Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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410
The
probability
to
obtain
a click in
an
ideal threshold
detector
placed
at
the
transmitted
output
(+) of polarimeter
Al
is
then
simply
the
integral
of
this
density
distribution
over
the
energy reaching
the
detector, from
the
threshold
<I>
to
Emax:
(3)
(4)
Similarly
the
probability
to
obtain
a click in
an
ideal threshold
detector
positioned
at
the
reflected
output
(-)
of polarimeter
Al
is
2 .
Eo
sin
('P
A - B
AI)
(5)
We
can
also consider a less ideal threshold
detector
,
that
would
be
more likely
to
resemble
the
characteristics of a real detector. In
order
to
keep
the
calculations of
the
integrals simple enough, we considered
the
case of threshold
detector
with
linear rise,
that
is, a threshold
detector
for
which
the
probability
to
generate a click increases linearly
starting
from
the
threshold
<I>,
possibly
with
a
saturation
value after which increasing
the
impinging energy no longer increases
the
probability
to
generate
a click.
In these cases,
the
analytical results
are
more complicated since
the
probability
to
get a click for
an
energy E +
dE
is
not
always equal
to
1,
but
the
principle of calculation remains
the
same: integrate
the
product
of
the
probability density
distribution
by
the
probability of
obtaining
a click for a
given energy.
The
analytical results
are
qualitatively similar
to
that
of
ideal
threshold detectors.
The
results in
the
case
Eo
=
2<I>
- which is leading
to
a violation of Bell inequalities exactly reproducing
the
predictions of
Quantum
Mechanics- are displayed in Fig.
3:
the
probability
to
get a click
in
the
polarimeter oriented along B
Al
depends on
I'P
A -
BAli
. It is
maximum
for
I'P
A - B
Al
I = 0 +
k7r
/2,
and
reaches zero for
I'P
A -
BAli
=
7r
/4+
k7r
/2.
Similar results would
be
obtained
for Alice's BAo polarimeter,
and
for
Bob's
polarimeters.
This
fair sampling
test
can
be
implemented very simply on Alice's side
by fixing
B
Al
= B Ao =
O.
The
random
switching in
Ekert
protocol (Fig. 1
and
Fig. 2) ensures
that
the
points
at
0
and
7r/4 are
both
scanned
automat-
ically. Any significant difference in
the
number
of single counts recorded
0.4
0.3
0.2
0.1
o
7r
4
7r
2
37r
4
411
OM predictions
Figure
3.
Analytical
Results
in
case
of
biased-sampling
attack
on
threshold
detectors
in
Alice's
Al
channel,
with
Eo =
2<1>.
The
blue
plots
represent
the
probability
to
get
a
click
in
detector
At,
whereas
the
purple
plots
represent
the
probability
to
get
a click
in
detector
Ai.
The
probability
to
get
a click
in
polarimeter
()
A,
thus
depends
on
I'P
A -
()
All·
By
contrast,
in
case
of
a
genuine
source
of
entangled
photons,
Quantum
Mechanics
predicts
independence.
when
'P
A = 0
and
'P
A =
7f
/ 4 would
betray
Eve's
attempt
to
bias
the
sam-
ple
through
a biased-sampling
attack
on
the
threshold
detectors. Similarly,
Bob
would chose e
B,
= e
Bo
=
7f
/8,
and
compare
the
number
of
singles
when
'PB
=
-7f/8
and
'PB
= 7f/8.
4.
Conclusion
This
fair sampling
test
can
be
implemented
during
the
production
of
the
key
and
together
with
the
violation
of
Bell inequality check, so
that
it
seems
hard
to
fool
it
without
reducing
the
visibility
of
the
violation. For instance,
increasing
the
energy
of
the
pulses
with
respect
to
the
threshold
would
tend
to
reduce
the
dip in
the
fair sampling test,
but
it
would reduce
the
visibility
of
the
correlation (weaker violation
of
Bell inequalities)
at
the
same
time,
and
it
would also give rise
to
double counts.
The
combination
of
a Bell inequality
test
with
a
monitoring
of
the
double
counts
and
our
local fair sampling
test
therefore
constitutes
a solid scheme
412
against
eavesdropping a E91
protocol
using a biased-sampling
attack.
