Statics and Dynamics of DNA in aNetwork of Nanofluidic Entropic Traps

Alexander R. Klotz

Master of Science

Department of Physics

McGill University

Montreal, Quebec

June 13, 2011

A thesis submitted to the Faculty of Graduate Studies in partial fulfilment ofthe requirements for the degree of Master of Science

c©Alexander R. Klotz, 2011

ACKNOWLEDGEMENTS

First and foremost I would like to thank my supervisor, Walter Reisner,

for introducing me to this project and guiding me through it. I also wish to

thank Hugo Brandao, from whose hard work I benefited immensely. I would

also like to thank the other members of the Reisner group, Robert Welch and

Yuning Zhang, for their support.

I am indebted to the technical staff in the McGill phyics department, partic-

ularly Mathieu Nanini, as well as Herve Berrier from Nikon. I also appreciate

Peter Grutter and Fernando Suarez-Sanchez for allowing me to use their lab-

oratory and equipment before ours had arrived. My work would have been

impossible without generous funding from McGill University and the Molson

and Hilton Hart Fellowship in Science.

Above all, I am grateful for my friends and family in Montreal, Toronto, and

around the world who kept me sane during this process.

2

ABSTRACT

A nanofluidic slit embedded with a lattice of square pits was used to en-

tropically trap polymers. DNA in the system was confined to two dimensions

and underwent self-assembly into discrete conformational states based on the

number of occupied pits. The molecules diffused by undergoing transitions to

higher or lower pit-occupancy states and relaxing to their equilibrium state.

A statistical mechanical model was used to predict the mean occupancy state

as a function of various geometric parameters. Experiments confirm many of

the predictions of the model. Regions of parameter space over which a sin-

gle state dominates were observed, indicating that entropic trapping can be

used to create stable self-assembled single polymer conformations. Measure-

ments of diffusion showed it to be geometry dependent, allowing a fine-tuning

of diffusivity. The diffusion showed non-monotonic behaviour: local minima

corresponding the stable equilibrium states were observed. This demonstrates

that the diffusion can be fine-tuned to a local resonance using entropic trap-

ping. The results show that polymers can self-assemble into entropically stable

structures, with implications for nanotechnology and biophysics.

3

ABREGE

Nous avons piege des polymeres avec un trellis de fosses nanofluidiques

de forme carree. Nous avons confine l’ADN en deux dimensions. Il s’est auto-

assemble en etats discrets et qui determine du nombre du fosses. Les molecules

one saute et sont tombe es dans un nouvel etat, et diffusaient. Nous avons

utilise un modele mecanique statistique pour une predire le nombre moyen de

fosses. Les experiences confirment le modele. Nous avons vu des plateaux

avec un unique etat dominant. Cela signifie que nous pouvons utiliser le piege

entropique pour la nanotechnologie : car les structures sont stables. Nous

avons mesure la diffusion. Elle depend de la geometrie et est non monotonique.

Elle a un minimum local vers la region des plateaux. Nous pouvons utiliser le

piege entropique pour controler la diffusion. Les resultats sont utiles pour la

nanotechnologie et biophysique.

4

TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . 2

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . 71.2 Polymer Physics . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Micro- and Nanofluidics . . . . . . . . . . . . . . . . . . . . 171.4 The Nanopit-Nanoslit System . . . . . . . . . . . . . . . . . 19

1.4.1 Table of Symbols and Values . . . . . . . . . . . . . . 28

2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Slit Free Energy Models . . . . . . . . . . . . . . . . . . . . 322.3 Extension to Diagonal Configurations . . . . . . . . . . . . . 352.4 Modelling of Diffusion . . . . . . . . . . . . . . . . . . . . . 392.5 Parameter Space Behaviour . . . . . . . . . . . . . . . . . . 44

3 Apparatus and Methods . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Lab-on-a-Chip: Fabrication and Geometry . . . . . . . . . . 473.2 Chuck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4 Buffer and DNA Preparation . . . . . . . . . . . . . . . . . . 503.5 Experimental Protocol . . . . . . . . . . . . . . . . . . . . . 51

3.5.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . 513.5.2 Experimentation . . . . . . . . . . . . . . . . . . . . . 53

3.6 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 543.7 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5

4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Qualitative Behavior . . . . . . . . . . . . . . . . . . . . . . 604.2 Occupancy Measurements . . . . . . . . . . . . . . . . . . . 614.3 Comparison to Theory . . . . . . . . . . . . . . . . . . . . . 674.4 Diffusion Measurements . . . . . . . . . . . . . . . . . . . . 734.5 Implications, Applications and Extensions . . . . . . . . . . 82

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6

CHAPTER 1Introduction

1.1 Background and Motivation

Richard Feynman in his 1959 speech There’s Plenty of Room at the Bottom

[14] envisioned a future in which machines could be created on the atomic

scale. In the subsequent half century, nanotechnology has progressed to the

point where nanoscale electronic devices are created routinely in commercial

fabrication plants, and single atoms can be manipulated and oriented using

techniques such as scanning tunnelling microscopy. The former, “top down”

fabrication, lacks resolution: it is increasingly difficult to fabricate structures

with length scales smaller than 10 nm [63]. The latter, “bottom up,” allows

atomic resolution but lacks speed and scalability (arranging atoms one by one

can take hours or days). The goal of nanotechnology is to build nanostructures

at the atomic or molecular scale with assembly-line efficiency. Using cues from

nature, biological materials that are routinely produced in organisms for small-

scale structures make an ideal substrate for nanotechnology. Polymer self-

assembly is proposed as a technique that satisfies both criteria. We investigate

deoxyribonucleic acid (DNA) as a potential substrate for self-assembly.

Polymer self-assembly in the area of nanotechnology is a relatively re-

cent field. A notable success is DNA “origami,” by which structures can be

created out of folded DNA molecules with nanometer resolution [52]. DNA

is a useful material for these constructions due to its self-binding properties

7

that result from Watson-Crick base pairing. This remarkable technology al-

lows self-assembled nanostructure both on surfaces and in three dimensions

[19]. Another use of polymers for bottom-up nanoscience is in the area of col-

loidal nanostructures. Nanoparticles tagged with DNA can self-assemble into

various supercrystal structures, and the characteristics of the supercrystal can

be tuned by varying, for instance, the length of the DNA [38]. Polymer self-

assembly has been used in surface science, for the assembly of complex surfaces

[43], the patterning of nanoparticles [28], and for fine tuning of surface prop-

erties [45]. Self-assembled structures of DNA have been used as a scaffold for

the creation of nanowires [16] and single carbon nanotube circuits [31].

Methods used to manipulate polymers on this scale include modifying the

chemical environment and deposition of oligonucleotide “staples” [52]. Rather

than actively dictating the structure of the polymer, we are interested in con-

trolling molecular conformation and diffusion by tailoring the equilibrium state

of the molecule to yield the desired structures. In the experiments described in

this thesis, a free energy landscape is created by varying the local topography,

which dictates the partitioning of polymer contour into the desired nanostruc-

tures. Ultimately, the polymer, at equilibrium, will self-assemble without ex-

ternal stimulus. The free energy landscape is created by entropic traps: regions

that allow greater conformational degeneracy and thus confine the molecule. A

polymer in an entropic trap can be envisioned as a particle in a potential well,

undergoing thermal excitations that may allow it to escape. Entropic traps

have previously been used to demonstrate various thermodynamic properties

of polymers [8], and entropic forces have even been proposed as a microscopic

mechanism of gravity [62]. The tool that allows polymers to be controlled

8

individually rather than in bulk is nanofluidics: confining the molecule to re-

gions below the 100 nanometer scale. In a nanofluidic region, smaller in one

dimension than the radius of gyration of the polymer, the molecule can only

orient itself and diffuse in two dimensions: the dimensionality of the system

has been essentially reduced to two. A network of entropic traps embedded in

this nanoslit allows the entire system to be studied within the focal plane of

an optical microscope.

DNA is an essential molecule in biological systems. Its code is translated

into ribonucleic acid (RNA) and transcribed to proteins: every protein that

a cell produces is assembled from the instructions carried in the DNA [11].

The structure of DNA was first determined by Watson and Crick in 1953 [65].

The structure is that of a double helix: two deoxyribose backbones joined

by nitrogenous bases. There are four bases: adenine, guanine, cytosine, and

thymine. The two helical strands are bound by hydrogen bonding between

the bases: adenine and thymine join with two hydrogen bonds, while guanine

and cytosine join with three. The bases, always found in pairs, are termed

base pairs. Within a cell, a long strand of DNA is folded with proteins into

a structure called a chromosome. Each chromosome is divided into a number

of genes, some containing the information for one protein and others playing

regulatory roles. The total set of chromosomes in an organism is termed a

genome. The length of the human genome is on the order of three billion base

pairs, while a single gene may contain several kilobase pairs.

Here, DNA is removed from its biological role and simply used as a model

polymer. A great deal of work in polymer science concerns solutions or melts

9

of many polymers. These experiments often involve simpler synthetic poly-

mers such as polystyrine or polyethylene glycol [6]. However, many studies

on the dynamics of single polymers are performed with DNA. The production

of DNA using the polymerase chain reaction (PCR) leads to monodisperse

molecules of constant length, as can extracting DNA from viral or bacterial

colonies. Since the length of the molecule strongly affects the dynamics (for

example, the diffusion coefficient [13]), it is ideal to have a monodisperse sam-

ple of molecules while experimenting. Furthermore, it is not difficult to stain

DNA with fluorescent dye, making fluorescent imaging of very small molecules

possible [7]. In particular, the dye YOYO-1 is a very powerful stain that was

invented in the 1990s; its fluorescence is greatly enhanced upon binding to

DNA. The results presented herein apply generally to semiflexible polymers,

but the focus of the experiments is DNA.

A polymer can be simplistically described as a chain of many discrete

segments. The number of total segments is related to the total length of the

polymer, known as the contour length. The length of each segment is known as

the Kuhn length, equal to twice the persistence length. The persistence length

can be thought of as the length scale at which a segment of a polymer acts rigid,

defined as the correlation halflife of tangent vectors along the polymer. The

persistence length may be referred to as the stiffness and is physically distinct

from the size of a monomer unit. For example, the persistence length of DNA

spans approximately 150 base pairs, or about 50 nanometers. The persitence

length can vary from less than three nanometers for some synthetic polymers

[2] to over 25 microns for certain biopolymers [51]. The characteristic size of

the polymer, when relaxed, is represented by the radius of gyration which is

10

a geometric average of the length and stiffness. The DNA primarily used in

these experiments is that of the lambda phage, a virus that infects E. coli,

and is referred to as λ-DNA. It is well studied both biologically as a model

organism [46] and physically as a model polymer [8]. It is approximately 48.5

kilobase pairs long and 16.5 microns extended end-to-end (although YOYO-1

intercalation can increase the contour length up to 22 microns). It has a radius

of gyration of apporoximately 700 nm.

The research described herein falls under many categories: statistical me-

chanics, polymer physics, nanotechnology, biophysics, and nanofluidics. These

fields, how they relate to this project, and the importance of this research will

be discussed in the following sections.

