Eindhoven University of Technology MASTER Unraveling ... · Unraveling single-bond kinetics in...

Eindhoven University of Technology

MASTER

Unraveling single-bond kinetics in tethered particle motion experiments using moleculardynamics simulations

Merkus, K.E.

Award date:2015

Link to publication

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https://research.tue.nl/en/studentthesis/unraveling-singlebond-kinetics-in-tethered-particle-motion-experiments-using-molecular-dynamics-simulations(401c8386-5df8-49ec-8ab8-5e6db0116c5c).html

Unraveling single-bond kinetics intethered particle motion experimentsusing molecular dynamics simulations

K.E. Merkus

July 2015

Eindhoven University of TechnologyDepartment of Applied PhysicsTheory of polymers and soft matter

Under supervision ofdr. C. Storm

Abstract

Tethered particle motion (TPM), the motion of a micro- or nanobead tethered to a substrate by amacromolecule, is widely studied to understand various properties related to the tether. However, wepropose a whole new way of looking at TPM: TPM as a probe for secondary bonds. Kinetic effectsin the motion pattern of a coated bead tethered to a coated substrate yield information about thebonding kinetics of the corresponding coating-molecules and/or solution. We use molecular dynamicssimulations to understand the relation between the bond kinetics and the observed motion patterns.Our results show that kinetic properties of a single bond can indeed be extracted and we presentexperimental optimization. We provide a proof of principle that is of both fundamental and techno-logical interest. The described measurement method may be used in the fundamental study of singlebond kinetics. Moreover, this principle may serve as the basis for a new kind of biosensing device.

Contents

1 Introduction 11.1 Tethered particle motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Tethered particle motion system 52.1 Components of a tethered particle system . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The sandwich assay principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Model system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Simulation methods 113.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Molecular dynamics basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Time integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.5 Implicit solvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.6 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.7 Bond creation and breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Hydrodynamic wall effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.2 Correction on parallel motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.3 Correction on perpendicular motion . . . . . . . . . . . . . . . . . . . . . . . . 213.4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Bead movement 244.1 Motion pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Experimental motion patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.2 Simulation motion patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 Geometrical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.2 Integration boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.3 Distribution results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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CONTENTS

4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Number of polymer beads in MD simulation . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Exclusion effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Influence of the engineering parameters R,L, lp . . . . . . . . . . . . . . . . . . . . . . 39

4.5.1 Varying the bead size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5.2 Varying the tether length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5.3 Varying the persistence length . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.6 Dynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6.1 Autocorrelation ~R(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6.2 Autocorrelation Z(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.7 Hitting and return times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7.1 Hitting times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7.2 Return times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 Step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.8.1 Tether length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.8.2 Bead radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Specific binding spots 495.1 Dots on the bead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Spots on the substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 Single Xdot and Xsubstr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Kinetic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4.1 Reaction process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.4.2 Quick rebinding events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.5 Analyzing bound patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.6 Relation to experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.6.1 Experimental situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.6.2 Sources of uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.6.3 Application to experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Optimization 616.1 Optimize contact area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3 Optimization in the sandwich assay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Conclusion & outlook 697.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

References 71

A Implementing anisotropic drag 75A.1 Diffusion of a bead near a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A.3 Control simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.3.1 Random forces without motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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CONTENTS

A.3.2 Movement parallel to surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.3.3 Movement towards surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A.4 Simulation times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B Tabulated values 84

3

Chapter 1Introduction

A key characteristic of modern health care is the increasing demand for point-of-care applications [1].The measurement of glucose levels by diabetic patients is a very successful example of point-of-careimmuno-biosensing that has been developed and abundantly used for decades [2]. Glucose is found inthe human blood at a concentration in the order of millimolars, which is a regime in which electro-chemical detection is well-suited.

One can think of many other biosensing applications that are yet to achieve such a level of commercialsuccess. Examples are: the detection of protein markers to diagnose cardiac diseases, the detection ofnucleic acid markers in case of infectious diseases, but also the screening for drugs of abuse. However,the concentration of the biomarkers that should be detected in these examples is much lower, typi-cal concentrations for these biomarkers are of the order of picomolars or even sub-picomolar [3]. Abiosensing device should be able to detect these biomarkers, even if very little biomarkers are present.In other words, the biosensor should have a high sensitivity.

Since such measurements are in general performed on body fluids, there are a lot of irrelevant moleculespresent at much higher concentrations. These other molecules should not interfere with the measure-ment, so the measurement should be highly selective. The high levels of sensitivity and selectivity thatare required, are generally not achieved by enzyme-based electrochemical detection.

Detections methods that are better suited for these applications exist. In particular, such a high levelof selectivity and sensitivity can be achieved by using antibodies in immunoassays. Several immunoas-say sensing technologies have been developed, but one that seems particularly suited for a lab-on-chipdevice is the optomagnetic immunoassay technology [4].

Optomagnetic immunoassay technology involves antibody-coated nanoparticles as well as an antibody-coated substrate. The nanoparticles are magnetically actuated and optically detected in a stationaryfluid. The eventual detection mechanism and resulting determination of the initial concentration ofa biomarker depends on the number of beads that is bound to the substrate, ideally by an antibody-biomarker complex for which the assay was designed. Sensitivity is of vital importance in theseapplications, therefore it is crucial to fully understand the binding mechanism that occurs at the sub-strate and how this binding mechanism affects the observed motion pattern of the nanoparticle.

1

CHAPTER 1. INTRODUCTION

Figure 1.1: A schematic representation of the system under investigation: a microsphere attached toa surface by a double-stranded DNA tether. The basic system that is described throughout this thesisconsists of a bead with radius R=500 nm and a tether with length L=50 nm. This picture is not drawnto scale. The dashed line represents the projection of the bead coordinates on the surface, which is theexperimentally observable value.

1.1. Tethered particle motion

To study the properties of a nanoparticle adhered to a substrate bound by an antibody-biomarkercomplex, we resort to a model system: a nanoparticle adhered to a substrate by a double-strandedDNA (dsDNA) tether [5], as is graphically represented in figure 1.1. This brings us into the fieldof tethered particle motion (TPM). Several studies have already been devoted to this subject, so wecan rely on literature to explore a lot of basic properties of this system. In previous research thefocus was on the properties and interactions of the tether (typically DNA), rather than the bindingproperties of coated nanobeads. For example, TPM has been used to determine the transcription ofRNA polymerase [6], the persistence length of DNA [7] and the looping kinetics of DNA [8].

Upon studying this TPM system, it is observed that a coated bead (nanoparticle) tethered to a coatedsubstrate is able to undergo temporary secondary binding events, which affect the motion pattern ofthe bead. This effect depends on the presence of binding molecules in the solvent. The idea hasrisen that this could be a biosensor by itself. Such a biosensor would consist of a coated nanoparticle

2


adhered to a coated substrate with one permanent tether. As a result of the symmetry of the sys-tem, the basic motion pattern would then be rotational symmetric, and have a circular outer border.Temporary changes in the motion pattern should involve the biomarker that one aims to probe andtherefore these changes are a measure of the concentration of this biomarker.

This principle would be a new type of biosensing that would exhibit three major advantages that arealso present in the optomagnetic immunoassay technology, namely: (1) it is suited for low concentra-tion, (2) the target biomolecule does not need to be labelled and (3) the presence of a nanoparticle isoptically detectable in a robust way.

It should be emphasized that the purpose of this research is not to determine the properties of DNAor any other tether-molecule, but rather to understand how the binding kinetics of several componentsof the system affect the observed motion pattern of a nanoparticle.

On a more fundamental level, this system may be a way to investigate single bond kinetics. Severaltechniques have already been exploited for this purpose, such as laminar flow chambers [9], atomicforce microscopy [10] and total internal reflection fluorescence microscopy [11]. Due to several sourcesof uncertainty in these measurements, a new way to measure single-bond kinetics would be a promisingcontribution [12]. This holds especially true since the biological relevance of such methods is widelyrecognized [12], [13], [14].

In order to fully understand the kinetics of a single bond in a TPM system, a distinction betweenhitting and binding should be made. The diffusive motion of the tethered bead, an example of aconfined Brownian motion, determines how often and how long a specific spot on the bead is near aspecific spot on the substrate. We will refer to this process as ‘hitting’. On the other hand, when twobinding spots are near each other, the actual molecular bond has to be formed. We will refer to thisprocess as ‘sticking’. Both processes happen at a different time and length scale and the combinedresult is experimentally observable in motion patterns.

In this report we use molecular dynamics (MD) simulations to acquire a better understanding of TPM.In particular, we review the effect of the bead radius R, the tether length L, and the tether persis-tence length lp on the hitting probability. On top of that we will develop a framework that enablesone to relate experimentally obtained motion patterns to the molecular bond properties. We concludeby determining the engineering parameters that provide the maximum number of observable bindingevents.

This research has been performed in close collaboration with the experimental group MolecularBiosensing for Medical Diagnostics (MBx), another group at the Eindhoven University of Technology.While this report describes how simulations can be used to understand TPM, Max Scheepers has beenfocussing on the experimental part of the same research. Throughout this report, we will refer severaltimes to Scheepers’ work [15].

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1.2. Outline of the thesis

We continue this thesis in chapter 2 with an extensive description of a TPM sytem. We start bydescribing the components of a TPM system. Subsequently we describe the principle that we aim touse in a biosensing device and we describe the model system that we use to start investigating TPM.

The simulation methods that we use are described in chapter 3. A short description of the previouslydeveloped Monte Carlo (MC) simulations is provided and an extensive treatment of the MolecularDynamics (MD) simulations that we developed is presented.

In chapter 4 equilibrium and dynamic properties of the bead can be found. We investigate whatfraction of the time the bead is close enough to the substrate to form a secondary bond and demonstratewhich effects are relevant in describing these distributions. Furthermore, the correlation times andhitting and returning times of the bead are computed.

Chapter 5 discusses the binding in a higher level of detail. In this chapter we zoom in on the actualbinding spot on the bead and on the substrate. We provide an upper bound for one of the involvedkinetic parameters. We finish chapter 5 by presenting a scheme that allows one to extract additionalkinetic parameters experimental data and we apply it to experimental data from Scheepers [15].

To optimize potential experimental systems, chapter 6 is dedicated to the determination of the optimalparameters in an experimental system. We provide the tether length L that leads to the maximumcontact area on which detectable bonds can be formed. Moreover, we outline a possible optimizationalgorithm for an actual biosensing device.

We finish in chapter 7 with a summary of our conclusions and possibilities for future research. At theend we provide a list of references to literature and appendices A and B.

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Chapter 2Tethered particle motion system

2.1. Components of a tethered particle system

Before considering any system-specific details about tethered particle motion (TPM), we should firsthave a clear picture of what TPM actually is. The method is in general based on a polymer (tether)with one end attached to a surface (substrate) and with the other end that is attached to an otherwisefree bead (particle). TPM then refers to the motion of this particle as well as the method of trackingthis motion. Clearly TPM requires two basic components: a polymer (tether) and a bead (particle).

The tethers that are usually considered in TPM experiments are either double-stranded DNA [6],single-stranded DNA [16] or RNA [17]. However, any polymer - or even, any macromolecule - thatis able to attach a bead to a surface could in principle be used for TPM. In this report we willconsider TPM systems based on double-stranded DNA (dsDNA), having a persistence length lp ofapproximately 50 nm [18].

In order to observe actual bead movement on experimental timescales the size of the bead shouldnot exceed the micrometer scale [19]. Several types of beads have been used in TPM experiments.These can be divided into three categories: metal beads (usually gold [6]), polystyrene beads [8] andfluorospheres [20]. Comparing metal beads to polystyrene beads, the advantage of metal beads is thestrong scattering of light. On the other hand, polystyrene beads enable the use of optical tweezerexperiments and enable magnetic control of the bead by including a magnetic core. One could alsouse fluorospheres, but for the application that we describe two major disadvantages come to mind:(1) the fluorescent molecules will occupy some of the available binding spots on the bead and (2)photobleaching should be controlled, since this would lead to a fading signal. In this report we willfocus on polystyrene beads with a radius of 500 nm.

The beads that are used in experiments are not always smooth, but may also have a patchy surface [15].We will refer to this principle as ‘bead roughness’. We have not explicitly modeled bead roughness,but we treat this in some of the discussions on the applicability of our results.

The configurations that are allowed for this system are restricted by three types of exclusion effects.The first effect is that the bead is not allowed to pass through the surface (bead-surface exclusion),the second effect is that the bead is not allowed to pass through the tether (tether-bead exclusion) andthe third effect is that the tether is not allowed to pass through the surface (tether-surface exclusion).Segall et al. already demonstrated in a theoretical paper that the first of these exclusion mechanisms(bead-surface exclusion) is indeed relevant ‘bead size matters’ [21]. On top of that, we will show in

5

CHAPTER 2. TETHERED PARTICLE MOTION SYSTEM

section 4.4 that both tether-bead exclusion and tether-surface exclusion have a significant effect onthe allowed configurations of the system.

In common experimental systems, the position of the bead is tracked by dark field microscopy [7],which results in data of the bead-center coordinates of X and Y parallel to the substrate i.e. the2D-projected motion of the bead. Although there are experimental systems in which the Z-coordinateof the bead can be tracked [20], these systems are less common. In the experimental system that westudy [15] the bead is tracked in two dimensions and it is thus important to keep in mind that everyresult should be translated into quantities that can be extracted from measuring X and Y .

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2.2. The sandwich assay principle

Figure 2.1: The initial configuration of a tethered particle in the setup of a sandwich assay. Notethat the bead is coated with a specific type of antibody (green) and the substrate is coated with anotherspecific type of antibody (blue). The molecule to be detected - the analyte - is represented by a red dot.Note: this picture is not drawn to scale.

The main technological motivation of our TPM study is the possibility of constructing a biosensingdevice based on TPM. For the detection of biomarkers with concentrations in the order of picomolarsor sub-picomolar, the sandwich immunoassay principle has proven useful [3].

An example of such a TPM sandwich assay is given by the figures 2.1 and 2.2. In figure 2.1 the initialconfiguration of a bead is schematically drawn. The bead is coated with a specific type of antibodiesand the substrate is coated with a second specific type of antibodies. The molecule that we wantto detect, the analyte, is represented by a red dot. Figure 2.2 schematically describes the four stepsinvolved in such a TPM sandwich assay. Here, only the antibodies that are involved in the bond thatis formed are drawn.

Figure 2.2 shows that the process of a tethered bead forming a secondary bond with the substrate inthe sandwich assay can be divided into three steps: (1) the binding of the analyte with an antibodyon the bead, (2) the bead moving towards the substrate so that two antibodies are within interactionrange (this process is defined as ‘hitting’) and (3) the molecular antibody-analyte-antibody-complexis formed, resulting in the actual sticking of the bead to the substrate.

We could describe this process using reaction formulas, let [FB] be the concentration of free (farfrom the substrate) beads with no analyte bound to it, [A] the concentration of analyte, [FBA] theconcentration of free beads bound by an analyte, [HBA] the concentration of hitting beads boundby an analyte and [SBA] the concentration of beads bound to the substrate by an antibody-analyte-

7


antibody complex. The reaction formula could then be written as

FB +Akcatch−−−−→←−−−−kdetach

FBAksep−−−−→←−−−−ksep

HBAkstick−−−−→←−−−−koff

SBA, (2.2.1)

where the reaction rates are determined by kcatch, kdetach, khit, ksep, kstick and koff as

∂[FB]

∂t=∂[A]

∂t= kdetach[FBA]− kcatch[FB][A] (2.2.2)

∂[FBA]

∂t= kcatch[FB][A]− kdetach[FBA]− (khit[FBA]− ksep[HBA]) (2.2.3)

∂[HBA]

∂t= khit[FBA]− ksep[HBA]− (kstick[HBA]− koff [SBA]) (2.2.4)

∂[SBA]

∂t= kstick[HBA]− koff [SBA]. (2.2.5)

Note that we implicitly made two assumptions while describing this system. First of all, we assumethat the bead is not hitting the substrate before the analyte has bound. Indeed, two antibodiescan already be within interaction range by chance, but this is a rare process in the systems that weconsider and therefore we do not take it into account in our model. Secondly, we have assumed thata bead is either bound by an analyte or not bound by an analyte. In reality, the bead can clearly bebound by several analytes. We aim to develop a biosensor for low concentrations, so the binding ofan analyte to the bead is a rare event. In this regime, only taking into account the single bonds is agood approximation.

kcatch

kdetach

khit

ksep

kstick

ko

Figure 2.2: A schematic picture of how tethered particle motion may be used to create a sandwichassay. The picture displays a bead that is tethered to the substrate by dsDNA, with one antibody onthe bead and one antibody on the substrate. The red dot represents the analyte: the molecule thatshould be detected. The picture describes the sandwich assay principle in four stages: (1) the initialconfiguration where no binding event has occurred. (2) The analyte has bound to the antibody on thebead. (3) The bead has moved so that it ‘hits’ the substrate, but it does not stick yet. (4) The twoantibodies have formed a molecular complex with the analyte and the bead is now stuck to the substrateat two spots. The probabilities to shift between one of these stages are governed by kcatch, kdetach, khit,ksep, kstick and koff , as displayed in the figure. A further description of these k-values is given inthe main text. In this picture only the antibodies relevant for the binding process are drawn, in realitymore antibodies are present, as figure 2.1 indicates. Note: this picture is not drawn to scale.

