Eindhoven University of Technology

MASTER

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

Merkus, K.E.

Award date: 2015

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

K.E. Merkus

July 2015

Under supervision of dr. C. Storm Abstract

Tethered particle motion (TPM), the motion of a micro- or nanobead tethered to a substrate by a macromolecule, is widely studied to understand various properties related to the tether. However, we propose a whole new way of looking at TPM: TPM as a probe for secondary bonds. Kinetic effects in the motion pattern of a coated bead tethered to a coated substrate yield information about the bonding kinetics of the corresponding coating-molecules and/or solution. We use molecular dynamics simulations 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 present experimental 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 single bond kinetics. Moreover, this principle may serve as the basis for a new kind of biosensing device. Contents

1 Introduction 1 1.1 Tethered particle motion ...... 2 1.2 Outline of the thesis ...... 4

2 Tethered particle motion system 5 2.1 Components of a tethered particle system ...... 5 2.2 The sandwich assay principle ...... 7 2.3 Model system ...... 9

3 Simulation methods 11 3.1 Monte Carlo ...... 11 3.2 Molecular dynamics ...... 12 3.2.1 Molecular dynamics basics ...... 12 3.2.2 Time integration algorithm ...... 12 3.2.3 Components ...... 13 3.2.4 Interactions ...... 13 3.2.5 Implicit solvent ...... 15 3.2.6 Timescales ...... 16 3.2.7 Bond creation and breaking ...... 17 3.3 Parameter values ...... 18 3.4 Hydrodynamic wall effects ...... 20 3.4.1 Introduction ...... 20 3.4.2 Correction on parallel motion ...... 20 3.4.3 Correction on perpendicular motion ...... 21 3.4.4 Implementation ...... 21 3.4.5 Discussion ...... 22

4 Bead movement 24 4.1 Motion pattern ...... 24 4.1.1 Experimental motion patterns ...... 24 4.1.2 Simulation motion patterns ...... 27 4.2 Analytic approach ...... 28 4.2.1 Geometrical framework ...... 28 4.2.2 Integration boundaries ...... 30 4.2.3 Distribution results ...... 32

1 CONTENTS

4.2.4 Conclusion ...... 34 4.3 Number of polymer beads in MD simulation ...... 35 4.4 Exclusion effects ...... 37 4.5 Influence of the engineering parameters R, L, lp ...... 39 4.5.1 Varying the bead size ...... 39 4.5.2 Varying the tether length ...... 40 4.5.3 Varying the persistence length ...... 41 4.5.4 Discussion ...... 42 4.6 Dynamic properties ...... 43 4.6.1 Autocorrelation R~(t)...... 43 4.6.2 Autocorrelation Z(t)...... 44 4.6.3 Conclusion ...... 44 4.7 Hitting and return times ...... 45 4.7.1 Hitting times ...... 45 4.7.2 Return times ...... 45 4.7.3 Discussion ...... 45 4.8 Step size ...... 47 4.8.1 Tether length ...... 47 4.8.2 Bead radius ...... 48

5 Specific binding spots 49 5.1 Dots on the bead ...... 49 5.2 Spots on the substrate ...... 51 5.3 Single Xdot and Xsubstr ...... 53 5.4 Kinetic regime ...... 54 5.4.1 Reaction process ...... 54 5.4.2 Quick rebinding events ...... 54 5.5 Analyzing bound patterns ...... 56 5.6 Relation to experiments ...... 57 5.6.1 Experimental situations ...... 57 5.6.2 Sources of uncertainties ...... 57 5.6.3 Application to experimental data ...... 58

6 Optimization 61 6.1 Optimize contact area ...... 61 6.2 Detectability ...... 64 6.3 Optimization in the sandwich assay ...... 67

7 Conclusion & outlook 69 7.1 Conclusions ...... 69 7.2 Outlook ...... 70

References 71

A Implementing anisotropic drag 75 A.1 Diffusion of a bead near a surface ...... 75 A.2 Implementation ...... 77 A.3 Control simulations ...... 78 A.3.1 Random forces without motion ...... 78

2 CONTENTS

A.3.2 Movement parallel to surface ...... 80 A.3.3 Movement towards surface ...... 82 A.4 Simulation times ...... 83

B Tabulated values 84

3 Chapter 1 Introduction

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-care immuno-biosensing that has been developed and abundantly used for decades [2]. Glucose is found in the 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 commercial success. Examples are: the detection of protein markers to diagnose cardiac diseases, the detection of nucleic 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]. A biosensing 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 molecules present 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 that are 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 level of 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-chip device 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 stationary fluid. The eventual detection mechanism and resulting determination of the initial concentration of a 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 these applications, 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 to a surface by a double-stranded DNA tether. The basic system that is described throughout this thesis consists of a bead with radius R=500 nm and a tether with length L=50 nm. This picture is not drawn to scale. The dashed line represents the projection of the bead coordinates on the surface, which is the experimentally observable value.

1.1. Tethered particle motion

To study the properties of a nanoparticle adhered to a substrate bound by an antibody-biomarker complex, we resort to a model system: a nanoparticle adhered to a substrate by a double-stranded DNA (dsDNA) tether [5], as is graphically represented in figure 1.1. This brings us into the field of tethered particle motion (TPM). Several studies have already been devoted to this subject, so we can rely on literature to explore a lot of basic properties of this system. In previous research the focus was on the properties and interactions of the tether (typically DNA), rather than the binding properties of coated nanobeads. For example, TPM has been used to determine the transcription of RNA 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 coated substrate is able to undergo temporary secondary binding events, which affect the motion pattern of the bead. This effect depends on the presence of binding molecules in the solvent. The idea has risen that this could be a biosensor by itself. Such a biosensor would consist of a coated nanoparticle

2 CHAPTER 1. INTRODUCTION 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 and therefore 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 are also 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 is optically detectable in a robust way. It should be emphasized that the purpose of this research is not to determine the properties of DNA or any other tether-molecule, but rather to understand how the binding kinetics of several components of 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. Several techniques have already been exploited for this purpose, such as laminar flow chambers [9], atomic force microscopy [10] and total internal reflection fluorescence microscopy [11]. Due to several sources of uncertainty in these measurements, a new way to measure single-bond kinetics would be a promising contribution [12]. This holds especially true since the biological relevance of such methods is widely recognized [12], [13], [14]. In order to fully understand the kinetics of a single bond in a TPM system, a distinction between hitting and binding should be made. The diffusive motion of the tethered bead, an example of a confined Brownian motion, determines how often and how long a specific spot on the bead is near a specific spot on the substrate. We will refer to this process as ‘hitting’. On the other hand, when two binding spots are near each other, the actual molecular bond has to be formed. We will refer to this process as ‘sticking’. Both processes happen at a different time and length scale and the combined result 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 enables one to relate experimentally obtained motion patterns to the molecular bond properties. We conclude by determining the engineering parameters that provide the maximum number of observable binding events. This research has been performed in close collaboration with the experimental group Molecular Biosensing 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 been focussing on the experimental part of the same research. Throughout this report, we will refer several times to Scheepers’ work [15].

3 CHAPTER 1. INTRODUCTION

1.2. Outline of the thesis

We continue this thesis in chapter 2 with an extensive description of a TPM sytem. We start by describing the components of a TPM system. Subsequently we describe the principle that we aim to use 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 previously developed Monte Carlo (MC) simulations is provided and an extensive treatment of the Molecular Dynamics (MD) simulations that we developed is presented. In chapter 4 equilibrium and dynamic properties of the bead can be found. We investigate what fraction of the time the bead is close enough to the substrate to form a secondary bond and demonstrate which effects are relevant in describing these distributions. Furthermore, the correlation times and hitting 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 actual binding spot on the bead and on the substrate. We provide an upper bound for one of the involved kinetic parameters. We finish chapter 5 by presenting a scheme that allows one to extract additional kinetic 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 optimal parameters in an experimental system. We provide the tether length L that leads to the maximum contact area on which detectable bonds can be formed. Moreover, we outline a possible optimization algorithm for an actual biosensing device. We finish in chapter 7 with a summary of our conclusions and possibilities for future research. At the end we provide a list of references to literature and appendices A and B.

4 Chapter 2 Tethered particle motion system

2.1. Components of a tethered particle system

Before considering any system-specific details about tethered particle motion (TPM), we should first have 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 otherwise free bead (particle). TPM then refers to the motion of this particle as well as the method of tracking this 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 - that is able to attach a bead to a surface could in principle be used for TPM. In this report we will consider TPM systems based on double-stranded DNA (dsDNA), having a persistence length lp of approximately 50 nm [18]. In order to observe actual bead movement on experimental timescales the size of the bead should not 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] and fluorospheres [20]. Comparing metal beads to polystyrene beads, the advantage of metal beads is the strong scattering of light. On the other hand, polystyrene beads enable the use of optical tweezer experiments and enable magnetic control of the bead by including a magnetic core. One could also use 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 will focus 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) and the 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 on the 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. the 2D-projected motion of the bead. Although there are experimental systems in which the Z-coordinate of the bead can be tracked [20], these systems are less common. In the experimental system that we study [15] the bead is tracked in two dimensions and it is thus important to keep in mind that every result should be translated into quantities that can be extracted from measuring X and Y .

