Biological and small molecule electron transfer:
some interesting connections
Spiros S. Skourtis
Department of Physics, University of Cyprus
Nicosia Cyprus
QuantSim09
Warwick, UK
Effects on ET of:
• Molecular structure
• Molecular dynamics
• Initial state preparation
• Control of ET
Main focus of talk is on tunneling ET reactions
in ET proteins
Biological electron transfer (ET) reactions
mediated by tunneling
• Often long distance ET:
The electron transfers from an initial
localized (donor) state to a final
localized (acceptor) state through
intervening molecular matrix (bridge).
D-A distance can be ~10Å
~10Å
De
A
Bridge:
(protein or
DNA)
solvent
• Bridge:
Protein and/or DNA,
hundreds of atoms between donor/acceptor
D e
A
protein
• Donor (acceptor) moieties:
metals atoms, small organic molecules,
aminoacids, DNA bases
protein
or DNA
solvent
MAIN EXP. OBSERVABLE: ET rate
biological rates: (psec)-1 or slower
Qualitative picture of molecular ET reactions
mediated by tunneling
• Thermal fluctuations of molecule+solvent
induce D-A resonance
Energy of ET molecule + solvent
excluding kinetic energy of atoms
UD
UA
• ET by tunneling takes place at
D-A resonance
• ET RATE:
U act
U Dmin
k ET  TDA e
2
U act / K BT
U Amin
Collective system
(Reaction) coord.
Assume classical
motion.
Tunneling matrix
element (coupling)
between D and A
Boltzmann factor
for activation to
resonance conformation
D
A
D
A
Crossing point
(resonance)
D
A
TDA: DonorDonor-Acceptor coupling
(tunneling matrix element)
Bridge atoms lower tunneling barrier and enhance the DonorDonor-Acceptor coupling
Pot. energy felt by e-
V0eff
Etun
-e
D
A
TDA  e
RDA
 2 m (Vo  Etun ) RDA 
D
A
B1
B2
BRIDGE
ATOMS
Biomolecules are very floppy:
floppy:
Fluctuating barrier tunneling
Bridge thermal motion  Fluctuating barrier/tunneling matrix element: TDA(t)
Pot. energy felt by e-
A
D
-e
D
A
B1
B2
BRIDGE
ATOMS
LongLong-distance ET rate for a fluctuating
tunneling matrix element in limit
of slow matrix element fluctuations
In high-temp limit:
kET 
2
2
TDA FC