1.2 Polymer Physics

A polymer is a molecule made up of a number of repeating “monomer”

units. The simplest model of a polymer in a freely jointed chain: a series of N

connected links of equal length b with independent orientations (figure 1–1).

The links of such a chain will undergo random walk throughout space, with a

mean-square end-to-end length determined by [20]:

⟨R2⟩

= Nb2

For a given end-to-end length, the degeneracy of the system is the number

of ways the nodes can be rearranged and yield the same value of R. As the

end-to-end length approaches the total length of the chain, the degeneracy

decreases (figure 1–2). Thus, entropy increases with decreasing end-to-end

distance. The entropy of this system can be described by the relationship of

the mean and actual square distances:

11

Figure 1–1: A polymer composed of N segments of length b, with an end-to-end distance of R.

S = −3

2

kbTr2

Nb2+ const

The corresponding Helmholtz free energy is

F =3

2

kbTr2

Nb2+ const′

Because the energy is quadratic with respect to the end-to-end distance, a

stretched polymer behaves as a spring displaced about its equilibrium position.

The elastic behaviour of the entropic polymer spring was demonstrated, among

others, by Bustamante et al [8].

In reality, the orientations of links in the polymer chain are not randomly

oriented, due to excluded volume and electrostatic effects. The correlation

in orientation angles between links is related to the dispersion of a polymer

about its centre of mass: a polymer whose links cannot bend arbitrarily takes

12

Figure 1–2: Extension reduces degeneracy.

up more space and is called “swollen.” A swollen polymer in d dimensions is

described by the Flory free energy:

FF ≈R2

Nb2+bdN2

Rd

The equilibrium radius is found by minimizing the energy with respect

to R, and is characterised by the Flory exponent, ν, which relates the mean

radius of a polymer to the number of chains and their length:

r = bN ν ν = 3/(d+ 2)

In the simple case the freely rotating chain described previously, known

as a theta solvent, ν = 1/2. With the assumption of excluded volume (the

good solvent case), Flory’s estimation puts ν = 3/5, while a more detailed

renormalisation group analysis reaches a value of 0.588 [20].

The Flory free energy is modified when the polymer is constrained to a

slit between parallel plates with separation h [21]:

FFslit ≈R2

(N/g)h2+h2(N/g)2

R2

13

Where g is the number of monomers in a blob within the slit. Minimizing

this function yields a mean radius of:

R ≈ N3/4

Which is the scaling behaviour expected in two dimensions. Thus, con-

straining the polymer in a two dimensional slit changes its scaling behaviour

towards that of a two dimensional system. Experimentally, the polymer can

only coil and undulate but not diffuse in the third dimension, thus its location

can be entirely determined from its two dimensional orientation. This dimen-

sional confinement generally occurs when the length scale is smaller than the

radius of gyration of the polymer [61]. A similar derivation shows that when

a polymer is confined to a cylinder or other quasi-one dimensional system, it

displays its one dimensional scaling behaviour.

The entropic elasticity of a single polymer was tested experimentally by

fixing a DNA molecule to an optically trapped bead [29]. By applying a force

to the molecule, either by hydrodynamic flow or electric field or by fixing

the free end to a surface and pulling the bead, a force-extension curve was

measured which corresponded to the predictions of polymer physics. The

elasticity of DNA was found to be of order 500 kbT/nm. The elasticity differs

from the concept of the harmonic spring constant in that the extension is

related to the length of the molecule with respect to its total length, rather

than displacement from equilibrium. The force-extension relationship for a

polymer with persistence length p stretched at a length z relative to its total

length L is:

14

F

kbT=

z

pL+

1

4p(1− z/L)2− 1

4p

Which contains the familiar elastic term as well as a higher order cor-

rection. The concept of the stretched entropic spring is useful in this thesis

because it concerns systems of polymers stretched between two nodes. While

the distance between the nodes is constant, the length of polymer linking them

is subject to fluctuation, and the entropic spring model allows us to predict

the equilibrium contour length of the link.

While the swollen chain describes the static behaviour of polymers, dy-

namics of a single polymer are often discussed in the context of Rouse’s bead-

spring model, which predicts a distribution of polymer relaxation times that

decay with increasing mode number [26]. The more complex Zimm model

includes hydrodynamic interactions between beads and weakens the effect of

polymer size on diffusivity. The diffusivity of a single polymer can be derived

from these theories, and is shown to scale with a negative power law with

respect to the molecular mass of the polymer: a -1 exponent in the Rouse

regime and a -1/2 exponent in the Zimm regime. A polymer entangled within

a network of other polymers displays a behaviour known as reptation, where

the polymer moves as a series of correlated blobs within a tube on the order of

the blob size [26]. Another mechanism of polymer diffusion is entropic trans-

port. A polymer will tend to move towards regions of higher entropy in order

to minimise its free energy. In a system with two distinct regions, one where

greater conformational degeneracy is possible, the second law of thermody-

namics dictates that the polymer tends to be found in the regions of higher

15

degeneracy. Polymers have been observed to move from regions of lower to

higher entropy [40], recoiling after being forced into the low-entropy region.

An early application of polymer entropic trapping was by Han and Craig-

head [17] who used a linear network of nanofluidic entropic traps to confine

DNA. There, an interface between microfluidic and nanofluidic regions was

used to separate different sizes of DNA into different microfluidic regions.

Because the probability of escape from an entropic trap was size dependent

(longer molecules were likelier to have contour at the trap interface), the bulk

mobility of polydisperse DNA was modified, causing strands of different lengths

to move at different velocities, effectively separating the molecules by size. The

entropic traps in that system were of a length scale larger than the character-

istic size of the polymer, while this thesis concerns entirely nanofluidic systems

where strong confinement is universal.

The region of higher entropy acts as a free energy potential well because

the Helmholtz free energy is spatially dependent. A Brownian particle in such

a well has a finite probability of escaping, known as a Kramers problem [49],

which can be extended to a polymer trapped in a potential well. A polymer

moving through a network of entropic traps is characterised by dwell times in a

single trap and transitions between traps that are of a shorter time scale than

the dwelling. The effect that such transport has on the diffusivity was discussed

theoretically by Muthukumar and Baumgartner [33]. Their main result as it

relates to this thesis is that the diffusivity is modified by the Boltzmann-

weighted free energy barrier of the trap, which has a strong damping effect.

This thesis will examine the effect of entropic trapping on polymer dy-

namics. Both the static effects of entropic and excluded volume free energy

16

interactions, as well as the effect that entropic trapping has on diffusion, will

be examined.

1.3 Micro- and Nanofluidics

The development and micro- and nanofluidics has lead to significant ad-

vancements in many areas of research. Due to the small volumes and con-

strained dynamics that it provides, it is useful for studying biological molecules

individual with excellent spatial and temporal resolution. Both fields are rel-

atively recent: microfluidics arising in the mid 1990s and nanofluidics around

the year 2000.

Microfluidics describes the behaviour of fluids in regions which are con-

fined in one or more dimensions at length scales below one millimetre. Nanoflu-

idics, in the most general sense, concerns fluid dynamical systems with length

scales under one micron. Others cite 100 nm as an upper bound of nanofluidic

systems [54], and the regime between 100 nm and 1 micron may fall under

either category.

Flows in microfluidic systems are often laminar (parallel layers of fluid do

not mix) due to the small length scales and low Reynolds numbers (the ratio

of inertial to viscous forces) involved [22]. At the low Reynolds numbers in

these systems, fluids do not mix normally but rather through diffusion. In

micro- and nanofluidic systems, Poiseuille flow is the dominant hydrodynamic

mechanism. This implies that the volumetric flow rate, Q, of a fluid in a

channel is linear with the pressure gradient across the channel [22]:

Q =∆P

RH

17

The constant of proportionality between flow rate and pressure is known

as the hydraulic resistance, which is dependent on the viscosity η, total length

L, and characteristic width scale, a, of the channel:

RH =ηL

a4

The small sizes associated with micro- and nanofluidics suggest very high

hydraulic resistances. The hydraulic resistance is an important concept when

calculating flow rates based on applied pressure, for instance by DelBonis-

O’Donnel [3] or Mikkelson [32]. In this research, however, the system is at

equilibrium and the applied pressure and volumetric flow rates are expected

to be zero.

Microfluidic systems are characterised by an electric double layer: the

walls bounding the fluid may be charged due to their chemical properties.

For instance, hydrogen atoms in silanol groups on the surface of glass can

dissociate, leaving a negative charge. The field from the charged walls affects

the ions in the fluid: ions with opposite charge to the walls are more likely to

be found in their vicinity. The Debye screening length represents the distance

at which the cumulated charge of the ions equals that from the walls. This

ionic behaviour allows electric fields to be used as a driver of fluid through the

channel, a process known as electroosmosis, in addition to applied pressure.

Conversely, the flow of fluid through such a system can result in net electric

fields [22].

While the Debye length in a microfluidic system is typically much smaller

than the lengthscale of the system, in a nanofluidic system they may be com-

parable. Thus, some authors characterise nanofluidics as having length scales

18

on the order of the Debye length of the fluid, rather than citing an abso-

lute length cutoff [22]. Due to the ionic environment in nanofluidic systems,

long range electrostatic interactions between molecules dominate over van der

Waals forces [54].

The experiments discussed in this thesis take place in nanofluidic systems,

while interface between the nanofluidic systems and the macroscopic world is

a microfluidic channel. Length scales of the experimental environments range

from 50 to 200 nanometers, while the Debye length is of nanometre order,

while the Reynolds numbers involved are of order 10−6. An understanding

of micro- and nanofluidic dynamics is necessary to perform the experiments

successfully and correctly interpret the results.

1.4 The Nanopit-Nanoslit System

This thesis discusses the behaviour of DNA in a nanopit-nanoslit system

(figure 1–3). The nanoslit is a quasi-two-dimensional region, which is smaller

in height than the radius of gyration of the polymer. Because the polymer

is confined in an effective 2D environment, its dynamics occur in two rather

than three dimensions. In addition, the molecule tends to spread out in addi-

tion to coiling, approaching its two-dimensional scaling behaviour [13]. This is

documented by Krishnan et al [25] who observed such scaling behaviour exper-

imentally in a nanoslit and by Tegenfeldt et al [61] in a channel. The nanoslits

used in this study are between 50 and 200 nm in thickness. Lambda-DNA has

a radius of gyration of approximately 700 nm [1]. The nanoslit is embedded

with a square lattice of square pits. The pits are on the order of hundreds

of nanometers in width (between 250 and 1000 nm) with lattice separations

between 500 and 1500 nm. The pits are double the depth of the nanoslit.

19

Figure 1–3: Schematic of the nanopit-nanoslit system (electron micrographinset). A DNA molecule in the background exhibits typical behaviour (actualfluorescent image inset). Typical values for geometric parameters are h=100nm, a=500 nm, l=1000 nm. The pits have a depth equal to h.