6 CHAPTER 2. TETHERED PARTICLE MOTION SYSTEM

2.2. The sandwich assay principle

Figure 2.1: The initial configuration of a tethered particle in the setup of a sandwich assay. Note that the bead is coated with a specific type of antibody (green) and the substrate is coated with another specific 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 biosensing device based on TPM. For the detection of biomarkers with concentrations in the order of picomolars or 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 initial configuration of a bead is schematically drawn. The bead is coated with a specific type of antibodies and the substrate is coated with a second specific type of antibodies. The molecule that we want to detect, the analyte, is represented by a red dot. Figure 2.2 schematically describes the four steps involved in such a TPM sandwich assay. Here, only the antibodies that are involved in the bond that is formed are drawn. Figure 2.2 shows that the process of a tethered bead forming a secondary bond with the substrate in the sandwich assay can be divided into three steps: (1) the binding of the analyte with an antibody on the bead, (2) the bead moving towards the substrate so that two antibodies are within interaction range (this process is defined as ‘hitting’) and (3) the molecular antibody-analyte-antibody-complex is 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 (far from the substrate) beads with no analyte bound to it, [A] the concentration of analyte, [FBA] the concentration of free beads bound by an analyte, [HBA] the concentration of hitting beads bound by an analyte and [SBA] the concentration of beads bound to the substrate by an antibody-analyte-

7 CHAPTER 2. TETHERED PARTICLE MOTION SYSTEM antibody complex. The reaction formula could then be written as

kcatch ksep kstick FB + A ←−−−−−−−−→ FBA ←−−−−−−−−→ HBA ←−−−−−−−−→ SBA, (2.2.1) kdetach ksep koff where the reaction rates are determined by kcatch, kdetach, khit, ksep, kstick and koff as

∂[FB] ∂[A] = = k [FBA] − k [FB][A] (2.2.2) ∂t ∂t detach catch ∂[FBA] = k [FB][A] − k [FBA] − (k [FBA] − k [HBA]) (2.2.3) ∂t catch detach hit sep ∂[HBA] = k [FBA] − k [HBA] − (k [HBA] − k [SBA]) (2.2.4) ∂t hit sep stick off ∂[SBA] = k [HBA] − k [SBA]. (2.2.5) ∂t stick off Note that we implicitly made two assumptions while describing this system. First of all, we assume that the bead is not hitting the substrate before the analyte has bound. Indeed, two antibodies can already be within interaction range by chance, but this is a rare process in the systems that we consider and therefore we do not take it into account in our model. Secondly, we have assumed that a bead is either bound by an analyte or not bound by an analyte. In reality, the bead can clearly be bound by several analytes. We aim to develop a biosensor for low concentrations, so the binding of an analyte to the bead is a rare event. In this regime, only taking into account the single bonds is a good approximation.

kcatch khit kstick

kdetach ksep ko

Figure 2.2: A schematic picture of how tethered particle motion may be used to create a sandwich assay. The picture displays a bead that is tethered to the substrate by dsDNA, with one antibody on the bead and one antibody on the substrate. The red dot represents the analyte: the molecule that should be detected. The picture describes the sandwich assay principle in four stages: (1) the initial configuration where no binding event has occurred. (2) The analyte has bound to the antibody on the bead. (3) The bead has moved so that it ‘hits’ the substrate, but it does not stick yet. (4) The two antibodies have formed a molecular complex with the analyte and the bead is now stuck to the substrate at 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 in the main text. In this picture only the antibodies relevant for the binding process are drawn, in reality more antibodies are present, as figure 2.1 indicates. Note: this picture is not drawn to scale.

8 CHAPTER 2. TETHERED PARTICLE MOTION SYSTEM

2.3. Model system

In this study we aim to investigate the effect of molecular bonds on the motion pattern of a tethered particle. The first step in figure 2.2 is in that sense irrelevant: the movement of the bead effectively does not change by the binding of an analyte. A similar system that is initially already in the second stage, corresponding to FBA, enables us to focus on the processes that indeed influence the motion pattern of the bead. In line with our experimental collaborators [15], we turn to a model system in which analytes are no longer present. In this system the molecules on the bead and the molecules on the substrate are coated with complementary oligonucleotide (short DNA) strands. We focus on the 8 basepair oligonucleotide −1 strands, with a length of 3 nm and a reported koff of 0.1 s [22]. Due to experimental convenience, in this system the bead is coated with streptavidin instead of an antibody. The substrate is again coated with an antibody. The streptavidin on the bead and the antibody on the substrate are thus coated with complementary oligonucleotide sequences. The principle of this system is outlined in figure 2.3. The simulations that are described in this report aim to describe this model system.

khit kstick

k ksep o

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

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

khit kstick FB ←−−−−−−−−→ HB ←−−−−−−−−→ SB, (2.3.1) ksep koff where FB indicates the free bead, HB the bead that is hitting the substrate and SB the bead that

9 CHAPTER 2. TETHERED PARTICLE MOTION SYSTEM sticks to the substrate. The changes in concentration are now described by

∂[FB] = k [HB] − k [FB] (2.3.2) ∂t sep hit ∂[HB] = k [FB] − k [HB] − (k [HB] − k [SB]) (2.3.3) ∂t hit sep stick off ∂[SB] = k [HB] − k [SB]. (2.3.4) ∂t stick off Hitting is defined as the process of the bead moving close enough to the substrate so that a bond can be formed. Taking Z as the z-coordinate of the bead center, R as the radius of the bead, and ZHIT the distance 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 solely on the diffusive properties of the bead. The time it takes for a sphere to diffuse its own radius, i.e. −1 the configurational relaxation time τcr [23] is in the order of 10 s for the beads that we consider. However, the distance that the bead needs to cover to hit the substrate is typically one order of magnitude smaller than the actual bead radius. So we expect relevant timescales for hitting to be in the order of τ ≈ 10−2 s. On the other hand, when two molecules, one on the bead and one on the substrate are within binding distance, 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-rate koff [12].

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

10 Chapter 3 Simulation methods

3.1. Monte Carlo

The code used for the Monte Carlo (MC) simulations was developed prior to this project and turned 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 equilibrium distribution of states that is found in a system. In other words, the MC simulations describe the distribution of states that will be found if the time window in which data is obtained is sufficiently large. This project focuses on the motion patterns that exhibit a shape and distribution of motion pattern 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, which do incorporate dynamical effects, i.e. we can review the motion of the bead in time and we can review 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 the MD simulations for sufficiently large simulation times. The MC simulations thus provide us a way to benchmark the long-term behavior of our system. The rest of this section provides a brief description of the MC simulations. The MC simulations were developed and executed using MATLAB [25]. The basic set up of the simulation is that first a tether is constructed, then a bead is attached at the end of it and then a check for the three exclusion mechanisms, discussed in section 2, is performed. The MC simulations require the input of four parameters: the bead radius R, the tether length L, the tether 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 set of Gaussian distributed random numbers so that in the limit of N → ∞ the distribution of a Worm-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 for the 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 then only keep the valid configurations to obtain a distribution of configurations. The long-time distributions 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 required to describe a time-dependent effect. The rest of this section provides a thorough description of the MD 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 communication times 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 solving Newton’s equation of motion for every particle. An MD algorithm divides the entire time span of a simulation into shorter intervals called simulation time 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 that particle [28]. Newton’s equation of motion for a particle of mass M at position ~r(t) for time t is given by [29]

∂2~r(t) F~ (t) = M , (3.2.1) ∂t2 where F~ (t) is the total force on the particle at time t. The force consists of several parts that we will discuss 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 particles after proceeding one time step in time. We have used velocity Verlet integration, which is the standard integration scheme of LAMMPS. This scheme is based on the Taylor expansion of the position vector ~r(t) around time t, namely 1 1 ~r(t + ∆t) = ~r(t) + ~v(t)∆t + ~a(t)∆t2 + ~b(t) + O(∆t4) (3.2.2) 2 6 1 1 ~r(t − ∆t) = ~r(t) − ~v(t)∆t + ~a(t)∆t2 − ~b(t) + O(∆t4), (3.2.3) 2 6 where ~v = ~r˙(t), the velocity, ~a = ~r¨(t) the acceleration and ~b the jerk (third derivative of position with respect to time t). Adding these two expansions leads to

F~ (t) ~r(t + ∆t) = 2~r(t) − ~r(t − ∆t) + ∆t2 + O(∆t4). (3.2.4) m Since the first and third order terms in ∆t cancel out, the estimate of the new position contains an error that is of order ∆t4 [28]. A simple concept - combining two Taylor expansions - leads to a significantly more accurate time integration algorithm!

12 CHAPTER 3. SIMULATION METHODS

3.2.3 Components There are two types of particles that play an important role in our simulations. The first type represents the bead. In most simulations the coordinates of the center of mass of the bead will be 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 which the radius is chosen to match the actual bead size. Implementing the tether requires slightly more care. We do not expect that the molecular details of the dsDNA tether significantly influence the motion of the bead. There are however four 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 has a mass and therefore inertia, (3) the polymer experiences hydrodynamic drag and (4) there is a steric interaction between the tether and the bead. Since we only want to include the effects that we expect to be relevant, we have looked for a model that efficiently implements these four effects. It turns out that this can be achieved by using a bead-spring model to represent the tether [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 to the former as ‘the bead’ and we will refer to the latter as ‘polymer beads’. The number of polymer beads that is required to produce good results is discussed in section 4.3. One end of the bead-spring string is rigidly connected with the bead so that it is effectively treated as one particle.

3.2.4 Interactions In MD simulations, interactions are included by defining relevant potentials for every particle in 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 effects will 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 a semiflexible 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

2 Ubond = Kb(r − r0) , (3.2.6) with r0 the equilibrium distance and bond coefficient Kb. One can think of r0 as being the length of the spring. Note that the usual factor 1/2 is included in Kb.

13 CHAPTER 3. SIMULATION METHODS

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

kBT Kb = 50 2 , (3.2.7) r0 with kB being the Boltzmann constant and T being the temperature. Double-stranded DNA is a polymer with limited flexibility, that may be described as a worm-like chain with persistence length lp. In a bead-spring model this translates into an angle-bending potential given by [31]

2 kBT lp 2 Uangle = Kaθ = θ , (3.2.8) 2r0 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. In MD this may be implemented using a steep potential. A widely used potential for this purpose is the repulsive part of the Lennard-Jones potential, also known as the WCA potential [33], given by

σ 12 σ 6 U (r) = 4 − , r < r (3.2.9) LJ r r c

1/6 where the energy  and distance σ are the parameters that define the potential and rc = 2 σ ≈ 1.12σ, so that only the repulsive part is used. A graphical representation is given in figure 3.1. Using this potential, r is usually the distances between the particle centers. We can also define it as the distance between a particle and the surface, which allows us to include bead-surface and tether-surface exclusion.

U Ε

6 H L 5

4

3

2

1

r Σ 0.95 1.00 1.05 1.10 -1 H L Figure 3.1: The repulsive part of the Lennard-Jones potential, given by equation 3.2.9. On the vertical axis the interaction energy is given in terms of  and on the horizontal axis the distance is given in terms of σ.