UD
Energy
UA
Average of TDA2 over diff. molecular conformations
Crossing point (region)
U act
FC factor
FC 
U Dmin
1
2  U
2
exp U act  BT 
U Amin
Collective system coord
rms fluctuation in the D-A energy gap: UD-UA
Dependence of electron tunneling on
biomolecular structure
Different proteins have different structures
sequences & folds)
Protein Data Bank @ http://www.pdb.org/
(AA
Protein ET systems (large rate scatter)
k ET  TDA
2
  2 2 m (V0eff  Etun ) 
TDA  e   RDA
2
1
k ET
β- sheet
  1.0Å 1
α- helix
  1.3Å 1
Gray H. B., Winkler J. R. PNAS 2005;102:3534-3539
Copyright © 2005, The National Academy of Sciences
 aver  1.1Å 1
Structure-function analysis of TDA
Input: experimental protein structure 
Hamiltonian/Fock matrix 
Quantum electronic structure calculations
• Computation of TDA
Effective Hamiltonian & Green’s function methods
Energy-splitting methods
Mulliken-Hush methods (excited state calcs)
• Analysis of TDA in
terms of bridge struct.
Green’s function tunneling pathways
Tunneling currents
Substitutions & pruning
Reviews
•
S.S. Skourtis and D.N. Beratan Adv. Chem. Phys. 106, 377 (1999)
•
M.D. Newton, in: Electron Transfer in Chemistry, Vol I, Ed. V. Balzani, Wiley-VCH Verlag
GmbH: Weinheim, Germany p. 3. (2001)
•
A.A. Stuchebrukhov Adv. Chem. Phys. 118, 1 (2001)
Tunneling pathway decomposition of TDA
D Tˆ A  D Vˆ Gˆ ( B )  Etun  Vˆ A
Bridge Green’s function operator
THROUGH-BRIDGE PROPAGATION
Bi
D
Bj
Vi , j
VD ,1
VN , A
Gˆ ( B )  Etun    Etun Iˆ  Hˆ ( B ) 
1
A
Dependence of tunneling on
molecular dynamics
Reviews
D. N. Beratan, S.S. Skourtis et. al.
Steering electrons on moving pathways
Acc. Chem. Res. ASAP. (2009)
S.S. Skourtis, J. Lin, and D.N. Beratan
The effects of bridge motion on electron transfer reactions mediated by tunneling
Modern methods for Theoretical Physical Chemistry of Biopolymers,
E. B. Starikov, S. Tanaka, and J. P. Lewis, editors, Elsevier, pp. 357-379 (2006)
A protein fluctuates around its structure
 FC
Time scales of structural fluctuations: tens of fsec to μsec/msec
Size of structural fluctuations: a few to tens of Angstroms
Question I:
Are static methods for computing TDA
transferable to time dep. systems?
• A. Teklos and S. S. Skourtis Chem. Phys. 319, 52-68 (2005)
• S. S. Skourtis Chem. Phys. Lett. 372, 224-231 (2003)
• S. S. Skourtis, G. Archontis and Q. Xie J. Chem. Phys. 115, 9444-9462 (2001)
• Q. Xie, G. Archontis and S. S. Skourtis Chem. Phys. Lett. 312, 237-246 (1999)
From practical point of view
Can one derive a reliable TDA(t) time series by:
• running MD
• saving MD snapshots of molecule
• for each snaphot computing TDA using a static method?
ANSWER IS YES FOR ELASTIC TUNNELING
IF THERE ARE NO STRUCTURAL MOTIONS
INVOLVING D, A, BRIDGE
WITH FREQUENCY ω
SUCH THAT
 ~
Tunneling barrier height
Question II:
Is the rate expression
kET 
2
2
TDA FC

valid for all types of fluctuating ET systems?
Time-domain formulation of ET rate
k ET
1
 2


 dt e
CFC  t   eihD t /  e ihA t / 
ˆ
iDAt
CTDA  t  CFC  t 

CTDA  t   TˆDA  t  TˆDA  0 
 FC
1
 coh
2

d
 1 2 d
CTDA  t  =CTDA  0   t   CTDA  0   + t   2 CTDA  0   + ....
 dt
 2
 dt

k ET
2
2

TDA  FC 1   k ET 

A. Troisi, A. Nitzan, M. Ratner JCP 119, 5782 (2003)
ˆ
etc.
D
B
Qualitative picture of TDA fluctuation effects
 rate   FC ,  coh
SLOW FLUC
FAST FLUC
 FC   coh
 FC   coh
TDA has time to fluctuate while D and
A are resonant 
For each D-A crossing event, D and A
see a time-dep. TDA
TDA does not have time to fluctuate
while D and A are resonant 
For each D-A crossing event, D and A
see a static TDA
 T2DA << TDA
2
 T2DA >> TDA
2
SMALL TDAfluc.
compared to TDA aver.
LARGE TDA fluc.
compared to TDA aver.
The values of TDA at
different D-A crossing
events are similar.
The values of TDA at
different D-A crossing
events differ.
k ET 
2
TDA

