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Entropic trapping of a singly charged molecule in solution Francesca Ruggeri, and Madhavi Krishnan Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b01011 • Publication Date (Web): 24 Apr 2018 Downloaded from http://pubs.acs.org on April 24, 2018
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Nano Letters
Entropic trapping of a singly charged molecule in solution Francesca Ruggeri† and Madhavi Krishnan∗ † ‡ , ,
†Department of Chemistry, University of Zu¨rich,Winterthurerstrasse 190, CH 8057 Zu¨rich,
Switzerland
‡Department of Physics, University of Zu¨rich,Winterthurerstrasse 190, CH 8057 Zu¨rich,
Switzerland
E-mail:
[email protected] Abstract
static double layer forces
We demonstrate the ability to con ne a single
perimental approaches to the spatial control
molecule in solution by spatial modulation of
and manipulation of single nanoscale entities in
its local con gurational entropy. Previously we
the uid phase remains an area of great cur-
established electrostatic trapping of a charged
rent interest.
macromolecule by geometric tailoring of a re-
ing externally applied optical elds,
pulsive electrical interaction potential in a par-
dependent electrical elds both determin-
allel plate system. However, since the lifetime
istic
Introduction.
810
mophoretic
the electrical charge of the molecule, the elec-
been
trostatic interaction alone is often insu cient in
need.
a net charge of magnitude
≤ 5 e.
as well as ther-
and hydrodynamic elds
reported
to
time-
address
this
13
have
experimental
technique to trap electrically charged molecular scale matter in solution.
uctuating molecule in a geometrically modu-
1416
The approach
utilized a geometry induced local minimiza-
lated system can be exploited to spatially con-
tion of an electrostatic interaction free energy
ne weakly charged molecules in solution. Mea-
for an object in solution, and thus circum3 vented the unfavorable a scaling of trap depth
surement of the con gurational entropy contribution reveals good agreement with theoreti-
with object size,
cal expectations. This additional translational
a,
common to polarizability-
dependent external- eld based approaches.
contribution to the total free energy facilitates
17
Here we demonstrate that an additional contri-
direct optical imaging and measurement of the
bution to the well depth from the translational
eective charge of molecules on the size scale
∼1
12
11
17
We recently introduced an external eld-free
Here we show
that the con gurational entropy of a thermally
of
A wealth of techniques exploit-
and stochastic
of the trapped state depends exponentially on
magnitude to stably con ne molecules carrying
The development of new ex-
entropy of the object itself may be harnessed to
nm, and charge as low as 1e physical
substantially enhance trap stability. Exploiting
properties comparable with those of a monova-
con gurational entropy enables us to extend the
lent ion in solution.
operation of the geometry-driven trapping concept into the regime of weakly charged entities,
e
Keywords
carrying a charge on the order of 1 , where the
con gurational entropy, single-molecule trap-
ergy is very small,
ping,
mann's constant
electrostatic contribution to the interaction en-
eective charge measurement,
electro-
ACS Paragon Plus Environment 1
∼1 kB T , where kB is and T is temperature.
Boltz-
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The use of entropy to spatially con ne long
Page 2 of 12
thermodynamic approach to trapping electri-
14,15
polymer molecules in a geometrically tailored
cally charged matter in aqueous solution.
landscape has been reported previously, pri-
The working principle of such a trap is based
marily in the context of polyelectrolyte sepa-
on the equilibrium repulsive electrostatic inter-
rations.
Local variations in height in a con-
action between a charged object in solution and
ned system modulate the conformational en-
like-charged con ning parallel plates (Fig. 1a).
tropy of a polymer by altering the number con-
Geometric tailoring of the parallel plates, by a
formational states accessible to the molecule.
nanostructured indentation of depth,
The deeper regions in the system give rise to
dius,
entropic traps that retain molecules for a period
sults in a local interaction energy minimum that
much longer than the typical diusive trans-
is capable of con ning an electrically charged
port timescale. In general one relevant con ning length scale has to be at least of the same
molecule for long periods. The Debye length p κ−1 = m 0 kB T /2cNA e2 , represents the range
order as the molecule's radius of gyration in or-
of the electrostatic interaction in solution and
der to eectively probe its conformational de-
is typically 10 nm in this work.
grees of freedom.