14 CHAPTER 3. SIMULATION METHODS

Since the bead and the tether-parts vary hugely in size, expression 3.2.9 , where r is the distance between the particle centers, is not sufficient to create a steep potential. As instead, the expanded Lennard-Jones potential is used " #  σ 12  σ 6 U (r) = 4 − , r < r + ∆, (3.2.10) LJ,exp r − ∆ r − ∆ c where the interaction distance r is shifted by a distance ∆ that allows us to tune the potential with the desired steepness.

3.2.5 Implicit solvent The actual motion in tethered particle motion experiments is a confined Brownian motion that results from the many solvent molecules that collide with the bead. The time- and lengthscales at which the solvent molecules collide with the bead are much smaller than the time- and lengthscales that describe the motion of the bead [23]. Effectively this results in a random force experienced by the bead. On top 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. This approach is usually referred to as ‘Langevin dynamics’ [34]. We are aware of the fact that our bead is not in free solution, but rather close to a surface. Hydrodynamics near a surface differ vastly from hydrodynamics 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 and is therefore unwanted. An implicit way to incorporate solvent effects in the MD simulation is by adding a random force and a drag force to every particle in the solution. At every time step each particle in the solution then experiences three types of forces

F~tot = F~c + F~r + F~d, (3.2.11) where F~tot is the total force, F~c is the conservative force, F~d is the drag force and F~r is the random force. The conservative force F~c is the result of the interactions as described in section 3.2.4 and is computed via equation 3.2.5. The drag force F~d and random force F~r are both the result of the interaction with the solvent. One may therefore expect that these forces are related. Indeed the fluctuation-dissipation theorem tells us that [35]

F~d = −MΓ~v (3.2.12)

hFr(t)i = 0 (3.2.13) D E F~r(t) · F~r(τ) = 6kBTMΓδ(t − τ), (3.2.14) where h...i denotes the ensemble average and Γ is the friction constant per mass. This should be translated into a force that is applied every time step, in MD this can be implemented by [36] r k TMΓ F ∼ B (3.2.15) r dt

F~d = −MΓ~v, (3.2.16) with dt being the size of the time step. The drag force F~d is identical to the expression given in equation 3.2.12, but now the velocity ~v refers to the velocity stored during simulations and the drag

15 CHAPTER 3. SIMULATION METHODS

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

The direction of F~r is random and the magnitude should be scaled by a prefactor. This prefactor depends on whether Gaussian random numbers are used or uniform random√ numbers. Uniform random numbers 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 the Stokes drag [38] F~d = −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 choose the 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 role in 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. Every harmonic spring has a characteristic vibrational frequency and a corresponding timescale. In order to correctly sample the whole movement of the tether, the time step that is used in the simulation should be well below this characteristic vibrational timescale. Using the potential of equation 3.2.6 and 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 r r Kb M ω = ⇒ τb = 2π (3.2.19) M Kb This is the relevant timescale for the bond vibrations and in order to not get into any problems with our simulations, our simulation time step should be well below this.

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

∂~v M = −MΓ~v ⇒ τ = 1/Γ. (3.2.20) ∂t mr

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

3πηR3 τcr = (3.2.21) kBT

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 smaller than the bead radius, so we expect the hitting mechanism to occur at timescales of order 0.1τcr. We neglect the effect of the surface on the drag on the tether.

16 CHAPTER 3. SIMULATION METHODS

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 from the previous and current time intervals [26], to enable longer time steps to be used. We checked for several 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/create and fix bond/break in LAMMPS [26]. The creation function is applied every predefined interval, which we choose to be 1 µs. If the distance between two particles is smaller than a certain cutoff radius Rc the bond 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 previously introduced.

17 CHAPTER 3. SIMULATION METHODS

3.3. Parameter values

In this section the actual values of several parameters are documented, along with the corresponding reasoning. When results are provided in this report, unless mentioned otherwise, the values reported in 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], with the values 500 nm, 50 nm and 50 nm respectively. The tether length in the experiments is actually 40 nm, but it is attached to the substrate by an antibody of approximately 15 nm [15]. We have chosen to take an effective tether length of 50 nm. The segment length ls is a parameter only used in MC simulations. We have not modified this parameter, but have kept it at 1 nm. 3 The bead consists of polystyrene, which is known to have a density of ρpol ≈ 1 g/cm . As we consider spherical particles, the mass is now given by 4 M = πR3ρ = 5.24 · 105 ag. (3.3.1) 3 pol Experiments are usually carried out at room temperature. Moreover, any point-of-care biosensing application should be able to function at room temperature. Therefore we take T = 293 K. Water is the 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 behavior should be considered carefully. A thorough discussion on this can be found in section 4.3 and this value is chosen to be 10.

Choosing Nbeads fixes the value of σ, m and ΓDNA. The polymerbead radius σ is simply given by L/Nbeads, 5 nm in this case. Nbeadsm should represent the mass of a dsDNA chain with length L. The mass 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. The DNA-chain itself is also continuously subject to hydrodynamic drag. Our DNA-chain is modeled as a bead-spring model. In this model the polymer beads are representing the inertia of the polymer as well 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 6πηR Γ = hy , (3.3.2) DNA m in which η is the dynamic viscosity of the solvent and Rhy is the so-called hydrodynamic radius of the beads. This hydrodynamic radius is thus an effective parameter which determines the amount of drag on the polymer. We should choose an appropriate value for this effective parameter. A way to determine 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]

2πηL ζ = mΓ = (3.3.3) k k ln(L/b) 4πηL ζ = mΓ = , (3.3.4) ⊥ ⊥ ln(L/b)

18 CHAPTER 3. SIMULATION METHODS where L is the length of the cylinder and b is the diameter of the cylinder. Since the tether can be viewed as being effectively trapped between two walls [8], the movement of the building blocks of the tether will be predominantly in the direction perpendicular to the tether itself. Therefore we use the perpendicular 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: 2 nm [43]. Equation 3.3.4 then gives the drag on the whole cylinder, which should be equal to the total drag on all Nbeads beads. This gives us 4πηL ΓDNA = , (3.3.5) mNbeads ln(L/b) 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 nm Tether length L 50 nm Tether persistence length lp 50 nm Tether segment length ls 1 nm Bead mass M 5.24 · 105 ag Bead friction per mass Γ 0.018 ns−1 Temperature T 293 K Solvent viscosity η 1.00 ag/(nm ns) Number of beads in polymer Nbeads 10 - Polymerbead radius σ 5 nm Polymerbead mass m 0.016 ag DNA width b 2 nm 3 −1 Polymerbead friction per mass ΓDNA 1.22 · 10 ns 2 Bond coefficient Kb 8.1 ag/ns 2 2 Angular coefficient Ka 20.23 ag nm /ns Time step dt 0.001 - 0.004 ns Characteristic bond time τb 0.016 ns Momentum relaxation time τmr 55.6 ns 8 Configurational relaxation time τcr 2.9 · 10 ns

19 CHAPTER 3. SIMULATION METHODS

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 the sphere parallel to the wall is indicated by vk and the perpendicular component is indicated by v⊥. The sphere will experience a larger drag than in free solution. Moreover the sphere will experience different drag 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 free solution, 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 results in increased drag on the sphere. Moreover, the drag does not increase isotropically, but displays a stronger 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, the bead has to ‘squeeze out’ the fluid between the sphere and the wall. Actually, the sphere experiences the same drag for the motion towards the wall as for the motion away from the wall. The drag on the sphere in the direction parallel to the wall is also increased, but not as much as the perpendicular drag.

3.4.2 Correction on parallel motion The drag on a sphere moving parallel to a single wall was worked out by Fax´enin his dissertation [45]. Fax´enapplied the method of reflections to a sphere near a wall, which is a method of iteratively

20 CHAPTER 3. SIMULATION METHODS applying boundary conditions at one spot and calculating the correction to the velocity field at the other spot. Furthermore, Fax´enexpressed the fundamental solution of Laplace’s equation in integral form [38].

This treatment yields a parallel drag per mass coefficient Γk of Γ Γ = 0 , (3.4.1) k 9 ∗ 1 ∗3 45 ∗4 1 ∗5 1 − 16 z + 8 z − 256 z − 16 z where Γ0 is the Stokes drag per mass on a sphere in free solution, given by Γ0 = 6πηR/M for a sphere with radius R in a liquid with dynamic viscosity η and z∗ is given by z∗ = R/z, with z the distance from 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]. Brenner applied bipolar coordinates to the Stokes equation to find an exact solution, given by the expression

∞ " ∗ ∗ # 4 ∗ X n(n + 1) 2 sinh(2n + 1)z + (2n + 1) sinh 2z Γ⊥ = sinh z − 1 Γ0, (3.4.2) 3 (2n − 1)(2n + 3) 4 sinh2(n + 1 )z∗ − (2n + 1)2 sinh2 z∗ n=1 2 where again z∗ = R/z, with z the distance from the center of the bead to the surface and R the radius of 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 Fax´en’slaw given by equation 3.4.1 is an approximation up to fifth order, we choose to use an interpolation for the perpendicular coefficient up to fifth order as well: Γ Γ = 0 . (3.4.3) ⊥ 9 ∗ 1 ∗3 57 ∗4 1 ∗5 1 − 8 z + 2 z − 100 z + 5 z

We can introduce the relative drag coefficients λk, λ⊥ to simplify the form of equation 3.4.1 and 3.4.3, by defining Γk = λkΓ0 and Γ⊥ = λ⊥Γ0. One may wonder if the effect of near-surface effects on the drag and diffusion of a bead is significant for our experiments. It should be pointed out that in a TPM system the width of the gap between the bead and the surface cannot exceed the tether length L and that L < R for all the systems that we consider. For gap widths smaller than the radius of the bead these effects are strongly pronounced and thus relevant. We will show in section 4.6 and 4.7 that the influence of this effect is indeed significant.

The coefficient λk 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 increases to a finite value.