2
 FC
k ET =
2

TDA
2
 FC
Static calculation of TDA is OK
Rate is determined by conformations with strongest TDA
(e.g., for minimum energy conformation)
Extensive sampling is necessary
Behaviour of rate
is more complex:
Inelastic tunneling &
Floquet resonances
TDA fluctuation effects in the ET protein azurin
S.S. Skourtis, I. Balabin, T. Kawatsu, and D.N. Beratan Proc. Natl. Acad. Sci. USA 102, 3552 (2005)
Dynamic correction to rate 1%
SLOW FLUC
 FC
< 0.1
 coh
RDA  17 A
LARGE FLUC
 T2DA
TDA
kET 
rms
TDA
 105 eV
1
2
> 10
2
2
TDA FC

k D  A  106 sec
RDA  26 A
rms
TDA
 108 eV
1
kD  A  102 sec
Fluctuations become important beyond critical distance
I. Balabin, D.N. Beratan and S.S. Skourtis Phys. Rev. Lett. 101, 158102 (2008)
Critical distance Rc = decay length of spatial TDA correl. func.
RDA < Rc
RDA > Rc
 T2DA < TDA
 T2DA > TDA
2
Water-mediated tunneling
Protein-mediated tunneling
2
Rc = 2-3 Angstroms
Rc = 6-7 Angstroms
Comparison of ET in azurin with ET in small
C-clamped molecules suggests similar couplingcouplingfluctuation regimes
• ET takes places by tunneling through
solvent in cavity
M.B. Zimmt and D.H. Waldeck J. Phys. Chem. A. 126, 3580 (2003)
Z
• Coupling fluctuation magnitudes and time
scales are similar for different solvents
Y
X
(acetonitile, benzene, 1-3 diisopropylbenzene)
A. Troisi, M. Ratner, M.B. Zimmt J. Am. Chem. Soc. 126, 2215 (2004)
• Dynamic corrections to rates ~ 1%
Fig. by
M. Zimmt
Why the similarity?
TDA fluctuations are “slow”
 coh   FC
The systems have reorg. energies of the order of eV
 FC ~   U DA  
2 k BT ~ 1 fsec for  ~ eV
τcoh is determined by the “localized” valence angle motions (typical periods: a few tens of fsec)
 coh ~ 30  100 fsec
TDA fluctuations are “large”
 T2DA  TDA
2
Both for azurin and for the C-clamp molecules the DA distances are much larger than the critical
distance for the tunneling medium
RDA  Rc
• Control of fluctuations/tunneling pathways
• Inelastic tunneling
Minimal model of molecular doubledouble-slit system
S.S. Skourtis, D.H. Waldeck and D.N. Beratan JPC B 108, 15511-15518 (2004)
n( y1 )
Initial
Initial bridge vibrational state (prior to ET)
y1
iB  n( y1 ) m( y2 )
D
-e
B1
A
D
Initial vibronic state (prior to ET)
D; iB  D n( y1 ) m( y2 )
B2
y2
m ( y2 )
Final bridge vibrational state (after to ET)
f B  n( y1 ) m( y2 )
Final vibronic state (after ET)
A; f B  A n( y1 ) m( y2 )
Total ET rate: sum over rates of vibronic transitions
k ET   i PiB  f k ET  D; iB  A; f B 
B
B
Initial vibronic state (before ET)
k ET  D; iB  A; f B    2 /  
D; iB Tˆ A; f B  D; iB Vˆ[ Etun Iˆ  Hˆ ( B ) ]1Vˆ A; f B
Vibronic tunneling matrix element
Final vibronic state (after ET)
D; iB Tˆ A; f B
2
  FC iB  f B 
 FC 
1
2  U
2
exp  U act (iB  f B )  BT 
Franck-Condon factor
(vibronic Donor-Acceptor coupling)
A. Troisi, A. Nitzan, M. Ratner JCP 119, 5782 (2003)
Elastic tunneling coupling :
iB  f B  0, 0
 VD , B1 VB1, A
VD , B 2 VB 2, A 
ˆ
D; 0, 0 T A; 0, 0  


E

E
E

E
 D ( A)
B1
D ( A)
B2 

y1
VD,B1
B1
VB1,A
A
D
VD,B2
B2
VB2,A
y2
Coherent Sum
Over Two Pathways
Inelastic tunneling coupling :
iB  1, 0 , f B  0, 0