20

There is greater conformational degeneracy for a polymer inside a pit

compared to the nanoslit. The pits thus act as entropic traps. While it is

entropically favourable in some circumstances for a polymer to fall into a

single pit, excluded volume interactions dictate an optimal fraction of contour

within a single pit. This has been observed by Reisner et al [49]. In a system of

multiple pits, the interplay between entropy and excluded volume dictates that

a molecule will span multiple pits. The molecule thus takes on a conformation

characterised by occupied pits connected lengths of polymer linking them,

termed linkers. These linkers act as entropic springs in order to minimise

conformational free energy.

A molecule in the system will have its contour divided into a number of

pits. The Helmholtz free energy of the system is determined by the amount of

contour that is in the pits, free in the slit, or spanning a linker. Depending on

the number of pits a molecule occupies, the equilibrium contour per pit as well

as the minimal free energy varies, creating a multiple-well potential: a free

energy landscape (figure 1–4). Thermal fluctuations drive lengths of contour

out of the pit, causing the molecule to adopt a new state. The molecule tends

to fall into the state with the lowest free energy.

At equilibrium, it is assumed that an equal amount of contour falls into

each occupied pit. The Helmholtz free energy of the system can be described

entirely by the length and rigidity of the polymer, the geometric properties of

the system, and the number of occupied pits. For a given system with all other

parameters unchanging, the free energy of the molecule is entirely determined

by the number of occupied pits.

21

Figure 1–4: Free energy landscape for a DNA molecule, comprised of the freeenergy potentials of multiple occupancy states. Many states are available,and as the amount of contour in a given pit is varied, different states becomeenergetically favourable. The relative time a molecule spends in a given state(inset) depends on the energy difference between states.

22

This is a significant experimental and theoretical simplification. Theo-

retically, it implies that the polymer can only exist in discrete energy states,

and a partition function of a single variable can be derived. Experimentally,

the occupancy state, and thus the free energy, can be easily determined by

pit-counting. There is no need to scrutinise the fluorescence of each pit or to

count the partition of Kuhn segments between each pit. It is easy to relate the

measured parameter to that predicted by theory, with minimal intermediate

reduction of data.

This system serves as a powerful tool for polymer physics experimentation.

The mean occupancy state of a given geometry is predicted based on various

theoretical assumptions. For instance, depending on the height of the slit

relative to the persistence length of the molecule, the DNA is expected either

to coil into blobs or to deflect from the walls (figure 2–1). These are the

de Gennes and Odijk regimes respectively (discussed in section 2-2), and the

underlying polymer theory allows a prediction of occupancy depending on

which regime the molecule finds itself. Measuring the number of occupied pits

allows one to determine at which slit height this transition occurs.

This thesis builds on work that was originated by Reisner and collabo-

rators in a 2009 PNAS paper [49]. They showed that an array of nanopits

in a nanoslit could promote self-assembly of DNA into various configurations.

They showed that the equilibrium conformation did not involve finding the

DNA entirely in the pit, and that metastable states would decay over time

into a ground state. Importantly, the essentials of the theory to predict oc-

cupancy was derived, and its predictions with respect to pit separation were

compared to observation. Fitting unknown parameters allowed the theory to

23

fit with four data points. However, four data points is insufficient to show the

strength of the theory. In this thesis, the occupancy theory is compared to a

much greater number of observations. In addition, the theory predicts regions

in which a single state dominates, known as plateaus. These plateaus were not

observed in the original PNAS paper; they are studied in this thesis.

In addition to the aforementioned linear separation experiment of Han

and Craighead [17], similar system was discussed by Mikkelson et al [32], who

observed different transport mechanisms exhibited by DNA to escape such a

trap under external driving. Mikkelson’s system differs from this one in that

the network of traps extends in only a single direction, and the behaviour is

observed under applied pressure rather than equilibrium. This system is also

of theoretical in addition to experimental interest. Streek et al. [57] analysed

the conformational behaviour of a molecule in such a system using a Brownian

dynamics simulation in order to calculate the mobility under electrical driving.

Such analyses may be useful when interpreting the results of these experiments,

for instance, with regards to how the geometry affects diffusion.

Subsequent work, mostly from former collaboraters of Walter Reisner, has

dealt with the behaviour of DNA in linear nanopit arrays. In addition to the

aforementioned Mikkelson paper, DelBonis-O’Donnel [3], which discussed the

rate of DNA transport through an array under hydrostatic or electrostatic

driving. While the results of those studies are of interest, they differ in the

dimensionality of the trap lattice and the driven rather than equilibrium condi-

tions. An undergraduate thesis by Levy [27] attempts to match the predictions

of the pit occupancy theory by a Monte Carlo simulation of polymer dynamics,

and agrees with the findings of the theory.

24

A key extension of this work with respect to Reisner’s original work is

the observation of diffusion between pits. In an unbounded fluid, a polymer

will undergo Brownian motion as modelled by Rouse or Zimm [26]. However,

when constrained to a number of entropic traps, the diffusion takes the form

of transitions between different pit occupancy states. For instance, a molecule

in equilibrium occupying two pits may be thermally activated into a three

pit state, changing the centre of mass slightly, and decay into a new two pit

equilibrium state, undergoing a diffusion out of and into the ground state. It

is predicted, both qualitatively and according to various theoretical models,

that in geometries with stable configurations, molecules are less likely to leave

their equilibrium state and the diffusivity will be smaller. If the diffusion

of molecules is measured throughout parameter space, it is expected to see

local minima where stable states dominate: plateaus in occupancy should

correspond to minima in diffusivity.

It has been demonstrated, both experimentally and via numerical simula-

tion, that the diffusion coefficient of a single polymer molecule between parallel

plates shows a -2/3 power scaling law with the reciprocal height of the plates

[1], which follows from deGennes blob theory [12]. The diffusion coefficients

measured in these experiments are not only slow compared to their bulk val-

ues, but also the dimensionally constrained values in parallel plates, due to

entropic trapping. While -2/3 is predicted from blob theory, Harden and Doi

derived a scaling exponent of -0.61, slightly smaller, due to confinement effects

of the correlation length of the molecule [18]. Furthermore, deviations from

the -2/3 law have been observed experimentally by Strychalki et al [58], who

observed that the scaling exponent for λ-DNA with respect to slit height was

25

Figure 1–5: Stable and unstable free energy landscapes. When multiple freeenergy minima are close to each other, the molecule is likely to undergo tran-sitions between them. When the states are separated by a greater energy, themolecules tend to stay in their ground state.

-0.47 ± 0.05. An even lower exponent was observed by Mathe et al [30], who

found it to be -0.23 ± 0.07.

The diffusion coefficient of a polymer in a lattice of entropic traps was

investigated by Muthukumar and Baumgartner[33] as well as Zimm [69] in the

late 1980s. It was found by simulation that up to a certain approximation,

the diffusion coefficient is modified by the Boltzmann-weighted energy bar-

rier of the trap. With regards to this system, this represents the free energy

difference between the lowest energy occupancy state that the molecule takes

in equilibrium and the state with the second-lowest energy, that the molecule

takes when thermally activated.

The system described herein is similar to that discussed in the thesis [36]

and paper [39] of Dmytro Nykypanchuk. Those experiments involved a two

26

dimensional network of enclosed spheres joined by circular apertures. DNA

was allowed to diffuse through the spheres, and the diffusion was tracked. The

primary independent variable in those experiments was the contour length of

the DNA, while only a single geometrical parameter, the size of the cavities,

was only varied by a factor of 2. For most of the experiments, the radius of

gyration of the DNA was less than the radius of the spheres, so different scaling

relationships apply. Nykypanchuk observed several trends that could not be

explained, including non-monotonic dependence of diffusion on system geom-

etry as well as the free energy minimum of the dimer state. These phenomena

are investigated in this thesis. His work serves as a basis of comparison for

many of the results found in this thesis. These experiments take place in a

regime of stronger confinement compared to Nykypanchuk, and we examine

more systematically the effect of the local geometry on the dynamics of the

system.

Our theoretical and experimental assumptions are as follows. While an

individual molecule of any length will obey the statistical behaviour described

herein, it is required that an ensemble of molecules be of monodisperse length

because the dynamics are length-dependent. Although DNA has excellent

monodispersity, the Lambda genome is known to break into two equal sized

fragments given sufficient shear due to the AT-rich region at the centre of the

genome [48] and thus care must be taken to avoid fragmentation. It is assumed

that after data analysis all molecules are equal length. Another primary as-

sumption is that at equilibrium, an equal amount of contour occupies each pit.

This is a requirement for the derived free energy and partition function to ac-

curately describe the system. This assumption can be verified by a fluorescent

27

measurement of a molecule spanning multiple pits. There will be fluctuations

of contour partitioning between pits, culminating in a transfer into or out of

a given pit. It is assumed that the time scale of these fluctuations is smaller

or much smaller than the frame rate of the time series. This is necessary to

ensure that the videos record the system in a stable state rather than during

a transition.

1.4.1 Table of Symbols and Values

The following table summarises the variables discussed in this paper and

their typical values.

Variable Meaning Values

a Pit width 250-1000 nm

l Pit separation 500-1500 nm

d Pit depth 50-200 nm

h Slit height 50-200 nm

L Contour length 21000 nm

p Persistence length 64 nm

w Effective width 9 nm

N Occupancy 1,2,3 etc

D Diffusivity ≈ 10000 nm2/s

28

CHAPTER 2Theoretical Background

2.1 Overview

This theory concerns the change in the Helmholtz free energy of a polymer

when it moves from a nanoslit to a nanopit. A nanoslit is defined as a quasi-

two-dimensional region with a height h smaller than the radius of gyration

of the polymer, not constrained in the other two dimensions. A nanopit is a

square region of width a in the nanoslit where the height is double that of

the slit; the pit has a depth d. When a polymer goes from the nanoslit to the

nanopit, it loses entropic free energy. When a molecule occupies a number N

of pits, there are three contributions to the free energy: the energy Fp gained

from the contour falling into the pit, the energy Fs lost when the contour is

removed from the slit, and the energy of the entropic springs of contour linking

each pit, Fspr. A molecule occupying N pits with N-1 linkers has a free energy:

∆F

kT= N(Fp − Fs) + (N − 1)Fspr (2.1)

T is the temperature and k is Boltzmann’s constant. The three energy

terms are defined geometrically. For a schematic of the geometry see figure 2.

The free energy of a polymer with a length Lp in a nanopit is given by geometric

arguments: there is a negative linear entropic term Ao and a positive quadratic

self-avoidance term B :

29

Fp = c1AoLp + c2BL2p (2.2)

The constants c1 and c2 are dimensionless prefactors of order unity. Ao

can be defined as Ap − As where p and s refer to pit and slit respectively.

As is discussed in section 2.2 and defined in equation 2.5 and the geometric

definition of Ap, assuming an ideal chain, is:

Ap = p

(1

(h+ d)2 +2

a2

)(2.3)

The B term is defined geometrically as the ratio of the effective width of

the molecule, w, to the total volume of the pit, including the channel above.

The entropic springs have a free energy given by:

Fspr =l2

2pLs

1 +1

2(1− l

Ls

) (2.4)

Where l is the separation between pits and Ls is the contour spanning the

linker between them, determined by conservation of total length.