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

21 CHAPTER 3. SIMULATION METHODS

Rel. drag coefficent 3.1

3.0

2.9

2.8

2.7

2.6

2.5

2.4 Distance nm 510 520 530 540 550

Figure 3.3: The parallel component λk of the drag as function of the distance of the bead center to the surface at z = 0 for R = 500. The relation displays a maximum of 3.08 at z = 500 nm (touchingH theL surface) and reduces to 1 for z → ∞. representation of the fluid by methods such as stochastic rotation dynamics (SRD) [47] or Lattice Boltzmann (LB) [48]. However, since the bead is much larger than the typical gap width, this would drastically increase computation times. Another way to implement this is to start from the implementation of an implicit solvent, as described in section 3.2.5 and add the prefactors that are described in this section. A way to include this was not readily available in LAMMPS, but we have adjusted the code to add this feature. A detailed description 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 experimental situations we consider. Let us discuss three potential objections. First of all, Fax´en’slaw is only a valid approximation when the bead is not too close to the surface [49] and λk ≤ 1.7. For small gap widths lubrication theory becomes the leading theory [47]. Clearly, this is not exactly the regime we are considering. The bead in our simulations is allowed to approach the surface and reduce the gap width to arbitrarily small distances. However, for gap widths larger than 0.013R (6.53 nm) the error does not exceed 10%. Moreover, in experiments Fax´en’slaw has shown correspondence up to λk ≤ 2.4 [44]. Another argument that one could make is that the rotational drag should be adapted as well. This is indeed the case, rotational dag anisotropy is also a physical effect. However, these effects are typically an order of magnitude lower than the translational drag effects [50].

22 CHAPTER 3. SIMULATION METHODS

Rel. drag coefficent

60

50

40

30

20

10

Distance nm 510 520 530 540 550

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

A third objection arises when the actual experiment is considered. The beads are not perfectly smooth spheres, but display a ‘roughness’. Although this effects the hydrodynamic forces and should be considered carefully, its integral force contribution is typically negligible compared to the total resistance [51]. The three aforementioned side-notes, - (1) Fax´en’slaw breaks down for too small gap widths, (2) rotational drag is also influenced and (3) bead roughness has an additional effect on bead drag - are all valid points. However, it is our vision that the implementation that we use captures the essence of this effect, while the computational complexity still remains tractable.

23 Chapter 4 Bead movement

In this chapter we will review the movement of the bead. We will first discuss the equilibrium distribution of positions that the bead attends and then turn to dynamic properties. After reviewing the typically observed motion patterns, we continue with an analytic approach. We will show that this analytic approach only provides valid results in some limiting cases. Therefore, we introduce simulation methods to fully describe our system. When the equilibrium distributions are known and understood, we continue by describing several dynamic properties, i.e. the correlation functions of bead 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 that our simulation produces similar output. As mentioned earlier, the only directly measurable quantity of interest is the position of the bead projected on the XY -plane. Our simulation results provide an extensive set of variables, amongst which the position of the bead. This enables us to create motion patterns 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 picture on the left a typical motion pattern is displayed without any traces of secondary binding events. On the other hand, in the picture on the right the bead is observed significantly more in a specific part of the plane. Our current understanding is that such effects are the result of the occurrence of one or more 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 determining what fraction of the time the bead is bound to the surface, another representation of the data is required. 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 step size - is plot against time. By averaging the step size over multiple frames one obtains an indicator to

24 CHAPTER 4. BEAD MOVEMENT

2D trajectory 2D trajectory 300 300

200 200

100 100

0 0 y position (nm) y position (nm)

−100 −100

−200 −200

−200 −100 0 100 200 300 −100 0 100 200 300 x position (nm) x position (nm)

Figure 4.1: Motion patterns of TPM experiments obtained by Scheepers [15]. These motion patterns are obtained for tether length L = 40 nm by measuring one minute with a frame rate of 30 Hz. In the left 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 to one 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] use this detection method along with another detection method based on the outer hull that the motion pattern spans within a time interval. We will be focusing on the step size method.

25 CHAPTER 4. BEAD MOVEMENT

Step size Step size 400 300 Step size Step size 350 Step size per window (window = 60 frames) Step size per window (window = 60 frames) 250

300

200 250

200 150

Step size (nm) 150 Step size (nm) 100

100

50 50

0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time (s) Time (s)

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

26 CHAPTER 4. BEAD MOVEMENT

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

250 250

150 150

50 50

Y (nm) Y (nm) Y -50 -50

-150 -150

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

Figure 4.3: Motion patterns obtained by simulations. In the left picture: a motion pattern obtained by combining 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 on the substrate. The results were obtained by combining the output of 7 simulations of 0.5 s with a frame rate of 300 Hz. Note: this figure was produced using isotropic drag.

27 CHAPTER 4. BEAD MOVEMENT

4.2. Analytic approach

4.2.1 Geometrical framework The initial experiments that triggered our interest in tethered particle motion were performed with a 40 nm long dsDNA tether, having a persistence length lp of roughly 50 nm. A first approximation to a semiflexible polymer with L < lp, like this dsDNA tether, could be to consider the polymer as a stiff rod, or mathematically: lp → ∞. This basic assumption allows us to obtain an analytical integral expression that we can solve numerically for the hitting probability including the effects of bead-tether exclusion and tether-surface exclusion. The results of this approximation are provided in this chapter. At the end we will compare these 6 results to simulation results for very high persistence length lp (1 · 10 nm) and we will review the effect 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 our approximation fails to yield valid results. We describe the tether as a stiff rod attached at one point at the surface so that the configuration of the 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 of bead and tether we need another polar and azimuthal angle. We define ψ as the polar angle and χ as the azimuthal angle both with respect to the axis in the direction of the tether. These angles are indicated in figure 4.4. This choice of angles turns out to be convenient when applying bead-tether exclusion.

R

φ ψ χ

θ 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 given by L and the radius of the bead by R. Note that L and R are not drawn to the scale that we typically consider.

28 CHAPTER 4. BEAD MOVEMENT

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 our system in one configuration. The vector that points along the tether is given by

~r1 = L (cos φ sin θ xˆ sin φ sin θ yˆ cos θ zˆ) , (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 the usual convention with spherical coordinates. Now for the vector that describes the bead, the derivation is slightly more sophisticated. We define a system {n,ˆ ˆb, rˆ1}, in whichr ˆ1 is the direction of the tether i.e. the unit vector corresponding to equation 4.2.1. The vector pointing from the top of the tether to the center of the bead is then given by   ~r2 = R cos χ sin ψ nˆ sin χ sin ψ ˆb cos ψ rˆ1 . (4.2.2)

In this equation R is the radius of the bead,n ˆ and ˆb are the so-called normal and binormal to obtain a new othogonal coordinate system. These unit vectors are defined by ∂rˆ nˆ = 1 (4.2.3) ∂θ ˆb =r ˆ1 × n.ˆ (4.2.4)

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

~r2 = R{( cos χ sin ψ cos φ cos θ − sin χ sin ψ sin φ + cos ψ cos φ sin θ)ˆx+ ( cos χ sin ψ sin φ cos θ + sin χ sin ψ cos φ + cos ψ sin φ sin θ)ˆy+ ( − cos χ sin ψ sin θ + cos θ cos ψ)ˆz}. (4.2.5)

Assuming every configuration is equally probable, the probability distribution of this system P (φ, θ, ψ, χ) = P obeys the relation ZZ ZZ 2 2 2 2 L R P (φ, θ, ψ, χ)dΩ1dΩ2 = L R P dΩ1dΩ2 = 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 = sin ψdψdχ (4.2.8) Z Z Z Z ⇒ L2R2 dφ dθ dχ dψP sin θ sin ψ = 1. (4.2.9)

Up to this point, the calculation has been rather straightforward. However, we have not yet discussed our integration boundaries, it turns out that these are not all trivial. In chapter 2 we discussed that 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 the determination of integration boundaries.

29 CHAPTER 4. BEAD MOVEMENT

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 be taken into account. Secondly, since θ is the angle between the surface and the tether, tether-surface exclusions leads to the boundaries 0 < θ < π/2. Similarly, ψ is the angle between the bead and the tether 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 χ and ψ in this geometry. Mathematically this exclusion corresponds to the fact that the z-coordinate of the 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 combinations of θ, χ and ψ than meet the condition given by equation 4.2.10.

Z = L cos θ + R(cos θ cos ψ − sin θ sin ψ cos χ) > R. (4.2.10)

Z-R nm 50 Χ H L H L 1 2 3 4 5 6 Ψ=0 -50 Ψ= -100 0.33 -150 Ψ=0.67

-200 Ψ=1.0 -250 -300

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

Z-R nm

Χ 1 2 3 4 5 6 H L H L Ψ=0 -100 Ψ=0.33

-200 Ψ=0.67

-300 Ψ=1.0

-400

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

30 CHAPTER 4. BEAD MOVEMENT

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 χ for several values of ψ and fixed value of θ = 0.3. It can be seen that for low ψ all values of χ are allowed, while for increasing values intersects with the horizontal axis appear. In figure 4.6 the same plot is shown for θ = 0.5. Here for low ψ all values of χ are forbidden, while again for higher values of ψ intersects with the horizontal axis appear. In both plots it can be seen that if ψ takes on even larger 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 specific angle θ∗, so that for θ < θ∗ all values of χ are allowed at ψ = 0, while for θ > θ∗ no values are allowed at ψ = 0. This value is given by  R  θ∗ = arccos . (4.2.11) R + L

Furthermore the value of ψ at which the intersects occur ψ− and the value at which those disappear ψ+ are given by   R − L cos θ ψ− = arccos − θ (4.2.12) R R − L cos θ  ψ = arccos + θ. (4.2.13) + R the intersects of the curves with the χ-axis occur at

∗ χ± = π ± χ (4.2.14) L cos θ + R(cos θ cos ψ − 1) χ∗ = π − arccos . (4.2.15) R sin θ sin ψ

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 nonzero # L a value for θ < π/2 exists at which ψ+ is equal to π/2. We will refer to this angle as θ , and its value is given by R − L θ# = 2 arctan . (4.2.16) R + L A graphical representation of θ# can be found in figure 4.7. Equation 4.2.11 and 4.2.16 inform us that θ∗ < θ# for small L/R, while for big L/R: θ∗ > θ#. The crossover θ∗ = θ# occurs at L/R ≈ 0.4. In this 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 CHAPTER 4. BEAD MOVEMENT