VD , B1 F1 / 2 M 11 /  VB1, A 
  0
D;1, 0 Tˆ A;0, 0  


( ED  EB1 )( E A  EB1 )


y1
VD,B1
B1
VB1,A
A
D
VD,B2
B2
VB2,A
y2
X
Single Pathway ONLY
Enters Coupling
Manipulation of ET pathways by IR
in doubledouble-slit type systems
S.S. Skourtis and D.N. Beratan AIP Proceedings Vol. 2, Part B, 809-812 (2007)
• D. Xiao, S.S. Skourtis, I. Rubtsov, D.N. Beratan, Nano Lett., 9 (5), 1818–1823 (2009)
3-pulse VISVIS-UV / midmid-IR / VIS transient spectroscopy:
Manipulate one of the pathways by
IR excitation of pathwaypathway-specific vibration
(created by isotopic substitution)
hν2
hν1
hν3
O
H
O
Donor
Acceptor
O
H
D O
Don*-Br-Acc
Don+-Br-Acc-
t
T
time
400 nm pump
e–
Mid-IR pump
k ET  vib   k ET  no vib 
3-5% decrease in ET rate
Z. Lin, C.M. Lawrence, D.Xiao, V.V. Kireev,
S.S. Skourtis, J.L. Sessler, D.N. Beratan, I.V. Rubtsov
Dependence of tunneling on
the initial state preparation
Example (I)
DNA photolyase
Electron transfer protein
that repairs UV-damaged DNA
(pyrimidine dimers)
upon absorption of a photon
The structure of a DNA photolyase
bound to a DNA oligomer (2004)
DNA
Repaired
Dimer
Donor
Antenna
A. Mees et al. Science 306, 1789-1793 (2004)
T. Prytkova, D.N. Beratan and S.S. Skourtis Proc.Natl. Acad. Sci. USA 104, 802 (2007)
Photo-selected tunneling pathways
In DNA Photolyase the photo-excitation itself
enhances the electronic coupling between FADHand the thymine dimer
DNA DIMER
FADH-
ET
-e
-e
h
ET rate
~(200)-1
psec-1
Photoexcitation
Example (II)
MOLECULES ON SURFACES
The transmission through helical molecular bridges
of electrons carrying angular momentum
depends on the bridge handedness
Hole donor
molecule
Helical
bridge
molecule
TRANSMISSION ASYMMETRY AS HIGH AS 10%
Experiment:
• K. Ray, S.P. Ananthavel, D.H. Waldeck, R. Naaman, Science 283, 814-816 (1999)
• J.J. Wei, C. Schafmeister, G. Bird, A. Paul, A.; R. Naaman, D.H. Waldeck, JPC B, 110,
1301-1308 (2006).
Reductionist model
• S.S. Skourtis, D.N. Beratan , R. Naaman, A. Nitzan, D.H. Waldeck, Phys. Rev. Lett. 101, 238103 (2008)
x
y
z
ML 
-e
-e
1 6
 exp[i(2 M L ( j  1) / 6)] j
6 j 1
1
1
6
2
5
3
6
2
3
5
4
4
Right handed helix
Left handed helix
N
in 
1
1  ei 2

2

in* 
1
1  e  i 2

2
N

Thanks to collaborators, postdocs and students
Cyprus
Israel
Prof. G. Archontis
Dr Q. Xie
A. Teklos
C. Aristi
M. Panagiotou
Tel
Tel--Aviv
Weizmann
Prof. A. Nitzan
Prof. R. Naaman
USA
Duke
Pitt
Tulane
Austin
Prof. D. N. Beratan
Dr. I. Balabin
H. Carias
Dr. S. Keinan
Dr. T. Kawatsu (ex-Duke)
Dr. J. Lin (ex-Duke)
Dr. T. Prytkova (ex-Duke)
D. Xiao (ex-Duke)
Prof. D. Waldeck
Prof. I. Rubtsov
Prof. J. Sessler
Funding
Cyprus Research Foundation
Leventis Foundation
University of Cyprus
NIH, NSF, Duke Univ. Theory Initiative