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2.3. Model system

In this study we aim to investigate the effect of molecular bonds on the motion pattern of a tetheredparticle. The first step in figure 2.2 is in that sense irrelevant: the movement of the bead effectivelydoes not change by the binding of an analyte. A similar system that is initially already in the secondstage, corresponding to FBA, enables us to focus on the processes that indeed influence the motionpattern of the bead.

In line with our experimental collaborators [15], we turn to a model system in which analytes are nolonger present. In this system the molecules on the bead and the molecules on the substrate are coatedwith complementary oligonucleotide (short DNA) strands. We focus on the 8 basepair oligonucleotidestrands, with a length of 3 nm and a reported koff of 0.1 s−1 [22].

Due to experimental convenience, in this system the bead is coated with streptavidin instead ofan antibody. The substrate is again coated with an antibody. The streptavidin on the bead andthe antibody on the substrate are thus coated with complementary oligonucleotide sequences. Theprinciple of this system is outlined in figure 2.3. The simulations that are described in this report aimto describe this model system.

kstickkhit

ksep ko

Figure 2.3: A schematic picture of the model system that we have used. The picture displays a bead thatis tethered to the substrate by dsDNA, with one streptavidin molecule on the bead and one antibody onthe substrate. The curved light blue lines represent complementary oligonucleotide sequences. Bindingin this model system is characterized by three stages, which are similar to the last four stages in thesandwich assay of figure 2.2. Similar to figure 2.2, the probabilities to shift between one of these stagesare governed by khit, ksep, kstick and koff , which is indicated at the arrows in the figure. In this pictureonly the antibody and streptavidin molecule relevant for the binding process are drawn, in reality morestreptavidin molecules are present on the bead and more antibodies on the substrate. Note: this pictureis not drawn to scale.

In this system no analytes are present and the reaction scheme simplifies to

FBkhit−−−−→←−−−−ksep

HBkstick−−−−→←−−−−koff

SB, (2.3.1)

where FB indicates the free bead, HB the bead that is hitting the substrate and SB the bead that

9


sticks to the substrate. The changes in concentration are now described by

∂[FB]

∂t= ksep[HB]− khit[FB] (2.3.2)

∂[HB]

∂t= khit[FB]− ksep[HB]− (kstick[HB]− koff [SB]) (2.3.3)

∂[SB]

∂t= kstick[HB]− koff [SB]. (2.3.4)

Hitting is defined as the process of the bead moving close enough to the substrate so that a bond canbe formed. Taking Z as the z-coordinate of the bead center, R as the radius of the bead, and ZHIT thedistance at which a bond can be formed, hitting can be mathematically defined as Z < (R+ ZHIT ).

Hitting is fully determined by the confined Brownian motion of the bead and thus depends solelyon the diffusive properties of the bead. The time it takes for a sphere to diffuse its own radius, i.e.the configurational relaxation time τcr [23] is in the order of 10−1 s for the beads that we consider.However, the distance that the bead needs to cover to hit the substrate is typically one order ofmagnitude smaller than the actual bead radius. So we expect relevant timescales for hitting to be inthe order of τ ≈ 10−2 s.

On the other hand, when two molecules, one on the bead and one on the substrate are within bindingdistance, the actual bond still has to be formed. This process happens on a much faster timescale,with attempt frequencies in the order of 109 - 1010 s−1 [24]. We will refer to this process as sticking,and this process is determined by the bond kinetics, expressed in the sticking rate kstick and off-ratekoff [12].

Using TPM to investigate single bond kinetics, which means ultimately determining kstick and koff fora single bond, it is crucial to make the aforementioned distinction. In this report we will use moleculardynamics (MD) simulations to isolate the hitting process and determine the dependence on systemparameters. Initially we will take into account every instance in which the bead is close enough to thesubstrate. Further on we will zoom in to a higher level of detail and look at hitting for specific spotson the bead as well as specific spots on the substrate.

We will not go into detail about the molecular processes that govern the sticking process. In thefirst place, the level of coarse-graining of our simulations does not allow for inclusions of such effects.Moreover, we aim to develop a method of characterizing single bond kinetics for a wide variety ofbond types. Understanding the relation between molecular structure and binding kinetics of two(bio-)molecules is a different area of expertise and could complement an eventual study of single-bondkinetics for a specific bond.

10

Chapter 3Simulation methods

3.1. Monte Carlo

The code used for the Monte Carlo (MC) simulations was developed prior to this project andturned out to be a valuable tool in explaining the different shapes and distributions of mo-tion patterns that were observed experimentally [5]. MC simulations provide the equilibriumdistribution of states that is found in a system. In other words, the MC simulations describethe distribution of states that will be found if the time window in which data is obtained issufficiently large.

This project focuses on the motion patterns that exhibit a shape and distribution of motionpattern that changes in time. Such dynamical effects are not incorporated in these MC simula-tions. For this reason we develop molecular dynamics (MD) simulations of this system, whichdo incorporate dynamical effects, i.e. we can review the motion of the bead in time and we canreview the way this motion is affected by binding events.

Nonetheless, in absence of any dynamical effects that influence the motion pattern, the distri-butions obtained by the MC simulations should match with the distributions obtained by theMD simulations for sufficiently large simulation times. The MC simulations thus provide us away to benchmark the long-term behavior of our system. The rest of this section provides abrief description of the MC simulations.

The MC simulations were developed and executed using MATLAB [25]. The basic set up ofthe simulation is that first a tether is constructed, then a bead is attached at the end of it andthen a check for the three exclusion mechanisms, discussed in section 2, is performed. The MCsimulations require the input of four parameters: the bead radius R, the tether length L, thetether persistence length lp and the tether segment length ls.

The tether is built up of N segments, the angle between every segment is extracted from a setof Gaussian distributed random numbers so that in the limit of N → ∞ the distribution of aWorm-Like Chain is reproduced.

Subsequently, the bead is attached at an angle extracted from a random distribution of angles.Bead-tether exclusion dictates that this angle cannot exceed π/2. These simulations allow forthe addition of protrusions on the bead, to incorporate the effect of bead roughness.

In summary, MC simulations produce quasi-random configurations of the system and thenonly keep the valid configurations to obtain a distribution of configurations. The long-timedistributions that follow from MD simulations should match these distributions.

11

CHAPTER 3. SIMULATION METHODS

3.2. Molecular dynamics

This section describes the molecular dynamics (MD) simulations. As opposed to the Monte Carlo(MC) method, this method does provide the evolution of the system in time, which is clearly requiredto describe a time-dependent effect. The rest of this section provides a thorough description of theMD simulations.

To perform these MD simulation, we use an open source simulation package called LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [26]. Despite the fact that our system is com-pletely bonded and therefore provides little opportunity for parallelization, (processor communicationtimes would rise drastically) this is still a very usable and accessible package for MD simulations.

3.2.1 Molecular dynamics basics

MD is widely employed to study kinetics and equilibrium properties of complex many body sys-tems [27]. In MD simulations the motion of all interacting particles are tracked by numerically solvingNewton’s equation of motion for every particle.

An MD algorithm divides the entire time span of a simulation into shorter intervals called simulationtime steps. The positions and velocities for every particle are obtained by adding to the initial po-sition the total displacement over one simulation time step, in response to all forces applied to thatparticle [28].

Newton’s equation of motion for a particle of mass M at position ~r(t) for time t is given by [29]

~F (t) = M∂2~r(t)

∂t2, (3.2.1)

where ~F (t) is the total force on the particle at time t. The force consists of several parts that we willdiscuss further on in this section.

3.2.2 Time integration algorithm

A time integration algorithm is required to compute the new velocities and positions of the particlesafter proceeding one time step in time. We have used velocity Verlet integration, which is the standardintegration scheme of LAMMPS. This scheme is based on the Taylor expansion of the position vector~r(t) around time t, namely

~r(t+ ∆t) = ~r(t) + ~v(t)∆t+1

2~a(t)∆t2 +

1

6~b(t) +O(∆t4) (3.2.2)

~r(t−∆t) = ~r(t)− ~v(t)∆t+1

2~a(t)∆t2 − 1

6~b(t) +O(∆t4), (3.2.3)

where ~v = ~r(t), the velocity, ~a = ~r(t) the acceleration and ~b the jerk (third derivative of position withrespect to time t). Adding these two expansions leads to

~r(t+ ∆t) = 2~r(t)− ~r(t−∆t) +~F (t)

m∆t2 +O(∆t4). (3.2.4)

Since the first and third order terms in ∆t cancel out, the estimate of the new position contains anerror that is of order ∆t4 [28]. A simple concept - combining two Taylor expansions - leads to asignificantly more accurate time integration algorithm!

12


3.2.3 Components

There are two types of particles that play an important role in our simulations. The first typerepresents the bead. In most simulations the coordinates of the center of mass of the bead willbe the relevant output of the simulation. The second type of particles represents the tether.Having a bead in the simulation is rather straightforward: We use a spherical particle of whichthe radius is chosen to match the actual bead size.

Implementing the tether requires slightly more care. We do not expect that the moleculardetails of the dsDNA tether significantly influence the motion of the bead. There are howeverfour properties of the tethers that we do expect to be relevant for the motion of the bead: (1)the polymer is an effective spring [30] that holds the bead near the surface, (2) the polymer hasa mass and therefore inertia, (3) the polymer experiences hydrodynamic drag and (4) there isa steric interaction between the tether and the bead. Since we only want to include the effectsthat we expect to be relevant, we have looked for a model that efficiently implements these foureffects. It turns out that this can be achieved by using a bead-spring model to represent thetether [31].

At this point the risk of confusion arises. We have been calling our polystyrene nanoparticle‘the bead’, but now our tether consists of several smaller beads. We will keep referring tothe former as ‘the bead’ and we will refer to the latter as ‘polymer beads’. The number ofpolymer beads that is required to produce good results is discussed in section 4.3. One end ofthe bead-spring string is rigidly connected with the bead so that it is effectively treated as oneparticle.

3.2.4 Interactions

In MD simulations, interactions are included by defining relevant potentials for every particlein the system. At every time step the force on a particle is then computed by

~F = −∇~U(~ri, ~rj , ...), (3.2.5)

where ~U(~ri, ~rj , ...) is the potential that typically depends on multiple interparticle distances.There are two main types of interactions in our simulations. Firstly, we will describe the interac-tions that prescribe the behavior of the tether. Secondly, the incorporations of exclusion effectswill be discussed. The formation of a secondary bond is not governed by an interaction poten-tial in our simulations. The implementation of these effects can be found in subsection 3.2.7.

The interactions acting between the polymer beads in a bead-spring model that represents asemiflexible polymer are governed by two types of potentials, i.e. bond potentials and angle-bending potentials.

An inherent property of a bead-spring model is the that between all subsequent beads a poten-tial is applied that may be considered a ‘spring’. For this we use a harmonic potential

Ubond = Kb(r − r0)2, (3.2.6)

with r0 the equilibrium distance and bond coefficient Kb. One can think of r0 as being thelength of the spring. Note that the usual factor 1/2 is included in Kb.

13


To ensure a fixed bond length in our simulation a large value of Kb should be chosen [32]

Kb = 50kBT

r20, (3.2.7)

with kB being the Boltzmann constant and T being the temperature. Double-stranded DNA is apolymer with limited flexibility, that may be described as a worm-like chain with persistence lengthlp. In a bead-spring model this translates into an angle-bending potential given by [31]

Uangle = Kaθ2 =

kBT lp2r0

θ2, (3.2.8)

where Ka is the angle-bending coefficient and θ is the angle between two adjacent springs.

To include the exclusion effects, bead-surface exclusion, tether-bead exclusion and tether-surface ex-clusion, certain system configurations should be accessible and certain configurations should be inac-cessible. This translates into the prohibition of certain interparticle or particle-surface distances. InMD this may be implemented using a steep potential.

A widely used potential for this purpose is the repulsive part of the Lennard-Jones potential, alsoknown as the WCA potential [33], given by

ULJ(r) = 4ε

[(σr

)12−(σr

)6], r < rc (3.2.9)

where the energy ε and distance σ are the parameters that define the potential and rc = 21/6σ ≈ 1.12σ,so that only the repulsive part is used. A graphical representation is given in figure 3.1. Usingthis potential, r is usually the distances between the particle centers. We can also define it as thedistance between a particle and the surface, which allows us to include bead-surface and tether-surfaceexclusion.

0.95 1.00 1.05 1.10r HΣL

-1

1

2

3

4

5

6

UHΕL

Figure 3.1: The repulsive part of the Lennard-Jones potential, given by equation 3.2.9. On the verticalaxis the interaction energy is given in terms of ε and on the horizontal axis the distance is given interms of σ.

14


Since the bead and the tether-parts vary hugely in size, expression 3.2.9 , where r is the distancebetween the particle centers, is not sufficient to create a steep potential. As instead, the expandedLennard-Jones potential is used

ULJ,exp(r) = 4ε

[(σ

r −∆

)12

−(

σ

r −∆

)6], r < rc + ∆, (3.2.10)

where the interaction distance r is shifted by a distance ∆ that allows us to tune the potential withthe desired steepness.

3.2.5 Implicit solvent

The actual motion in tethered particle motion experiments is a confined Brownian motion that resultsfrom the many solvent molecules that collide with the bead. The time- and lengthscales at which thesolvent molecules collide with the bead are much smaller than the time- and lengthscales that describethe motion of the bead [23]. Effectively this results in a random force experienced by the bead. Ontop of that, a bead with finite velocity will also experience a drag force due to the solvent.

This section describes the way the interactions with the bulk solvent are commonly implemented. Thisapproach is usually referred to as ‘Langevin dynamics’ [34]. We are aware of the fact that our beadis not in free solution, but rather close to a surface. Hydrodynamics near a surface differ vastly fromhydrodynamics in free solution. The way we incorporate this is discussed in section 3.4

Explicitly taking every solvent molecule into account will significantly increase computation time andis therefore unwanted. An implicit way to incorporate solvent effects in the MD simulation is byadding a random force and a drag force to every particle in the solution. At every time step eachparticle in the solution then experiences three types of forces

~Ftot = ~Fc + ~Fr + ~Fd, (3.2.11)

where ~Ftot is the total force, ~Fc is the conservative force, ~Fd is the drag force and ~Fr is the randomforce. The conservative force ~Fc is the result of the interactions as described in section 3.2.4 andis computed via equation 3.2.5. The drag force ~Fd and random force ~Fr are both the result of theinteraction with the solvent. One may therefore expect that these forces are related. Indeed thefluctuation-dissipation theorem tells us that [35]

~Fd = −MΓ~v (3.2.12)

〈Fr(t)〉 = 0 (3.2.13)⟨~Fr(t) · ~Fr(τ)

⟩= 6kBTMΓδ(t− τ), (3.2.14)

where 〈. . .〉 denotes the ensemble average and Γ is the friction constant per mass. This should betranslated into a force that is applied every time step, in MD this can be implemented by [36]

Fr ∼√kBTMΓ

dt(3.2.15)

~Fd = −MΓ~v, (3.2.16)

with dt being the size of the time step. The drag force ~Fd is identical to the expression given inequation 3.2.12, but now the velocity ~v refers to the velocity stored during simulations and the drag

15


force ~Fd is the force that a particle experiences at every time step during simulation.

The direction of ~Fr is random and the magnitude should be scaled by a prefactor. This prefactordepends on whether Gaussian random numbers are used or uniform random numbers. Uniform randomnumbers are computationally more efficient and require the prefactor

√24 [37]. This is how implicit

solvent is implemented in the function fix langevin in LAMMPS [26].