Ψmax 1.6

1.4

1.2

1.0

0.8

0.6 Θð Θ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 

Figure 4.7: The angle ψ+ that gives the maximum angle at which a solution for Z = R exists. If ψ+ < π/2, then ψ+ should be the integration boundary for ψ, while for ψ+ > π/2, the integration boundary 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

∗ " + # Z 2π Z θ Z ψ− Z 2π Z ψ Z π+χ∗ PL2R2 dφ dθ dψ dχ sin θ sin ψ + dψ dχ sin θ sin ψ ∗ 0 0 0 0 ψ− π−χ # + ! Z θ Z ψ Z π+χ∗ Z 2π Z π/2 Z π+χ∗ + dθ dψ dχ sin θ sin ψ + dθ dψ dχ sin θ sin ψ = 1 (4.2.17) ∗ ∗ # ∗ θ ψ− π−χ θ ψ− π−χ

4.2.3 Distribution results Now that we have set up a geometric framework and analyzed what the relevant integration boundaries are, we can look at the spatial distribution of states and, even more relevant, the hitting probability and 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-plane radius ~r = (x, y). This is a relevant distribution when comparing with experiments, since usually only the movement in the (x, y)-plane can be observed. We define hitting as Z < (R + ZHIT ) and the hitting probability is the number of configurations that are a hit divided by the total number of configurations. The spatial hitting distribution is again the distribution of states for varying in-plane vector ~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 obtained by numerically solving this integral. First of all, the constant P should be determined. We numerically obtain P = 8.11 · 10−10 nm−4. We represent these spatial distributions as P (r)/r, to account for the fact that the surface of a ring with radius r and width dr, Aring = 2πrdr, varies. By dividing by r we find the probability per surface area. The spatial distribution P (r) is numerically obtained by integrating over the integral given in equation 4.2.17 with a resolution of dr = 1 nm, by inserting the

32 CHAPTER 4. BEAD MOVEMENT

#10-3 8 l = 106 nm p 7 l = 200 nm p l = 50 nm p 6 analytic at l p ! 1

) 5 -1

4

P(r)/r (nm P(r)/r 3

2

1

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

Figure 4.8: The spatial distribution of the bead for the analytic expression and for several values of 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 other lines result from MC simulations. It can be observed that there are only minor deviations between the different curves. block function into the integral u(r) − u(r − dr), where u(x) is the Heaviside step function. The result is 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 normalized distributions, the area under both lines is a measure for how well the distributions match. For perfect matching this value will be equal to 1. We will call this the ‘overlap’ between two curves. 6 It can be observed that for high persistence length lp = 1 · 10 nm the spatial distributions are in rather good agreement: there is 0.983 overlap between the curves. Moreover, at the real persistence length the lp = 50 nm there is still an overlap of 0.945, indicating that spatial distribution obtained by 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, we calculate the overall hitting probability by inserting the Heaviside step function u(ZHIT − Z) in the integral. The overall hitting probability fhit is 0.36 for both the numerically integrated result as well 6 as the simulation with lp = 1 · 10 nm. For lp = 200 nm we find fhit = 0.33 and for lp = 50 nm we also find fhit = 0.33.

33 CHAPTER 4. BEAD MOVEMENT

Finally, we consider the spatial hitting distribution. We obtain Phit(r) numerically with a resolution of dr = 1 nm, by inserting the two previous ingredients: (u(r) − u(r − dr))u(ZHIT − Z) into the integral of equation 4.2.17. The result is represented in figure 4.9 along with simulation results. Again, there 6 is a rather good agreement for lp = 1 · 10 nm: the curves overlap 0.983, but for lp = 50 nm this is reduced to only 0.714.

0.012 l = 106 nm p l = 200 nm p 0.01 l = 50 nm p analytic at l p ! 1

0.008

) -1

0.006

(r)/r (nm (r)/r

hit P 0.004

0.002

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

Figure 4.9: The spatial hitting distribution of the bead for the analytic expression and for several values of 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 other lines result from MC simulations. It can be observed that the numerical integration corresponds well for 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 physical interpretation 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 gets spatially extended in the direction perpendicular to the tether. We will come back to the importance of these exclusion effects in section 4.4. At this point we conclude that the stiff-rod approximation is not sufficient to describe the configurations of the bead near the surface exhaustively.

34 CHAPTER 4. BEAD MOVEMENT

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 be determined. If the tether is represented by a small number of segments, the simulation might not take the effect of bead-tether and surface-tether interactions correctly into account. Moreover, the bead-spring approximation of the configurations of the worm-like chain will get less accurate for fewer segments. On the other hand, if more segments are taken into account simulation times will increase due to two effects: (1) there are more segments for which the equations have to be solved every time step of the simulation and (2) a smaller time step is required to correctly sample the bonds between the segments of 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 main contributions 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 to choose the minimal number of segments so that the correct radial hitting probability distribution is still obtained. The radial hitting probability distribution is obtained in the Monte Carlo simulations with a 50 nm tether consisting of 50 segments. The results in figure 4.10 that the Molecular Dynamics results converge towards the Monte Carlo results for an increasing number of tether segments. The show that for a tether consisting of 2 polymer beads, exclusion effects are not significantly taken into account. However, for 5 polymer beads the result 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 shorter tethers we have scaled this number down so that the distance per polymer bead remained 5 nm bead and for longer tethers we have kept it at 10 polymer beads to preserve computational achievability.

35 CHAPTER 4. BEAD MOVEMENT

0.012 2 atoms 5 atoms 10 atoms 0.01 30 atoms 50 atoms MC

0.008

0.006

/r (1/nm) /r

hit P

0.004

0.002

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

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

36 CHAPTER 4. BEAD MOVEMENT

4.4. Exclusion effects

We already mentioned in section 2.1 that there are three exclusion principles that govern the allowed configuration 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 in describing 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 results obtained by MD simulations can be found in figure 4.11, both MD and MC simulations yielded similar results. In figure 4.11 it can be observed that both tether-bead and tether-surface exclusion alter the system in 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 of tether-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 line in figure 4.11) shows that both effects change the hitting distribution in a similar way and that both effects are of the same order. The fact that both effects are approximately equally relevant is something that 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 and 0.33 for both exclusion mechanisms. One could state that these exclusion mechanisms not only drive the bead radially outward, but also away from the surface. Although the influence on the overall distribution of the bead is minor, the effects become significant when reviewing the hitting distribution. Physically this result implies that when the bead is near the surface (hitting), there is a higher probability of the tether intersecting with the bead or with the surface for small in-plane radius r than for lange r.

37 CHAPTER 4. BEAD MOVEMENT

0.01 No tether exclusions

0.009 Tether-bead exclusions Tether-surface exclusions Both exclusions 0.008

0.007

0.006 /r

0.005

hit P

0.004

0.003

0.002

0.001

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

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 and wall-tether exclusion have a significant effect on the hitting distribution and both effects decrease the likelihood of a hitting event at small r.

38 CHAPTER 4. BEAD MOVEMENT

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

When constructing a tethered particle system, the three main parameters that the designer is more or 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. The distributions obtained by Molecular Dynamics simulation agree with the Monte Carlo simulations with 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 700 nm in addition to the usual 500 nm. We evaluate the fraction of distribution of which the bead hits the 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 nm to 1.7 · 102 nm and 2.6 · 102 nm for a bead radius of 300 nm and 700 nm respectively. This is of course a direct result of the geometrical constraint√ that the tether imposes on the bead, which can be approximated for small tether lengths as Rmax ≈ 2RL. Aside from this effect, we observe little change in equilibrium distribution when changing the bead radius. The overall hitting fraction remained at 0.33 for all three cases. The shape of the hitting distribution 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 TPM system.

0.012 R=300 nm R=500 nm R=700 nm 0.01

0.008 /r

0.006

hit P

0.004

0.002

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

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

39 CHAPTER 4. BEAD MOVEMENT

4.5.2 Varying the tether length To access the influence of the tether length, simulations were performed for tether lengths 25 nm and 100 nm. The results can be found in figure 4.13. Similar to the effect of a varying√ bead radius, varying the 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 the fraction of frames in which the bead hits the surface. For L = 25 nm the hitting fraction fhit equals 0.62, for the usual L = 50 nm, fhit = 0.33 and for L = 100 nm, fhit = 0.17. It is expected that a bead attached to a shorter tether will hit the surface more often and vice versa. The results indeed confirm this 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 this comes at the cost of decreasing the overall hitting fraction.

0.012 L=25 nm L=50 nm L=100 nm 0.01

0.008 /r

0.006

hit P

0.004

0.002

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

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

40 CHAPTER 4. BEAD MOVEMENT

4.5.3 Varying the persistence length Although it is experimentally somewhat less straightforward to vary the persistence length of the tether, it is a parameter for which different values can be chosen. Identifying the way this alters the behavior 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 were fhit = 0.34, 0.33, 0.32. We observe that for higher persistence length, ‘stiffer polymers’, the hitting distribution changes to higher in-plane radius. Given this data, we conclude that the persistence length lp, compared to bead raduis R and bead length L, has a relatively small influence on the hitting process and that adjusting it will not yield big advantages.

0.012 l =25 nm p l =50 nm p l =100 nm 0.01 p

0.008 /r

0.006

hit P

0.004

0.002

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

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

41 CHAPTER 4. BEAD MOVEMENT

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 tether length leads to a higher part of the surface covered, but a lower overall hitting fraction. This is a tradeoff that we further discuss in section 6. Changing the radius of the bead is another possibility. Beads are more challenging to fabricate on demand, but several sizes are commercially available. One aspect that should be kept in mind is visibility: too small beads may be harder to visualize. Based on the results in this section, the bead size should be maximized. However, we will come back to this point when considering dynamics as well.

The most challenging would be the alteration of the persistence length lp of the tether. Although several 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 parameters at 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 influenced by R. We have determined the effect of each parameter separately when it is varied in the regime of the actual experimental values and therefore it provides us with information about how changing a single parameter will change the experimental system.