Avogadro's number,
1820
2123
R,
constant of water,
Here we show that the entropic principle may
d
and ra-
both larger than the Debye length, re-
c
m = 78.5
0
Here
NA
is
is the dielectric
is the permittivity of free
be applied to spatially trap hard-sphere enti-
space, and
ties such as globular macromolecules, and even
the experiment. In the limit of strong electro-
small organic molecules, that possess no rel-
static interactions (e.g., high molecular charge)
evant internal conformational degrees of free-
we have measured molecular residence times in
dom.
the trap as long as
This is due to the
con gurational
en-
is the bulk salt concentration in
∼30 min. 16
Residence times
tropy of the hard-sphere object, arising from
can be tuned by the geometry of the trapping
translational freedom in the axial dimension,
nanostructure and salt concentration in solu-
which can change substantially as a function
tion.
16,32,33
of spatial position in a corrugated landscape.
In this work we use free-energy landscapes
We nd that with appropriate choice of dimen-
created in a parallel plate slit of typical height,
sions, well depths of up to 5
kB T
2h = 70 − 80
may be ex-
∼100
nm.
One of the con ning sur-
faces carries lattices of nanostructured indenta-
pected due to con gurational entropy alone. In
R = 200 − 400 d = 100 − 330 nm.
tions of radius,
nm (Fig. 1a)
hancement of the residence time given by free
and depth,
Fluorescently
diusion.
The eect has been previously ex-
labeled macromolecules in solution are intro-
amined in theoretical studies on particle trans-
duced into the lattice at a concentration of 150
port in corrugated channels, and the local en-
pM by capillary ow in a buer containing 1
hancement of states accessible by the particle is
mM Tris and 0.25 - 2 mM NaCl.
sometimes interpreted in terms of a reduction
is arrested and molecular motion in the lattice
in local or overall diusion coe cient.
Al-
is imaged under purely diusive conditions by
though con gurational entropy in many body
wide- eld uorescence microscopy as previously
systems has been examined extensively in ther-
described.
modynamic measurements at the macroscopic
such as short DNA fragments, namely 10 and
scale,
terms of timescales this implies
2731
fold en-
2426
16
16
The ow
We study charged macromolecules
few experiments thus far have probed
20 b ssDNA and and 60 bp dsDNA (Microsynth
the translational entropy associated with the
AG, Switzerland), intrinsically disordered pro-
spatial uctuations of a single entity, e.g., an
teins,
Starmaker-like
(Stm-l)
atom, or molecule in solution.
mosin
α
and also examine small
Experimental set-up and measurement approach. In order to explore the possibil-
(ProTα),
35,36
34
and
Prothy-
uorescent organic dye molecules with dier-
qstr , namely = −1e), ATTO 532ATTO 542-carboxy (−4e)
ent nominal structural charges,
ity of using con gurational entropy to trap a
ATTO 532-maleimide (qstr
single molecule, and to experimentally measure
carboxy (−2e) and
this quantity, we employ our recently developed
(ATTO-Tec, Germany).
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Prior to the experi-
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Nano Letters
Figure 1: time,
tesc ,
(a) Schematic representation of the experimental setup. We measured the average escape of 60 ds DNA in a device where nanoslits of height
2h
are patterned in an alternating
fashion with lattices of nanostructured circular indentations of depths, nm, and radius,
R = 200nm.