A final note on the drag parameter Γ. The drag on a spherical particle in a liquid is given by theStokes drag [38]

~Fd = −6πηR~v, (3.2.17)

where R is the radius of the bead and η is the dynamic viscosity of the solvent. We use this to choosethe correct value for Γ.

3.2.6 Timescales

One of the biggest challenges of simulating this system is the large range of timescales that play a rolein TPM. The most relevant timescales are discussed in this section.

In the bead-spring model that represents the tether, the beads are bonded by harmonic springs. Everyharmonic spring has a characteristic vibrational frequency and a corresponding timescale. In orderto correctly sample the whole movement of the tether, the time step that is used in the simulationshould be well below this characteristic vibrational timescale. Using the potential of equation 3.2.6and Newton’s second law we find that

Mr = −2Kb(r − r0). (3.2.18)

Solving this differential equation we obtain a characteristic frequency ω and a corresponding charac-teristic timescale τb

ω =

√Kb

M⇒ τb = 2π

√M

Kb(3.2.19)

This is the relevant timescale for the bond vibrations and in order to not get into any problems withour simulations, our simulation time step should be well below this.

Other relevant timescales are the momentum relaxation time τmr and the configurational relaxationtime τcr of the bead [23]. The momentum relaxation time, is the characteristic time at which the dragforce decreases momentum, so given by

M∂~v

∂t= −MΓ~v ⇒ τmr = 1/Γ. (3.2.20)

The configurational relaxation time τcr is the time it takes for the bead to have an average meansquared displacement equal to its own radius R.

τcr =3πηR3

kBT(3.2.21)

Typically, this value is many orders of magnitude larger than the momentum relaxation time τmr.Distances that the bead needs to cover to hit the surface are typically one order of magnitude smallerthan the bead radius, so we expect the hitting mechanism to occur at timescales of order 0.1τcr. Weneglect the effect of the surface on the drag on the tether.

16


To increase the maximum time step that may be used in the simulations, we have used the Grønbech-Jensen/Farago time-discretization of the Langevin model to the polymer beads for some simula-tions [39]. The principle of this method is to change the way in which the random force is ap-plied, so that it is composed of the average of two random forces representing half-contributions fromthe previous and current time intervals [26], to enable longer time steps to be used. We checked forseveral cases that the obtained dynamic properties corresponded to simulations without the Grønbech-Jensen/Farago time-discretization.

3.2.7 Bond creation and breaking

The secondary bonds that are formed in our simulations are created by the functions fix bond/createand fix bond/break in LAMMPS [26]. The creation function is applied every predefined interval, whichwe choose to be 1 µs. If the distance between two particles is smaller than a certain cutoff radius Rc thebond is formed with a certain probability, which corresponds to the kstick we previously introduced.

The breaking of the bond is simulated in a similar fashion. We apply this function every 1 ms.Every millisecond, the bond is broken with a probability, which corresponds to the koff we previouslyintroduced.

17


3.3. Parameter values

In this section the actual values of several parameters are documented, along with the correspondingreasoning. When results are provided in this report, unless mentioned otherwise, the values reportedin this section are used. All parameter values can be found in table 3.1.

The engineering parameters R, L and lp are chosen to match the experimental conditions [15], withthe values 500 nm, 50 nm and 50 nm respectively. The tether length in the experiments is actually 40nm, but it is attached to the substrate by an antibody of approximately 15 nm [15]. We have chosento take an effective tether length of 50 nm. The segment length ls is a parameter only used in MCsimulations. We have not modified this parameter, but have kept it at 1 nm.

The bead consists of polystyrene, which is known to have a density of ρpol ≈ 1 g/cm3. As we considerspherical particles, the mass is now given by

M =4

3πR3ρpol = 5.24 · 105 ag. (3.3.1)

Experiments are usually carried out at room temperature. Moreover, any point-of-care biosensingapplication should be able to function at room temperature. Therefore we take T = 293 K. Water isthe usual solvent, of which the dynamic viscosity is equal to 1.00 ag/(nm ns) at room temperature [40].The drag on the bead should match the Stokes drag for a spherical particle, given by equation 3.2.17.This leads to a friction constant per mass Γ of 0.018 ns−1.

The number of beads Nbeads that is chosen to make the bead-spring model yield the correct behaviorshould be considered carefully. A thorough discussion on this can be found in section 4.3 and thisvalue is chosen to be 10.

Choosing Nbeads fixes the value of σ, m and ΓDNA. The polymerbead radius σ is simply given byL/Nbeads, 5 nm in this case. Nbeadsm should represent the mass of a dsDNA chain with length L. Themass of one nucleotide pair is on average equal to 6 · 102 Da with 3 nucleotide pairs per nm [41].

The polystyrene bead is not the only element in our system that undergoes hydrodynamic drag. TheDNA-chain itself is also continuously subject to hydrodynamic drag. Our DNA-chain is modeled asa bead-spring model. In this model the polymer beads are representing the inertia of the polymer aswell as the spots at which the drag acts on the polymer.

Each polymer bead is in turn subject to a Stokes drag, which is given by

ΓDNA =6πηRhym

, (3.3.2)

in which η is the dynamic viscosity of the solvent and Rhy is the so-called hydrodynamic radius ofthe beads. This hydrodynamic radius is thus an effective parameter which determines the amount ofdrag on the polymer. We should choose an appropriate value for this effective parameter. A way todetermine this, is by taking the drag on a cylinder as a starting point.

The drag on a cylinder is different in the parallel and perpendicular direction [42]

ζ‖ = mΓ‖ =2πηL

ln(L/b)(3.3.3)

ζ⊥ = mΓ⊥ =4πηL

ln(L/b), (3.3.4)

18


where L is the length of the cylinder and b is the diameter of the cylinder. Since the tether can beviewed as being effectively trapped between two walls [8], the movement of the building blocks of thetether will be predominantly in the direction perpendicular to the tether itself. Therefore we use theperpendicular component given by equation 3.3.4.

For L we take the length of our DNA molecule and for b we take the width of a dsDNA molecule: 2nm [43]. Equation 3.3.4 then gives the drag on the whole cylinder, which should be equal to the totaldrag on all Nbeads beads. This gives us

ΓDNA =4πηL

mNbeads ln(L/b), (3.3.5)

yielding the numerical value of 1.22 · 103 ns−1.

Table 3.1: The standard values for several parameters that are used in the simulations. Unless men-tioned otherwise, results reported are obtained using these parameter values.

Parameter description symbol value units

Bead radius R 500 nmTether length L 50 nmTether persistence length lp 50 nmTether segment length ls 1 nmBead mass M 5.24 · 105 agBead friction per mass Γ 0.018 ns−1

Temperature T 293 KSolvent viscosity η 1.00 ag/(nm ns)Number of beads in polymer Nbeads 10 -Polymerbead radius σ 5 nmPolymerbead mass m 0.016 agDNA width b 2 nmPolymerbead friction per mass ΓDNA 1.22 · 103 ns−1

Bond coefficient Kb 8.1 ag/ns2

Angular coefficient Ka 20.23 ag nm2/ns2

Time step dt 0.001 - 0.004 nsCharacteristic bond time τb 0.016 nsMomentum relaxation time τmr 55.6 nsConfigurational relaxation time τcr 2.9 · 108 ns

19


3.4. Hydrodynamic wall effects

3.4.1 Introduction

A striking feature of a relatively big bead on a small tether is that it is at all times close to the surface,i.e. the distance between the surface and the edge of the bead is smaller than the radius of the bead(in fact, much smaller). This is the regime in which hydrodynamic wall effects become important [44].

v

v

Figure 3.2: Schematic representation of a sphere near a wall. The component of the velocity of thesphere parallel to the wall is indicated by v‖ and the perpendicular component is indicated by v⊥. Thesphere will experience a larger drag than in free solution. Moreover the sphere will experience differentdrag in the parallel and perpendicular direction.

Consider a sphere moving in the proximity of a wall as in figure 3.2. A sphere moving through freesolution, i.e. in the absence of any walls, creates a flow field and experiences a resulting Stokes drag.However, in this case a wall is present which perturbs the flow field around the sphere. This resultsin increased drag on the sphere. Moreover, the drag does not increase isotropically, but displays astronger increase in the direction perpendicular to the surface than the direction parallel to the surface.We will refer to this principle as drag anisotropy.

Intuitively the increase in drag when the sphere moves towards the walls is the easiest to grasp, thebead has to ‘squeeze out’ the fluid between the sphere and the wall. Actually, the sphere experiencesthe same drag for the motion towards the wall as for the motion away from the wall. The drag onthe sphere in the direction parallel to the wall is also increased, but not as much as the perpendiculardrag.

3.4.2 Correction on parallel motion

The drag on a sphere moving parallel to a single wall was worked out by Faxen in his dissertation [45].Faxen applied the method of reflections to a sphere near a wall, which is a method of iteratively

20


applying boundary conditions at one spot and calculating the correction to the velocity field at theother spot. Furthermore, Faxen expressed the fundamental solution of Laplace’s equation in integralform [38].

This treatment yields a parallel drag per mass coefficient Γ‖ of

Γ‖ =Γ0

1− 916z∗ + 1

8z∗3 − 45

256z∗4 − 1

16z∗5 , (3.4.1)

where Γ0 is the Stokes drag per mass on a sphere in free solution, given by Γ0 = 6πηR/M for a spherewith radius R in a liquid with dynamic viscosity η and z∗ is given by z∗ = R/z, with z the distancefrom the center of the bead to the surface.

3.4.3 Correction on perpendicular motion

The drag on a sphere moving perpendicular to a single wall was worked out by Brenner [46]. Brennerapplied bipolar coordinates to the Stokes equation to find an exact solution, given by the expression

Γ⊥ =4

3sinh z∗

∞∑n=1

n(n+ 1)

(2n− 1)(2n+ 3)

[2 sinh(2n+ 1)z∗ + (2n+ 1) sinh 2z∗

4 sinh2(n+ 12)z∗ − (2n+ 1)2 sinh2 z∗

− 1

]Γ0, (3.4.2)

where again z∗ = R/z, with z the distance from the center of the bead to the surface and R the radiusof the bead.

This infinite sum series including several hyperbolic functions is not very efficiently implementable.Fortunately, several interpolation functions can be found in literature [44]. Since Faxen’s law givenby equation 3.4.1 is an approximation up to fifth order, we choose to use an interpolation for theperpendicular coefficient up to fifth order as well:

Γ⊥ =Γ0

1− 98z∗ + 1

2z∗3 − 57

100z∗4 + 1

5z∗5 . (3.4.3)

We can introduce the relative drag coefficients λ‖, λ⊥ to simplify the form of equation 3.4.1 and 3.4.3,by defining Γ‖ = λ‖Γ0 and Γ⊥ = λ⊥Γ0.

One may wonder if the effect of near-surface effects on the drag and diffusion of a bead is significantfor our experiments. It should be pointed out that in a TPM system the width of the gap between thebead and the surface cannot exceed the tether length L and that L < R for all the systems that weconsider. For gap widths smaller than the radius of the bead these effects are strongly pronounced andthus relevant. We will show in section 4.6 and 4.7 that the influence of this effect is indeed significant.

The coefficient λ‖ can be found in figure 3.3 and both drag coefficients can be observed in figure 3.4.It should be noted that the perpendicular component diverges, while the parallel component increasesto a finite value.

3.4.4 Implementation

Since we aim to develop Molecular Dynamics simulations to better understand the behavior of sucha TPM system, we need a method to include these anisotropic effects. We have considered explicit

21


510 520 530 540 550Distance HnmL2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1Rel. drag coefficent

Figure 3.3: The parallel component λ‖ of the drag as function of the distance of the bead center to thesurface at z = 0 for R = 500. The relation displays a maximum of 3.08 at z = 500 nm (touching thesurface) and reduces to 1 for z →∞.

representation of the fluid by methods such as stochastic rotation dynamics (SRD) [47] or LatticeBoltzmann (LB) [48]. However, since the bead is much larger than the typical gap width, this woulddrastically increase computation times.

Another way to implement this is to start from the implementation of an implicit solvent, as describedin section 3.2.5 and add the prefactors that are described in this section. A way to include this wasnot readily available in LAMMPS, but we have adjusted the code to add this feature. A detaileddescription of the implementation of this function can be found in Appendix A.

3.4.5 Discussion

It is debatable whether this implementation truly captures all hydrodynamic effects in the experimentalsituations we consider. Let us discuss three potential objections.

First of all, Faxen’s law is only a valid approximation when the bead is not too close to the surface [49]and λ‖ ≤ 1.7. For small gap widths lubrication theory becomes the leading theory [47]. Clearly, thisis not exactly the regime we are considering. The bead in our simulations is allowed to approach thesurface and reduce the gap width to arbitrarily small distances. However, for gap widths larger than0.013R (6.53 nm) the error does not exceed 10%. Moreover, in experiments Faxen’s law has showncorrespondence up to λ‖ ≤ 2.4 [44].

Another argument that one could make is that the rotational drag should be adapted as well. This isindeed the case, rotational dag anisotropy is also a physical effect. However, these effects are typicallyan order of magnitude lower than the translational drag effects [50].

22


510 520 530 540 550Distance HnmL

10

20

30

40

50

60

Rel. drag coefficent

Figure 3.4: The parallel (blue) and perpendicular (red) components of the drag as function of thedistance of the bead center to the surface at z = 0 for R = 500. Both coefficients increase as thebead approaches the surface, but the perpendicular component diverges while the parallel componentincreases to a finite value.

A third objection arises when the actual experiment is considered. The beads are not perfectlysmooth spheres, but display a ‘roughness’. Although this effects the hydrodynamic forces and shouldbe considered carefully, its integral force contribution is typically negligible compared to the totalresistance [51].

The three aforementioned side-notes, - (1) Faxen’s law breaks down for too small gap widths, (2)rotational drag is also influenced and (3) bead roughness has an additional effect on bead drag - areall valid points. However, it is our vision that the implementation that we use captures the essence ofthis effect, while the computational complexity still remains tractable.

23

Chapter 4Bead movement

In this chapter we will review the movement of the bead. We will first discuss the equilibriumdistribution of positions that the bead attends and then turn to dynamic properties. After reviewingthe typically observed motion patterns, we continue with an analytic approach. We will show thatthis analytic approach only provides valid results in some limiting cases. Therefore, we introducesimulation methods to fully describe our system. When the equilibrium distributions are known andunderstood, we continue by describing several dynamic properties, i.e. the correlation functions ofbead positions and the distribution of hitting and rebinding times.

4.1. Motion pattern

In order to convincingly link our simulations to the experiments, we should in the first place show thatour simulation produces similar output. As mentioned earlier, the only directly measurable quantityof interest is the position of the bead projected on the XY -plane. Our simulation results provide anextensive set of variables, amongst which the position of the bead. This enables us to create motionpatterns comparable to the experimentally obtained patterns [15].

4.1.1 Experimental motion patterns

In figure 4.1 two motion patterns that are experimentally obtained by Scheepers [15]. In the pictureon the left a typical motion pattern is displayed without any traces of secondary binding events. Onthe other hand, in the picture on the right the bead is observed significantly more in a specific partof the plane. Our current understanding is that such effects are the result of the occurrence of one ormore secondary binding events.

The motion patterns in figure 4.1 are constructed by tracking the position of the bead in time. How-ever, the time dependence is no longer visible in this representation. Therefore, when determiningwhat fraction of the time the bead is bound to the surface, another representation of the data isrequired.

A method that enables one to determine whether a secondary bond has formed is graphically repre-sented in figure 4.2. Here, the projected distance that the bead travels between two frames - the stepsize - is plot against time. By averaging the step size over multiple frames one obtains an indicator to

24

CHAPTER 4. BEAD MOVEMENT

−200 −100 0 100 200 300

−200

−100

0

100

200

300

2D trajectory

x position (nm)

y po

sitio

n (n

m)

−100 0 100 200 300

−200

−100

0

100

200

300

2D trajectory

x position (nm)

y po

sitio

n (n

m)

Figure 4.1: Motion patterns of TPM experiments obtained by Scheepers [15]. These motion patternsare obtained for tether length L = 40 nm by measuring one minute with a frame rate of 30 Hz. In theleft picture: the uniform motion pattern without any observable binding events. In the right picture:a motion pattern in which the bead was observed more often in a distinct region, presumably due toone or more binding events.

determine whether the bead is bound or not.