42 CHAPTER 4. BEAD MOVEMENT

4.6. Dynamic properties

Now that we have constructed our simulation model for the tethered particle motion and we understand the equilibrium distributions that we obtain, we can turn to the dynamic properties of the system. In describing the dynamic properties of the system it is also relevant to reveal the 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 D E 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, hZ(t)Z(t + τ)i, the average hitting and rebinding time 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 the in-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. D E 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 experimentally obtained values, that range from 0.1 - 0.3 s [15]. If we do not apply anisotropic drag we find a value of (0.039 ± 0.001) s, which does not agree with experimentally obtained values.

#104 From simulations exponential fit

2

)

2

m

n

(

E

) =

+ 1

t

(

~

R

"

)

t

(

~

R D

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 = (s) D E Figure 4.15: R~(t) · R~(t + τ) in the case of anisotropic drag. The result was fit by a function of 4 2 the form y(t) = C exp(−τ/τ1), with values C = (2.0±0.1)·10 nm and τ1 = (0.09±0.01) s. The deviations at the right side of the graph are the result of poor averaging: a larger τ corresponds to less data points available.

43 CHAPTER 4. BEAD MOVEMENT

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 time at which the bead moves from and to the substrate. In figure 4.16 hZ(t)Z(t + τ)i obtained by simulation is shown together with an exponential fit. This graph starts at Z(t)2 and relaxes towards hZ(t)i2.

We obtain a value of τ2 = (21 ± 1) ms. If we do not apply the anisotropic drag we find a value of τ2 = (2.0 ± 0.5) ms. Since the experiments do not have a Z-resolution, we are not able to compare these values to experimental values. However, this value gives us an indication of how fast 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 the bead and the substrate. The provided uncertainties are the 95% confidence intervals provided by the exponential fit function of MATLAB [25].

From simulations 700 exponential fit

600

) 2

500

m

n

(

i )

= 400

+

t (

Z 300

)

t

(

Z h 200

100

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

Figure 4.16: hZ(t)Z(t + τ)i in the case of anisotropic drag averaged over 10 simulations. The result was fit by a function of the form y(t) = C0 + C1 exp(−τ/τ2), with fit parameters C0 = 2 2 2 (3.2 ± 0.2) · 10 nm , C1 = (150 ± 5) nm and τ2 = (21 ± 1) ms.

4.6.3 Conclusion The conclusion of this section is twofold. Firstly, we show that we indeed need anisotropic drag to 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. In the direction parallel to the substrate this turns out to be in the order of 10−1 s and in the direction perpendicular to the substrate this is roughly one order of magnitude smaller.

44 CHAPTER 4. BEAD MOVEMENT

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 the gap between the bead and the substrate becomes smaller than ZHIT . Define t = τ when Z = ZHIT upon approaching the substrate and then Z = ZHIT again at t = τ + ∆t while moving away from the substrate, ∆t is then the hitting time. The hitting time is also a parameter in which the influence of the anisotropic drag is reflected. A first interesting quantity would be the average hitting time. For the isotropic drag system this quantity is equal 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 is well-captured by a power-law fit. An example of such a fit is given in figure 4.17. Averaging over all our fits we obtain a power of (1.51 ± 0.06) in the case of isotropic drag and (1.55 ± 0.09) in the case of 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 is smaller than ZHIT . Define t = τ when Z = ZHIT upon moving away from the substrate and then Z = 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 anisotropic drag leads to longer hitting times. Indeed the average hitting time increases when the anisotropic drag 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 power law fits yield a power that corresponds to −3/2 within uncertainty. The anisotropic drag does not influence the power corresponding to this distribution.

45 CHAPTER 4. BEAD MOVEMENT

105 Fit with power law: p=-1.52

104

103

102 Probability (a.u.) Probability

101

100 102 103 104 105 Hitting time in ns

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

46 CHAPTER 4. BEAD MOVEMENT

4.8. Step size

One of the most critical values that enables us to determine whether the bead is bound or not is 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 influence of the tether length and the bead radius. The results may help in choosing the frame rate for experiments. Moreover, the results provide us with information about the detectability of binding 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 bead is 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 this is 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 obtained average step size equals 132 ± 8 nm, which agrees with experimental data as well.

150 L=20 nm 140 L=50 nm 130 L=100 nm L=200 nm 120 L=330 nm

110

100

) 90

m n

( 80

i p

e 70

t s

R 60 h

50

40

30

20

10

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

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

47 CHAPTER 4. BEAD MOVEMENT

4.8.2 Bead radius The theory of Brownian motion tells us that big beads move slower, as the diffusion coefficient D ∼ 1/R. On the other hand, as we have observed in section 4.5.1, a bigger bead is allowed to move 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, a bigger beads corresponds to a smaller step size. This implies that binding events will be more challenging to detect when working with larger beads.

140 R=300 nm 130 R=500 nm

120 R=700 nm

110

100

90

m 80

n i

p 70

e

t s

60

R h

50

40

30

20

10

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

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

48 Chapter 5 Specific 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 rather a very specific spot on the bead. Similarly, the actual bond is not formed with the whole substrate, but a specific spot on the substrate. In this section we zoom in one level deeper to understand 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. A schematic representation of this is given in figure 5.1, where the dots are colored red. We initialize the system in an upright configuration with Xdot = R sin θ and Ydot = 0. Due to axisymmetry, the bead is free to rotate with respect to the tether, the equilibrium properties of a dot will be completely fixed by the angle θ.

R

θ

Figure 5.1: Examples of specific dots on the surface. The location of a dot is determined by angle θ. Since the system is axisymmetric we conveniently take Ydot = 0 for every dot, so that Xdot = 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 geometrical arguments, we can predict that the maximum angle at which a dot is able to hit the substrate is     R R−ZHIT given by θmax = arccos R+L + arccos 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 the maximum angle θ that hits the substrate is slightly lower. In this case, no hits are observed beyond θ = 0.60.

0.04

0.03

0.02

hit f

0.01

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 3

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

50 CHAPTER 5. SPECIFIC BINDING SPOTS

5.2. Spots on the substrate

The other way around, we may also look for a spot at a specific position ~rspot = Xsubstrxˆ + 0ˆy + 0ˆz on the substrate as in figure 5.3. Again, due to the axisymmetry of our system, we can conveniently choose 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, with an x-coordinate of Xsubstr. Since the system is axisymmetric, we conveniently take Ysubstr = 0 for every 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 of Xsubstr. This approach corresponds to defining an ‘imprint’ when the bead hits the substrate. Similar to 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 the bead given in figure 5.2. This can be explained by the fact that the motion of the bead is largely limited by a tether of length L = 50 nm.

51 CHAPTER 5. SPECIFIC BINDING SPOTS

0.04

0.03

hit 0.02 f

0.01

0 0 50 100 150 200 250 300 350 X (nm) substr

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

52 CHAPTER 5. SPECIFIC BINDING SPOTS

5.3. Single Xdot and Xsubstr

Now that we have reviewed the variation of hitting probabilities at different spots on the surface of the substrate as well as different spots on the bead, we should combine these results. The eventual binding pattern will indeed be the result of the specific location on the bead and the surface. In figure 5.5 the hitting probability of a dot on the bead at every positions on the substrate are shown for several values of θ. The figure shows that for θ = 0 there is a narrow range of r where the dot hits the surface, which we attribute to the steric effect of the tether. For larger θ, the range of r that correspond to hitting the surface remains fairly constant at approximately 100 nm. Furthermore higher values of θ lead to higher values of r where the dots hit the substrate. The trend of figure 5.2 corresponds to the relative heights of these peaks.

#10 -3 8 3 = 0 3 = 0.15 7 3 = 0.30 3 = 0.45 3 = 0.6

6

5 (r)

4

hit P

3

2

1

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

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

53 CHAPTER 5. SPECIFIC BINDING SPOTS

5.4. Kinetic regime

The actual bond formation between an oligonucleotide on the bead and an oligonucleotide on the substrate can also be incorporated in the simulations. Although our simulation times remained several orders of magnitudes below the typical experimental times, these simulations provide us with a tool to estimate the kinetic regime of our bonds. In this section we use the implementation of single-bonds to 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 dissociation rate koff . While this framework is well-suited for the description of binding between molecules in free solution, the application of the concept of kon to a secondary binding event in a TPM system is not straightforward. 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

khit kstick FB ←−−−−−−−−→ HB ←−−−−−−−−→ SB, (5.4.1) ksep koff 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 an antibody-oligo-streptavidin complex. The k-values describe rates that govern the dynamics of these processes. These rates are all per unit of time. However, it should be noted that we have also defined a 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, the overlap of two spheres with radius Rc, and subsequently an effective concentration. However, since both oligonucleotides are attached to a surface with one end, it is not straightforward to relate this effective 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, the distribution of rebinding times enables us to determine the value of kstick. Indeed, in the simulations only one binding spot on the bead and one binding spot on the substrate are implemented.

5.4.2 Quick rebinding events

koff In our model system it is entirely possible that a bond releases (HB ←−−−− SB) and reattaches kstick (HB −−−−→ 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 for kstick, koff and Rc in our simulation, we can evaluate for which values the observed koff is significantly altered by quick rebinding events. The equation ∂[SB] = k [HB] − k [SB] (5.4.2) ∂t stick off reduces to ∂[SB] = −k [SB] (5.4.3) ∂t off

54 CHAPTER 5. SPECIFIC BINDING SPOTS

for kstick → 0. From this we deduce that if we take kstick small enough, the number of bound beads will decrease exponentially with a characteristic time 1/koff . 2 −1 2 −1 5 −1 We have chosen our koff to be equal to 1.0 · 10 s and varied kstick from 10 s to 3 · 10 s . Once the binding distance is larger than the bond length +5 nm for 0.01 seconds, we count the bond as released. The characteristic binding times were extracted by making exponential fits. The results can be found in figure 5.6. 5 It may be observed that for kstick ≥ 1·10 significant deviations in the effective bond time are present. The oligonucleotide sequences used in the experiments have a known value of koff [22]. The extracted values of koff , obtained by exponentially fitting the observed binding times in the experiments agree with previously documented values. From this we draw the conclusion that the actual value of kstick 5 −1 in the experiments is upper bounded by kstick < 10 s .

