Entropic trapping of a singly charged molecule in solution

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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 conne a single

perimental approaches to the spatial control

molecule in solution by spatial modulation of

and manipulation of single nanoscale entities in

its local congurational 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 insucient 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 congurational 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 congurational entropy of a thermally

of

A wealth of techniques exploit-

and stochastic

of the trapped state depends exponentially on

magnitude to stably conne 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-

congurational 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

congurational 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 conne 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 conning 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 conning an electrically charged

port timescale. In general one relevant conning 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

congurational

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 conning sur-

faces carries lattices of nanostructured indenta-

pected due to congurational 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 coecient.

Al-

is imaged under purely diusive conditions by

though congurational 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 congurational 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).


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 conned 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

Briey, 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 congu-

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 congurational 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 congurational en-

pr (z)Fr (z)dz 0

tropy of single molecules in a conned spatially modulated system.


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


16

).

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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 congurational entropy. Intially we probed the contribution of

in the system

qe

fS to the total trap W , by measuring escape times of a given

congurational entropy,

β,

depth,

(Fig. 2), we nd an interesting de-

molecular species in traps created by nanos-

pendence of the congurational 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 connement

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 congurational 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 congurational 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.

Congurational 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 congurational 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 congurational 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

conne

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.

congurational 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


tesc ,

Page 9 of 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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 congurational 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. Congu-

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-specic 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 congurational 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 congurational 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

conned in a slit such

Supporting Movie


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Entropic trapping of a singly charged molecule in solution

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