This approach is similar to the method employed by [52]. In this article the steps size is called Brow-nian motion amplitude and is used to determine the looping state of DNA. Our collaborators [15] usethis detection method along with another detection method based on the outer hull that the motionpattern spans within a time interval. We will be focusing on the step size method.

25


0 10 20 30 40 50 600

50

100

150

200

250

300

350

400

Time (s)

Ste

p si

ze (

nm)

Step size

Step sizeStep size per window (window = 60 frames)

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (s)

Ste

p si

ze (

nm)

Step size

Step sizeStep size per window (window = 60 frames)

Figure 4.2: Time traces of the step size in TPM experiments obtained by Scheepers [15]. These timetraces are obtained for tether length L = 40 nm by measuring one minute with a frame rate of 30Hz. The blue lines represent the steps size between measured frames, the green line represents the stepsized averaged over 60 frames. In the left picture: the time trace corresponding to the uniform motionpattern without any observable binding events. In the right picture: a motion pattern in which bindingevents were observed. The bead binds to the surface around t = 20s, than detaches around t = 40 sand binds again around t = 50 s.

26


4.1.2 Simulation motion patterns

As a first step in using simulations to better understand this system, we should be able to reproducethe experimental motion patterns such as the ones mentioned in the previous section. Figure 4.3 givesexamples of such motion patterns. On the left is the motion of a free bead. On the right is a beadthat is able to form a bond on the substrate. We will discuss this in more detail, at this point themessage is that we can indeed produce similar motion patterns with our simulations.

X (nm)-250 -150 -50 50 150 250

Y (nm)

-250

-150

-50

50

150

250

X (nm)-250 -150 -50 50 150 250

Y (nm)

-250

-150

-50

50

150

250

Figure 4.3: Motion patterns obtained by simulations. In the left picture: a motion pattern obtained bycombining the output 10 simulations of 0.7 s with a frame rate of 150 Hz. In the picture on the right:a motion pattern that is influenced by the possibility to create a secondary bond with an antibody onthe substrate. The results were obtained by combining the output of 7 simulations of 0.5 s with a framerate of 300 Hz. Note: this figure was produced using isotropic drag.

27


4.2. Analytic approach

4.2.1 Geometrical framework

The initial experiments that triggered our interest in tethered particle motion were performed witha 40 nm long dsDNA tether, having a persistence length lp of roughly 50 nm. A first approximationto a semiflexible polymer with L < lp, like this dsDNA tether, could be to consider the polymer as astiff rod, or mathematically: lp →∞. This basic assumption allows us to obtain an analytical integralexpression that we can solve numerically for the hitting probability including the effects of bead-tetherexclusion and tether-surface exclusion.

The results of this approximation are provided in this chapter. At the end we will compare theseresults to simulation results for very high persistence length lp (1 · 106 nm) and we will review theeffect of decreasing the persistence length to the actual value. We will show that in the limit of lp →∞a valid expression can be obtained, but that as lp ↓ l exclusion effects become so pronounced that ourapproximation fails to yield valid results.

We describe the tether as a stiff rod attached at one point at the surface so that the configuration ofthe tether can be described by one polar and one azimuthal angle, which we call θ and φ respectively.Moreover, the bead is free to rotate with respect to the tether, so to describe the full configuration ofbead and tether we need another polar and azimuthal angle. We define ψ as the polar angle and χas the azimuthal angle both with respect to the axis in the direction of the tether. These angles areindicated in figure 4.4. This choice of angles turns out to be convenient when applying bead-tetherexclusion.

R

θ

φ ψ χ

L

Figure 4.4: The idealized system of a bead attached to the surface by a stiff rod. The complete con-figuration of the system is described by the four angles φ, θχ and ψ. The length of the tether is givenby L and the radius of the bead by R. Note that L and R are not drawn to the scale that we typicallyconsider.

28


We now have a system in which each configuration is given unambiguously by the coordinates (φ, θ, χ, ψ).We assume that every allowed configuration is equally probable.

Before we turn to a description of boundaries of the system and resulting probabilities, it seems worth-while to provide the reader with a mathematical description of the vectors and angles that fix oursystem in one configuration.

The vector that points along the tether is given by

~r1 = L (cosφ sin θ x sinφ sin θ y cos θ z) , (4.2.1)

where L is the length of the tether, and the angles produce a vector in the (x, y, z) system as is theusual convention with spherical coordinates. Now for the vector that describes the bead, the derivationis slightly more sophisticated. We define a system n, b, r1, in which r1 is the direction of the tetheri.e. the unit vector corresponding to equation 4.2.1. The vector pointing from the top of the tetherto the center of the bead is then given by

~r2 = R(

cosχ sinψ n sinχ sinψ b cosψ r1

). (4.2.2)

In this equation R is the radius of the bead, n and b are the so-called normal and binormal to obtaina new othogonal coordinate system. These unit vectors are defined by

n =∂r1∂θ

(4.2.3)

b = r1 × n. (4.2.4)

Using these definitions we can express the vector ~r2 in the initial (x, y, z) unit coordinates, which leadsto the expression

~r2 = R( cosχ sinψ cosφ cos θ − sinχ sinψ sinφ+ cosψ cosφ sin θ)x+

( cosχ sinψ sinφ cos θ + sinχ sinψ cosφ+ cosψ sinφ sin θ)y+

(− cosχ sinψ sin θ + cos θ cosψ)z. (4.2.5)

Assuming every configuration is equally probable, the probability distribution of this system P (φ, θ, ψ, χ) =P obeys the relation

L2R2

∫ ∫P (φ, θ, ψ, χ)dΩ1dΩ2 = L2R2

∫ ∫PdΩ1dΩ2 = 1. (4.2.6)

Here we have introduced solid angles Ω1 and Ω2 corresponding to ~r1 and ~r2 respectively, so that

dΩ1 = sin θdθdφ (4.2.7)

dΩ2 = sinψdψdχ (4.2.8)

⇒ L2R2

∫dφ

∫dθ

∫dχ

∫dψP sin θ sinψ = 1. (4.2.9)

Up to this point, the calculation has been rather straightforward. However, we have not yet discussedour integration boundaries, it turns out that these are not all trivial. In chapter 2 we discussedthat three types of exclusion mechanisms determine the allowed configurations of the system, i.e.bead-surface exclusion, tether-bead exclusion and tether-surface exclusion. Incorporating these ex-clusion effects corresponds to applying the correct integration boundaries. We will now turn to thedetermination of integration boundaries.

29


4.2.2 Integration boundaries

Bead-surface exclusion, tether-bead exclusion and tether-surface exclusion should be taken into ac-count and this can be done by choosing the right integration boundaries for φ, θ, χ and ψ. First of all,φ does not influence any of these exclusion effects and therefore all possibilities 0 < φ ≤ 2π should betaken into account. Secondly, since θ is the angle between the surface and the tether, tether-surfaceexclusions leads to the boundaries 0 < θ < π/2. Similarly, ψ is the angle between the bead and thetether and tether-bead exclusion only allows for 0 < ψ ≤ π/2.

However, bead-surface exclusion has a less straightforward effect on the integration boundaries of χand ψ in this geometry. Mathematically this exclusion corresponds to the fact that the z-coordinate ofthe bead center Z should not get smaller than the bead radius R. Z is the sum of the z-coordinate of ~r1,given by equation 4.2.1 and the z-coordinate of ~r2, given by equation 4.2.2, the allowed combinationsof θ, χ and ψ than meet the condition given by equation 4.2.10.

Z = L cos θ +R(cos θ cosψ − sin θ sinψ cosχ) > R. (4.2.10)

1 2 3 4 5 6Χ

-300

-250

-200

-150

-100

-50

50

HZ-RL HnmLΨ=0

Ψ=0.33

Ψ=0.67

Ψ=1.0

Figure 4.5: The Z-coordinate of the bead minus the radius of the bead as a function of azimuthalangle χ for several polar angles ψ. Bead-surface exclusion dictates that only positive values of Z-R areallowed. Values used for this plot are L=50 nm, R = 500 nm and θ = 0.3.

1 2 3 4 5 6Χ

-400

-300

-200

-100

HZ-RL HnmLΨ=0

Ψ=0.33

Ψ=0.67

Ψ=1.0

Figure 4.6: The Z-coordinate of the bead minus the radius of the bead as a function of azimuthalangle χ for several polar angles ψ. Bead-surface exclusion dictates that only positive values of Z-R areallowed. Values used for this plot are L=50 nm, R = 500 nm and θ = 0.5.

30


In other words, bead-surface exclusion demands that Z − R > 0, our goal is now to find the combi-nations of θ, χ and ψ that allow for this. In figure 4.5 Z-R has been plot as function of the angle χfor several values of ψ and fixed value of θ = 0.3. It can be seen that for low ψ all values of χ areallowed, while for increasing values intersects with the horizontal axis appear. In figure 4.6 the sameplot is shown for θ = 0.5. Here for low ψ all values of χ are forbidden, while again for higher valuesof ψ intersects with the horizontal axis appear. In both plots it can be seen that if ψ takes on evenlarger values, not a single value of χ is allowed anymore.

Looking at figure 4.5 and 4.6 and then turning back to equation 4.2.10 we find that there is a specificangle θ∗, so that for θ < θ∗ all values of χ are allowed at ψ = 0, while for θ > θ∗ no values are allowedat ψ = 0. This value is given by

θ∗ = arccos

(R

R+ L

). (4.2.11)

Furthermore the value of ψ at which the intersects occur ψ− and the value at which those disappearψ+ are given by

ψ− =

∣∣∣∣arccos

(R− L cos θ

R

)− θ∣∣∣∣ (4.2.12)

ψ+ = arccos

(R− L cos θ

R

)+ θ. (4.2.13)

the intersects of the curves with the χ-axis occur at

χ± = π ± χ∗ (4.2.14)

χ∗ = π − arccos

(L cos θ +R(cos θ cosψ − 1)

R sin θ sinψ

). (4.2.15)

There is one last complication that we should take into account. As stated before, bead-tether

exclusion dictates that ψ can never exceed π/2. This means that integrating ψ from ψ− to ψ+ is fine,as long as ψ+ does not exceed π/2. However, from equation 4.2.13 we can deduce that for any nonzeroL a value for θ < π/2 exists at which ψ+ is equal to π/2. We will refer to this angle as θ#, and itsvalue is given by

θ# = 2 arctan

(R− LR+ L

). (4.2.16)

A graphical representation of θ# can be found in figure 4.7. Equation 4.2.11 and 4.2.16 inform us thatθ∗ < θ# for small L/R, while for big L/R: θ∗ > θ#. The crossover θ∗ = θ# occurs at L/R ≈ 0.4. Inthis section we will discuss a system with L/R = 0.1, and therefore assume from now on that θ∗ < θ#.

To summarize, it is useful to make the distinction between four regimes that describe allowed config-urations:

(I) 0 < θ < θ∗ and 0 < ψ < ψ−, so that all 0 < χ ≤ 2π are allowed.

(II) 0 < θ < θ∗ and ψ− < ψ < ψ+, so that π − χ∗ < χ < π + χ∗.

(III) θ∗ < θ < θ# and ψ− < ψ < ψ+, so that π − χ∗ < χ < π + χ∗.

(IV) θ# < θ ≤ π/2 and ψ− < ψ ≤ π/2, so that π − χ∗ < χ < π + χ∗.

31


Θð

0.2 0.4 0.6 0.8 1.0 1.2 1.4Θ

0.6

0.8

1.0

1.2

1.4

1.6

Ψmax

Figure 4.7: The angle ψ+ that gives the maximum angle at which a solution for Z = R exists. Ifψ+ < π/2, then ψ+ should be the integration boundary for ψ, while for ψ+ > π/2, the integrationboundary should be π/2, dictated by bead-tether exclusion. The value for θ at which this shifts is θ#.Values used for this plot are L=50 nm and R=500 nm.

Finally, we can now include the boundary conditions in the integral of equation 4.2.9 to find

PL2R2

∫ 2π

0dφ

(∫ θ∗

0dθ

[∫ ψ−

0dψ

∫ 2π

0dχ sin θ sinψ +

∫ ψ+

ψ−

dψ

∫ π+χ∗

π−χ∗dχ sin θ sinψ

]

+

∫ θ#

θ∗dθ

∫ ψ+

ψ−

dψ

∫ π+χ∗

π−χ∗dχ sin θ sinψ +

∫ 2π

θ#dθ

∫ π/2

ψ−

dψ

∫ π+χ∗

π−χ∗dχ sin θ sinψ

)= 1 (4.2.17)

4.2.3 Distribution results

Now that we have set up a geometric framework and analyzed what the relevant integration boundariesare, we can look at the spatial distribution of states and, even more relevant, the hitting probabilityand spatial hitting distribution. Let us first carefully define these results.

The spatial distribution describes the probability that the bead will be found at a certain in-planeradius ~r = (x, y). This is a relevant distribution when comparing with experiments, since usuallyonly the movement in the (x, y)-plane can be observed. We define hitting as Z < (R + ZHIT ) andthe hitting probability is the number of configurations that are a hit divided by the total number ofconfigurations. The spatial hitting distribution is again the distribution of states for varying in-planevector ~r = (x, y), but in this case only for the hits.

We have not succeeded in analytically solving equation 4.2.17, as instead we provide results obtainedby numerically solving this integral. First of all, the constant P should be determined. We numericallyobtain P = 8.11 · 10−10 nm−4. We represent these spatial distributions as P (r)/r, to account for thefact that the surface of a ring with radius r and width dr, Aring = 2πrdr, varies. By dividing byr we find the probability per surface area. The spatial distribution P (r) is numerically obtained byintegrating over the integral given in equation 4.2.17 with a resolution of dr = 1 nm, by inserting the

32


r (nm)0 50 100 150 200 250 300

P(r

)/r

(nm

-1)

#10-3

0

1

2

3

4

5

6

7

8lp = 106 nm

lp = 200 nm

lp = 50 nm

analytic at lp ! 1

Figure 4.8: The spatial distribution of the bead for the analytic expression and for several values ofthe persistence length lp. The probability P (r) is divided by r to get the probability per surface area.The red line displays the result for the numerical integration, corresponding to lp →∞, while the otherlines result from MC simulations. It can be observed that there are only minor deviations between thedifferent curves.

block function into the integral u(r)−u(r−dr), where u(x) is the Heaviside step function. The resultis represented in figure 4.8 along with simulation results with various persistence lengths.

The area under a normalized distribution function is equal to 1. When comparing two normalizeddistributions, the area under both lines is a measure for how well the distributions match. For perfectmatching this value will be equal to 1. We will call this the ‘overlap’ between two curves.

It can be observed that for high persistence length lp = 1 · 106 nm the spatial distributions are inrather good agreement: there is 0.983 overlap between the curves. Moreover, at the real persistencelength the lp = 50 nm there is still an overlap of 0.945, indicating that spatial distribution obtainedby the stiff-rod approximation only slightly deviates from the actual spatial distribution.

However, this is not the case when considering hitting. Using a hitting distance ZHIT of 10 nm, wecalculate the overall hitting probability by inserting the Heaviside step function u(ZHIT − Z) in theintegral. The overall hitting probability fhit is 0.36 for both the numerically integrated result as wellas the simulation with lp = 1 · 106 nm. For lp = 200 nm we find fhit = 0.33 and for lp = 50 nm wealso find fhit = 0.33.

33


Finally, we consider the spatial hitting distribution. We obtain Phit(r) numerically with a resolution ofdr = 1 nm, by inserting the two previous ingredients: (u(r)− u(r− dr))u(ZHIT −Z) into the integralof equation 4.2.17. The result is represented in figure 4.9 along with simulation results. Again, thereis a rather good agreement for lp = 1 · 106 nm: the curves overlap 0.983, but for lp = 50 nm this isreduced to only 0.714.

r (nm)0 50 100 150 200 250 300

Phi

t(r)/

r (n

m-1

)

0

0.002

0.004

0.006

0.008

0.01

0.012lp = 106 nm

lp = 200 nm

lp = 50 nm

analytic at lp ! 1

Figure 4.9: The spatial hitting distribution of the bead for the analytic expression and for several valuesof the persistence length lp. The probability P (r) is divided by r to get the probability per surface area.The red line displays the result for the numerical integration, corresponding to lp →∞, while the otherlines result from MC simulations. It can be observed that the numerical integration corresponds wellfor high persistence length, but fails to describe the system at lower,more realistic, persistence lengths.