0.025

0.02

0.015

(s)

off =

0.01

0.005

0

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

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

55 CHAPTER 5. SPECIFIC BINDING SPOTS

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 this data to the properties of the system. In this section we demonstrate how the position of the binding molecules on the bead and the substrate can be extracted from a bound motion pattern. In the next section we combine all results of this chapter and we outline how this may be applied to experimental data. 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 given time window [15]. Consider. Several values can be extracted from this motion pattern during this time window. In particular, the he length L, width W and mean distance to the origin M, as can be seen in figure 5.7. The expected values of the length L, width W and mean distance to the origin M can be obtained for 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 a bond is detected in an experiment, a value L0, W 0 and M 0 may be extracted from the data. We may then define a matching function Φ

0 0 0 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 the experimental 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.

250

200 M: Distance 150 L: Length 100 W: Width 50

0 Y (nm) -50

-100

-150

-200

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

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 M can be extracted.

56 CHAPTER 5. SPECIFIC BINDING SPOTS

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 coating molecules are abundant on the substrate, or to sparsely coat the substrate, so that molecules on the substrate are rarely found. Likewise, one could either fully or sparsely coat the bead used in an experiment. 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 reasonable assumption that all observed binding events involve the same binding molecule as sparsely coated. We assume that if a surface is not sparsely coated, it is fully coated. We can then distinguish four types of 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 our purposes. Situation IV would truly allow us to monitor single bonds, but the downside of this setup is that experimental observation times drastically rise. The performed experiments [15] used situation III, 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 is not. Therefore, in situation III all bound motion patterns are found in a specific region of the motion pattern, 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 conclude that the time between two binding events is only dependent on the a priori probability of the bead to be near that specific binding spot on the substrate. Figure 5.8 schematically represents the processing of experimental data. First a bound motion pattern is analyzed as described in the previous section, and then the information about the specific binding spots is extracted. Depending on whether situation II, III or IV is used, either Xdot, Xsubstr or both Xdot and Xsubstr should be extracted.

5.6.2 Sources of uncertainties Several sources of uncertainties exist in experiments. When extracting information from experimental data it is necessary to have an idea about the relative magnitude of uncertainties. Four sources of uncertainties 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 bead during the image capturing time. This is a relatively small effect, that leads to uncertainties in the order of nanometers.

57 CHAPTER 5. SPECIFIC BINDING SPOTS

Length

Xdot Isolate Compare Look up TPM binding Width tabulated kstick data events values fhit X Distance substr to center

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

A much larger effect is caused by drift: the movement of the sample as a whole. The drift velocity may vary during a measurement, which makes correcting for it non-trivial. Drift may lead to several tens of nanometers uncertainties in determining the mean distance to the origin M, width W and length L of a bound motion pattern. An accurate correction could be applied if beads were permanently fixed on 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 perfectly smooth, but may show a finite surface roughness. The beads that Scheepers used in combination with a short tether have a roughness of 150 nm [15]. It is not straightforward to relate the roughness to the actual experimental error, but the shape of a bound motion pattern is likely affected by protrusions on the bead. A final source of uncertainty is the fact that some binding events are too fast to detect. This means that if we extract the rebinding time, the time it takes for a free bead to bind again, that possibly the bead has bound and unbound undetectably within the rebinding time. This leads to the erroneous determination 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 the bonds 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 data sequence 75 hitting events were observed within 60 minutes. The performed experiments used the experimental 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 the substrate. Part of the data that we used for this is graphically represented in figure 5.9. As we have outlined in section 5.5, the next step is to determine the mean distance to the origin M, the width W and the length L of the bound motion pattern.

58 CHAPTER 5. SPECIFIC BINDING SPOTS

We isolate 6 bound motion patterns that exhibit small drift. For every bound pattern we extract the 2 values of Xdot and Xsubstr and then we average over 6 results to find X¯substr = (2.8 ± 0.3) · 10 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 the substrate for 2.2 ms per second. The full data set showed 75 binding events and thus 74 rebinding times. Since the typical rebinding times are significantly larger than the correlation time of the system and binding events are thus uncorrelated, this situation corresponds to starting out with 74 free beads and observe the number of free beads in time. The result can be found in figure 5.10. The result was fit by an exponential formula of the form N(t) = N0 exp(−κt). As mentioned in subsection 5.6.2, the rebinding times are increasingly erroneous for 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, this 1 −1 5 −1 leads to a kstick of 1.7 · 10 s . This is well below the upper bound of 10 s that we provided in section 5.4 and thus a consistent result.

300 300

200 200

100 100

0 0

Y (nm) Y (nm) Y

-100 -100

-200 -200

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

Figure 5.9: In the left figure: the motion pattern obtained by measuring for 6 minutes. This is only a part of the actual experimental data set, that contained 60 minutes of data. To limit the visible influence of drift, only a part of the data set is shown. By analyzing the time trace, 75 binding events were 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 motion pattern. This motion pattern corresponds to M = 151 nm, W = 64 nm and L = 108 nm.

59 CHAPTER 5. SPECIFIC BINDING SPOTS

80 Measured rebinding times Exponential fit 70

60

50

40

30 Number of free beads free of Number 20

10

0 0 50 100 150 200 250 Time (s)

Figure 5.10: The number of free beads as a function of time for the experimental data set. The rebinding time between every binding event is used to determine how the number of free beads decreases with 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 6 Optimization

One of the motivations to investigate this system using simulations is that it enables us to find the optimal set of experimental parameters for a system with a given purpose. The purpose being either the investigation of single bond kinetics or a biosensing application. In both cases the contact area plays a central role in optimizing the system parameters. We will first introduce the contact area in section 6.1 and then discuss the role of detectability in section 6.2. Finally, we will discuss the optimization 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 the substrate, the contact area between the bead and the substrate determines how many bonds will be formed. We define the contact area ζ as the fraction of time for which a specific dot on the bead is within interaction range of a specific spot on the substrate summed over all dots on the bead and all spots on the substrate.

In figure 6.1 a schematic representation of a dot on the bead and a spot on the substrate can be found. 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 a fraction 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 same fhit(Xdot,Xsubstr). The area of both rings is given by

2 dAring,bead = 2πR 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 the substrate. In equation 6.1.1 and 6.1.2 both rings are of infinitesimal width. Note that these equations are related to Xdot and Xsubstr by Xdot = R sin θ and Xsubstr = Rs. The contact area ζ is now given by Z Z ζ = fhit(Xdot,Xsubstr)dAring,beaddAring,substr. (6.1.3)

Asubstr Abead

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 and Xsubstr 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 amount of surface area that is able to enter the interaction range on the bead as well as the substrate. On the other hand, on average it decreases the hitting probability fhit(Xdot,Xsubstr). In figure 6.2 we observe that ζ decreases for higher tether lengths L, which leads us to understand that the latter of both effects is more pronounced. ‘Optimizing’ the contact area then leads to the trivial solution of L = 0. Clearly, this is not a valid solution for a device based on tethered particle motion (TPM): for L = 0 the bead would not be moving and would in fact not be tethered, but rather simply bound. In the next sections we will extend our optimization procedure to come up with non-trivial solutions.

62 CHAPTER 6. OPTIMIZATION

#105 18 1

16

14

12

) 10 4

(nm 8 1

6

4

2

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

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

63 CHAPTER 6. OPTIMIZATION

6.2. Detectability

When investigating molecular interactions it is not so much the number of bonds that one wants to maximize, but rather the number of bonds that one can observe. This is where detectability enters the 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 whether the bead is bound or not. In section 4.8 we have shown how the step size varies with tether length and bead radius. The influence of a specific bond on the step size determines whether this bond can be 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 nm at 30 Hz can be found. We have to define a threshold step size difference for which bonds are visible. There are two principles that give rise to uncertainties when measuring hRstepi averaged over a specific time window. (1) Rstep is distributed with a finite standard deviation, so a measurement in a predefined time interval will have 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 threshold step size should not be defined based on simulation results solely. After discussing this with our experimental 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 determination of the minimum Xsubstr that leads to a visible bond. Using linear fits for L = 20 nm and L = 50 nm and using interpolation for L = 70 nm we estimate the value of Xsubstr,min. Moreover, from section 4.8 we 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 yielded Xsubstr,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 events are detectable. We can now define a new optimization parameter ξ that represents the contact area that leads to detectable bonds, in other words, ξ corresponds to ζ without the undetectable bonds. This corresponds to the equation X substr,maxZ Z ξ = fhit(Xdot,Xsubstr)dAring,beaddAring,substr, (6.2.2)

Xsubstr,min Abead 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 be found in figure 6.4. It turned out that ξ has a maximum at L = 60 nm. In other words, for a bead of radius R = 500 nm a tether of 60 nm leads to the maximum contact area that leads to visible bonds.

64 CHAPTER 6. OPTIMIZATION

50 L=20 nm L=50 nm L=70 nm 40

30

)

m

n

( i

p 20

e

t

s

R

h " 10

0

0 50 100 150 200 250 X (nm) substr

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

65 CHAPTER 6. OPTIMIZATION

#105 9

6

) 4

4

(nm 9

2

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

Figure 6.4: The new optimization parameter ξ that accounts for detectability. The maximum contact area is found for L = 60 nm at R = 500 nm and a frame rate of 30 Hz. For L → 0 ξ vanishes since none of the bonds are detectable. As L increases, an increasing fraction of binding events is detectable and in the regime L > 60 nm ξ converges to ζ.

66 CHAPTER 6. OPTIMIZATION

6.3. Optimization in the sandwich assay

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

kcatch khit kstick

kdetach ksep ko

Figure 6.5: A schematic picture of how tethered particle motion may be used to create a sandwich assay. 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 for the first step of figure 6.5, in which as many analytes as possible should be caught. Then one minute remains 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 does not limit the detection principle. In this regime of ‘hitting is sticking’, the optimal design ensures that the surface area of the bead that hits the substrate within one minute is maximized. One might think of ‘coloring’ the bead at every spot 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 length L = 50 nm after 0.3 seconds. The color scheme of the bead itself has no physical meaning.

