pattern

ideal

Sperlonga 2010

A.Meller et al. PNAS 2000;97:1079-1084

©2000 by The National Academy of Sciences

Sequencing & spectroscopy (II)

IB: Blockage current

tD: translocation time

Well defined clustering

Time and current are sensitive to:length, composition, structure,temperature, channel propertiesit means �“spectroscopy�”!!

characterization of single strandedDNA [poly(dA) and poly(dC)]at different temperatures

STATISTICS

Sperlonga 2010

Common feature: skewnesssimple explanation? Model?

DNA strands

1) No translocation2) Fast translocation3) Slow translocation

MBP Protein

G. Oukhaled et al. PRL 98 (2007)Kasianowicz et al. PNAS 93 (1993)

Distribution of blockage times

Sperlonga 2010

Model (physicist�’s perspective)

Cylindrical channel of finite length L

homogeneous importing force F

Coarse-Grained protein model (united atom) to C -backbone

(also Makarov 2006)

(collecting many events)(preserving UBI structure)

UBI

GS of force field

ubiquitin

Sperlonga 2010

Simplest pore actions

V (ri ) =1 tanh[ x(x L)]

2

yi2

+ zi2

Rp

2

q

Confining effect

Average importing force along x (acting only inside the pore)

Vimp(rk ) = Fxk

L

x

(smooth step)

Sperlonga 2010

Translocation Simulations

Folding equilibrium (setting of parameters)

Runs of translocations to measure

Vel(F), ProbTr(F), time(F) , (t;F) (time distrib.)

At different pulling forces (F)

and temperatures Tref , Tph

Constant temperature MD (Langevin)

Umbrella Sampling + Multiple histograms

mk

d2rk

dt2= m

k r k+ F

kV +R

k

Sperlonga 2010

Examples of Trajectories

analyze trajectories via the reaction(collective) coordinate:center of mass XCM

F=2.5f F=3.0f

Put Ubiquitin near pore entrance and drag it

Pathway: Unfolding transport refolding

Sperlonga 2010

Capture process

monitoring the occupation

of the channel p(tc)

#cis #in #trans

dPc

dt=

c(1 P

c)

pc(t) =dPc

dt= ce

ct

exponential PdF

Sperlonga 2010

Current (Mobility)

=

vix

i=1

N

NF

Non Ohmic behavior of = (F)

indicates the presence of barriers

�“First�” Transport observable

Dominated by fluctuationsinduced by the pore

v = F

TphTref

No Transl

(free) (pore)

Sperlonga 2010

Translocation probability

Given a time window [0,TW], translocation may or

may not occur, thus, we can define aProb. to translocate = #successes/#attempts

Sigmoid shape indicatesCritical Force

Free Energy-Barriers

PTr (Fc) =1 PTr (Fc)

PTr (Fc) =1/2

Fc decreases with Temp.

Tph Tref

Sperlonga 2010

Blockage (translocation) times

Arrhenius-like only on high force regime

(F) = tfp

Aexp( Fa)

Translocation time is given by simulations of firstarrival time at trans side (Mean First Passage Time)

tfp =min{t : XCM(t) L}

cis trans

Random variableTref

Tph

Sperlonga 2010

F =1.38f

Blockage time distribution

first passage time (FPT) at the end of the channel (experiments: time interval of current drop)

not Gaussian with exponential tails controlled by F

F=3.0f

Simple explanation ?

Sperlonga 2010

Free-Energy Profile

G(X) = Gs{1 tanh[

s(X L /2)

2ls

2]}

s=1

3

Transport observables indicated barriers: energylandscape in the reaction coordinate: center of mass X

G(X) = RT lnP(X)

Umbrella simulations restraining the protein in the channel P(X)

Free-en.= U - TS

includes conform.

Sperlonga 2010

Mathematical model (Drift-Diffusion)

Simulation results (exp.) on translocation have a natural interpr. within FPT-problem in terms of a particle undergoing a driven diffusion on a free-energy profile G(X)

J(X,t) = D0eU (X )

Xe

U (X )P(X,t)

U(X) =G(X) FX

+ boundary at channel ends, the most general ones:Radiation BC (Berezhkowskii, Gophic B.J. 2003)

tP = D0 X

eU (X )

Xe

U (X )P

current

L

J(0,t) = R0P(0,t)

J(L,t) = RLP(L,t)

tD

L2

D0

tB

L

0F

tR

L

R

Metzler & Klafter

Anom. Translocation B.J. 2003.