4.2.4 Conclusion

In conclusion, the stiff-rod approximation does a good job at describing the overall spatial distribu-tion of the bead, but turns out to be a rather poor approximation near the surface. Our physicalinterpretation of this is that a stiff rod allows for many states that will be excluded by either bead-tether exclusion or tether-surface exclusion once the tether becomes more ‘wiggly’, and therefore getsspatially extended in the direction perpendicular to the tether. We will come back to the importanceof these exclusion effects in section 4.4. At this point we conclude that the stiff-rod approximation isnot sufficient to describe the configurations of the bead near the surface exhaustively.

34


4.3. Number of polymer beads in MD simulation

Before the simulations are set up, the number of beads that represent the DNA chain has to bedetermined. If the tether is represented by a small number of segments, the simulation might nottake the effect of bead-tether and surface-tether interactions correctly into account. Moreover, thebead-spring approximation of the configurations of the worm-like chain will get less accurate for fewersegments.

On the other hand, if more segments are taken into account simulation times will increase due to twoeffects: (1) there are more segments for which the equations have to be solved every time step of thesimulation and (2) a smaller time step is required to correctly sample the bonds between the segmentsof the chain (since masses are decreasing and distances are decreasing, see equation 3.2.19).

In summary, we aim to find the number of beads in the chain that is large enough to capture the maincontributions of the tether, while still leading to reasonable simulation times.

Simulations have been performed for a tether consisting of 2, 5, 10, 30 and 50 segments. We aim tochoose the minimal number of segments so that the correct radial hitting probability distribution isstill obtained. The radial hitting probability distribution is obtained in the Monte Carlo simulationswith a 50 nm tether consisting of 50 segments.

The results in figure 4.10 that the Molecular Dynamics results converge towards the Monte Carloresults for an increasing number of tether segments. The show that for a tether consisting of 2 polymerbeads, exclusion effects are not significantly taken into account. However, for 5 polymer beads theresult is significantly better and for 10 polymer beads we have good agreement (0.96 overlap).

In our simulations we have used a number of 10 polymer beads for a tether of 50 nm. For shortertethers we have scaled this number down so that the distance per polymer bead remained 5 nm beadand for longer tethers we have kept it at 10 polymer beads to preserve computational achievability.

35


r (nm)0 50 100 150 200 250 300

Phit/r (1/nm)

0

0.002

0.004

0.006

0.008

0.01

0.012

2 atoms5 atoms10 atoms30 atoms50 atomsMC

Figure 4.10: The hitting distribution that was obtained by MD simulations for an increasing numberof poylmerbeads that represent the tether. It can be observed that the distribution converges to the MCresult for an increasing number of beads.

36


4.4. Exclusion effects

We already mentioned in section 2.1 that there are three exclusion principles that govern the allowedconfiguration of the TPM system: (1) bead-surface exclusion, (2) tether-bead exclusion and (3) tether-surface exclusion. It is widely acknowledged that the first one of these effects is indispensable indescribing TPM [21] and this effect is therefore implemented in our simulations.

The influence and significance of the other two mechanisms is not as readily accessible in literature.Therefore, we have implemented each effect separately and reviewed the significance. The resultsobtained by MD simulations can be found in figure 4.11, both MD and MC simulations yielded similarresults.

In figure 4.11 it can be observed that both tether-bead and tether-surface exclusion alter the systemin such a way that the hitting distribution increases for higher r and decreases for lower r. For tether-bead exclusion this aligns with our intuition: if the bead hits the surface close to the tethering-point(r = 0) it will experience more steric hindrance from the tether. Perhaps less intuitive is the result oftether-surface exclusion, which also effectively shifts the hitting distribution towards higher r.

The fact that both these mechanisms yield such similar hitting distributions (the red and yellow linein figure 4.11) shows that both effects change the hitting distribution in a similar way and that botheffects are of the same order. The fact that both effects are approximately equally relevant is somethingthat we did not predict and is indeed valuable knowledge when describing this system.

If we review the overall fraction of frames in which the bead hits the surface fhit, then we find fhit =0.40 for no exclusions, 0.37 for only tether-surface exclusions, 0.36 for only tether-bead exclusions and0.33 for both exclusion mechanisms. One could state that these exclusion mechanisms not only drivethe bead radially outward, but also away from the surface.

Although the influence on the overall distribution of the bead is minor, the effects become significantwhen reviewing the hitting distribution. Physically this result implies that when the bead is near thesurface (hitting), there is a higher probability of the tether intersecting with the bead or with thesurface for small in-plane radius r than for lange r.

37


r (nm)0 50 100 150 200 250 300

Phit/r

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

No tether exclusionsTether-bead exclusionsTether-surface exclusionsBoth exclusions

Figure 4.11: The normalized hitting distribution per in-plane radius r plot against r for several ex-clusion mechanisms. These results are obtained by MD simulations. Both bead-tether exclusion andwall-tether exclusion have a significant effect on the hitting distribution and both effects decrease thelikelihood of a hitting event at small r.

38


4.5. Influence of the engineering parameters R,L, lp

When constructing a tethered particle system, the three main parameters that the designer is moreor less free to choose are the bead radius R, the tether length L and the tether persistence length lp.We will therefore investigate the influence of these parameters on the equilibrium distribution. Thedistributions obtained by Molecular Dynamics simulation agree with the Monte Carlo simulationswith an accuracy of at least 0.95. (The hitting distribution curves have an overlap of at least 0.95).

4.5.1 Varying the bead size

To asses the effect of a varying bead size, simulations were performed with bead radii 300 nm and 700nm in addition to the usual 500 nm. We evaluate the fraction of distribution of which the bead hitsthe surface as well as the change in spatial distribution of hitting the surface.

The most noteworthy difference that can be observed when changing the bead size, is that the max-imum in-plane radius at which we locate the bead is significantly altered from about 2.2 · 102 nmto 1.7 · 102 nm and 2.6 · 102 nm for a bead radius of 300 nm and 700 nm respectively. This is ofcourse a direct result of the geometrical constraint that the tether imposes on the bead, which can beapproximated for small tether lengths as Rmax ≈

√2RL.

Aside from this effect, we observe little change in equilibrium distribution when changing the beadradius. The overall hitting fraction remained at 0.33 for all three cases. The shape of the hittingdistribution is also comparable for different bead radii as can be seen in figure 4.12.

All in all, by changing the bead radius, we can change the surface area that is covered by the bead,while omitting any other significant changes to the equilibrium distribution behavior of the TPMsystem.

r (nm)0 50 100 150 200 250 300

Phit/r

0

0.002

0.004

0.006

0.008

0.01

0.012

R=300 nmR=500 nmR=700 nm

Figure 4.12: The normalized hitting distribution per in-plane radius r plotted against r for severalbead radii R. The results represented are obtained by MD simulations. While the shape of the hittingdistribution and the overall hitting fraction are not significantly altered, the covered surface area is.

39


4.5.2 Varying the tether length

To access the influence of the tether length, simulations were performed for tether lengths 25 nm and100 nm. The results can be found in figure 4.13. Similar to the effect of a varying bead radius, varyingthe tether length influences the covered surface are, corresponding to Rmax ≈

√2Rl, the maximum

radius that the bead is geometrically allowed to reach.

However, as opposed to the case of varying bead radii, the tether length does significantly alter thefraction of frames in which the bead hits the surface. For L = 25 nm the hitting fraction fhit equals0.62, for the usual L = 50 nm, fhit = 0.33 and for L = 100 nm, fhit = 0.17. It is expected that a beadattached to a shorter tether will hit the surface more often and vice versa. The results indeed confirmthis and fhit even approximately scales as ∼ 1/L.

In conclusion, changing the tether length allows a larger part of the surface to be covered, but thiscomes at the cost of decreasing the overall hitting fraction.

r (nm)0 50 100 150 200 250 300 350

Phit/r

0

0.002

0.004

0.006

0.008

0.01

0.012

L=25 nmL=50 nmL=100 nm

Figure 4.13: The normalized hitting distribution per in-plane radius r plotted against r for severaltether lengths L. The results represented are obtained by MD simulations. The distribution displays asimilar trend, but the overall hitting fraction as well as the covered surface are are significantly alteredby changing L.

40


4.5.3 Varying the persistence length

Although it is experimentally somewhat less straightforward to vary the persistence length of thetether, it is a parameter for which different values can be chosen. Identifying the way this alters thebehavior of the system is thus of importance.

The results for lp = 25, 50, 100 nm can be found in figure 4.14. The corresponding hitting fractions werefhit = 0.34, 0.33, 0.32. We observe that for higher persistence length, ‘stiffer polymers’, the hittingdistribution changes to higher in-plane radius. Given this data, we conclude that the persistencelength lp, compared to bead raduis R and bead length L, has a relatively small influence on thehitting process and that adjusting it will not yield big advantages.

r (nm)0 50 100 150 200 250 300

Phit/r

0

0.002

0.004

0.006

0.008

0.01

0.012

lp=25 nm

lp=50 nm

lp=100 nm

Figure 4.14: The normalized hitting distribution per in-plane radius r plotted against r for severaltether persistence lengths lp. The results represented are obtained by MD simulations. With increasingpersistence length the bead hits on average at higher r.

41


4.5.4 Discussion

It should be carefully considered whether R, L and lp can indeed be changed in the actual system.Changing the tether length L is relatively straightforward, thanks to the polymerase chain reaction,TPM studies in which tether lengths are varied have been published before [53], [54]. Increasing tetherlength leads to a higher part of the surface covered, but a lower overall hitting fraction. This is atradeoff that we further discuss in section 6.

Changing the radius of the bead is another possibility. Beads are more challenging to fabricate ondemand, but several sizes are commercially available. One aspect that should be kept in mind isvisibility: too small beads may be harder to visualize. Based on the results in this section, the beadsize should be maximized. However, we will come back to this point when considering dynamics aswell.

The most challenging would be the alteration of the persistence length lp of the tether. Althoughseveral molecules can be used and have been used, the vast majority uses double-stranded DNA.Assuming dsDNA is used allows us to rely on all the literature already available on this subject.Moreover, we have shown that the influence of the persistence length on the hitting process is small.

As a last point it should be stressed that even though we have changed one of the three parametersat a time, it is by no means a given that the influences of R,L and lp are independent. For example,in the limit of long tethers and small beads, the surface area covered will only slightly be influencedby R. We have determined the effect of each parameter separately when it is varied in the regime ofthe actual experimental values and therefore it provides us with information about how changing asingle parameter will change the experimental system.

42


4.6. Dynamic properties

Now that we have constructed our simulation model for the tethered particle motion and weunderstand the equilibrium distributions that we obtain, we can turn to the dynamic propertiesof the system. In describing the dynamic properties of the system it is also relevant to revealthe influence of the anisotropic drag, discussed in section 3.4.

The parameters that we use to characterize the dynamics of the system are the autocorrelation

of the in-plane vector ~R(t) that describes the positions of the bead,⟨~R(t) · ~R(t+ τ)

⟩, the auto-

correlation of the Z-component of the bead, 〈Z(t)Z(t+ τ)〉, the average hitting and rebindingtime and the distribution of hitting and rebinding times.

4.6.1 Autocorrelation ~R(t)

We will refer to the characteristic time corresponding to the autocorrelation function of thein-plane vector ~R(t) that describes the positions of the bead bead as τ1. In experiments ~R(t)is directly measured, so we may compare our obtained value of τ1 with experimental values.

In figure 4.15⟨~R(t) · ~R(t+ τ)

⟩obtained by simulations is shown together with an exponential

fit. The value for τ1 that we obtain is (0.09±0.01) s. This agrees with the lowest experimentallyobtained values, that range from 0.1 - 0.3 s [15]. If we do not apply anisotropic drag we find avalue of (0.039± 0.001) s, which does not agree with experimentally obtained values.

= (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

D ~ R(t

)"~ R(t

+=)E (n

m2)

#104

0

1

2

From simulationsexponential fit

Figure 4.15:⟨~R(t) · ~R(t+ τ)

⟩in the case of anisotropic drag. The result was fit by a function of

the form y(t) = C exp(−τ/τ1), with values C = (2.0±0.1)·104 nm2 and τ1 = (0.09±0.01) s. Thedeviations at the right side of the graph are the result of poor averaging: a larger τ correspondsto less data points available.

43


4.6.2 Autocorrelation Z(t)

We will refer to the characteristic time corresponding to the autocorrelation function of the Z-coordinate of the bead Z(t) as τ2. τ2 provides us with information about the characteristic timeat which the bead moves from and to the substrate. In figure 4.16 〈Z(t)Z(t+ τ)〉 obtained bysimulation is shown together with an exponential fit. This graph starts at

⟨Z(t)2

⟩and relaxes

towards 〈Z(t)〉2.

We obtain a value of τ2 = (21± 1) ms. If we do not apply the anisotropic drag we find a valueof τ2 = (2.0 ± 0.5) ms. Since the experiments do not have a Z-resolution, we are not able tocompare these values to experimental values. However, this value gives us an indication of howfast the bead moves from and to the substrate.

The fact that τ2 is significantly lower than τ1 is the result of the small distance between thebead and the substrate. The provided uncertainties are the 95% confidence intervals providedby the exponential fit function of MATLAB [25].

= (s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

hZ(t

)Z(t

+=)i

(nm

2)

0

100

200

300

400

500

600

700From simulationsexponential fit

Figure 4.16: 〈Z(t)Z(t+ τ)〉 in the case of anisotropic drag averaged over 10 simulations. Theresult was fit by a function of the form y(t) = C0 + C1 exp(−τ/τ2), with fit parameters C0 =(3.2± 0.2) · 102 nm2, C1 = (150± 5) nm2 and τ2 = (21± 1) ms.

4.6.3 Conclusion

The conclusion of this section is twofold. Firstly, we show that we indeed need anisotropic dragto obtain dynamic properties that are in line with experimental results. In addition to that,this section provides us with insight on the correlation times of the position of the bead. Inthe direction parallel to the substrate this turns out to be in the order of 10−1 s and in thedirection perpendicular to the substrate this is roughly one order of magnitude smaller.

44


4.7. Hitting and return times

4.7.1 Hitting times

The hitting time is defined as follows: consider the bead that approaches the substrate, so that thegap between the bead and the substrate becomes smaller than ZHIT . Define t = τ when Z = ZHITupon approaching the substrate and then Z = ZHIT again at t = τ + ∆t while moving away from thesubstrate, ∆t is then the hitting time.

The hitting time is also a parameter in which the influence of the anisotropic drag is reflected. A firstinteresting quantity would be the average hitting time. For the isotropic drag system this quantity isequal to τav = (11± 2)µs. In the case of the anisotropic drag this is equal to τav = (39± 8)µs. Again,we observe the significant impact of anisotropic drag in our system.

Even more interesting is the full distribution of hitting times. It turns out that this distribution iswell-captured by a power-law fit. An example of such a fit is given in figure 4.17. Averaging over allour fits we obtain a power of (1.51 ± 0.06) in the case of isotropic drag and (1.55 ± 0.09) in the caseof anisotropic drag.

4.7.2 Return times

A similar procedure may be applied to the return times. The return time is similarly defined as:consider the bead that is near the substrate, so that the gap between the bead and the substrate issmaller than ZHIT . Define t = τ when Z = ZHIT upon moving away from the substrate and thenZ = ZHIT again at t = τ + ∆t while approaching the substrate, ∆t is then the return time.

The average value of the return time is given by (19±2)µs for the isotropic drag system and (24±6)µs.Averaging over all our fits we obtain a power of (1.48±0.06) in the case of isotropic drag and (1.6±0.1)in the case of anisotropic drag.

4.7.3 Discussion

Again we observe that the anisotropic drag effects significantly affect the properties of the system.Since the drag on a sphere increases upon approaching a wall, it may be expected that the anisotropicdrag leads to longer hitting times. Indeed the average hitting time increases when the anisotropicdrag is taken into account. The average return time increases as well, but this is a weaker effect.

The probability distribution of the first return time T in Brownian motion scales as ∼ T−3/2 [55].The hitting time and return time as we defined can be viewed as first return times. Indeed, all powerlaw fits yield a power that corresponds to −3/2 within uncertainty. The anisotropic drag does notinfluence the power corresponding to this distribution.

45


Hitting time in ns102 103 104 105

Probability (a.u.)

100

101

102

103

104

105

Fit with power law: p=-1.52

Figure 4.17: A typical picture of a fit of the distribution of hitting times. The distribution was fit bythe function y = Ct−pt, with a value p of 1.52.