As mentioned, the free energy of the molecule in the slit is regime de-

pendent. When the height of the slight is much smaller than the persistence

length, the Odijk model is appropriate. When it is much greater than the

persistence length, the Brochard-Wyart model is appropriate. In the crossover

regime where both are on the same order, the Chen-Sullivan model is appro-

priate.

30

As =

ph2

h << p(1.2865( p

h)2 +0.992 ( p

h) + 1

)−2/3 (π2p6h2

)h ≈ p

1.104p1/3h2/3

p << h

(2.5)

When the molecule falls into N equally sized pits on a square lattice, we

assume an equilibrium state where there is an equal amount of contour in each

pit.

For a given N, Lp can be determined by differentiating F with respect to

Lp and calculating the minimal Lp numerically. An analytical value only exists

for the N=1 state. A partition function can be calculated using the multiplicity

values ΩN for self-avoiding nearest neighbour walks of N steps, given by [59].

An assumption of this iteration of the model is that only nearest neighbour

configurations occur and there are no diagonal linkers.

Z =∞∑N=1

ΩNe−F (N)

kT (2.6)

Realistically, higher occupancy states will be so energetically unfavourable

that they have no numeric effect on the partition function; the sum is only

taken to N=10 in our calculations.

Conversely, the probability of each occupancy state can be calculated from

the partition function:

PN =ΩNe

−F (N)kT

Z(2.7)

The mean occupancy, defined as 〈N〉 =∑N PNN , can be calculated based

on this theory, and is an experimentally testable quantity.

31

This theory, as mentioned, was derived in the original nanopit paper [49].

Its validity was based on four data points, after having the two free parame-

ters fit to the data. Although good agreement is found, four data points are

insufficient to assess the validity of a theory. For instance, the theory predicts

the existence of plateaus at integer occupancies, but Reisner et al did not ob-

serve any data points clustered around plateaus, and only half the measured

occupancies were near integer value. The portion of this thesis that deals with

statics attempts to further establish the validity of Reisner et al’s theoretical

framework, although certain modifications and improvements will be explored.

It is noteworthy that several of the defined energies contain unknown

prefactors. These prefactors may be determined by performing experiments

and attempting to fit the results to the theoretical predictions with a floating

prefactor. This may be used to confirm or falsify the definitions of the various

energies, which is discussed in a later section.

2.2 Slit Free Energy Models

The free energy of a polymer between two parallel plates is dependent

on the spacing of the plates relative to the characteristic length scales of the

polymer. The three models discussed are those of Brochard-Wyart (BW) [5],

Odijk [41], and Chen & Sullivan (CS) [10]. The BW regime, also known as

the deGennes regime, aims to describe the free energy per unit length when

the spacing between the slits is much greater than the persistence length. In

such systems, the polymer has room to coil and forms spherical “blobs” with

radii on the order of the slit height. The polymer acts as a series of these

correlated blobs, while within a blob the molecule orients itself as if free. The

Odijk regime applies when the slit height is much less than the persistence

32

Figure 2–1: Schematic of a polymer in the Brochard-Wyart (or de Gennes)regime (top) and Odijk regime (bottom). The length of the deflections in theOdijk regime is the persistence length.

length. In this regime, the polymer does not have room to coil and thus is

reflected from the walls, stretching the polymer. A transition between these

two regimes was observed by Reisner et al [50].

Stained λ-DNA has a persistence length of near 64 nm given the salt

conditions in these experiments, while our slits range from 50 to 200 nm. In

both extremes, the limit of much greater or much less does not strictly apply.

The Chen-Sullivan regime aims to describe the “crossover” between the other

two regimes, using numerical estimates to derive an analytical free energy per

unit length. In the narrow-slit limit, it follows the Odijk prediction, while in

the wide-slit limit it follows the same scaling law as the BW prediction but is

separated by a constant, as seen in figure 2–2.

With regards to the measurable predictions of each model, the expected

mean occupancy of the three models is similar around the persistence length,

33

Figure 2–2: Log-log plot of slit free energy per unit length according to theBrochard-Wyart, Odijk, and Chen-Sullivan models. The persistence length isnoted with a vertical line, while the experimental range is shaded gray.

34

Figure 2–3: Mean occupancy of λ-DNA in 500 nm wide pits separated by 1000nm, as a function of slit height, according to three models of slit free energy.

while the CS and Odijk models agree at low slit height and the CS and BW

models agree at large slit height, as expected. This is seen in figure 2–3. By

comparing the measured mean occupancy and comparing it to the predicted,

a transition between various confinement regimes can be observed.

2.3 Extension to Diagonal Configurations

An assumption of the Reisner model is that only nearest-neighbour config-

urations occur: that all filled pits on vertices of a square lattice are connected

by linkers that traverse a distance equal to the lattice constant. However,

it has been observed experimentally that on occasion next-nearest-neighbour

35

Figure 2–4: Free energy with respect to nearest-neighbour and diagonal springcontours. Example configurations are shown on the right.

configurations occur (figure 4–2), where linkers span the diagonals between

pits. The theory was modified to account for these configurations.

The theory was expanded to include diagonal linkers. Maintaining the

assumption that the system at equilibrium has equal contour in each occupied

pit, the Helmholtz free energy was defined in a system of N pits, M diag-

onal linkers, and N-M-1 nearest-neighbour linkers (figure 2–4, right). The

free energy with respect to contour length was minimised with respect to two

variables: the contour per nearest-neighbour linker and the contour per di-

agonal linker (figure 2–4, left). Minimising a function with respect to two

variables is much more computationally intensive than one. To keep compu-

tation times realistic, the optimal pit contour length was calculated to the

nearest 10 nanometers, rather than the nearest nanometer, and a maximum

occupancy limit of 5 was imposed. Calculations took on the order of 100 times

longer assuming diagonal linkers.

36

The partition function was defined as follows:

Z =∞∑N=1

N−1∑M=0

ΩNMe−F (N,M)

kT (2.8)

The degeneracy of a self-avoiding random walk of N steps on a square

lattice with M diagonal steps has been derived by Oitmaa [42]. For the sake

of computation time, only energies for N=1 to N=5 are calculated. The

degeneracy is as follows:

M

0 1 2 3 4

N 1 1

2 4 4

3 12 32 12

4 36 136 160 36

5 100 528 936 672 100

The enumeration of states used by Reisner et al [49] is based on a cer-

tain symmetry that demands the first walk be in the positive x-direction, and

the first vertical walk be in the positive y-direction. Given an enumeration

Q without such symmetry, the number with such symmetry is Q/8+1. In-

cluding diagonals, such a relationship applies to cases where M=0 or M=N-1,

but does not apply for 0 < M < N − 1. Such a relationship has not been

published to this author’s knowledge, but two options remain. The first is to

discard such a symmetry relationship: using the larger degeneracy values for

the original theory yields similar but not identical predictions of occupancy

37

Figure 2–5: Probabilities of the molecule being found in various states, includ-ing those with diagonal configuration.

and probabilities. The second is to reduce the number of possibilities by as-

suming dihedral symmetry with certain conditions, which is beyond the scope

of this project. When calculating the partition function, the full enumeration

without symmetry reductions was used.

Analysis of the probabilities for various geometries shows that while the

nearest neighbour states dominate in certain domains, there are regions in

which diagonal configurations are relatively common (figure 2–5).

The deviations between the diagonal and nearest neighbour models (figure

2–6) are most significant when the free energy differences between states are

greatest: when the slit depth is large or the pit separation is small. For many

of the systems studied herein the difference is irrelevant.

38

Figure 2–6: Expected occupancy as a function of pit size for a 100 nm slit with1000 nm pit spacing, for both nearest neighbour and diagonal configurations.

Configurations beyond the next-nearest neighbour were not observed ex-

perimentally and thus are not treated theoretically.

2.4 Modelling of Diffusion

The diffusion coefficient of a polymer in an array of entropic traps is diffi-

cult to derive theoretically. A Monte Carlo simulation was created to attempt

to model the behaviour of the diffusion coefficient as a function of the geomet-

ric parameters. The model assumes a polymer occupying an initial number

of pits, and then randomly perturbs the molecule into a higher or lower occu-

pancy state. Using the Metropolis method, the molecule will either adopt the

new state or remain in its current state (figure 2–7). Given enough iterations

the Monte Carlo method can show where in parameter space the expected

minima and maxima of diffusion lie. However, the absolute magnitude cannot

be deduced without specifying a time-scale for the perturbations.

39

Figure 2–7: Visualisation of Monte Carlo algorithm developed to model thedynamics of a polymer in this system.

Another method of calculating the diffusivity involves solving a Kramers-

style problem of a particle trapped in a potential well while subject to thermal

fluctuations. This is outlined by Reisner et al [49]. A sample solution for the

N=2 to N=1 transition is described in that paper. The potential well is based

on the free energy of confinement as a function of contour length. The height

represents the free energy difference between equilibrium in the dimer state,

and a transition state where the linker contains all the contour formerly in the

pit. The rate is given by

R =√

2πkbT

ζU ′(0)

√U ′′(〈Lp〉)e−Ub (2.9)

U represents the confinement free energy potential and a prime indicates

a derivative with respect to contour. Ub represents the absolute free energy

depth of the potential, and ζ is an intrinsic friction factor, approximated by

double the product of the viscosity and Kuhn length.

40

The diffusion coefficient is related to the hopping rate (or its reciprocal,

the mean dwelling time) by the square of the lattice constant:

D = R · (a+ l)2 =(a+ l)2

τ(2.10)

The diffusivity may also be calculated directly by assuming a purely en-

tropic confinement. The diffusion coefficient of a polymer in periodic entropic

traps was presented by Muthukumar and Baumgartner [33]. In that derivation,

the diffusivity was modified by the Boltzmann-weighted free energy barrier of

the trap:

Dtrap

Dbulk

= e−∆F/kbT (2.11)

This can be applied to the nanoslit system by approximating any given

geometry as a two state system. While this is often the case, it should be

noted that there are exceptions in geometries where multiple states lie near

each other in the free energy landscape. The free energy barrier represents

the separation in free energies of the two lowest-energy states, according to

equation 2.1. These are termed the equilibrium and transition states. The

bulk diffusion coefficient is modified by a -2/3 power scaling relationship with

the height of the slit, as described by Balducci [1]. Thus, the diffusivity as

predicted is:

D = Dbulk

(Rg

h

)−2/3

e(Feq−Ftr)/kbT (2.12)

This formulation predicts that the diffusivity is maximal in regions of

parameter space where the mean occupancy lies between two integer states,

41

and near zero along the plateaus. The locations of the diffusivity peaks cor-

respond to those predicted by the Monte Carlo simulation. The advantage of

the deterministic model is that it can predict the magnitude of the diffusion

coefficient as well as the trends. A limitation is that it assumes a two-state

system. In reality, multiple states may overlap in probability space, requiring

a more nuanced model. This may be achieved by assuming that each possible

transition is a potential barrier that must be crossed. This is implemented

by summing the contributions to diffusion of the multiple possible transitions.