Lubensky & Nelson 99

Sperlonga 2010

First Passage Time theory

Survival prob. S(t) = dX P(X, t)0

L

Blockage Time PdF (t) =dP

out

dt=

dS(t)

dt

(t) = R0P(0,t) + RLP(L, t)

�ˆ (s) = R0�ˆ P (0,s) + R

L�ˆ P (L,s)

Flux at boundaries

Not escaped the channel (time < t)

Recalling Rad-BC

Pout(t) =1 S(t)

Sperlonga 2010

Solving Smoluchovski Equation

Green Function method for s=0 lucky case

PTr (F) = RL

dt P(L,t)0

= RL

�ˆ P (L,0)

(F) = dt dX P(X, t)0

L

0

= dX �ˆ P (X,0)0

L

V (F)L

(F)+ ....

Transloc Prob.

Average time

D0 Xe

U (X )

Xe

U (X )P(X,s){ } sP(X,s) = (X X0)

Laplace Transform �ˆ P (X,s) = dt0

estP(X,t)

Velocity

Sperlonga 2010

Time distribution (unlucky s=0)

0(t;F) =L

4 D0t3exp

[ (F)t L]2

4D0t

t3/2

e(F )t /(4D0 )

strategy to improve the fitting to data

�ˆ (s) = R0�ˆ P (0,s) + R

L�ˆ P (L,s)

Absorption0 L

source

the shape of PdF is reasonable, but the fitting of tails can be often unsatisfactory, can we do better?

Inverse Gaussian

Sperlonga 2010

Re-absorbe approximation into BC (mimicking thepresence of barriers at the channel ends)

J = FP D0

P

xP

G

x

J(0,t) = [R0 + R0 ]P(0,t) R0 G'(0)

J(L,t) = [RL+ R

L]P(L, t) R

LG'(L)

now an analytical expressionof Laplace transf. can be derived

Simplest driftdiffusion eq.

�“Dirty trick�” attempt

XJ(X,s){ } sP(X,s) = (X X0 )

Sperlonga 2010

) (s;F) =

RZ(s)evL /2D0

RZ(s)cosh( L)+ (vR + 2D0s)sin( L)

Z(s) = v2 + 4D0s = Z(s) /2D0 v = F

RBC(t;F) = a(t)exp( t)

only numerical inversion: large timebehaviour is controlled by the 1st Pole

helps the fitting procedure: constrain

Sperlonga 2010

Conclusions

C.G. Model of Protein Translocation reproduces basic

phenomenology.

Support to 1dim Driven-Diffusion model: D0, 0,RL Free-energy profile G(X)=-RT log P(X) MESSAGE

Driven-Diffusion and FTP theory work: conceptualflexible framework to interpret translocation data:a step farther than mere fitting.

Limitation: reliable reaction coordinates and

some knowledge on free-energy landscape

(Ammenti et al. JPCB 113, 10348 (2009))

Sperlonga 2010

Go model: outline

Folding ruled by topology of native state

Force Field: Native Structure bias

No need of sequence = Ideal Sequence

(Minimal Frustration)

Support: Folding time vs contact order

K.W. Plaxco et al. (1998) JMB

Go-model confers protein-like properties to chains

Sperlonga 2010

Go-model for Ubiquitin

Dihedral potential (torsion)

V = k(1)[1 cos( )]+

k(3)[1 cos3( )]

V ( ) =k

2( )

2

Bending potential (elasticity) Chain potential

Backbone of C carbon (Native interactions)

Preserves Protein-like properties

V (r) =kh

2(ri,i+1 R

i,i+1)2

Clementi et al. JMB 2000

Sperlonga 2010

Given a Protein Native state

Choose distance cutoff Rc and

Define Native Interactions Rij < Rc

Vnat = 5Rij

rij

12

6Rij

rij

10

i, j> i+1

Vnnat =10

3i, j> i+1

rij

12

PDB structure is the ground state of Go Force-Field