46


4.8. Step size

One of the most critical values that enables us to determine whether the bead is bound or notis the in-plane distance Rstep = |~R(t + ∆t) − ~R(t)| that the bead travels between two frames,where ∆t is the frame time.

In this section we determine the step size for several detection frequencies and the influenceof the tether length and the bead radius. The results may help in choosing the frame ratefor experiments. Moreover, the results provide us with information about the detectability ofbinding events. We will describe this principle in more detail in section 6.2.

4.8.1 Tether length

First, we consider the effect of different tether lengths on the average step size. When a beadis attached by a longer tether, there is a larger volume that the bead is allowed to occupy.Moreover, on average the bead will be further away from the surface with a longer tether,leading to lower drag. One may expect that a longer tether leads to bigger step sizes and thisis indeed the result, as can be seen in figure 4.18.

The average step size we obtained for a tether of 50 nm and a frame rate of 30 Hz is 92± 4 nm(SEM), which agrees with experimental data [15]. For a larger tether of 330 nm the obtainedaverage step size equals 132± 8 nm, which agrees with experimental data as well.

Frame time (s)0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

hRst

epi(

nm

)

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

L=20 nmL=50 nmL=100 nmL=200 nmL=330 nm

Figure 4.18: The average step size (displacement) of the bead between two frames as a functionof the frame time for several tether lengths. Smaller tethers results in smaller average stepsizes. The results have been compared with experimental results for L = 50 nm and L = 330nm at 30 Hz and are in agreement. Error bars indicate SEM.

47


4.8.2 Bead radius

The theory of Brownian motion tells us that big beads move slower, as the diffusion coefficientD ∼ 1/R. On the other hand, as we have observed in section 4.5.1, a bigger bead is allowed tomove to higher in-plane radii, so a bigger bead can move further without being constrained.

The results are represented in figure 4.19. We observe that at the frame rates we consider, abigger beads corresponds to a smaller step size. This implies that binding events will be morechallenging to detect when working with larger beads.

Frame time (s)0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

hRst

epin

m

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

R=300 nmR=500 nmR=700 nm

Figure 4.19: The average step size (displacement) of the bead between two frames as a functionof the frame time for several bead radii. Bigger beads result in smaller step sizes at these frametimes. Error bars indicate SEM.

48

Chapter 5Specific binding spots

Now that we have determined several characteristics of the movement of the bead as a whole,we should remind ourselves that the actual bond does not involve the whole bead, but rathera very specific spot on the bead. Similarly, the actual bond is not formed with the wholesubstrate, but a specific spot on the substrate. In this section we zoom in one level deeper tounderstand the role of the specific location of the binding molecules.

5.1. Dots on the bead

To examine the exact location of binding spots on the bead, we place dots on the bead. Aschematic representation of this is given in figure 5.1, where the dots are colored red. Weinitialize the system in an upright configuration with Xdot = R sin θ and Ydot = 0. Due toaxisymmetry, the bead is free to rotate with respect to the tether, the equilibrium propertiesof a dot will be completely fixed by the angle θ.

Rθ

Figure 5.1: Examples of specific dots on the surface. The location of a dot is determined byangle θ. Since the system is axisymmetric we conveniently take Ydot = 0 for every dot, so thatXdot = R sin θ. Note: tether length and bead radius are not drawn to typical scale.

49

CHAPTER 5. SPECIFIC BINDING SPOTS

In figure 5.2 we find the hitting probability of a dot as function of the angle θ. From geometricalarguments, we can predict that the maximum angle at which a dot is able to hit the substrate is

given by θmax = arccos(

RR+L

)+ arccos

(R−ZHIT

R

)= 0.65. However, the radial distribution of the

end-to-end length of the tether vanishes as the end-to-end-length approaches the contour length [56].In other words, it is highly improbable that the tether is fully stretched out. Therefore, effectively themaximum angle θ that hits the substrate is slightly lower. In this case, no hits are observed beyondθ = 0.60.

30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

f hit

0

0.01

0.02

0.03

0.04

Figure 5.2: The fraction of frames in which a dot hits the surface as function of the angle on the beadθ. The dots around θ = 0.3 have the highest probability to hit the substrate and the dots at θ > 0.6will in practice not hit the subtrate at all.

50


5.2. Spots on the substrate

The other way around, we may also look for a spot at a specific position ~rspot = Xsubstrx + 0y + 0zon the substrate as in figure 5.3. Again, due to the axisymmetry of our system, we can convenientlychoose our spots on the x-axis. If the the vector pointing to the center of the bead is called ~rbead,hitting is now defined as |~rspot − ~rbead| ≤ |R+ rHIT |, with R the radius of the bead.

Xsubstr

Figure 5.3: Examples of a specific spot on the substrate. We take our spots to lie on the x-axis, withan x-coordinate of Xsubstr. Since the system is axisymmetric, we conveniently take Ysubstr = 0 forevery spot. Note: tether length and bead radius are not drawn to typical scale.

We now consider the fraction of frames in which the bead hits a specific spot fhit as a function ofXsubstr. This approach corresponds to defining an ‘imprint’ when the bead hits the substrate. Similarto previous results, we take rHIT = 10 nm.

The result is represented in figure 5.4. The curve is remarkably similar to the curve for dots on thebead given in figure 5.2. This can be explained by the fact that the motion of the bead is largelylimited by a tether of length L = 50 nm.

51


Xsubstr

(nm)0 50 100 150 200 250 300 350

f hit

0

0.01

0.02

0.03

0.04

Figure 5.4: The fraction of frames in which a dot hits the substrate as function of position on thesubstrate Xsubstr. The shape of the curve is similar to the shape of the curve in figure 5.2, whichreflects the fact that movement of the bead with respect to the substrate is largely constrained by thetether.

52


5.3. Single Xdot and Xsubstr

Now that we have reviewed the variation of hitting probabilities at different spots on the surface ofthe substrate as well as different spots on the bead, we should combine these results. The eventualbinding pattern will indeed be the result of the specific location on the bead and the surface. Infigure 5.5 the hitting probability of a dot on the bead at every positions on the substrate are shownfor several values of θ.

The figure shows that for θ = 0 there is a narrow range of r where the dot hits the surface, whichwe attribute to the steric effect of the tether. For larger θ, the range of r that correspond to hittingthe surface remains fairly constant at approximately 100 nm. Furthermore higher values of θ lead tohigher values of r where the dots hit the substrate. The trend of figure 5.2 corresponds to the relativeheights of these peaks.

r(nm)0 50 100 150 200 250 300 350 400

Phit(r)

#10 -3

0

1

2

3

4

5

6

7

8

3 = 03 = 0.153 = 0.303 = 0.453 = 0.6

Figure 5.5: The hitting probability of a dot at angle θ for several values of θ as function of the in-planedistance to the origin r.

53


5.4. Kinetic regime

The actual bond formation between an oligonucleotide on the bead and an oligonucleotide on thesubstrate can also be incorporated in the simulations. Although our simulation times remained severalorders of magnitudes below the typical experimental times, these simulations provide us with a toolto estimate the kinetic regime of our bonds. In this section we use the implementation of single-bondsto determine an upper bound for the sticking rate kstick.

5.4.1 Reaction process

In literature often binding kinetics are expressed in terms of an association rate kon and a dissociationrate koff . While this framework is well-suited for the description of binding between molecules in freesolution, the application of the concept of kon to a secondary binding event in a TPM system is notstraightforward. We have therefore chosen to refrain from describing our system in terms of kon.

Instead, we will describe the system, as outlined in chapter 2, by the two-step reaction process

FBkhit−−−−→←−−−−ksep

HBkstick−−−−→←−−−−koff

SB, (5.4.1)

where FB stands for free bead, not within reaction range, HB stands for hitting bead, within re-action range and SB stands for stuck bead, a bead that is actually bound to the substrate by anantibody-oligo-streptavidin complex. The k-values describe rates that govern the dynamics of theseprocesses. These rates are all per unit of time. However, it should be noted that we have also defineda molecular interaction range Rc: when the distance between both antibodies lies within the interac-tion range, the sticking process may occur. This Rc allows one to define an interaction volume, theoverlap of two spheres with radius Rc, and subsequently an effective concentration. However, sinceboth oligonucleotides are attached to a surface with one end, it is not straightforward to relate thiseffective concentration to the bulk concentration in free solution.

In the ideal situation of only one binding spot on the bead and one binding spot on the surface, thedistribution of rebinding times enables us to determine the value of kstick. Indeed, in the simulationsonly one binding spot on the bead and one binding spot on the substrate are implemented.

5.4.2 Quick rebinding events

In our model system it is entirely possible that a bond releases (HBkoff←−−−− SB) and reattaches

(HBkstick−−−−→ SB) before the bead has significantly moved. How often this happens depends on the

value of kstick and the chosen cutoff for the molecular bond Rc. Since we can freely choose a value forkstick, koff and Rc in our simulation, we can evaluate for which values the observed koff is significantlyaltered by quick rebinding events.

The equation∂[SB]

∂t= kstick[HB]− koff [SB] (5.4.2)

reduces to∂[SB]

∂t= −koff [SB] (5.4.3)

54


for kstick → 0. From this we deduce that if we take kstick small enough, the number of bound beadswill decrease exponentially with a characteristic time 1/koff .

We have chosen our koff to be equal to 1.0 · 102 s−1 and varied kstick from 102 s−1 to 3 · 105 s−1. Oncethe binding distance is larger than the bond length +5 nm for 0.01 seconds, we count the bond asreleased. The characteristic binding times were extracted by making exponential fits. The results canbe found in figure 5.6.

It may be observed that for kstick ≥ 1 ·105 significant deviations in the effective bond time are present.The oligonucleotide sequences used in the experiments have a known value of koff [22]. The extractedvalues of koff , obtained by exponentially fitting the observed binding times in the experiments agreewith previously documented values. From this we draw the conclusion that the actual value of kstickin the experiments is upper bounded by kstick < 105 s−1.

kstick

(7s)-110-5 10-4 10-3 10-2 10-1 100

= off (

s)

0

0.005

0.01

0.015

0.02

0.025

Figure 5.6: The characteristic binding time for koff for several values of kstick. The red dashed linerepresents the expected characteristic binding time for kstick = 0. At kstick = 0.1µs−1 significantdeviations become visible. The used value of Rc was 3.0 nm. The location of the bond was Xdot = 80nm, Xsubstr = 120 nm. Error bars indicate 95% confidence bars of the fit parameter.

55


5.5. Analyzing bound patterns

We aim to develop an algorithm that processes the motion of the bead in time and is able to relate thisdata to the properties of the system. In this section we demonstrate how the position of the bindingmolecules on the bead and the substrate can be extracted from a bound motion pattern. In the nextsection we combine all results of this chapter and we outline how this may be applied to experimentaldata. We end this chapter by applying this algorithm to actual experimental data.

A bound pattern can be recognized by evaluating the step size of the positions of the bead in a giventime window [15]. Consider. Several values can be extracted from this motion pattern during thistime window. In particular, the he length L, width W and mean distance to the origin M , as can beseen in figure 5.7.

The expected values of the length L, width W and mean distance to the origin M can be obtainedfor several combinations of Xdot and Xsubstr. These results are tabulated in appendix B.

However, the actual measured values will never perfectly match the values in these tables. When abond is detected in an experiment, a value L′, W ′ and M ′ may be extracted from the data. We maythen define a matching function Φ

Φ = |L′ − L(Xdot, Xsubstr)|+ |W ′ −W (Xdot, Xsubstr)|+ |M ′ −M(Xdot, Xsubstr)|. (5.5.1)

The combination of Xdot and Xsubstr that yields the minimum value of Φ is the best match to theexperimental data. So if the shape of a bound motion pattern is known, the most likely values of Xdot

and Xsubstr can be determined by minimizing Φ.

X (nm)-250 -200 -150 -100 -50 0 50 100 150 200 250

Y (n

m)

-250

-200

-150

-100

-50

0

50

100

150

200

250

M: DistanceL: LengthW: Width

Figure 5.7: A motion pattern obtained from simulations for a bead bound at Xdot = 160 nm, Xsubstr =140 nm. From the bound motion pattern the length L, width W and mean distance to the origin Mcan be extracted.

56


5.6. Relation to experiments

5.6.1 Experimental situations

In an experimental situation, one could either choose to fully coat the substrate, so that the coatingmolecules are abundant on the substrate, or to sparsely coat the substrate, so that molecules on thesubstrate are rarely found. Likewise, one could either fully or sparsely coat the bead used in anexperiment.

Clearly, there is an intermediate regime in which several molecules are present on the bead or substrate,but the molecules are not yet abundantly present. We define the regime in which it is a reasonableassumption that all observed binding events involve the same binding molecule as sparsely coated. Weassume that if a surface is not sparsely coated, it is fully coated. We can then distinguish four typesof experimental situations

I) A fully coated bead and a fully coated substrate.

II) A sparsely coated bead and a fully coated substrate.

III) A fully coated bead and a sparsely coated substrate.

IV) A sparsely coated bead and a sparsely coated substrate.

Since we are interested in the investigation of single bond kinetics, situation I is not usable for ourpurposes. Situation IV would truly allow us to monitor single bonds, but the downside of this setupis that experimental observation times drastically rise. The performed experiments [15] used situationIII, while a typical biosensing device would correspond to situation II.

Situation II and III are in a sense similar. However, the bead is free to rotate, while the substrate isnot. Therefore, in situation III all bound motion patterns are found in a specific region of the motionpattern, while for situation II several binding events can occur at all accessible positions.

The experimentally observed times between binding events are typically in the order of tens of seconds,which is much larger than the correlation time of the position of the bead. Therefore, we concludethat the time between two binding events is only dependent on the a priori probability of the bead tobe near that specific binding spot on the substrate.

Figure 5.8 schematically represents the processing of experimental data. First a bound motion patternis analyzed as described in the previous section, and then the information about the specific bindingspots is extracted. Depending on whether situation II, III or IV is used, either Xdot, Xsubstr or bothXdot and Xsubstr should be extracted.

5.6.2 Sources of uncertainties

Several sources of uncertainties exist in experiments. When extracting information from experimentaldata it is necessary to have an idea about the relative magnitude of uncertainties. Four sources ofuncertainties and their magnitudes are discussed in this subsection.

One of the sources of uncertainty is the motion blur. This is caused by the movement of the beadduring the image capturing time. This is a relatively small effect, that leads to uncertainties in theorder of nanometers.

57


TPM data

Isolate binding events

Length

Width

Distance to center

Comparetabulatedvalues

Xdot

Xsubstr

Look up fhit

kstick

Figure 5.8: The schematic representation of the processing of TPM data. This scheme is applicable foran experiment performed with the conditions of experimental situation IV. In the case of experimentalsituation II or III, only Xdot or only Xsubstr has to be obtained respectively. The tabulated values maybe found in appendix B.

A much larger effect is caused by drift: the movement of the sample as a whole. The drift velocity mayvary during a measurement, which makes correcting for it non-trivial. Drift may lead to several tensof nanometers uncertainties in determining the mean distance to the origin M , width W and length Lof a bound motion pattern. An accurate correction could be applied if beads were permanently fixedon the substrate, but these were not present in the experimental data obtained so far.

Thirdly, as mentioned in section 2.1, the beads that are used in experiments are not always perfectlysmooth, but may show a finite surface roughness. The beads that Scheepers used in combination witha short tether have a roughness of 150 nm [15]. It is not straightforward to relate the roughness to theactual experimental error, but the shape of a bound motion pattern is likely affected by protrusionson the bead.

A final source of uncertainty is the fact that some binding events are too fast to detect. This meansthat if we extract the rebinding time, the time it takes for a free bead to bind again, that possibly thebead has bound and unbound undetectably within the rebinding time. This leads to the erroneousdetermination of rebinding time, with a larger error for longer rebinding times. Since koff is deter-mined by an exponential fit, it can be approximated from experimental data that 10% to 20% of thebonds release too fast to be detected [15].

5.6.3 Application to experimental data

We could apply this scheme to actual experimental data provided by Scheepers [15]. In this datasequence 75 hitting events were observed within 60 minutes. The performed experiments used theexperimental situation of type III: a fully coated bead and a sparsely coated substrate. Therefore,we should use the data to extract the value of Xsubstr, the position of the binding molecule on thesubstrate.

Part of the data that we used for this is graphically represented in figure 5.9. As we have outlined insection 5.5, the next step is to determine the mean distance to the origin M , the width W and thelength L of the bound motion pattern.