In
principle,
the
same
idea
could
be
implemented
in
other
QKD
protocol,
by
replacing passive
detectors
in each channel
by
a device
with
the
same
efficiency,
that
would analyze
further
whichever degree
of
freedom is used
to
encode
the
key,
instead
of
simply
feeding
detectors
with
it
b.
Acknowledgments
We
are
grateful
to
Hoi-Kwong Lo,
Jan-Ake
Larsson, Takashi
Matsuoka
and
Masanori
Ohya
for useful discussions
on
Quantum
Key
Distribution.
References
1. A. K.
Ekert,
Phys. Rev. Lett.
67,
661(Aug 1991).
2.
N. Gisin, G. Ribordy,
W.
Tittel
and
H. Zbinden, Rev. Mod. Phys. 74,
145(Mar
2002).
3. G.
Jaeger,
Quantum Information (Springer New-York, 2007).
4. V. Scarani, H.
Bechmann-Pasquinucci,
N.
J.
Cerf, M. Dusek, N.
Ltitkenhaus
and
M. Peev, Rev. Mod. Phys.
81,
1301(Sep 2009).
5. H.-K. LO
and
Y. Zhao,
Quantum
cryptography
(2008).
6. A.
S.
H.
Z.
J.
G. R.
T.
W.
D. Stucki, G. Ribordy, Journal
of
Modern Optics
48,
p. 1967 (2001).
7.
P. M. Pearle, Phys. Rev. D
2,
1418(Oct 1970).
8. A.
Garg
and
N. D.
Mermin,
Phys. Rev. D
35,
3831(Jun
1987).
9.
P. H.
Eberhard,
Phys. Rev. A
47,
R747(Feb 1993).
10.
J.
Ake Larsson, Physics Letters A
256,
245 (1999).
11. N. Gisin
and
B. Gisin, Physics Letters A
260,
323 (1999).
12. A.
Ekert,
Less reality,
more
security(September,
2009).
13. Y. Zhao, C.-H.
F.
Fung, B. Qi, C.
Chen
and
H.-K. Lo, Phys. Rev. A
78,
p.
042333(Oct 2008).
14. G.
F.
Knoll, Radiation Detection and Measurement (Wiley &Sons, 1999).
15. G. Adenier,
Violation
of
bell inequalities
as
a violation
of
fair
sampling
in
threshold
detectors
FOUNDATIONS
OF
PROBABILITY
AND
PHYSICS5
1101
(AlP, 2009).
16. K.
J.
Resch,
J.
S.
Lundeen
and
A. M.
Steinberg,
Phys. Rev. A
63,
p.
020102(Jan
2001).
17. H.-K. L.
Bing
Qi, Li
Qian,
A
brief
introduction
of
quantum
cryptography
for
engineers
(2010).
bThe
use
of
four
detectors
on
each
side
can
also serve
other
purpose,
like
shielding
Alice
and
Bob
from a
time-shift
attack
17
Quantum
Bio-Informatics
IV
eds. L.
Accardi,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co. (pp.
413-426)
BROWNIAN DYNAMICS SIMULATION OF
MACROMOLECULE DIFFUSION
IN A PROTOCELL
TADASHI ANDO
Center
for
the Study
of
Systems Biology, School
of
Biology, Georgia Institute
of
Technology250 14th Street
NW,
Atlanta,
GA
30318-5304,
USA
JEFFREY
SKOLNICK
Center
for
the Study
of
Systems Biology, School
of
Biology, Georgia Institute
of
Technology250 14th Street NW, Atlanta,
GA
30318-5304,
USA
The interiors
of
all living cells are highly crowded with macromolecules, which differs
considerably the thermodynamics and kinetics
of
biological reactions between
in
vivo
and
in
vitro. For example, the diffusion
of
green fluorescent protein (GFP) in
E.
coli
is
-IO-fold slower than in dilute conditions. In this study, we performed Brownian
dynamics (BD) simulations
of
rigid macromolecules in a crowded environment
mimicking the cytosol
of
E.