Accounting for less-than-likely transition modes, the diffusion becomes:

D = Dbulk

(Rg

h

)−2/3 ∑N 6=Neq

eFeq−FN

kbT (2.13)

The difference between the two-state assumption and the full summation

is minimal except in parameter spaces of weak confinement such as small pits

or wide channels. This is similar to the difference between the diagonal and

nearest-neighbour assumptions. Indeed, in these geometries diagonal configu-

rations may serve as a transition state between two nearest-neighbour states.

The analytic formula and the Monte Carlo simulation predict similar

trends: low diffusion when the occupancy is near an integer, fast diffusion

peaks during transition periods in the occupancy. A comparison of the two

methods is seen in figure 2–9.

As mentioned, the general behaviour of the diffusivity is that it is minimal

when the occupancy is at a plateau (near-integer average state), and increases

in the transition regions between plateaus. If the diffusivity for a given ge-

ometric system is compared to the mean occupancy in the same geometry,

42

Figure 2–8: Diffusivity according and mean occupancy as a function of pitwidth. Diffusivity assumes free energy confinement [33] based on the energiespredicted by Reisner et al [49]. Diffusivity is minimal when occupancy is at aplateau.

Figure 2–9: Diffusivity as a function of pit size as predicted by equation 2.12and the Monte Carlo simulation. The vertical scale of the Monte Carlo ischosen to match that of the analytic prediction.

43

Figure 2–10: Parametric plot of diffusivity and average occupancy at the samegeometry. The varied parameter is the pit width.

a series of peaks and troughs between half-integers and integers is expected

(figure 2–10).

2.5 Parameter Space Behaviour

Within the confines of this model, seven parameters dictate the behaviour

of the molecule. These are the polymer’s properties: the effective width, persis-

tence length and contour length of the molecule; and the geometric parameters:

the slit height, pit depth, pit width, and pit spacing. To provide a sense as

to how the system behaves with respect to these parameters, the mean occu-

pancy is shown as a function of these parameters. In each of the following

graphs (figures 2–11 to 2–13), the following parameters, with the exception

of the varied one, are used: 9 nm effective width, 64 nm persistence length,

21 µm contour length, 500 nm pit width, 1000 nm pit spacing, and 100 nm

44

Figure 2–11: Mean occupancy of λ-DNA with respect to pit width and sepa-ration. The non-varied parameters are 100 nm pit and slit depth, 500 nm pitwidth, 1000 nm pit separation.

Figure 2–12: Mean occupancy of with respect to polymer length and stiffness.The non-varied parameters are 100 nm pit and slit depth, 500 nm pit width,1000 nm pit separation.

slit height and pit depth. For maximum generality, the Chen-Sullivan model

is used. Because the slit height and pit depth are often kept equal to each

other, their combined scaling behaviour, as well as the behaviour with respect

to their ratio, will be shown.

At small pit widths, it is predicted by theory that the occupancy decreases

below a point of maximum occupancy (figure 2–11). Qualitatively, this is due

to the fact that it becomes energetically favourable for the polymer to occupy

45

Figure 2–13: Mean occupancy of of λ-DNA with respect to shared pit depthand slit height, as well as the ratio of the two. The non-varied parameters are100 nm pit and slit depth, 500 nm pit width, 1000 nm pit separation.

Figure 2–14: Maxmimum occupancy and pit width at maximum occupancy asa function of slit height for λ-DNA

the slit rather than the pit. As the slit height is varied, the pit width that

corresponds to maximum occupancy, as well as the maximum occupancy itself,

vary accordingly (figure 2–14).

The theory that has been discussed in this section will be used as a basis

of comparison for the results of the experiments.

46

CHAPTER 3Apparatus and Methods

3.1 Lab-on-a-Chip: Fabrication and Geometry

Micro- and nanofluidics are often associated with lab-on-a-chip technol-

ogy, which refers to the process of reducing laboratory functions to a small

piece of glass. The experiments in this thesis took place in a lab on a chip:

the nanopit lattice represents the lab, and it is found in a square piece of fused

silica (glass) approximately two centimetres per side and 0.5 mm thick.

The topography created for the experiments consisted of three parallel

channels with a length of 450 microns and a width of 50 microns, etched to

between 50 and 200 nanometres depth. On each end of the channels was

a microchannel with a depth of approximately one micron. A reservoir was

situated at each end of the microchannels for a total of four reservoirs (figure

3–1).

One inner surface of each nanoslit was embedded with a square lattice of

square pits. The pits used in the experiments ranged in size from 250 nm to

1000 nm, with lattice spacings between 500 and 1500 nm. The depth of the

pits was kept equal to the depth of the slits, such that over a pit the distance

between the two walls of the slit was doubled. Each nanoslit had seven or eight

lattices of varying pit size, with the pit spacing varying between nanoslits.

Nanoslits with depths of 50, 70, 100 and 200 nm were used. The spacing of

the lattices in the experiments was 500, 1000, or 1500 nm. The pits had sizes of

47

250, 300, 350, 400, 500, 600, 750, and 1000 nm. Each combination of geometric

parameters represented a different physical system for experimentation, for a

total of approximately 100 systems.

The devices used in this research were fabricated at the Danchip facility

at the Danish Technical University by Walter Reisner. Because I did not

fabricate the devices used for the bulk of the experiments, fabrication will

not be a focus of this thesis. To summarise, the pits were patterned with

electron lithography in zep520A resist and etched using CF4:CHF3 reactive

ion etching. The slits and channels were patterned in S1813 photoresist with

UV lithography and etched with CF4:CHF3 reactive ion etching. Resist was

removed with Piranha (a 3:1 mixture of sulphuric acid and hydrogen peroxide).

Each wafer contained nine chips, which were separated using a dicing saw. The

chips were treated with an RCA clean and bonded to silica cover slips. The

bonded chips were baked at 1000 degrees to convert the hydrogen bonds to

covalent silicon-oxide bonds and cement the bonding process. RCA cleaning

and bonding was performed in the McGill Nanotools Microfabrication facility,

while annealing was performed in-house.

3.2 Chuck

During experimentation the chip was fastened by a metal retaining ring to

a plastic chuck, which was connected to the positioning stage of a microscope.

The chuck contained larger reservoirs that supplied the smaller reservoirs of the

chip, as well as connections for applying hydrostatic or electrostatic pressure

to each channel.

The chuck has been used for many research projects involving DNA in

nanofluidics (e.g. [3], [49], [32], [48]). It is a plastic square prism with a large

48

Figure 3–1: Schematic of the basic layout of the chip. Microchannels areconnected by nanoslits and are terminated by circular reservoirs.

hole bored through the centre. Eight smaller holes are drilled adjacent to and

surrounding the large hole. An indented square groove on the bottom allows

the chip to be aligned flush with the chuck such that the reservoirs of the

chip are aligned with the small holes. The interfaces between the holes and

reservoirs are sealed with rubber O-rings. The chip is fastened to the chuck

by attaching a metal retaining ring. Substrate may be injected into the small

holes, which when sealed act as larger reservoirs supplying those on the chip.

Side channels connecting to the small holes connect to an air pump via luer

connection, which can be used to arbitrarily apply pressure to a given reservoir.

3.3 Optics

A Nikon Eclipse Ti inverted microscope was used for visualisation. A

Nikon Plan Apo VC 100x oil immersion objective lens was used for magnifi-

cation. Fluorescence illumination was provided by a X-Cite Series 120 lamp,

focused with a condenser-aperture through a dichroic filter. Absorption of

49

Figure 3–2: Photograph of the chuck in the microscope positioning stage.

light by the dye peaked at 491 nm, while emission was at 510 nm. The iris

shutter was a Lambda SC controller. The fluorescence was modified with a

UV filter as well as optional neutral density (4x and 8x) filters. An Andor

iXon digital camera with a cooled electron-mulitplying charge-coupled device

(EMCCD) was used to acquire the signal and feed it to the computer. NIS

Elements software was used for data acquisition. The chuck was attached to

the positioning stage which was controlled by a Prior Proscan III.

3.4 Buffer and DNA Preparation

The DNA was stained for visualisation with YOYO-1 (a dimer of oxa-

zole yellow) fluorescent die with a 10:1 intercalation ratio. The staining used

a 10 mM TRIS (tris(hydroxymethyl)aminomethane) with 1 mM of EDTA

(ethylenediaminetetraacetic acid). After being heated at 60 degrees and flash-

cooled to prevent dimerisation, the stock DNA was diluted to 100 µg/mL.

Stock YOYO-1 was diluted to 100 µM then vortexed and centrifuged to en-

sure homogeneity. Pure buffer, dye, and DNA were added in a 177:3:20 ratio

to ensure a 10:1 staining ratio. DNA was mixed with the buffer by repeatedly

50

drawing it into and releasing it from a large orifice pipette tip, to distribute it

evenly without breaking it.

λ-DNA was primarily used in these experiments, although in some exper-

iments T4 or Charomid DNA was used. The molecules, when stained, were

assumed to have a length of 21 microns, a persistence length of 64 nm, and an

effective width of 9 nm, taking into consideration the extension due to staining

[49].

50 mM Tris with a pH near 8.0 was used as a buffer for experiments, mixed

49:1 with β-mercaptoethanol (BME) to reduce photobleaching, as described

in [49]. Care was taken to avoid skin contact and inhalation when handling

BME. The Tris-BME solution was vortexed until mixed. Stained DNA was

diluted approximately 1:17 with the buffer to create the substrate.

3.5 Experimental Protocol

3.5.1 Preparation

Approximately 2 microlitres of substrate were inserted via micropippete

into each reservoir of the device. Time was given to allow wetting of fluid from

one side of reservoirs to the nanoslit region before the other side was filled,

in order to avoid bubble formation. The reservoirs of the device were lined

up with the O-rings of the chuck, and the device was fastened with a metal

retaining ring. Between 15 and 20 microlitres were pipetted into the reservoirs

of the chuck, which were then sealed. A line from an air pump was attached to

each reservoir on the chuck. The chuck was fastened to the positioning stage

of the microscope. A schematic of the loading chuck can be seen in figure 3–3.

51

Figure 3–3: Schematic cross section of the experimental setup.

Figure 3–4: Valve operation for flowing DNA from reservoirs to microchannels(left), forcing DNA from the microchannels into the nanoslits (middle), andobserving the system in equilibrium (right). Arrows indicate the direction offlow.

52

3.5.2 Experimentation

The nanofluidic features of the chip were centred in the microscope’s field

of view. The desired geometry was moved to the centre using the positioning

stage. By controlling the pressure to each reservoir, DNA could be forced into

the nanoslit region and over the desired geometry. Typically, a pressure on the

order of 0.1 bar was applied to two reservoirs in order to accumulate DNA at a

microchannel-nanoslit interface. When a sufficient level of DNA was present,

a greater pressure (about 0.7 bar) was applied to force the DNA into the

nanoslit. The pressure was controlled by partially occluding the outlet valve

of the pump-luer connection: an open value left no pressure on the reservoirs,

while a closed value pressed over two atmospheres. In the nanoslit, low pressure

(on the order of 0.02 bar) was used to flow it to the desired geometry. When an

ensemble of molecules was in the region of interest, the system was placed in

equilibrium by removing pressure from the reservoirs (figure 3–4). Videos were

taken of the molecules, with a frame rate and total capture time dependent on

the timescale of the dynamics, usually 1 Hz for 10 minutes. Exposure time of

an individual frame was 100 ms in most cases. After each movie was taken, the

channels were flushed by applying pressure of order 0.1 bar, and new molecules

were brought into the region of interest. For each geometry, enough movies

were recorded such that a sufficient number (of order 30) of molecules and

transitions were observed.