58


We isolate 6 bound motion patterns that exhibit small drift. For every bound pattern we extract thevalues of Xdot and Xsubstr and then we average over 6 results to find Xsubstr = (2.8 ± 0.3) · 102 nm(SEM).

From figure 5.4 we can determine that the hitting fraction for Xsubstr = 280 nm is equal to 0.0022.In other words, we expect the bead to be within reaction distance of this specific binding spot on thesubstrate for 2.2 ms per second.

The full data set showed 75 binding events and thus 74 rebinding times. Since the typical rebindingtimes are significantly larger than the correlation time of the system and binding events are thusuncorrelated, this situation corresponds to starting out with 74 free beads and observe the number offree beads in time.

The result can be found in figure 5.10. The result was fit by an exponential formula of the formN(t) = N0 exp(−κt). As mentioned in subsection 5.6.2, the rebinding times are increasingly erroneousfor larger values. Therefore, we have chosen to only fit the first 30 seconds of the data. As expected,for larger times the data deviates towards a slower decay of free beads.

The fit yields a value of κ = 0.038 s−1. Given the fact that the hitting fraction was 2.2 · 10−3, thisleads to a kstick of 1.7 · 101 s−1. This is well below the upper bound of 105 s−1 that we provided insection 5.4 and thus a consistent result.

X (nm)-300 -200 -100 0 100 200 300

Y (

nm)

-300

-200

-100

0

100

200

300

X (nm)-300 -200 -100 0 100 200 300

Y (

nm)

-300

-200

-100

0

100

200

300

Figure 5.9: In the left figure: the motion pattern obtained by measuring for 6 minutes. This is onlya part of the actual experimental data set, that contained 60 minutes of data. To limit the visibleinfluence of drift, only a part of the data set is shown. By analyzing the time trace, 75 binding eventswere isolated in the total data set. In the right figure: the motion pattern of one of the binding events.The outer curve of the bound motion pattern is drawn to indicate the width and length of the motionpattern. This motion pattern corresponds to M = 151 nm, W = 64 nm and L = 108 nm.

59


Time (s)0 50 100 150 200 250

Num

ber

of fr

ee b

eads

0

10

20

30

40

50

60

70

80

Measured rebinding timesExponential fit

Figure 5.10: The number of free beads as a function of time for the experimental data set. Therebinding time between every binding event is used to determine how the number of free beads decreaseswith time. The first 30 seconds of the data was fit by a function of the form N(t) = N0 exp(−κt),which yielded the value κ = 0.038 s−1.

60

Chapter 6Optimization

One of the motivations to investigate this system using simulations is that it enables us to find theoptimal set of experimental parameters for a system with a given purpose. The purpose being eitherthe investigation of single bond kinetics or a biosensing application. In both cases the contact areaplays a central role in optimizing the system parameters. We will first introduce the contact areain section 6.1 and then discuss the role of detectability in section 6.2. Finally, we will discuss theoptimization algorithm for an eventual sandwich assay biosensing application in section 6.3

6.1. Optimize contact area

When using TPM to investigate interactions between molecules on the bead and molecules on thesubstrate, the contact area between the bead and the substrate determines how many bonds will beformed. We define the contact area ζ as the fraction of time for which a specific dot on the bead iswithin interaction range of a specific spot on the substrate summed over all dots on the bead and allspots on the substrate.

In figure 6.1 a schematic representation of a dot on the bead and a spot on the substrate can befound. First, we consider both along the x-axis, so that the positions area characterized by Xdot

and Xsubstr. For a given system, every combination of Xdot and Xsubstr is within contact range for afraction fhit(Xdot, Xsubstr) of the time.

Subsequently, we use the fact that our system is axisymmetric and that the bead is free to rotate,so there is in fact a ring on the bead and a ring on the substrate that all correspond to the samefhit(Xdot, Xsubstr). The area of both rings is given by

dAring,bead = 2πR2 sin θdθ (6.1.1)

dAring,substr = 2πRsdRs, (6.1.2)

where R is the radius of the bead, θ is the polar angle of the bead and Rs is the in-plane radius on thesubstrate. In equation 6.1.1 and 6.1.2 both rings are of infinitesimal width. Note that these equationsare related to Xdot and Xsubstr by Xdot = R sin θ and Xsubstr = Rs. The contact area ζ is now givenby

ζ =

∫Asubstr

∫Abead

fhit(Xdot, Xsubstr)dAring,beaddAring,substr. (6.1.3)

61

CHAPTER 6. OPTIMIZATION

R

Xdot

Xsubstr

Figure 6.1: A schematic illustration of the specific locations of dots on the bead, characterized by Xdot

(red) and spots on the substrate, characterized by Xsubstr (blue). For every combination of Xdot andXsubstr the two points are within hitting range for a fraction fhit(Xdot, Xsubstr) of the time. Note:tether length and bead radius are not drawn to typical scale.

This relation has been used to determine the contact area ζ for several tether lengths L. Clearly,when the data is extracted from simulations we can no longer integrate over infinitesimal elements,but rather sum over finite elements. The result can be found in figure 6.2.

Increasing the tether length L has two opposite effects on ζ. On the one hand, it increases the amountof surface area that is able to enter the interaction range on the bead as well as the substrate. Onthe other hand, on average it decreases the hitting probability fhit(Xdot, Xsubstr). In figure 6.2 weobserve that ζ decreases for higher tether lengths L, which leads us to understand that the latter ofboth effects is more pronounced.

‘Optimizing’ the contact area then leads to the trivial solution of L = 0. Clearly, this is not a validsolution for a device based on tethered particle motion (TPM): for L = 0 the bead would not bemoving and would in fact not be tethered, but rather simply bound. In the next sections we willextend our optimization procedure to come up with non-trivial solutions.

62


L (nm)0 50 100 150 200 250 300 350

1 (n

m4 )

#105

0

2

4

6

8

10

12

14

16

18

1

Figure 6.2: The contact area ζ for several tether lengths L obtained by MD simulations and calculatedwith equation 6.1.3. ζ decreases for increasing tether length.

63


6.2. Detectability

When investigating molecular interactions it is not so much the number of bonds that one wants tomaximize, but rather the number of bonds that one can observe. This is where detectability entersthe optimization algorithm. The detectability of a single molecular bond depends on Xdot and Xsubstr.The next objective is to only take the observable combinations of Xdot and Xsubstr into account.

In section 4.1 we explained how the average step size of the bead can be used to determine whetherthe bead is bound or not. In section 4.8 we have shown how the step size varies with tether lengthand bead radius. The influence of a specific bond on the step size determines whether this bond canbe detected or not.

As a first approximation we assume that only the position of the binding spot on the substrate Xsubstr

influences the step size. In figure 6.3 the influence of a bond for a tether of 20 nm, 50 nm and 70 nmat 30 Hz can be found.

We have to define a threshold step size difference for which bonds are visible. There are two principlesthat give rise to uncertainties when measuring 〈Rstep〉 averaged over a specific time window. (1) Rstepis distributed with a finite standard deviation, so a measurement in a predefined time interval willhave an ‘intrinsic spread’. (2) Experimental uncertainties, in particular motion blur and drift.

Since the experimental uncertainties are not incorporated in the simulations, the value of the thresholdstep size should not be defined based on simulation results solely. After discussing this with ourexperimental collaborators we choose a threshold of 30 nm.

As can be seen in figure 6.3, the minimum Xsubstr for which a binding event is detectable Xsubstr,min

decreases with tether length. The uncertainty in the step size difference complicate the determinationof the minimum Xsubstr that leads to a visible bond. Using linear fits for L = 20 nm and L = 50 nmand using interpolation for L = 70 nm we estimate the value of Xsubstr,min. Moreover, from section 4.8we know that the difference in step size between L = 20 nm and L = 100 nm is larger than 30 nm.Since a secondary bond is approximately 18 nm in length, for L = 100 nm every bond is detectable,so Xsubstr,min = 0.

A linear fit of these values yieldedXsubstr,min = X0 − cL, (6.2.1)

with X0 = 303 nm and c = −2.9. This is the relation that we use to determine which binding eventsare detectable.

We can now define a new optimization parameter ξ that represents the contact area that leads todetectable bonds, in other words, ξ corresponds to ζ without the undetectable bonds. This correspondsto the equation

ξ =

Xsubstr,max∫Xsubstr,min

∫Abead

fhit(Xdot, Xsubstr)dAring,beaddAring,substr, (6.2.2)

where Xsubstr,max is the maximum distance on the substrate that would lead to a detectable bond.This value may be arbitrarily large, but for Xsubstr &

√2RL the hitting fraction fhit(Xdot, Xsubstr) is

equal to zero.

We have calculated ξ for several values of L and constant bead radius R = 500 nm. The result can befound in figure 6.4. It turned out that ξ has a maximum at L = 60 nm. In other words, for a bead ofradius R = 500 nm a tether of 60 nm leads to the maximum contact area that leads to visible bonds.

64


Xsubstr

(nm)0 50 100 150 200 250

"hR

stepi(n

m)

0

10

20

30

40

50

L=20 nmL=50 nmL=70 nm

Figure 6.3: The difference in step size between a bead bound at a distance on the substrate Xsubstr anda free bead as function of Xsubstr for tether lengths L =20 50, 70 nm. The dashed red line indicatesthe minimum value that is required for a bond to be detectable.

65


L (nm)0 10 20 30 40 50 60 70 80 90 100

9 (n

m4 )

#105

0

2

4

6

9

Figure 6.4: The new optimization parameter ξ that accounts for detectability. The maximum contactarea is found for L = 60 nm at R = 500 nm and a frame rate of 30 Hz. For L → 0 ξ vanishes sincenone of the bonds are detectable. As L increases, an increasing fraction of binding events is detectableand in the regime L > 60 nm ξ converges to ζ.

66


6.3. Optimization in the sandwich assay

In an actual sandwich immunoassay biosensing device, the optimal design is determined by differentproperties. The schematic outline of the mechanism of such a biosensor is represented in figure 6.5.

kcatch

kdetach

khit

ksep

kstick

ko

Figure 6.5: A schematic picture of how tethered particle motion may be used to create a sandwichassay. An extensive description is given in section 2.2.

One of the demands of ease of use biosensing devices is a limited detection time, in the order of min-utes [3]. In a time window of 10 minutes, a biosensing device could for example take 9 minutes forthe first step of figure 6.5, in which as many analytes as possible should be caught. Then one minuteremains for step 2 and 3.

Since the sandwich assay we have in mind is aimed at the detection of low concentrations, high speci-fity is required. Ideally, we would be in the regime kstick → ∞, so that the process of sticking doesnot limit the detection principle.

In this regime of ‘hitting is sticking’, the optimal design ensures that the surface area of the bead thathits the substrate within one minute is maximized. One might think of ‘coloring’ the bead at everyspot that hits the substrate. An example of this is given in figure 6.6. This optimization principle

Figure 6.6: An example of a bead colored red at every spot that hit the substrate for a tether of lengthL = 50 nm after 0.3 seconds. The color scheme of the bead itself has no physical meaning.

67


is governed by a different tradeoff principle. On the one hand, with a longer tether, a larger portionof the bead surface is able to hit the substrate. On the other hand, if the tether is too long, then thebead will only be very rarely near the substrate, and thus hit a smaller portion in the given time.

We have not been able to simulate 1 minute. With the simulation method we used, it takes approx-imately 10 days to simulate 0.3 seconds. This implies that a simulation of 1 minute would take fiveyears, which is beyond the scope of this project. However, we were able to simulate this process forshorter timescales and we may observe the general trend.

Figure 6.7 displays the resulting fraction of bead colored for several tether lengths. We observe forexample that for a time window of 0.18 s, a tether of 30 nm is more effective than a tether of 10 nm.

In a real biosensor kstick will have a finite value. The idea of coloring the bead could then be adaptedin such a way that the bead is only colored at one spot if that spots hits the substrate at least a timeτmin within one minute. A natural choice would be τmin > τstick. This approach would decrease thefraction of the bead that is colored in a given time window, but we expect that the trends in figure 6.7will still be present.

time (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fra

ctio

n be

ad c

olor

ed

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

L=10 nmL=30 nmL=50 nmL=70nm

Figure 6.7: The fraction of the bead colored after a certain time plot for several tether lengths. Resultswere obtained by averaging over 5 simulations. We observe that for t=0.18 s the tether L = 30 nm isalready better than L = 10 nm.

68

Chapter 7Conclusion & outlook

7.1. Conclusions

In this thesis we used Molecular Dynamics (MD) simulations to investigate secondary bonds in teth-ered particle motion (TPM) systems. We described our TPM system and introduced the extraordinaryapplication that we envision.

We reported the way that the MD simulations were set up to capture the equilibrium behavior of thebead and we showed that the exclusion mechanisms bead-surface exclusion, tether-bead exclusion andtether-bead exclusion are all three significant in the description of the equilibrium distribution of thebead close to the substrate.

After producing the correct equilibrium distributions, we turned our attention to the dynamic prop-erties of the bead. We determined that hydrodynamic wall effects are indispensable in the character-ization of the dynamic properties of a TPM system with the dimensions we consider.

The significance of three exclusion effects and hydrodynamic wall effects make an analytic approachto the movement of a tethered particle near a substrate prohibitively difficult. Therefore, we concludethat MD simulations can provide us with properties of the system that are not obtainable analytically.

Using MD simulations, we found that the correlation time of the in-plane vector ~R(t) is (0.09± 0.01)s and the correlation time of the Z-coordinate of the bead is (21 ± 1) ms. We have shown how thestep size, the in-plane distance that the bead travels in one frame, varies with the tether length L, thebead radius R and the frame rate.

Subsequently, we focused on specific binding spots on the bead and/or the substrate. We showed howthe hitting probability varies for different positions on the bead and the substrate. The location ofthe binding molecules can be reconstructed from the motion pattern of a bound bead. Using thisprinciple, we have constructed a scheme that allows one to interpret experimental data and extractkinetic parameters for several experimental situations. As a proof of principle, we have applied thisto actual experimental data to find a kstick of 1.7 · 101 s−1.

To investigate the regime of ‘quick rebinding events’, the event of a bond releasing and reattachingbefore significant bead movement has occurred, we have performed simulations with several values ofkstick for a fixed value of koff . We found that quick rebinding events contribute insignificantly in theregime of kstick < 105 s−1.

69

CHAPTER 7. CONCLUSION & OUTLOOK

Exploiting the ease of adjusting parameters in a simulation, we have sought for an optimal TPMsystem design. This optimization is discussed separately for two different applications: (1) experi-ments to determine single-bond kinetics and (2) a biosensing application. For the former we found anoptimal tether length of 60 nm, for the latter the dynamic range of our simulations turned out to beinsufficient.

Our results may be applicable in fundamental research as well as technological applications. In funda-mental research our results may be used to optimize experiments and to interpret data in a single-bondkinetics experiment. Technologically, our results may be used to design a TPM-based biosensor.

7.2. Outlook

In this research we have used molecular dynamics (MD) simulations to gain insight on a whole newway of using a TPM system. Now that we have developed a basic framework for the understandingof secondary binding effects in TPM experiments, several opportunities for future research lie ahead.

We have determined the optimal length of the tether L to maximize the contact area between thebead and the substrate that leads to detectable bonds for a fixed bead radius R and fixed frame rateω. In these calculations we have assumed that the detectability of a bond is solely determined by thebinding spot on the substrate Xsubstr. This approach may be too simplistic, as figure 6.3 indicates. Anext step would be to determine the detectability as function of both Xsubstr and Xdot.

The bead radius R and frame rate ω each have a non-trivial effect on the detectability of bindingevents, but the optimization principle outlined in section 6.2 can still be applied. A logical next stepin the optimization of this system would be to determine the optimal combination of L, R and ω.Along these lines, one could even include the persistence length lp in the optimization algorithm,although we expect this parameter to be of smaller influence.

Another way to increase the contact area between the bead and the substrate might be to use anon-spherical bead. Elongated gold nanoparticles are already of interest in certain areas of biosensingresearch [57] and may also be useful for a biosensing device based on TPM.

The simulation methods discussed in this project require relatively large simulation times. Due tothese constraints, some of the properties of the system have not been evaluated in as much detail aswe would have liked, e.g. the last optimization method in section 6.3. Future investigators should beaware of the restrictions that are raised by the limitations in simulation times. On the other hand,one of the goals of a future project may be to develop a way to parallelize simulations or to developa coarse-graining simulation method, in order to increase the dynamic scope of the simulations.