(b) While the electrostatic well depth,
d1 = 130
nm and
d2
= 330
qe ψm is the same in both cases, ∆f = f2 −f1 , by
the trap created by the deeper indentation entails a larger uctuation contribution,
kB T in this case. (c) Probability density distributions of experimentally recorded ∆t, t with a form P (∆t) = At exp(−∆t/t), where A ≈1. (d) Measured average escape time tesc , well depth W , uctuation contribution f and measured values of eective charge, qm for 60ds DNA for the two cases. The ratio of the measured escape times, tesc,2 /tesc,1 = 2.66 ± 0.44 is in ∆f excellent agreement with the expected value, exp = 2.72 (see text for details and Supporting kB T
approximately 1 escape times,
Movie).
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Page 4 of 12
in buer was reduced with 2 fold molar excess of
Free energy of a single molecule in a geometrically tailored landscape. The single
2-Mercaptoethanol and sonicated extensively,
particle partition function serves as an appro-
in order to minimize aggregation.
priate starting point for a complete thermody-
ment, the ATTO 532-maleimide dye dissolved
namic analysis of a particle in a spatially mod-
For an object con ned in a potential well
40
in the uid phase, overdamped diusive cross-
ulated free energy landscape.
ing of a barrier is well described by Kramers'
partition function for a point particle whose
theory in the regime
W > 5kB T ,
center is located at r(x, y) in the landscape as Rz qr = 0 max exp(−Fr (z)/kB T )dz where zmax denotes the maximum axial extent of the gap at
where the
average time to escape the potential well is given by
tesc = tr exp(W/kB T ). 16,37
Here
tr
We write the
is
a time scale representing the position relax-
r.
ation time of the molecule. Brownian Dynam-
pocket region,
ics simulations that take into consideration the used to convert the measured average escape
cal axial occupation probability of the particle exp(−Fr (z)/kB T ) is given by pr (z) = qr In our analysis, Fr (z) = Ur (z) − T Sr (z) rep-
time, tesc of a trapped molecule to a well depth,
resents the electrostatic interaction free energy
W
of the particle located at
full 3D morphology of the potential well, are
16,33
(see Supporting Information for further
zmax = 2h, while in the zmax = 2h+d (Fig. 2a). The lo-
In the slit region,
(r, z).
Where needed
Fr (z) can be
details). Since the well depth in turn depends
for comparison with measurement,
directly on the eective charge of the molecule,
calculated for a given set of experimental pa-
qe ,
rameters as previously described.
38
we have previously achieved highly pre-
16,41
Brie y, it
of the ef-
is a volume integral over the whole system, in-
fective charge of a variety of biomolecules using
cluding contributions from both the electrical
the escape-time based measurement approach
eld-energy,
cise measurements (precision
∼ 1%)
U
as well as the entropy of mix-
described above, which we term escape-time
ing of the ions in solution,
electrometry (ET ).
rived by Overbeek.
e
16
The depth of the wells,
42
S,
as originally de-
As has been shown al-
W , in previous work is typically 5-6 kB T , yielding trap residence times of ∼50-200 ms. For molecules of eective charge |qe | < 5e,
ready, for all practical purposes the electro-
under comparable experimental conditions, the
as
electrostatic interaction alone contributes not
represents the eective charge of a molecule car-
more than about 1
kB T
to the trap depth,
static interaction free energy for an object in the landscape may be written in a simpler form
Fr (z) = qe ψr (z). 38
Here, the parameter
rying a structural charge,
W.
qstr ,
This is often too small to yield molecular resi-
electrical potential in the slit at
dence times of about 5-10 ms - a minimum to
absence of the particle. The total free energy,
facilitate observation of trapping and long-term
w
ψr (z) is (r, z), in
and
function of its location in the landscape,
rophore labels.
thus
We demonstrate that con gu-
wr = −kB T lnqr .
the the
of the particle as a
imaging using conventional detectors and uo-
39
qe
r,
is
This total free energy
rational entropy can be used to greatly enhance
includes the contribution of axial position uc-
trapping times in the regime of weak electro-
tuations of the particle and can be decomposed
static interactions.