Stained DNA has a tendency to photobleach. Measures taken to prevent

this include limiting the fluorescence exposure time by closing the iris of the

aperture when data was not being recorded by the CCD. The experiment was

ideally performed in a dark room. In addition, fresh DNA was periodically

53

Figure 3–5: Time series of DNA forced from the microchannel to the nanoslit.a) pressure is applied to bring DNA to the microchannel-nanoslit interface(marked by red line). b) pressure is increased to force DNA into nanoslit. c)pressure is removed, the molecules self-assemble into their configurations.

brought from the reservoirs by applying about 0.1-0.2 bar to a single reservoir

on each side.

3.6 Image Analysis

Image processing was initially performed using ImageJ. The NIS Elements

software saved the time series images in a .ND2 format. An ImageJ plugin,

“ND to ImageJ Converter” was used to convert the movies to .AVI format.

ImageJ was also used to align the axes of the nanopit lattice to the orthogonal

axes of the screen, necessary due to the slight angle of the CCD with respect

to the system. If necessary, contrast was enhanced to better visualise the

occupied pits.

54

Figure 3–6: Digitisation of a time series. a) raw data in .avi form. b) superim-pose grid over movie. c) register super-threshold intensities within each box.d) convert to intensity map.

The software used to analyse the raw data was developed primarily in

MATLAB by Hugo Brandao in a summer research project. It is further elab-

orated upon in his report, which was published in the McGill Science Under-

graduate Research Journal [4]. The data was analysed by superimposing a

digital matrix over the physical lattice. The matrix was aligned by choosing

the size of each box and the spacing between them, and tuning the horizontal

and vertical offsets until the boxes aligned with the data. When the software

was signalled to analyse, the movie was converted to a three dimensional ma-

trix of intensities. If the registered intensity within a box on the grid was

above a specified threshold then it registered as an occupied pit. Each frame

was discretised to the box level. The analysis procedure can be seen in figure

3–6.

The result of an analysis was a data structure for each molecule: for each

frame of the time series, the x and y locations of each pit of each molecule was

recorded. This allowed the centre of mass and occupancy state to be tracked

55

Figure 3–7: A single molecule tracked over 10 minutes. Throughout this time-frame it transitions primarily between the N=3 and N=4 states and diffusesapproximately one micron.

over time (figure 3–7). The simplest measure of the molecule’s behaviour was

the average occupancy: the number of occupied pits averaged over the whole

movie. An ensemble of molecules in a given geometry was used to calculate

an average occupancy for each given system.

Care was taken to eliminate fragmented molecules from the data. A

histogram of mean occupancy, as well as an occupancy-intensity scatterplot

(figure 3–8), was used to determine if any molecules were indeed fragments.

Obvious outliers were removed. After this concern had been realised, the ap-

plication of external pressure was modified to avoid quick burst of maximal

shear, and fragmentation was greatly reduced.

3.7 Diffusion

The aforementioned digitising software was used to index the location

of a given molecule at a given time. This was used to track its centre of

mass, assuming the centre of mass corresponded to the geometric centre of

the pits that it occupied. This allows the location of the centre of mass to

be determined. The diffusion coefficient was calculated by an oversampling

56

Figure 3–8: Histrogram of mean occupancies of many molecules in a singlegeometry, and a scatter plot of intensity and occupancy. The obvious outliersare fragments and ignored from analysis.

method outlined by Wang et al [64]. The mean square displacement over a

time series was calculated for various time step lengths, n. For example, the

mean displacement between t = to and t = to + 1 for the entire movie, then

between t = to and t = to + 2, etc.

⟨(∆x)2

⟩=

∑Ni=1(∆xi)

2

N(3.1)

The mean squared displacement is expected to be linear with the corre-

lation length up to a certain value before this relationship breaks down. The

slope of this linear relationship can be used to determine the diffusion coeffi-

cient:

D ≈ 〈(∆x)2〉2∆t

(3.2)

57

Figure 3–9: Mean square displacement as a function of correlation length foran ensemble of molecules on a log-log scale. The derivative of the low-n portionof these curves can be used to calculate diffusivity.

58

Given perfect measurements of the center of mass, the diffusion coefficient

is equal to the value for a correlation length of one time-step. However, limi-

tations of resolution and signal-to-noise made this unrealistic in some cases, so

the oversampling method was used to better estimate the diffusion coefficient

and its error.

3.8 Fitting

For each geometry, the mean occupancy and diffusion of the observed

molecules was calculated. MATLAB was used calculate the theoretical free

energy of each occupancy state and to calculate the expected average. A non-

linear fitting algorithm, lsqcurvefit, was used to fit the theory to the data. This

was accomplished by using the measured values for each geometry as points

of comparison and varying different theoretical parameters. This returned, for

instance, the values of the prefactors in equation 2.5 that yielded the minimal

sum of squares. These fit values were used to further refine the theory, as

discussed in the results session.

The data was fit to several systems of floating parameters: the prefactors

of the linear and quadratic terms for the three models (Brochard-Wyart, Chen-

Sullivan, and Odijk), as well as for model-free assumptions: fitting Ao and Bo

assuming B = Bo/a2 and a simple Ap − As = Ao, as well as a more complex

version of the linear parameter: Ap − As = A1 + A2/a2. In addition to fitting

to the traditional model, the data was also fit to the model assuming diagonal

linkers.

59

CHAPTER 4Results and Discussion

4.1 Qualitative Behavior

Several occupancy states were available for a polymer in a given geometry.

The molecules tended to occupy their lowest energy equilibrium state, and

undergo thermal activations to higher or lower occupancy states (figure 4–1).

As the size of the pits was varied, the equilibrium state tended to change, as

seen in figure 4–2. In systems with the largest widths the molecules tended

to occupy a single pit: the monomer state. As the pits decreased in width

the molecule tended to occupy two (dimer) then three and four pits. Even

the smallest pits were many times larger than the excluded volume of a single

polymer: this behaviour suggests an interplay between excluded volume and

entropic effects. The height of the slit played a role in the behaviour of the

DNA. Wider slits decrease the strength of the confinement and molecules more

likely to maximise entropy in a single pit rather than minimise excluded volume

in multiple pits. Thus, the mean occupancy generally decreases with increasing

slit depth. Again, due to tighter binding, a greater damping of diffusion is seen

in narrow slits compared to wider ones.

From an experimental perspective, narrow nanoslits were associated with

greater pressure required to flow the DNA from the microchannel. That be-

haviour is beyond the scope of this project but is discussed, for instance, by

Yeh et al [66]. In narrow slits, care must be taken to avoid shearing the DNA

60

Figure 4–1: A λ-DNA molecule undergoing a transition in 400 nm pits in a100 nm slit with 1000 nm pit spacing. a) the molecule is at equilibrium inthe N=2 state. b) the molecule has been activated to the N=3 state and hasgained approximately 2 kbT of free energy. c) the molecule has relaxed to anew equilibrium state and has been displaced by approximately 1 µm.

Figure 4–2: Various occupancy regimes of a λ-DNA in various pit sizes. Asthe pit size is decreased, the mean number state tends to increase.

at the interface, while in wider slits it may be difficult to build up a large

amount of substrate for experimentation.

4.2 Occupancy Measurements

For each molecule, the average occupancy state over the course of the time

series was calculated. For an ensemble of molecules in a given geometry, the

average for the entire system was calculated. The trend of average occupancy

as a function of slit height, pit size, and pit separation could be examined.

By measuring the mean occupancy of an ensemble of polymers as a func-

tion of pit width for a single slit depth and lattice spacing, the behaviour of

the system with respect to pit size was observed. Experimental occupancy

data can be seen in figures 4–3, 4–5, 4–6, 4–11, 4–13 and 4–14. The behaviour

with respect to lattice spacing was discussed in [49]. In addition, analysis [4]

61

Figure 4–3: Mean occupancy of λ-DNA in nanopit systems as a function ofpit width. 100 nm slit depth and 1500 nm spacing (top), 50 nm slit depthand 1000 nm pit spacing (bottom). Plateaus in the occupancy are visible,for instance, at the N=2 state. The theoretical curve is based on a fit tomodel-independent linear and quadratic energy terms.

of digested DNA of varying lengths in the system showed an increase in pit

occupancy with fragment size.

The mean occupancy tended to decrease with increasing pit size. In addi-

tion, theory predicts regions of parameter space over which a single state dom-

inates. These are termed plateaus due to their appearance in an occupancy-

geometry plot. Within a system that exists on a plateau, the molecules are

expected to be found predominately in a single state, suggested by a near-

integer mean occupancy. In addition, relatively few transitions to higher or

lower states are expected. Plateaus were observed at the dimer state in several

data sets (figures 4–3 and 4–5).

In addition to plateaus being observed with respect to pit width, plateaus

at the N=1 and N=2 states were observed as a function of contour length, as

seen in Brandao’s report [4]. As demonstrated by figure 2–12, this is predicted

by theory. To test this, an experiment was performed using T4 DNA, which

62

Figure 4–4: A histogram of the distance of the occupancy to the nearest integerfor DNA fragments of various lengths in a single geometry. A plurality of themolecules display occupancies close to integer values. A theoretical predictionusing prefactors fit to the 70 nm system is overlaid.

is approximately four times as long as λ-DNA. The buffer was vortexed for

approximately ten seconds in order to fragment the polymers into various sizes.

All molecules were observed in a 70 nm slit with 300 nm pits separated by 1000

nm, in order to observe the effect of varying length. Rather than estimating

the total length via the fluorescent intensity, the occupancy of each molecule

was measured and the results compared in a histogram. It is expected that

molecules be found most often with integer or near-integer occupancies, so the

rounded occupancy of each molecule was subtracted, leaving a distribution of

numbers from -0.5 to 0.5. The data shows that they cluster near the zero point

of the histogram, which is matched by theory.

63

Figure 4–5: Mean occupancy of λ-DNA in nanopit systems as a function of pitwidth with 1500 nm pit separation and 50 (left) and 70 (right) nm slit depthand 1500 nm pit separation. The lack of error bars at the N=1 and N=2 datapoints on the 50 nm plot indicate that 100 % of molecules were observed inthat state.

While stable states may manifest themselves in the form of plateaus, the

selection of geometries in a given chip may not have allowed such a plateau

to be observed. However, the stability of a given state may be deduced not

only from plateau-like behaviour, but from a single data point if an ensemble

of molecules is observed entirely in a given state, with no observed transitions.

This is the case for the N=1 state in most systems, and, for example, the

N=2 state in figure 4–5. While a plateau does not obviously pass through

the dimer state, the molecules were observed entirely in that state with no

observed transitions over ten minutes of observation. This was also the case

for the monomer state, although that is a more trivial result.