coli
to
study the motions
of
macromolecules. The simulation
systems contained
35
70S ribosomes, 750 glycolytic enzymes,
75
GFPs, and 392 tRNAs
in a
100 nm x
100
nm x 100 nm simulation box, where the macromolecules were
represented by rigid-objects
of
one bead per amino acid or four beads per nucleotide
models. Diffusion tensors
of
these molecules in dilute solutions were estimated by using
a hydrodynamic theory
to
take into account the diffusion anisotropy
of
arbitrary shaped
objects in the BD simulations. BD simulations
of
the system where each macromolecule
is
represented by its Stokes radius were also performed for comparison. Excluded
volume effects greatly reduce the mobility
of
molecules in crowded environments for
both molecular-shaped and equivalent sphere systems. Additionally, there were no
significant differences in the reduction
of
diffusivity over the entire range
of
molecular
size between two systems. However, the reduction in diffusion
of
GFP in these systems
was still 4-5 times larger than for the
in
vivo experiment.
We
will discuss other plausible
factors that might cause the large reduction in diffusion
in vivo.
1.
Introduction
One
of
the most characteristic features
of
the interiors
of
all living cells
is
the
extremely high total concentration
of
biological macromolecules. Typically,
20-30%
of
the total volume
of
cytoplasm
is
occupied by a variety
of
proteins,
nucleic acids and other macromolecules. Under these conditions, the distance
between neighboring proteins
is
comparable to the protein size itself, though the
molar concentration
of
each protein ranges from
nM
to
11M.
In this crowded,
heterogeneous environment, biomolecules work
to
maintain living systems and
413
414
they have evolved over several billion years. Therefore, modeling the crowded
cellular environment
is
not only an important first step toward whole cell
simulation but also a crucial factor in understanding the nature
of
living
systems.
In this study, we performed Brownian dynamics (BD) simulations
of
rigid
macromolecules in a crowded environment mimicking the cytosol
of
E.
coli to
study the motions
of
macromolecules. BD simulations using an equivalent
sphere system, where macromolecules were represented by their Stokes radius
were also performed.
It
has been reported that the diffusion
of
green fluorescent
protein
(GFP) in
E.
coli is
~10-fold
slower than in dilute conditions (1, 2). Our
aim is to investigate the mechanism(s) that causes this large reduction in
diffusion
in vivo.
2.
Methods
2.1. Estimation
of
diffusion tensor
of
a macromolecule
from
atomic
structure
To account for the diffusion anisotropy
of
macromolecules in our simulation,
the diffusion tensors
of
macromolecules were calculated
by
using the
rigid-particle formalism method (3-5). Here, we will describe this approach
briefly. The diffusion
of
an arbitrarily shaped object undergoing Brownian
motion is expressed by a 6
x 6 diffusion tensor,
D,
which is related to a
frictional or resistance tensor,
S, through the generalized Einstein relationship,
D = kBT S·l. Both D and S can be partitioned into 3 x 3 sub-matrices, which
correspond to translation
(tt), rotation (rr), and translation-rotation coupling (tr
and rt) tensors:
(
DII
D=
D"
where
D,,)
= k
T(SII
D
B -
.:I
rr
tr
_
)-1
.:I
"
- '
~rr
(1)
(2)
Here, the superscript
T indicates transposition. Translational and rotational
diffusion coefficients in a dilute solution are given by
Do"
=
1/3
Tr{D,,),
Do"
=
1/3
Tr{n,,),
where
Tr
is the trace
of
the tensor.
(3)
(4)
415
The components
of
E can be obtained by the following procedure. From the
Cartesian coordinates
of
the object consisting
of
N beads with the same radius, a,
the 3 x 3 hydrodynamic interaction tensors between beads i and},
Tij
(i,)
=
1,
...
,
N) are calculated using the expression formulated by Rotne and Prager (6) and
Yamakawa
(7), the so-called RPY tensor,
T.=
'J
l";j
~
2a,
(5)
Here,
1]
is
the viscosity
of
the solvent and
rij
is
the distance vector between
beads
i and
j.
It
is
important
to
note that the radius
of
bead
is
the only parameter
to be optimized to reproduce hydrodynamic properties in dilute conditions. In
what follows, we ignore intermolecular hydrodynamic interactions.