Interestingly, inclusion of
into an electrostatic interaction energy part and
the con gurational entropic contribution also
and a spatial uctuation entropy component,
enables measurements of the eective charge
such that
on single molecules in the weak electrostatic
to the average electrostatic interaction energy,
regime, albeit with comparatively lower preci-
and entropy, respectively, of an axially uctu-
sion (5-20%) in a single measurement.
ating particle at
Sub-
wr = ur − T sr
r,
. Here
u
and
s
refer
and are given by
tracting the electrostatic contribution to the trap depth, characterized in previous work, we
Z ur =
report measurements of the con gurational en-
pr (z)Fr (z)dz 0
tropy of single molecules in a con ned spatially modulated system.
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zmax (1)
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Figure 2: system.
Nano Letters
(a) Cross-sectional view of a trapping nanostructure.
β
denotes the ratio of heights in the
Representative axial probability distributions of particle position,
P (z),
in the pocket
(b) Calculated axial uctuation entropy, fS for various combinations β , system size, κh, and qe . qe = 0 depicts the ideal-gas case. (c) fS vs. ln β for values of qe = −1e (grey), −10e (light blue), −30e, −50e and −80e (green) from bottom to top, and κh = 3. The behavior between qe = −10e (light blue) and qe = −80e (green), in the regime β > 4, is ακ−1 +d where α ' 1 − 2 better captured by a phenomenological expression of the form kB T ln ακ−1 (dashed lines). (d) Normalized probability distributions, Pn (z) in the slit half-space for various qe and β = 10. Calculations yield the magnitude of eective slit height, he , underlying the obtained value of fS . ∼ 98% of the total axial sampling probability is contained within he . Note that in −1 the limit of large particle charge, 2he = ακ in the slit, and the corresponding eective height −1 in the pocket region is ακ + d. (e) fS vs. qe for various combinations of β and κh. The ts |qe /e|b shown are of the form fS = kB T ln( + d), where b = 0.5 for larger values of κh and β (circles: c c = 0.18, d = 5.6; squares: c = 0.4, d = 4.1); b = 0.3 for small κh (inverted triangles: c = 0.24, d = 1.8). For small values of both κh and β , f is essentially independent of qe (upright triangles, inset). κh = 4, β = 6 (squares) is typical for these experiments. and slit domains (blue).
of
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Page 6 of 12
and
zmax
Z sr = −kB
pr (z)lnpr (z)dz
(2)
0 For a given value of
qe
the above integrals
are all a function of spatial electrical poten-
ψ(x, y, z)
tial,
alone, which is readily obtained
by solving the non-linear Poisson-Boltzmann equation in the nanostructure as previously described.
14
We use constant charge boundary
conditions on the slit walls that correspond to a value of
ψs = −2.8kB T /e
for the surface
potential of the walls, which we have found to hold under our experimental conditions.
33
From Eq.1 we further note that uctuations render the particle's mean electrostatic energy, Rz ur = qe 0 max pr (z)ψr (z)dz slightly larger than its electrostatic energy at the midplane of the
qe ψm,r ,
slit,
ψm,r
typically by about 5%.
Here
denotes the electrical potential midway be-
tween the parallel surfaces at any lateral location
r(x, y)
(Fig. 1b).
wr in a more physias wr = qe ψm,r + fr ,
We may thus write cally intuitive form
where the rst term indicates the total particle free energy at
r
in the absence of par-
ticle position uctuations.
fr = fS,r + fU,r ,
The second term,
fS,m fS,c values of the axial uctuation entropy, fS for three molecular species, 60bp dsDNA (circles), ProTα (squares) and Stm-l (triangles) using values of β from 2 to 6. (b) Measurements of average escape times, tesc for 10b ssDNA, 20b ssDNA, ProTα and 60ds
Figure 3:
denotes the uctuation con-
tribution, consisting of a larger entropic part,
fS,r = −T sr and a smaller nent, fU,r = ur − qe ψm,r .
energetic compoFinally, we write
the total well depth as the dierence between the particle free energy inside and outside the
W = wr |slit − wr |pocket = qe ψm + f . Here f = fS + fU , and the absence of the subscript r denotes a quantity that is the dierence between values at two in-plane spatial locations r ,
trap,
DNA, under the same experimental conditions (β
ψm
= 5.6
and
κh ∼ 4).