If an increase of computational efficiency in the order of 102 could be achieved, the actual experimentaltimes could be simulated, to determine the value of kstick with a higher level of accuracy.

An important next step in the understanding of a TPM-based biosensor is the incorporation ofanalytes. The process of catching analytes, the first step in figure 6.5, can also be included in thesimulations.

Eventually, a future study may provide the values of the rates kcatch, kdetach, khit, ksep, kstick and kofffor a given system, so that the initial concentration of analytes in a biosensor can actually be relatedto the observed number of bound beads. The regimes in which one specific step is the limiting stepin the binding process could be determined. This may be investigated for several bonds, so that theoptimal biosensor for a specific detection can be designed.

70

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74

Appendix AImplementing anisotropic drag

This appendix describes the implementation of hydrodynamic wall effects in LAMMPS via the ad-justment of the function fix langevin that generates Langevin dynamics. We also describe several testsimulations that we have performed to validate the implementation. We are contented to find thatthat this implementation does not significantly alter our simulation times.

A.1. Diffusion of a bead near a surface

A commenly used approach to simulate a Brownian motion in the canonical ensemble is to applyLangevin dynamics. In Langevin dynamics, two forces are added to the conservative force field - adrag force ~Fd proportional to the velocity with drag coefficient γ ≥ 0 and thermal white noise ~Fr [39].Explicitely, the Langevin equation of motion is given by

M~v = ~Ftot = ~Fc + ~Fr + ~Fd, , (A.1.1)

where ~Fc is the sum of the conservative forces in the system and ~Ftot is the sum of all the forces inthe system, m is the mass of a particle and ~v is the velocity. The thermal white noise or random forceobeys the relations ⟨

~Fr(t)⟩

= 0 (A.1.2)⟨~Fr(t) · ~Fr(τ)

⟩= 6γkBTδ(t− τ). (A.1.3)

In the main text we have described this principle in terms of the drag per mass coefficient Γ, which isclearly related to the drag coefficient γ by Γ = γ/M .

In LAMMPS the usual implementation of Langevin dynamics applies a friction force ~Ff and a random

force ~Fr every time step dt given by [36]

~Ff = −γ~v (A.1.4)

~Fr =

√6kBTγ

dt~ξ(t). (A.1.5)

In theory, ~ξ(t) should be a Gaussian distributed random vector, but, as Dunweg et al. demontsrated[37], in simulations the same result can be achieved by constructing vectors out of uniformly distributed

75

APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

random numbers. Since this is computationally more efficient, LAMMPS uses uniform random num-bers.

In the main text of section 3.4 we have introduced the concept of hydrodynamic wall effects. We haveintroduced adjusted the parallel and perpendicular drag coefficient. For the parallel component theresult is provided by Faxen’s law [45] and for the perpendicular component we use an interpolationformula [44] based on Brenners exact result [38]

γ‖ =γ0

1− 916z∗ + 1

8z∗3 − 45

256z∗4 − 1

16z∗5 (A.1.6)

γ⊥ =γ0

1− 98z∗ + 1

2z∗3 − 57

100z∗4 + 1

5z∗5 , (A.1.7)

where γ0 is the usual Stokes drag on a sphere, given by γ0 = 6πηR for a sphere with radius R in aliquid with viscosity η and z∗ is given by z∗ = R/z, with z the distance from the center of the beadto the surface.

We can introduce the relative drag coefficients λ‖, λ⊥ to simplify the form of equation 3.4.1 and 3.4.3,by defining γ‖ = λ‖γ0 and γ⊥ = λ⊥γ0.

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A.2. Implementation

To implement the anisotropic drag in any Molecular Dynamics simulation, we need to adjust the gen-erated drag force, given by equation A.1.4, and the generated random force, given by equation A.1.5.As we can see in the equations, the random force scales linearly with the drag coefficient, while therandom fluctuating force scales with the square root of the drag coefficient and thus the anisotropiceffects should be incorporated in line with that. In the code below the essential lines of code arerepresented.

//For loops loops over all atoms in bin

//and applies random force, after that

//drag force is applied.

for (int i = 0; i < nlocal; i++)

//Assign the perpendicular and parrallel

//component a value depending on the z-coordinate

zs = R_bead/(R_bead+x[i][2]);

//This is a rescaled version of z, convenient for the expressions of the parr and perp drag

//Use Faxen’s law and an interpolation formula given by Schaffer et al.

//to determine both coeffs

cparr = 1/(1-0.5625*zs+0.125*pow(zs,3)-0.175781*pow(zs,4)-0.0625*pow(zs,5));

cperp=1/(1-1.125*zs+0.5*pow(zs,3)-0.57*pow(zs,4)+0.2*pow(zs,5));

if (mask[i] & groupbit)

if (Tp_TSTYLEATOM) tsqrt = sqrt(tforce[i]);

if (Tp_RMASS)

gamma1 = -rmass[i] / t_period / ftm2v;

gamma2 = sqrt(rmass[i]) * sqrt(24.0*boltz/t_period/dt/mvv2e) / ftm2v;

gamma1 *= 1.0/ratio[type[i]];

gamma2 *= 1.0/sqrt(ratio[type[i]]) * tsqrt;

else

gamma1 = gfactor1[type[i]];

gamma2 = gfactor2[type[i]] * tsqrt;

fran[0] = sqrt(cparr)*gamma2*(random->uniform()-0.5);

fran[1] = sqrt(cparr)*gamma2*(random->uniform()-0.5);

fran[2] = sqrt(cperp)*gamma2*(random->uniform()-0.5);

fdrag[0] = cparr*gamma1*v[i][0];

fdrag[1] = cparr*gamma1*v[i][1];

fdrag[2] = cperp*gamma1*v[i][2];

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A.3. Control simulations

A.3.1 Random forces without motion

We can evaluate the forces that are applied every time step to a bead at a certain distance z from thesurface at z = 0. To this end, we calculate the forces with our anisotropic langevin thermostat but wedo not actually integrate the equation of motion. The result is that the bead remains fixed, but weobtain a distribution of forces.

We perform the simulation at z = 550, so that z∗ = 0.91. As mentioned in section A.1 LAMMPS usesuniform random numbers to generate a random force. We know from equation 3.4.1 and 3.4.3 thatat this point γ⊥/γ‖ = 4.83. We know that the random force scales with the square root of the drag

coefficient, therefore we expect F⊥/F‖ =√

4.83 = 2.20.

In figure A.1 a component-wise line histogram of the resulting force of this simulation can be found.As we expected, the forces are uniformly randomly distributed. Moreover, if we review the maximumoccurring fz divided by the maximum occurring fx we obtain 2.20, which corresponds exactly to thevalues we expected.

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Force (0/<)-150 -100 -50 0 50 100 150

Occ

uren

ce

0

500

1000

1500

2000

2500

fxfyfz

Figure A.1: A component-wise line histogram of the occuring forces. This graph shows how oftencertain values occur for fx (blue), fy (red) and fz (yellow). The simulation was run for 2 · 105 timesteps.

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A.3.2 Movement parallel to surface

To test the implemention of the parallel drag coefficient we perform the following simulation: aparticle starts out with a relatively large velocity parallel to the surface and the relaxation of thevelocity is recorded. One expects the drag to decelerate the velocity in an exponential fashion with acharactaristic timescale τ = 1/γ‖. These simulations have performed with a bead radius of R = 500and a z-coordinate of z = 900, 550, 501, so that z∗ = 0.56, 0.91, 0.99. The results for 1/γ‖ can be foundin figure A.2 together with a plot of Faxen’s law. The simulations agree with the analytical formula.

To demonstrate that the decay of the velocity is indeed exponential, we plot the velocity against timefor five times the charactaristic time in a semilogarithmic graph in figure A.3. We observe a straightgraph, which indicates that the velocity indeed decays exponentially. The wiggles that appear in theat the last part of the graph are the result of the influence of the random fluctuating force becomingmore visible for lower values on a logarithmic axis.

æ

æ

æ

0.2 0.4 0.6 0.8 1.0Rz

1

2

3

4

Figure A.2: The results for 1/γ‖ at z = 900, 550, 501 (blue circles) plot together with Faxen’s law givenby equation 3.4.1.

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Time(=)0 0.5 1 1.5 2 2.5 3 3.5

v x(</=

)

100

101

102

103

104

Figure A.3: The decay of the velocity represented in a semilogarithmic graph for z = 900, R = 500,v(0) = 2000.

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A.3.3 Movement towards surface

Another simulation that we can perform is the movement of a bead towards a surface. We define aninitial velocity in the direction of the surface. The drag coefficient should now increase as the beadmoves closer to the surface and the velocity in a semilogarithmic plot should no longer have a constantslope.

In figure A.4 the result of such a simulation can be found. Indeed, the slope of the magnitude of thevelocity on a vertical logarithmic axis is not constant, but is increasingly negative. This is a result ofthe fact that the perpendicular drag coefficient increases when the bead approaches the surface.

Time(=)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

v z(</=

)

100

101

102

103

104

105

Z(<

)

500

600

700

800

900

1000

Figure A.4: The evolution of the magnitude of the velocity towards the surface with a logarithmicvertical axis (blue) plot together with the evolution of the z-coordinate of the center of the bead ona linear vertical axis (red). The values used for this simulation were v(0) = −2000, R = 500. Thefact that the slope of the blue line is not contsant is reminiscent of the fact that the drag coefficientincreases.

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A.4. Simulation times

To find the influence of our adapted fix on the simulation times we run several simulations with theregular ‘fix langevin’ as well as our tailor-made ‘fix langevin/aniso’. We run five simulations with2 · 107 time steps for both fixes. For the regular fix we obtain a simulation time of (15.5± 0.4) seconds(Standard Error of Mean), while for the adapted fix we obtain simulation times (15.5± 0.3) seconds.In other words, our program did not significantly slow down due to the adapted fix, which is a majoradvantage compared to explicit hydrodynamic simulations near surfaces.

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Appendix BTabulated values

To perform the procedure outlined in section 5.5, the tabulated values for the mean distance to theorigin, the width and the length of the pattern are required. This appendix provides these tables forthe system with the usual parameter values. The tabulated mean distances to the origin can be foundin table B.1, the widths can be found in table B.2 and the lengths in table B.3.

When Xdot and Xsubstr are extracted using tables B.1, B.2 and B.3, the corresponding hitting fractionmay be found in table B.4.

Table B.1: The mean distance(in nm) to the center of the motion pattern as function of the binding position on the bead Xdot and the position on thesubstrate Xsubstr. The overall trend is that for higher Xdot and Xsubstr the mean distance to the origin is higher.

Xsubstr (nm)20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

Xdot

20 - - 77.5 70.3 - - - - - - - - - - - - -40 - - - 124.8 - - - - - - - - - - - - -60 34.7 - - - 117.3 - - - - - - - - - - - -80 24.2 62.7 - - 137.4 122.0 102.4 - - - - - - - - - -

100 - - 73.0 112.1 140.7 150.8 128.9 110.7 - - - - - - - - -120 - - - - 120.7 145.6 153.5 136.8 120.1 - - - - - - - -140 - - - - 83.0 121.8 149.0 155.6 143.6 - - - - - - - -

(nm) 160 - - - - 55.5 87.3 126.0 154.3 157.5 149.5 140.2 - - - - - -180 - - - - - 64.8 93.9 124.9 153.3 162.7 158.2 - - - - - -200 - - - - - - - 100.1 130.4 157.4 169.4 167.7 - - - - -220 - - - - - - - - - - 162.4 175.7 178.1 - - - -240 - - - - - - - - - - 140.7 168.5 184.6 188.0 - - -260 - - - - - - - - - - 126.7 148.8 174.5 191.5 198.6 201.1 -280 - - - - - - - - - - - - - 180.6 199.3 208.5 -300 - - - - - - - - - - - - - 166.1 187.5 206.3 218.0320 - - - - - - - - - - - - - - - 194.7 213.1

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APPENDIX B. TABULATED VALUES

Table B.2: The width of the motion pattern (in nm) as function of the binding position on the bead Xdot and the position on the substrate Xsubstr. Theoverall trend is that for higher Xdot and Xsubstr the width is lower.

Xsubstr (nm)20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

Xdot

20 - - 251.1 204.2 - - - - - - - - - - - - -40 - - - 183.5 - - - - - - - - - - - - -60 261.2 - - - 224.4 - - - - - - - - - - - -80 220.9 259.2 - - 247.5 202.3 185.5 - - - - - - - - - -

100 - - 243.3 247.4 244.1 226.7 196.0 153.6 - - - - - - - - -120 - - - - 235.5 224.9 197.9 180.7 139.9 - - - - - - - -140 - - - - 205.8 124.1 205.9 187.9 165.8 - - - - - - - -

(nm) 160 - - - - 135.0 190.3 184.1 177.1 162.4 140.8 116.8 - - - - - -180 - - - - - 106.7 153.0 163.1 170.2 142.0 140.7 - - - - - -200 - - - - - - - 135.2 137.6 157.6 142.1 120.6 - - - - -220 - - - - - - - - - - 142.8 101.2 89.4 - - - -240 - - - - - - - - - - 193.8 100.2 86.9 70.6 - - -260 - - - - - - - - - - 79.9 95.6 90.2 78.8 68.4 49.0 -280 - - - - - - - - - - - - - 75.9 69.3 53.7 -300 - - - - - - - - - - - - - 62.8 63.6 49.1 36.2320 - - - - - - - - - - - - - - - 40.9 37.7

Table B.3: The length of the motion pattern (in nm) as function of the binding position on the bead Xdot and the position on the substrate Xsubstr. Theoverall trend is that for higher Xdot and Xsubstr the length is lower.

Xsubstr (nm)20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

Xdot

20 - - 260.4 220.1 - - - - - - - - - - - - -40 - - - 257.9 - - - - - - - - - - - - -60 269.3 - - - 257.9 - - - - - - - - - - - -80 234.5 270.2 - - 260.4 259.9 218.9 - - - - - - - - - -

100 - - 260.9 264.9 259.9 268.7 26- 228.6 - - - - - - - - -120 - - - - 271.7 268.2 262.7 258.2 216.7 - - - - - - - -140 - - - - 255.2 269.0 269.5 265.5 257.9 - - - - - - - -

(nm) 160 - - - - 192.9 247.9 265.0 264.4 262.4 247.0 204.4 - - - - - -180 - - - - - 170.9 236.0 248.2 258.0 252.7 255.5 - - - - - -200 - - - - - - - 212.9 246.1 264.2 264.9 246.1 - - - - -220 - - - - - - - - - - 252.3 227.5 216.1 - - - -240 - - - - - - - - - - 159.9 224.0 216.3 187.2 - - -260 - - - - - - - - - - 134.2 182.9 203.9 198.3 175.8 123.8 -280 - - - - - - - - - - - - - 176.1 185.4 150.3 -300 - - - - - - - - - - - - - 115.8 136.3 139.7 128.2320 - - - - - - - - - - - - - - - 94.5 108.9

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APPENDIX B. TABULATED VALUES

Table B.4: The hitting fraction for a single Xsubstr and a single Xdot. The provide values are given in fraction ·105. Severalcombinations of Xsubstr Xdot do not occur at all. This table may be used when a single binding molecule is present on thebead and a single molecule is present on the substrate.

Xsubstr (nm)20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320

Xdot

20 0.0 0.1 15.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.040 0.5 3.4 6.1 8.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.060 17.0 7.1 3.9 4.0 7.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.080 0.0 9.6 4.6 3.6 4.3 8.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

100 0.0 0.0 8.2 4.7 4.5 5.3 8.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0120 0.0 0.0 0.0 7.7 6.0 5.7 6.9 7.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0140 0.0 0.0 0.0 0.0 6.9 7.7 7.3 8.2 5.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0

(nm) 160 0.0 0.0 0.0 0.0 0.0 5.5 8.9 8.5 8.7 4.3 0.0 0.0 0.0 0.0 0.0 0.0180 0.0 0.0 0.0 0.0 0.0 0.0 4.0 8.9 8.9 7.5 2.8 0.0 0.0 0.0 0.0 0.0200 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.7 7.7 7.5 5.8 1.7 0.0 0.0 0.0 0.0220 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 4.5 5.4 4.1 1.3 0.0 0.0 0.0240 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 2.7 3.7 2.4 0.4 0.0 0.0260 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 1.8 1.1 0.2 0.0280 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.7 0.4 0.0300 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1320 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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