Now, consider a
3N
x
3N
supermatrix, B, consisting
of
N x N
Bij
blocks at
an arbitrary origin
0
[
B."
:..
B:N
1
B=
; .. ; ,
BNl
...
BNN
(6)
Bij
=6ij
6:17
a
+
(1-6ij)rij.
Here,
~ij
is
the Kronecker delta function. This supermatrix
is
then inverted
to
obtain a
3N
x
3N
supermatrix, C,
...
C 1
IN
. . ,
...
C
NN
(7)
in which each
of
the
Cij
block is a 3 x 3 matrix. Now, the elements
ofE
can be
written
as
Ell = LLCij'
E" =
LLU
i
·Cij'
(8)
E"
=-LLUi·Cij.U
j
,
where
416
U,
~[
~
-Zi
y, ]
0
-;;
.
(9)
-Y;
x;
Here,
Xi,
Yi,
and ziare the components
of
the position vector
of
bead i at origin
0.
So far the choice
of
the origin
of
the coordinates has been left arbitrary.
However, the diffusion tensor, D, especially translational and
translation-rotation coupling tensors, depends on the origin. At a certain origin,
the so-called center
of
diffusion
Q,
the translational diffusion coefficient reaches
a minimum. The position
of
Q with respect
to
the arbitrary origin
0,
is
calculated with the diffusion tensor obtained at
0,
Do,
as
XOQ
Tr,D Tr,O
_D
XY
Tr,O
-D"
D
Y
'
-D'Y
Tr,O
fr,O fr,O
[
1
[
DYY
+
D"
ROQ
= Y
OQ
= -
D;;,o
D
XX
+
D"
Tr,O Tr,O
]
_1[
]
-
D:'~o
Dt~O
-
Dt;~O
. (10)
D
xz
Z -
Oil
Tr,O
D
YY
+ D
XX
DXY
- D
YX
Tr,O Tr,O
tr,O tr,O
D at the center
of
diffusion Q are calculated by
where
D
=D
-U
·D
·U
+D
·U
-U
·D
!t,Q
It,D
OQ
rr,O
OQ
rt,O
OQ OQ
tr,O'
D
=D
-U
·D
rl,!)
rl,O
OQ
Tr,O'
D
=D
rr,Q
Tr,O'
U
OQ
=(
Z~Q
l-YoQ
o
YOQ]
-~OQ
.
(11)
(12)
A volume correction term for rotational and intrinsic viscosity estimation
is
applied in Eq. 8 in some studies (3, 4). However, significant deviations
of
calculated diffusion properties from experimental values were not observed
even without the correction.
2.2. Brownian dynamics
for
arbitrarily shaped objects
BD
is
one
of
the most important simulation approaches
to
investigate the
Brownian motion
of
arbitrary shaped objects, in which solvent molecules are
treated implicitly and the influence
of
solvent on solute particles is incorporated
through frictional and stochastic forces (8). In the high-friction limit, where it
is
assumed that momentum relaxation
is
much faster than position relaxation, and
when we treat the diffusion tensor
as
a constant, a BD propagation scheme for
an arbitrarily-shaped object can be written
as
(9)
417
Xi =
X~
+
kl:,.~
Di
·F
i
P
+gi(M),
B
(13)
where
I:,.t
is the time step and Xi is the vector describing the position
of
the center
of
diffusion and orientation
of
the i-th object,
(14)
Here,
rJ,
r2, and r3 are the position
of
the center
of
diffusion, and
qJ
I,
qJ2,
qJ3
describe its orientation.
FP
is a generalized system force having two components,
the force acting on the center
of
diffusion (f) and the torque
Cr):
(15)
g(l:,.t)
is a 6 x 1 random displacement vector during time step
I:,.t
due to the
Brownian noise, which satisfies the following relations:
(16)
Here
Di is the 6 x 6 diffusion tensor
of
object i at the center
of
diffusion.