(c) The table presents
the fractional contribution of well depth
namely the slit and the pocket (Fig. 2a). In general we use
(a) Comparison of measured,
and calculated,
W,
fS,m
to the total
the number of escape events,
N,
recorded for each species and the average # measured escape time, tesc . denotes that qm
to simply refer to the electri-
cal potential at the midplane in the slit region,
for 60bp DNA was set equal to
as the corresponding electrical potential in the
permitting a calibrated measurement of all the
pocket region is zero in most experimental
other species in the same experiment. Measure-
situations.
ment uncertainties are standard error of the mean (s.e.m.).
qc = −43.4e,
The values obtained for
qm
in
The Gibb's entropy equation Eq.2 gives the
the uctuation entropy dominated regime are
additional entropy due to position uctuations
in excellent agreement with previous measure-
of the center of mass of the particle.
ments (last column
Note
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).
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Nano Letters
fS
that as we are interested in free energy dier-
behavior of
ences we have dropped a multiplicative constant
tive charge, over a 2 order of magnitude range
in the arguments of the logarithms.
in
We nd
qe
essentially independent of eec-
(Fig. 2e, inset). Finally we note that par-
that in the ideal-gas limit of a point parti-
ticle position uctuations in the radial dimen-
cle with no interactions or equivalently, for
sion in the trap are explicitly accounted for in
an uncharged system the well depth,
W = kB T lnβ as β = 2h+d 2h
BD simulations of the 2D escape process (see
wr |slit − wr |pocket = fS ,
Supporting Information, Section I).
reduces to
expected, where the parameter denotes the ratio of (Fig. 2a). of
fS
κh,
heights
Analyzing further the dependence
on experimental parameters, namely,
and
Probing the eect of con gurational entropy. Intially we probed the contribution of
in the system
qe
fS to the total trap W , by measuring escape times of a given
con gurational entropy,
β,
depth,
(Fig. 2), we nd an interesting de-
molecular species in traps created by nanos-
pendence of the con gurational entropy on the
tructured indentations of two dierent depths
charge of the object, rameter,
κh.
qe , and the system size pa-
(β1
= 2.85, β2 = 5.71)
in the same device, and
We note that for strongly charged
observed a ratio of average escape times that
entities the translational entropy contribution
agreed well with the theoretical expectation
can be substantially larger that the ideal-gas
(Fig. 1d). In order to perform a broader quan-
expectation.
This essentially implies that the
eective height,
he ,
titative comparison of measurements of
fS
with
available for particle uc-
the theoretical expectation, we measured tesc on
tuations in the slit region is much smaller than
dierent molecules 60bp dsDNA, ProTα and
the physical hard-wall slit height (Fig. 2a,d). In
Stm-l and obtained measured values of
fact we nd that in the regime of strong elec-
values of
trostatics, the slit region provides con nement
from 3.8 to 5.6 (see Supporting Information).
within an axial extent on the order of the De−1 −1 bye length, κ . Since in this work κ ∼ 10
The calculated values of eective charge,
nm is much smaller than the slit height
kB T
−31e
qc
Furthermore, under a given set of experithat
fS
and
β,
and
qc , −43.4e,
respectively under our ex-
16,32
We assumed that
qe =
in each case, and determined the theoret-
using Eq.2 in the expression fS,c = T sr |pocket − T sr |slit . The experimentally measured value of fS,m in turn was obtained using the measured value of W and Eq.1 in the rela tion fS,m = W − ur |slit − ur |pocket . Note that
The strongest p |qe /e| in the intermediate value of κh
dependence we encounter is a ln
β
ranging
tropy
charge of the object (Fig. 2e).
∼3.
−89.6e
for
ically expected value of the uctuation en-
we nd
also displays some dependence on the
regime of large
κh
the molecules carry an eective charge
object (Fig. 2c).