In systems where the free energy difference between molecules was small,

plateaus were not expected because diffusion was always present. This was

apparent is systems with 500 nm lattice spacing (figure 4–6).

With respect to the slit height, the occupancy is expected to decrease

with increasing slit height. While this is for the most part observed (figure

64

Figure 4–6: Mean occupancy of λ-DNA in nanopit systems as a function ofpit width with 70 nm slit depth and 500 nm pit separation. Plateaus are notobserved due to the weak confinement of the entropic traps.

65

Figure 4–7: Mean occupancy of λ-DNA in nanopit systems as a function ofslit height for various pit widths with 1000 nm atd 1500 pit separation.

4–7) there are notable deviations. The 1000 nm lattice shows non-monotonic

behaviour centered at 70 nm slits for large pits, a behaviour that vanishes for

smaller pits. In the 1500 nm lattice, there again appears to be non-monotonic

behaviour for larger pits, with occupancy increasing from 50 to 70 nm for large

pits but decreasing for small pits. The non-monotonic behaviour is predicted

by theory and is hinted at by the non-monotonic behaviour in figure 2-13.

These plateaus indicate that the free energy difference between neighbour-

ing states is relatively high compared to the room temperature energy. The

experimental observation of the plateaus suggests that the nanopit-nanoslit

system of entropic confinement can be used to create stable nanostructures

at equilibrium. Previous work in polymer self-assembly has proposed using

aggregates of polymers that assemble due to chemical modification [44]. The

structures that these plateaus dictate are of a single naturally occuring poly-

mer, in an unaltered state. More recent polymer self-assembly techniques,

such as DNA origami [52] requires the careful deposition of DNA staples on a

66

surface and their examination by atomic force microscopy. In contrast, these

results show that a lithographically patterned surface is sufficient for creat-

ing self-assembled structures, which are form at equilibrium without external

stimulus.

The stability of higher-occupancy states was also observed by Nykypanchuk

in the strong confinement regime [36], who found a local free energy minimum

when a λ-DNA molecule was stretched equally between two traps. We find

agreement with the interpretation of the results therein: self-exclusion effects

counteract entropic effects to create a lower energy dimer state. A method

of using oligonucleotide fragments of DNA as a basis for self-assembly, [55],

noted that higher order structures of many dendrimers exhibited greater stabil-

ity than expected, leading to an increased melting temperature. The stability

of the structures was due to chemical attraction between sub-units, rather than

a free energy minimum due to entropic trapping. These structures are created

using significant chemical treatment and joining of many units, which differs

from the structures observed in these experiments, which are composed of a

single polymer.

4.3 Comparison to Theory

Qualitatively, the linear fits shown in figure 4–3 display good agreement

between experiment and theory given two free parameters. Theoretical curves

appear to be within error bars and the validation that was observed in the

original paper [49] is upheld.

The model used to calculate free energy assumes a quadratic self-exclusion

term based on the ratio of the effective size of the molecule to the volume of

the pit. The prefactors of A and B can be elucidated using a least squares

67

Figure 4–8: Fit quadratic prefactor as a function of inverse slit height. A linearrelationship is expected from theory.

regression fit. It is expected that if polymers show the expected self-exclusion

behaviour then the fit B coefficient will be linear with the reciprocal height.

Such a pattern is indeed observed. The behaviour of the quadratic coefficient

with reciprocal slit height can be seen in figure 4–8. For the fitting parameters

from the 1000 nm pit spacing system and assuming the geometric definition

of B, a linear regression yields an effective width of 8.30 ± 0.77 nm, similar to

the expected value of 9 nanometers.

An initial comparison was performed, comparing the fit values of the lin-

ear energy term to the geometric definitions. While the definitions tended to

overestimate the values, the trends showed that the Odijk model was more

appropriate for smaller slits, while the Brochard-Wyart model was more ap-

propriate for wider slits (figure 4–9).

By assuming geometric definitions as defined in the theory section, pref-

actors can be fit to the linear and quadratic energy terms. By comparing the

68

Figure 4–9: Ratio of fitAo term to geometric definition for Odijk and Brochard-Wyart models.

fits to various models (BW, Odijk, and CS), it can be ascertained which model

is appropariate for which regime. For the Ao prefactor, all values are below

unity the dimensionless prefactor increased as a function of slit height for the

BW model but decreased for the CS and Odijk models (figure 4–10). This is

appropriate given the regime each model attempts to describe. Assuming that

a prefactor of unity indicates that the geometric definition is accurate, the

model describing a narrow slit (Odijk) has a prefactor approaching unity for

narrower slits, while the BW model for wider slits has a prefactor approaching

unity in that limit. The fit prefactors of the Chen-Sullivan model appeared

linear. Given that the CS model describes the crossover of the two regimes,

it is not unreasonable to assume that its prefactor should be unity when the

two regimes cross over. According to linear extrapolation of the data, this is

expected to occur at 29±5 nm. Extrapolating from the Brochard-Wyart fit

prefactor data, the regime is expected be ideal at 119±11 nm. Reisner et al

69

Figure 4–10: Dimensionless prefactors for the linear Ao free energy term whendata is fit to the three models.

[50] expect a transition to occur in slits that are twice as wide as the persis-

tence length. In the case of these experiments, twice the persistence length is

128 nm, within error range of the fitting estimate.

The energy terms as described by Reisner et al [49] do not explicitly con-

tain dependence on the pit spacing or additional dependence on slit height, but

it appears to be present. While there appears to be a difference in slope be-

tween the two pit spacings for the Chen-Sullivan fits, the Brochard-Wyart and

Odijk fits appear to be more or less spacing-independent. Further investigation

is required to determine the exact dependencies of the prefactors.

The fit data for 200 nm slit depth does not corroborate the theoretical

model. However, this system is still of interest. A prediction of theory is that

when the pit size is varied, there should be a pit width at which maximum

occupancy occurs, below which the occupancy decreases. This is due to the

fact that it becomes energetically favourable for the polymer to be in the

slit rather than the pits. This was observed in the 200 nm data. Because

the molecule becomes disengaged from the lattice in the regime of very weak

confinement, the standard analysis method of pit counting was not optimal.

However, qualitatively it can be seen be examining stills from the experiment

70

Figure 4–11: Images from an experiment of λ-DNA in a 200 nm slit with pitsseparated by 1000 nm, for various pit widths. Quantitative results are inset.

that such a maximum occupancy occurs around 350 nanometer pit width,

and the measured data corroborates this (figure 4–11). While there may be

modifications of the theory necessary to take into account the behaviour in the

regime of wide slits, the general prediction that occupancy decreases below a

certain pit width has been confirmed experimentally.

Another theoretical framework deserving of mention is that of Saltzman

and Muthukumar [53]. They consider a chain of Lennard-Jones interacting

segments in a one dimensional array of chambers connected by pores, such

that only one segment can pass through a pore at a given time. The Langevin

71

equation for each segment was simulated numerically to determine the static

and dynamic properties of the system with respect to chain length and chamber

size. A crucial flaw in the model was the discounting of the free energy cost of

a segment translocating through a pore.

They provide scaling predictions of occupancy based on chain length and

pore size. In larger chambers in which the chain can coil, they find an R−5/3

scaling law. Given that R ≈ V 1/3 and V ≈ a2, we can substitute the pit

width for the radius to compare their prediction to these experiments. With

those substitutions, it is predicted that < N >≈ a−10/9. They also derive the

scaling relation for small cavities in which molecules cannot coil: < N >≈

R−1 ≈ a−2/3.

However, because they do not consider the energy cost of translocation

the cavity-pore system does not act as an entropic trap, and chains will not

be confined to specific occupancy. The direct implication is that plateaus will

not be observed when varying a given parameter. Plateaus are observed in

the data in these experiments, and a power law would not be an appropriate

fit. In systems were plateaus are not observed, for instance the 200 nanometer

nanoslit system, a power law can be fit to the occupancy data to show that

< N >≈ a−0.61±0.05, which is not in agreement with the -10/9 prediction. It

is within error range of the -2/3 prediction, but does not correspond to that

particular regime. In other data where plateaus are not observed, for instance

the 70 nm deep slit with 500 nm pit spacing, a −0.91±0.07 occupancy scaling

law is observed. Similarly, a 500 nm lattice with a 100 nm slit height displays

a −0.96 ± 0.01 scaling behaviour. This is closer to the theoretical prediction

of -10/9, but still not within error range.

72

4.4 Diffusion Measurements

The diffusivity of a single polymer was measured by tracking the centre of

mass of the occupied pits. The hopping rate was calculated using the method

outlined by Wang et al [64], and the absolute quantity was calculated by

multiplying this rate by the square of the lattice constant, the sum of the pit

separation and length.

The diffusion coefficient of a single molecule is relevant in various fields

ranging from biochemistry [23] to materials science [60]. The effect of con-

finement on the diffusivity has been shown to influence the glass transition

temperature of polymer films [15]. While a polymer in a free solution un-

dergoes Brownian motion, a polymer in this system is confined to a discrete

number of pits, and displaces its centre of mass by thermal activation to and

relaxation from a higher energy state, as seen in figure 4–1. This modified dif-

fusion mechanism has a damping effect on the diffusivity. Results from these

nanopit experiments indicate that the diffusivity is geometry dependent. In

weakly bound systems where the free energy difference between states is rel-

atively small, there is a monotonic decrease in diffusivity when the pit size is

increased: in a 200 nm nanoslit with 1 µm pit spacing, the diffusion coeffi-

cient in a system with 1000 nm pit width was a factor of 90 smaller than that

in a system of 350 nm pits (figure 4–12). This allows an additional degree

of freedom in the fine tuning of the diffusivity of a single polymer, beyond

confinement due to slit height.

More interestingly, in systems with greater separation of states, corre-

sponding to the aforementioned plateaus, the diffusion shows non-monotonic

behaviour. Local minima exist when the molecule is strongly bound to a

73

Figure 4–12: Diffusion as a function of pit width for λ-DNA in a 200 nm deepnanoslit with 1 µm lattice spacing. The diffusion is damped by a factor of 90over the course of this graph.

certain state, because there are fewer thermal transitions to higher or lower

states. These local minima are termed resonances. Resonances were observed

in several systems at geometries corresponding to integer values of the mean

occupancy. Such a resonance can be seen in figure 4–13, where local minima in

diffusion correspond to near-integer values of mean occupancy. An anomalous

result reported by Nykypanchuk [36] indicates that diffusivity resonances were

observed in that system as well. While those results were unexplained, the

experiments in this paper and the theory behind it suggest that stable higher

occupancy structures play a role in creating a diffusion resonance.

Diffusion resonances were difficult to measure due to the requirement of

stable equilibrium states. Because the equilibrium states on either side of the

minimum were still relatively stable, very long observation times were required.

Multiple diffusivity minima are expected in chips with multiple plateaus. For

a slit of 70 nm depth with lattice spacing of 1000 nm, plateaus were observed

74

Figure 4–13: Diffusion and occupancy as a function of pit width for λ-DNAin a 100 nm deep nanoslit with 1 µm lattice spacing. Minima in diffusioncorrespond to integer values of occupancy.