Once this diffusion tensor is calculated as described above, we can compute the
random displacement vector using a Cholesky decomposition technique (9). The
Cholesky decomposition
of
the diffusion tensor D is determined as
D=S·ST,
(17)
where
S is a lower triangle matrix. The desired vector geM) is then obtained
by
the following:
(18)
where Z
is
a 6 x 1 vector, which has elements chosen from a Gaussian
distribution so that
(19)
In BD simulations, quatemions, q
=
(qQ,
ql,
q2,
q3),
were used for handling
rotations
of
rigid objects (8). Diffusion tensors
of
objects were evaluated in
body-fixed frames only once at the start
of
the simulation. The force and torque
on each object calculated in the laboratory or space-fixed frame
(f'
or T
S
)
were
converted to their body-fixed frame
(f' or T
b
) using the rotation matrix Q
obtained with quatemions,
(20)
418
For each step, quatemions were scaled usmg Lagrange's method
of
undetermined multipliers to satisfY
q2
= I (10).
2.3. Potential function
In this study, we considered only repulsive interactions between intermolecular
particles in BD simulations using a soft-sphere potential described by
(21)
where
rij
is
the distance between particles i
andj,
and
kss
is a force constant.
rm
is
ai
+
aj
+ fl, in which
ai
and
aj
are radii
of
particles i and j and fl is an arbitrary
parameter representing buffer distance between particles. In this study,
fl
of
2 A
and
kss
of
5k
B
TI
fl2
was used, which means
Vss
= 5k
B
T at the distance
rij
=
ai
+
aj.
2.4. Simulation conditions and analysis
All simulations were performed at 298 K with periodic boundary conditions.
For all simulation systems, ten independent simulations were run with different
initial configurations.
35
I-ls
simulations were performed with time step
of
0.5 ps.
Configurations
of
the systems were sampled every 1 ns. Trajectories for the first
5
I-lS
were discarded for analysis. The translational diffusion coefficient
of
a
particle in three dimensions is estimated by
([r(t
+ T)-
r(t
)]') =
6D
T , (22)
where
ret) is position
of
the particle at time t and T is time interval.
()
indicates the ensemble average over the same particle type and time
t.
3. Results
3.1. Estimation
of
diffusion tensor
of
a macromolecule from atomic
structure
0.18
0.16
,;<11
0.14
·0
8
0.12
s:-
"0
9
-g
0.10
0
Q
v
0.08
0.06
0.04
5.0
5.5 6.0
Radius
(A)
Translational
--B--
Rotational
-e-
6.5
7.0
419
Figure
1.
Errors in translational and rotational diffusion coefficients calculated by the rigid-particle
formalism as function
of
bead radius.
Do"l
and
Doex
p
represent the translational or rotational
diffusion constant in a dilute condition calculated by theory and estimated by experiment,
respectively. Values for translational (open squares) and rotational (open circles) diffusion
coefficient were averaged over the twelve proteins listed in Table 2
of
Ref. (4) except for lactose
dehydrogenase. Lines are drawn to guide the eye.
To estimate diffusion tensors
of
macromolecules from their atomic structures,
the rigid-particle formalism was used (3-5).
As
described in Methods, the
particle radius
is
the only one parameter optimized to reproduce the
experimental translational and rotational diffusion coefficients
of
macromolecules at infinite dilution. The bead radius was optimized using the
twelve different proteins whose molecular weight ranges from 6 kDa to
230 kDa,
which are the same
as
those used in Ref. (4) except for lactose dehydrogenase.
Proteins were represented by Ca beads. Figure I shows that the difference
between experimental and calculated diffusion coefficients average over the
twelve proteins
of
Ref. (4)
as
a function
of
bead radius. Radius of6.1 A gave the
minimal error in both translational and rotational diffusion coefficients. For
example, the method provided translationla diffusion coefficient for
GFP
of
8.9
A2/ns
at
293
K, which
is
excellent agreement with experimental value
of
8.7
A2/ns
for this protein in dilute conditions at room temperature (11,
12).
In order
to simulate the inside
of
cells, we treat not only proteins but also nucleic acids.
Thus, diffusion coefficients
of
a Phe-tRNA (PDB
ID:
lEHZ) were also
evaluated using the same method. Nucleic acids were represented by
P,
C4',
NI,