κh
and
perimental conditions.
for a highly charged
mental conditions, namely
ranging from 2 to 6 and
for 60ds DNA, ProTα and Stm-l are
2h ∼ 70
nm, the con gurational entropic contribution increases by about 2
β
W
This behavior can be explained by the
previous measurements of the eective charge
fact that in the pocket region the molecule en-
of these molecular species agreed well with the
counters an approximately square-well poten-
calculated
tial of axial extent
2h + d,
regardless of its
qc
values.
16
Fig. 3a displays a comparison of measure-
charge (Fig. 2a). In the slit however the same
ments and theoretically expected valued of
molecule encounters an parabolic electrical in2 teraction energy, F (z) = qe ψ(z) ∝ qe (z − h) ,
for the three molecular species for various val-
resulting in the observed overall ln pendence. screening, tio,
|qe /e|
ues of
de-
κh = 1.5,
β.
We note good agreement between
theory and experiment with an r.m.s.
Interestingly in the limit of weak
β = 3,
are strong
p
fS
devia-
tion over all measurements considered within
∼10 %.
and moderate height ra-
although electrostatic interactions
We point out that the measured val-
ues presented are from single experiments using
per se, the con gurational entropy in
the nominal value of
2h
in Eq.1 and Eq.2.
16,32
both spatial domains responds in a similar fash-
Since the nominal slit height in a given mea-
ion to molecular charge, rendering the overall
surement may dier from the true value by up
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Page 8 of 12
to 3 nm, the agreement could in fact be much better than shown.
16,32
Eective charge measurements on weakly charged molecules. Using our model described above, which we nd correctly accounts for the additional particle uctuation entropy,
we demonstrate that it is possible
to operate in the entropy-governed trapping regime and still determine the unknown effective charge of a molecule using the ETe approach.
Con gurational entropy enhances
molecular residence times in the trap by at least a factor 3 compared to the electrostatic limit. Fig. 3b reports measurements of the effective charge of various biomolecular species in the regime where the trap depth is dominated by con gurational entropy, i.e.,
qe
The obtained
fS /W ≥ 0.4.
values compare well with
previous measurements in the electrostatically dominated limit (Fig. 3c).
16
The ability to use
ETe to measure molecular eective charge in a regime dominated by con gurational entropy suggests the feasibility of applying the approach not
only
(|qe |
to
> 5e)
highly
charged
macromolecules
but also to weakly charged, small
organic molecules or even ions in solution. To conclude, we demonstrate the ability to spatially
con ne
and
measure
the
eective
charge of single organic molecules that are typically about 1 nm in diameter and carry a net structural charge of (Fig. 4).
qstr = −1e, −2e
and
−4e
Since these molecules have a hynm, both their size
Figure 4: (a) Chemical structures of the ATTO
and charge are reminiscent of hydrated mono
532 dye molecules with maleimide and car-
and multivalent ions in solution.
boxylic acid functional groups. (b) Probability
drodynamic radius
∼ 0.5
43,44
Perform-
P (∆t) of recorded escape
ing ETe on these molecular species, we note
density distributions,
very comparable average escape times despite
times for ATTO 542-carboxy (orange squares)
the large disparity in charge in these molecules
and ATTO 532-carboxy (green circles), mea-
(Fig. 4b-d).
sured in the same experiment (β
W
This is due to the fact that
3.5),
in these measurements arises largely from
and for ATTO 532-maleimide (empty tri-
κh = 2.
con gurational entropy and is thus essentially
angles) measured using
charge independent. Nonetheless the weak elec-
of the measured eective charge,
trostatic contribution to
W
nal structural charge,
permits eective
= 5.6, κh =
qstr
(c) Comparison
qm
and nomi-
for the three molecu-
We
lar species. Error bars are s.e.m. and represent
nd that our measured values are indeed close
the statistical measurement uncertainty alone
charge measurements on these molecules.
values. The measurements
(See Supporting Information, Section II). (d)
also agree with the calculated eective charge
Details on the experimental conditions for the
to the nominal
qstr
obtained by modelling each dye as a
three species and their mean escape times,
sphere of radius given by the hydrodynamic ra-
averaged over 10-20 molecules per species.
values,
qc
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tesc ,
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Nano Letters
h − a = 10 nm with a nanostructure depth d = 600 nm, well-depths of over 4kB T can be
dius, carrying a uniformly distributed total net charge of
qstr ,
as previously described.
that
38
achieved due to con gurational entropy alone.