67 CHAPTER 6. OPTIMIZATION is governed by a different tradeoff principle. On the one hand, with a longer tether, a larger portion of the bead surface is able to hit the substrate. On the other hand, if the tether is too long, then the bead 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 five years, which is beyond the scope of this project. However, we were able to simulate this process for shorter timescales and we may observe the general trend. Figure 6.7 displays the resulting fraction of bead colored for several tether lengths. We observe for example 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 adapted in such a way that the bead is only colored at one spot if that spots hits the substrate at least a time τmin within one minute. A natural choice would be τmin > τstick. This approach would decrease the fraction of the bead that is colored in a given time window, but we expect that the trends in figure 6.7 will still be present.

0.02 L=10 nm L=30 nm 0.018 L=50 nm L=70nm 0.016

0.014

0.012

0.01

0.008

0.006 Fraction bead colored bead Fraction

0.004

0.002

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

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

68 Chapter 7 Conclusion & 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 extraordinary application that we envision. We reported the way that the MD simulations were set up to capture the equilibrium behavior of the bead and we showed that the exclusion mechanisms bead-surface exclusion, tether-bead exclusion and tether-bead exclusion are all three significant in the description of the equilibrium distribution of the bead 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 approach to the movement of a tethered particle near a substrate prohibitively difficult. Therefore, we conclude that 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 the step size, the in-plane distance that the bead travels in one frame, varies with the tether length L, the bead radius R and the frame rate. Subsequently, we focused on specific binding spots on the bead and/or the substrate. We showed how the hitting probability varies for different positions on the bead and the substrate. The location of the binding molecules can be reconstructed from the motion pattern of a bound bead. Using this principle, we have constructed a scheme that allows one to interpret experimental data and extract kinetic parameters for several experimental situations. As a proof of principle, we have applied this 1 −1 to actual experimental data to find a kstick of 1.7 · 10 s . To investigate the regime of ‘quick rebinding events’, the event of a bond releasing and reattaching before significant bead movement has occurred, we have performed simulations with several values of kstick for a fixed value of koff . We found that quick rebinding events contribute insignificantly in the 5 −1 regime of kstick < 10 s .

69 CHAPTER 7. CONCLUSION & OUTLOOK

Exploiting the ease of adjusting parameters in a simulation, we have sought for an optimal TPM system 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 an optimal tether length of 60 nm, for the latter the dynamic range of our simulations turned out to be insufficient. 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-bond kinetics 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 new way of using a TPM system. Now that we have developed a basic framework for the understanding of 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 the bead and the substrate that leads to detectable bonds for a fixed bead radius R and fixed frame rate ω. In these calculations we have assumed that the detectability of a bond is solely determined by the binding spot on the substrate Xsubstr. This approach may be too simplistic, as figure 6.3 indicates. A next 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 binding events, but the optimization principle outlined in section 6.2 can still be applied. A logical next step in 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 a non-spherical bead. Elongated gold nanoparticles are already of interest in certain areas of biosensing research [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 to these constraints, some of the properties of the system have not been evaluated in as much detail as we would have liked, e.g. the last optimization method in section 6.3. Future investigators should be aware 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 develop a 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 experimental times 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 of analytes. The process of catching analytes, the first step in figure 6.5, can also be included in the simulations.

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

70 Bibliography

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74 Appendix A Implementing 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 test simulations that we have performed to validate the implementation. We are contented to find that that 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 apply Langevin dynamics. In Langevin dynamics, two forces are added to the conservative force field - a drag force F~d proportional to the velocity with drag coefficient γ ≥ 0 and thermal white noise F~r [39]. Explicitely, the Langevin equation of motion is given by ˙ M~v = F~tot = F~c + F~r + F~d,, (A.1.1) where F~c is the sum of the conservative forces in the system and F~tot is the sum of all the forces in the system, m is the mass of a particle and ~v is the velocity. The thermal white noise or random force obeys the relations D E F~r(t) = 0 (A.1.2) D E F~r(t) · F~r(τ) = 6γkBT δ(t − τ). (A.1.3)

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

In LAMMPS the usual implementation of Langevin dynamics applies a friction force F~f and a random force F~r every time step dt given by [36]

F~f = −γ~v (A.1.4) r 6k T γ F~ = B ξ~(t). (A.1.5) r dt

In theory, ξ~(t) should be a Gaussian distributed random vector, but, as D¨unweg 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 have introduced adjusted the parallel and perpendicular drag coefficient. For the parallel component the result is provided by Fax´en’slaw [45] and for the perpendicular component we use an interpolation formula [44] based on Brenners exact result [38] γ γ = 0 (A.1.6) k 9 ∗ 1 ∗3 45 ∗4 1 ∗5 1 − 16 z + 8 z − 256 z − 16 z γ γ = 0 , (A.1.7) ⊥ 9 ∗ 1 ∗3 57 ∗4 1 ∗5 1 − 8 z + 2 z − 100 z + 5 z where γ0 is the usual Stokes drag on a sphere, given by γ0 = 6πηR for a sphere with radius R in a liquid with viscosity η and z∗ is given by z∗ = R/z, with z the distance from the center of the bead to the surface.

We can introduce the relative drag coefficients λk, λ⊥ to simplify the form of equation 3.4.1 and 3.4.3, by defining γk = λkγ0 and γ⊥ = λ⊥γ0.

76 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

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 the random fluctuating force scales with the square root of the drag coefficient and thus the anisotropic effects should be incorporated in line with that. In the code below the essential lines of code are represented.

//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];

77 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

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 the surface at z = 0. To this end, we calculate the forces with our anisotropic langevin thermostat but we do not actually integrate the equation of motion. The result is that the bead remains fixed, but we obtain a distribution of forces. We perform the simulation at z = 550, so that z∗ = 0.91. As mentioned in section A.1 LAMMPS uses uniform random numbers to generate a random force. We know from equation 3.4.1 and 3.4.3 that at this point γ /γ = 4.83. We know that the random force scales with the square root of the drag ⊥ k √ coefficient, therefore we expect F⊥/Fk = 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 maximum occurring fz divided by the maximum occurring fx we obtain 2.20, which corresponds exactly to the values we expected.

78 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

2500 fx fy fz

2000

1500

1000 Occurence

500

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

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

79 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

A.3.2 Movement parallel to surface To test the implemention of the parallel drag coefficient we perform the following simulation: a particle starts out with a relatively large velocity parallel to the surface and the relaxation of the velocity is recorded. One expects the drag to decelerate the velocity in an exponential fashion with a charactaristic timescale τ = 1/γk. These simulations have performed with a bead radius of R = 500 ∗ and a z-coordinate of z = 900, 550, 501, so that z = 0.56, 0.91, 0.99. The results for 1/γk can be found in figure A.2 together with a plot of Fax´en’slaw. The simulations agree with the analytical formula. To demonstrate that the decay of the velocity is indeed exponential, we plot the velocity against time for five times the charactaristic time in a semilogarithmic graph in figure A.3. We observe a straight graph, which indicates that the velocity indeed decays exponentially. The wiggles that appear in the at the last part of the graph are the result of the influence of the random fluctuating force becoming more visible for lower values on a logarithmic axis.

4

3 æ

æ 2

æ

1

R z 0.2 0.4 0.6 0.8 1.0

Figure A.2: The results for 1/γk at z = 900, 550, 501 (blue circles) plot together with Fax´en’s law given by equation 3.4.1. 

80 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

104

103

)

= / 2

< 10

(

x v

101

100 0 0.5 1 1.5 2 2.5 3 3.5 Time(=)

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

81 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

A.3.3 Movement towards surface Another simulation that we can perform is the movement of a bead towards a surface. We define an initial velocity in the direction of the surface. The drag coefficient should now increase as the bead moves closer to the surface and the velocity in a semilogarithmic plot should no longer have a constant slope. In figure A.4 the result of such a simulation can be found. Indeed, the slope of the magnitude of the velocity on a vertical logarithmic axis is not constant, but is increasingly negative. This is a result of the fact that the perpendicular drag coefficient increases when the bead approaches the surface.

105 1000

104 900

103 800

)

=

)

/

<

<

(

Z(

z v

102 700

101 600

100 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time(=)

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

82 APPENDIX A. IMPLEMENTING ANISOTROPIC DRAG

A.4. Simulation times

To find the influence of our adapted fix on the simulation times we run several simulations with the regular ‘fix langevin’ as well as our tailor-made ‘fix langevin/aniso’. We run five simulations with 2 · 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 major advantage compared to explicit hydrodynamic simulations near surfaces.

83 Appendix B Tabulated values

To perform the procedure outlined in section 5.5, the tabulated values for the mean distance to the origin, the width and the length of the pattern are required. This appendix provides these tables for the system with the usual parameter values. The tabulated mean distances to the origin can be found in 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 fraction may 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 the substrate 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 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 ------Xdot 140 - - - - 83.0 121.8 149.0 155.6 143.6 ------160 - - - - 55.5 87.3 126.0 154.3 157.5 149.5 140.2 ------(nm) 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 ------166.1 187.5 206.3 218.0 320 ------194.7 213.1

84 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. The overall 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 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 ------Xdot 140 - - - - 205.8 124.1 205.9 187.9 165.8 ------160 - - - - 135.0 190.3 184.1 177.1 162.4 140.8 116.8 ------(nm) 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 36.2 320 ------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. The overall 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 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 ------Xdot 140 - - - - 255.2 269.0 269.5 265.5 257.9 ------160 - - - - 192.9 247.9 265.0 264.4 262.4 247.0 204.4 ------(nm) 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 ------115.8 136.3 139.7 128.2 320 ------94.5 108.9

85 APPENDIX B. TABULATED VALUES

5 Table B.4: The hitting fraction for a single Xsubstr and a single Xdot. The provide values are given in fraction ·10 . Several combinations of Xsubstr Xdot do not occur at all. This table may be used when a single binding molecule is present on the bead 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 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 0.0 40 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 0.0 60 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 0.0 80 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 0.0 120 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 0.0 Xdot 140 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 0.0 160 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 0.0 (nm) 180 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 0.0 200 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 0.0 220 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 0.0 240 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 0.0 260 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 0.0 280 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 0.0 300 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 0.1 320 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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