75

Figure 4–14: Diffusion and occupancy as a function of pit width for λ-DNAin a 70 nm deep nanoslit with 1 µm lattice spacing. A local minimum is seenwhen the occupancy is close to integer values.

at the N=1, N=2, and N=3 states. Diffusion measurements were carried out

on λ-DNA in that system, and diffusivity minima were observed at each of the

plateaus. While the diffusivity at the N=3 plateau was not as low as at N=2,

it still serves as evidence that a such configurations are entropically stable and

that higher order structures can show damped diffusion. The diffusivity mea-

surement at 750 nm is due to a single molecule undergoing a 1-2-1 transition,

while all other observed molecules stayed in the monomer state, and the error

in the diffusivity is equivalent to the magnitude. Longer measurements with

more molecules in this geometry to further describe this point.

In addition to comparing the mean diffusion and occupancy as a function

of some geometric parameter, it is worthwhile to compare the two directly.

When the two data sets are shown on a scatter plot, clear peaks and troughs

76

can be seen, with the troughs occurring near integer values of occupancy. This

can be compared to a theoretical prediction, based, for instance, on Muthuku-

mar entropic trapping. In figure 4–15, it can be seen that the theoretical and

experimental values correspond. This lends further support to the statement

that the diffusion is damped by entropic confinement phenomena.

While the distinction between different geometries is not made in this

plot, the local minima behaviour can still be seen. The variation between dif-

ferent pit widths in these data is not continuous or uniform. Given enough

molecules observed for a long enough time, it is expected that data points

would clump around certain diffusion and occupancy values rather than be

distributed somewhat continuously. A notable discrepency is the increasing

magnitude of subsequent diffusion peaks that is not predicted by the entrop-

ing trap theory alone. In conjunction with the more recent theoretical work

of Saltzman and Muthukumar, [53], diffusivity is expected to decrease with

increasing trap size, which may explain the smaller magnitude of the diffusion

peaks for larger pits.

An experiment was performed to test the effect of varying molecule length.

Lambda Mono digest was used in a chip with a 70 nm slit, 500 nm lattice spac-

ing, and 300 nm pit size. As seen when comparing diffusivity to occupancy

with varying pit width, molecules with a near-integer occupancy had lower dif-

fusivity compared to molecules that frequently underwent transitions (figure

4–16). This indicates that within the nanopit-nanoslit system, there plateaus

and non-monotonic diffusion resonance with respect to molecule length as well

77

Figure 4–15: Measured diffusion coefficient of λ-DNA in a 70 nm deep nanoslitwith 1 µm lattice spacing. The diffusion is smoothed with a fourth-orderSavitzky-Golay fit, and each point is matched with its corresponding averageoccupancy. A theoretical prediction based on equation 2.12 is overlaid.

78

Figure 4–16: Measured diffusion coefficient of λ digest in a 70 nm deep nanoslitwith 500 nm lattice spacing and 300 nm pits. The diffusion is smoothed with afifth-order Savitzky-Golay fit, and each point is matched with its correspondingaverage occupancy.

as the systems geometric parameters. This may find application where trap-

ping of a specific biomolecule is desired: the geometry can be tailored to damp

the diffusion of a specific sized polymer or protein.

As expected, wider nanoslits lead to a weaker damping of diffusion due to

a smaller strength of confinement. This is seen in figure 4–17. The 200 nm slit

displays much greater diffusion in the same lattice compared to the 70 nm and

100 nm cases. While the trend here is monotonic, this may not always be the

case because plateaus and local minima may occur with varying height as well

as varying pit width. Ignoring the trapping effect of the slits, the diffusivity

79

Figure 4–17: Diffusivity as a function of slit height for λ-DNA in 400 nm pitsseparated by 1000 nm.

is expected to scale with the 2/3 power with respect to slit height [1]. The

observed behaviour clearly does not correspond to that scaling law.

Other literature tends to suggest that the scaling exponent is smaller

rather than larger than 2/3. Harden and Doi [18] found a 0.61 exponent

theoretically, and Mathe et al observed in experiments on slit confined DNA

an even smaller scaling exponent of 0.23 ± 0.07 for diffusion with respect to

slit height [30]. The data observed here points to a larger rather than smaller

exponent. The deviation from slit scaling is due to unequal trapping strengths:

the free energy barrier of the pit does not scale the same way with respect to

height. Both the damping of diffusion due to the slit as well as the entropic

confinement must be taken into account. Thus, it is not unexpected that the

pure-slit scaling exponent is not observed.

80

The non-monotonic dependence of diffusion is one of the key findings of

this thesis. Other studies found non-monotonic diffusion in other contexts,

both through theoretical analysis and experimentation. The diffusion of a

polymer in a network of entropic traps was studied computationally by Nixon

and Slater [35] [34]. In an analysis of a polymer in a two dimensional sys-

tem of periodic obstructions with a larger period of gaps, a non-monotonic

diffusion was predicted with respect to the density of polymer chains. In an

analysis of a polymer in a two dimensional system with one dimensional peri-

odicity, a local maximum of diffusion was shown to correspond with a critical

electrophoretic driving field. Indeed, diffusion in systems of entropic traps is

characterised by non-monotonic behaviour with respect to the parameters of

the traps [56]. Thus, it is not unexpected that the minima are observed. How-

ever, while many theoretical and computational studies have predicted such

monotonic behaviour, experimental evidence has been sparse [35]. In a thesis

by Nykypanchuk, the diffusivity of DNA in a network of spherical traps was

observed to show non-monotonic behaviour with respect to the dimensions of

the trap. Chapman et al [9] showed that the diffusivity of a DNA molecule

in a mixed network of linear and circular DNA is at a local minimum at a

critical fraction of linear and circular DNA, although this diffusion behaviour

can better be described with reptation rather than entropic trapping. Zheng

et al [68] examined the diffusion of polystyrene through a polymer matrix near

a surface and observed that a local minimum of diffusion occurs at a critical

value of the matrix molecular mass. Reguera et al examine a polymer in a

linear network of entropic traps and predict a local maximum in the diffusion

81

with respect to the length scale of the trap [47]. A local maximum in mobil-

ity, rather than diffusion, with respect to applied electric field was noted by

Zeng and Harrison when driving DNA through a system of spherical entropic

traps, similar to those of Nykypanchuk [67]. They posit that above the critical

field, the DNA is electrically trapped in the cavity and undergoes an entropic

trapping behaviour in order to escape.

The few experimental reports on transport through entropic traps, such as

those by Nykypanchuk [36], Mikkelson [32], or Zeng and Harrison [67] indicate

that the mechanism of polymer transport is different than in a free or entan-

gled system. The non-monotonic behaviour of diffusion that is caused when

multiple-trap states become energetically favourable serves as evidence towards

this. While other work has reported non-monotonic diffusion behaviour, this

is the first to predict and measure it systematically with entropic traps.

4.5 Implications, Applications and Extensions

These experiments have demonstrated that entropic traps can be used to

confine macromolecules: the simple case of 1000 nm pits showed very little dif-

fusion. Such trapping is useful in single molecule biophysics and may be used

as a replacement for optical trapping. A similar nanopit system was developed

by Krishnan et al [24] to trap gold, polystyrene, and lipid nanoparticles. Be-

cause spheres lack configurational entropy, the system was described in terms

of electrostatic trapping due to the interaction between the charged fluidic

walls, buffer solution, and charged particles. By analysing the behaviour of

the spheres within these pits, they were able to measure an effective spring

constant of the trap that was similar to that attainable by optical trapping.

While the analysis in this thesis looked at pit occupation as a binary event,

82

analysis of a monomer’s behaviour within a pit could allow a measurement

of the effective spring constant. In addition to studying trapped nanoscale

objects in a single pit, stable multiple pit occupancy can be used to study

macromolecules with non-trivial structure. Such higher-order trapping may

be useful in biochemistry, allowing the study of proteins that have been de-

formed by a network of entropic traps.

The ability to dictate polymer conformation into complex structures at

equilibrium has many uses in nanoscience. While in these experiments the only

structures made stable are dimers or trimers on a square lattice, more detailed

structures can be envisioned. For example, the complex structures of DNA

origami may be created simply by allowing the molecules to self-assemble on a

patterned surface. The surface scaffolds used to direct assembly of other nanos-

tructures such as nanotubes can be made more efficient in a properly designed

entropic trap system. It has been demonstrated that nanowires can be tem-

plated onto DNA [16], and a system of entropic traps may allow the connection

of self-assembled polymer based circuits. Complex surfaces or structures cre-

ated by block co-polymers may gain an additional degree of complexity from

proper surface topography.

The entropic control of single molecule diffusion has wide ranging scien-

tific applications. A network of traps on a lengthscale shorter than the poly-

mers studied may aide in the research of confinement effects on bulk polymer

physics, such as the glass transition temperature confinement effect. Rates of

reaction in macromolecular or biological chemistry may be tailored by modi-

fying the free energy landscape of the reaction environment.

83

This thesis has presented experimental results of occupancy and diffusion

for varying length, pit size, pit spacing, and joint slit and pit height. The

full parameter space may be explored by modifying the persistence length and

effective width of the molecule, either by modifying the buffer concentration or

by binding the DNA to proteins to increase its stiffness. The exact mechanisms

of pit transitions were not examined. The experiments may be repeated with

greater temporal resolution to observe the partition of fluorescence leading to

a transition. Whether a transition occured in and end pit or in the middle

was not examined, and performing the experiments with circular DNA to

ensure hernia-style transport may provide interesting experimental results and

theoretical challenges. Another such study found perplexing results with a

similar circular DNA experiment, [37], and the nanoslit-nanopit system may

provide insight. While the results were not ready for description in this thesis,

the experiment can be performed at lower or higher temperature to observe

the potential effect on conformation and diffusion, perhaps even as the buffer

is frozen.

Further control over the molecule’s behaviour may be achieved by func-

tionalizing the pits, either by deposited chemicals, surface chemistry, or electro-

static trapping. Chemical reactions, for instance between proteins and DNA,

may be observed in the nanoslit-nanopit system.

84

CHAPTER 5Conclusion

A nanofluidic slit embedded with a network of entropic traps was used

to observe the statistical dynamical properties of single DNA polymers. DNA

molecules self-assembled into pit occupancy states within the system, and were

observed in equilibrium. By measuring the mean occupancy of molecules in

varying geometries, the predictions of the theoretical model [49] were largely

confirmed, including effects that the original paper failed to identify. Notably,

plateaus in parameter space where single states dominated were observed, in-

dicating that the nanoslit-nanopit system can be used to create stable nanos-

tructures. Scaling predictions of a generalised theory [53] failed to describe

the observations. The diffusion coefficient was measured for DNA molecules

in various geometries. While diffusion generally increased with decreasing

confinement, the diffusivity showed non-monotonic behaviour, with resonance

minima corresponding to the stable occupancy states in the plateaus. The

results of this research provide insight into polymer physics and may be used

to guide self-assembly in nanotechnology and to study biophysical systems.

Future work will further expand the experimental parameter space and at-

tempt to exert finer control over the static and dynamic behaviour of a single

polymer.

85

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