Despite the fact that the measurement uncertainty on tesc is statistically limited, given 1 by √ , where N is the number of escape N events recorded, we note a rather large uncertainty on
qm
of around
10 − 15%
Our ndings also carry strong implications for the electrostatic trap-based biomolecular charge measurement principle we recently in-
in a sin-
troduced. This work establishes the applicabil-
gle measurement on weakly charged molecules
ity of the ETe approach for molecular eective
(Fig. 4d).
charge measurements of magnitude
This is due to the fact that in the
low well depth regime,
W < 5kB T , and partesc is close
ing from 1 to 100
e.
|qe |
rang-
The ability to measure
ticularly for small molecule where
eective charge in the regime of very weak elec-
to the sampling time, simulations show that
trostatics (qe ψm
the measured
tesc
exp(W/kB T )
suggests that the
W
ETe measurement principle can be readily ap-
(see Support-
plied to charged biomolecules in solutions with
displays a dependence on
much weaker than
< 1kB T )
It follows that
higher salt concentrations, where electrostatic
the fractional uncertainty of a single measure-
interactions are typically diminished. Con gu-
ment in this regime can be approximated as
rational entropy also greatly enhances the dy-
ing Information, Fig.S2(b)).
√ kB T / 0.6qe ψm N
ranging from 0.4 to
namic range of the measurement in a single
∼ 5 − 20%, with qe ψm 1 kB T (see Table in Sup-
porting Information).
experiment.
measuring in real-time, changes over an order
The overall accuracy
of magnitude (Fig. 3) in the eective charge
can be improved either by including a cali-
of a single diusing entity using e.g., the pre-
bration molecule" in a single measurement, or
viously described lattice diusion approach.
by averaging over several independent measurements.
16,32
ber of dynamic inter- and intramolecular processes that strongly impact biomolecular elec-
possible in future to trap and measure the ef-
trostatics, including binding, nucleation and ag-
fective or renormalized charge of inorganic ions
45,46
33
This ability would be useful in studying a num-
Our observations suggest that us-
ing ion sensitive uorescent dyes it should be
in solution.
This opens up the possibility of
gregation, folding and conformational changes,
It may also be possible to use
and ion-speci c eects.
these measurements to infer the spatial distri-
Acknowledgement
bution of charge or to better understand the in-
We gratefully acknowl-
terplay of individual ionizable groups in small
edge the Swiss National Science Foundation,
molecules in solution.
the European Research Council and University
Conclusions.
A quantitative view of the role
of Zurich for nancial support. We thank Ben
of con gurational entropy could play an impor-
Schuler and Andrzej Oz˙ yhar for gifts of the pro-
tant role in optical microscopy-based measure-
teins ProTα and Stm-l.
ments of interaction energies of particles and
carried out at FIRST Center for Micro- and
molecules in solution.
Nanoscience, ETH Zurich.
Moreover, the ability
Nanofabrication was
to trap and visualize single molecules in the regime of negligible electrostatic repulsion (as low as about 0.4
kB T ,
Supporting Information Avail-
Fig. 4d and Table in
able
Supporting Information) strongly suggests that trapping based on con gurational entropy alone should be possible in a completely uncharged
The following les are available free of charge.
system - where neither the molecule nor the surfaces carry electrical charge.
For instance
Details on Brownian Dynamics simula-
in an entropic uidic trap composed of slit
tions and Error Propagation, Fig S1 and
surfaces coated with a neutral lipid bilayer, for
S2.
a molecule of radius,
a
con ned in a slit such
Supporting Movie
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